# Inventory — the Lebesgue line at its measured strength

Generated from the 68 recomputation reports (`reports/` of the evidence tree, at the commit the paper pins). Each item restates what the source document asserts, the strength at which the recomputation measured it, and a quotation from the report; the quotation can be found in the report named in the run's heading. Nothing here is stronger than its report. The strength column was audited against the reports for every item first classified as certified or proof-checked; an item the audit moved says so in data/inventory.json.

## Counts

- runs: 68; items: 1218; corrections: 236; open items: 300

| strength | items |
| --- | ---: |
| analytic (proof checked) | 136 |
| cited theorem | 52 |
| exact-arithmetic certificate | 115 |
| numerical only | 389 |
| corrected | 124 |
| refuted | 27 |
| open | 42 |
| not measured | 40 |
| label only | 293 |

## Corrections (source value against recomputed value)

| run | what | the source says | the recomputation finds |
| --- | --- | --- | --- |
| RUN-003 | Round 00 §21.6 claim-status rule | 每個重要結果只能使用下列狀態之一 (each important result may use only one of ten statuses) | Census v5: 758 judgment statements, 89 use one of the ten; PASS 118 and PROVED 72 are not among them; 614 statements in 238 phrases outside the vocabulary. |
| RUN-003 | This report's own census (self-correction across versions v1 to v5; not a source statement) | Census v1: 495 judgment statements, 53 using one of the ten, 389 outside in 187 phrases, PASS 37. | Census v5: 758 judgment statements, 89 using one of the ten, 614 outside in 238 phrases, PASS 118, PROVED 72, registry 251 phrases; verdicts unchanged. |
| RUN-003 | Span of COMPUTE-DEFERRED as a judgment (this report's own earlier statement) | v1 said COMPUTE-DEFERRED runs through Round 25. | v2: as a judgment through Round 19; v5: from Round 00 to Round 22 and in DLMVC. |
| RUN-004 | §22 branch-type list | branch types restricted to one-active degeneracy; two-active balance; three-active balance | the one-active type is empty (conv{u} ∌ 0), so every minimiser has at least two active directions; the corpus itself records this in Round 03 §8 as R01-CORRECTION-001 |
| RUN-005 | §32 displayed quantized curvature maximum | 0.77710000… | 0.7771000125… (package record, exact recomputation 0.77710001254259385, and symmetry 1 − 0.2228999875…); no consequence for the 0.2229 legality margin |
| RUN-007 | §33 recorded sanity polygon (package numbers) | polygon area 0.8337053574119659, Hausdorff excess 0.0006459849700081 | the polygon §7–8 define has area 0.8336391643823979 and excess 0.0006224936758761; the record is reproduced to 10⁻¹² by pushing six on-level heights (h = 0.525 and h = 0.515) one extra level, from the package's 10⁻¹⁴ rounding guard; all §33 assertions still hold |
| RUN-007 | This report's own first-committed paragraph on the floating-point quotients (self-correction, not a source statement) | the quotient at h = 0.505 "falls just below its integer"; quotient at 0.525 given as 50.000000000000044 | the quotient at 0.505 is above its integer (overshoot 9 × 10⁻¹⁵, absorbed by the guard); at 0.525 it is 50.00000000000004 |
| RUN-008 | Package's recorded LB_env (quadrature vs §21 integral) | recorded LB_env, e.g. 0.7936755224944383 at M = 64 (a 200,000-point rectangle rule) | exact ¼∫ℓ_b is 0.7936755210933901 at M = 64 (recorded − exact = +1.40×10⁻⁹); four of five recorded values exceed the exact integral; harmless here, not certified |
| RUN-008 | This report's own exact envelope routine (self-correction made during RUN-009, not a source statement) | envelope routine cut one grid step early when a crossing fell exactly on a grid angle | routine rewritten (bisection on the maximizing piece); every check passes and every logged number unchanged to 2×10⁻¹⁵ |
| RUN-009 | Citation of Xie (reference 5) | A Reproducible Certificate for the Brass–Sharifi Lower Bound in Lebesgue's Universal Cover Problem (title of v1/v2) | retitled in v3 (13 Jun 2026, bound 0.83201) to 'A Certified Lower Bound for Lebesgue's Universal Cover Problem'; v4 (8 Jul) raised the bound to 0.833 — both before Round 06's date; no Round 06 number depends on it |
| RUN-009 | Reference list | Pál 1920 and Gibbs 2014 used in the text (§2, §3, §35) | both are missing from the reference list |
| RUN-010 | Fixed-seed lift of B_13 (§22–23, package) | 0.83597 (package sampled 0.8359676), recorded as the lift minimum | exact minimum 0.8359056; a placement 6.2×10⁻⁵ below the recorded value exists (package differential evolution, population 12, 80 generations, stopped in a local minimum); the ranking still holds |
| RUN-011 | R_K used for sanity body S1 in sanity.py | R_K = 0.55 | true value 0.52; a larger R_K only enlarges δ, so S1 stays valid |
| RUN-012 | 結論 J (depth conclusion) wording | a positive margin at "some representative" closes at finite depth | the proof (§26) needs m to be a uniform lower bound on the margin of every representative the refinement meets; a margin at one centre promises nothing about the children |
| RUN-012 | RUN-011's inner-point bound (this tree's earlier run) | RUN-011's inner-point bound left out the cores' corners | it fell 0.95–3.7×10⁻⁵ below the exact core hull; with corners it comes within 1.3×10⁻⁷; RUN-011 verdicts stand |
| RUN-012 | RUN-011's widest-closing-cell estimate near the four-body optimum | about 4×10⁻⁴ (linear extrapolation from two widths) | with the exact core, 5.1×10⁻⁴ (5.08×10⁻⁴) |
| RUN-013 | §6 verifier independence | the verifier re-implements the core reconstruction | the verifier's core construction (erode_reuleaux, Body, contact) is the emitter's line for line with cosmetic edits; only the hull routine is independent |
| RUN-013 | §4 split rule specification | split by argmin of δ_7 | x_7 and y_7 tie from the second cut on; the tie-break (earlier axis) and child order (lower child first) exist only in the code; with ties to y 108 of 1,417 leaves of tree 00 fail, higher child first 88 fail |
| RUN-014 | §19 depth forecast | r_pred is a "theorem-derived upper-depth forecast" | at the global 0.835 root the centre margin 3.99×10⁻³ gives r_pred = 9 cycles, but the root contains Mishra's seed q† with base area 0.8347809 < 0.835, so no depth closes the cells around it on the base alone; not an upper bound (scheduling only) |
| RUN-014 | Package spec merge checks and manifest fields | §22 lists thirteen merge checks; §30–31 list minimum shard and global fields | the spec lists eleven of thirteen merge checks (missing the global manifest's own hash and target-and-policy agreement); the reference payloads omit most §30–31 fields, including split_policy_hash |
| RUN-015 | Stored aggregates of checkpoints V3, V4, V5 | V3 and V4 store theorem_ready: true; V4 and V5 list claim-shard-B-v1 as active | by the round's own rules V3 and V4 are not theorem-ready, V4's active claims are A1, C1 and V5's are A1, B2, C1; the demo copied V1's aggregate unchanged into V2–V5 |
| RUN-015 | V5 correction of shard B | an appended correction of B makes V5 theorem-ready | the corrected claim carries the same certificate digest as the rejected one; its A1 PASS is a literal with no replay metrics; the A2 FAIL stays unanswered |
| RUN-015 | Audit protocol vs text | §36 publication gate has ten items; the text's ledger states | the protocol lists nine of ten (drops "source and certificate hashes") and adds REJECTED-PENDING-REVIEW and STALE-AGGREGATE |
| RUN-016 | Reference merger and stored aggregates | the merger has sole authority over current_frontier, active_claim_ids, theorem_ready and pending_seed_paths | the reference merger updates only the frontier; all four race checkpoints keep V1's aggregate (theorem_ready: true; active A-v1, B-v1, C-v1) even while seeds are pending and naming rejected B-v1 |
| RUN-016 | Duplicate certification (§15–16) vs validator | two valid certificates of one seed are kept, primary chosen by least canonical claim hash | the validator counts a seed with two valid claims as covered twice and blocks readiness; the selector is not implemented |
| RUN-016 | Merge journal fields | §22 lists six merge-journal fields | the orchestration ledger lacks base_checkpoint_id, current_checkpoint_id_at_merge, reason and result_checkpoint_id |
| RUN-017 | Reported global capacity in the failing scenario | global capacity 0.8350009475 (min over claims that passed) | frontier capacity (least over seeds of the best covering claim) is 0.8350003445377508, the text's own T_cap^ref, below the failing target |
| RUN-017 | Higher-target scenario readiness at 0.8350001 (found by RUN-018; also corrects this report's first version) | ready at 0.8350001 with budget 2×10⁻⁸ | each claim's capacity is also bounded by its a-priori exclusion (B7 square sized at t_7(0.835)), which gives 0.835 with no slack; a sliver 3.04×10⁻⁷ wide is proved by nothing (exclusion gives 0.83500005); the case is unsupported though the actual hull there is about 1.005 |
| RUN-017 | Hot core slack for B-v2 | B-v2 slack 9.5749×10⁻⁷ | that figure was audited under the rejected B-v1; B-v2's own A1 record has none; §17 says a rejected claim never lends its standing |
| RUN-017 | Migration rule names in the hot core | §32 registers TARGET-MONOTONICITY-V0.1, SCALAR-ERROR-MARGIN-CREDIT-V0.1, WITNESS-CATALOG-SUPERSET-V0.1, ANCESTOR-DOMAIN-COVERAGE-V0.1 | the hot core cites TARGET_MARGIN_CREDIT_V0.1, SCALAR_ERROR_BUDGET_MIGRATION_V0.1, WITNESS_CATALOG_EXTENSION_V0.1, DOMAIN_ANCESTOR_COVERAGE_V0.1; none registered |
| RUN-018 | Local proof capacity / free promotion to 0.83500035 | Capacity (root) 0.8350003545377508; 0.83500035 is a free promotion, READY. | Local proof capacity is min(0.835, 0.8350003545) = 0.835, because the B7 lift root square of half-width t_7(0.835) leaves a sliver of width 1.06×10⁻⁶ at T = 0.83500035 whose exclusion gives only 0.83500017. |
| RUN-018 | Round 14 migration to 0.8350001 (passed by RUN-017) | Round 14's higher-target scenario at 0.8350001 is ready. | Unsupported: sliver width 3.04×10⁻⁷ with exclusion 0.83500005 in the sliver; RUN-017 corrected. |
| RUN-018 | Planner's root capacity | root_domain_capacity = INFINITE-WITHIN-FIXED-LOCAL-SCOPE; free promotion compares target with evidence capacity only. | The B7 lift root is a second root whose exclusion capacity is 0.835. |
| RUN-019 | §7 reading of APR | APR contained-center bound is not yet close enough to 0.835; APR/REP = 0 does not mean they are never useful. | f(d) exceeds π/4 only for d > ½ and every body centre lies within √2·t_n <= 0.287, so APR is π/4 ≈ 0.785 in every cell at every depth. |
| RUN-019 | Near-frontier profile (§9–11) | Least HARD centre 0.8356828; 8 cells within +0.002; 70 within +0.005; 8 need 7 cycles; 142 need at least 5. | With exact centres the least is 0.8385552; 0 within +0.002; 12 within +0.005; none needs 7 cycles; 92 need at least 5; sampled centres lie 1.6×10⁻³ to 5.6×10⁻³ below exact (median 3.5×10⁻³). |
| RUN-019 | Cycle proxy conservativeness (§12) | The cycle proxy is conservative ("保守"). | 19 cells holding configurations below 0.835 (witness areas 0.834781 to 0.834971) get 3 to 8 cycles though no base refinement closes them; the proxy is a good heuristic, not conservative. |
| RUN-019 | Share of HARD cells within 3 cycles (§13) | About 90%. | 95.9%; 90% is the quantile's level. |
| RUN-019 | Provenance of the shipped summary | global_pilot.py is the reproduction code for the package. | It runs depths 16 and 18 only and writes none of the summary keys scheduler_table, cycle_tail_ge, b7_probes, cap, top8, near_counts, priority_tiers; it times one run where the summary records two (13.77 s and 13.51 s). |
| RUN-019 | Typo in the official weight rule | The rule closes with `\right\\}`. | It does not typeset. |
| RUN-020 | Quotient gain (§6) | 47.32% fewer nodes and 32.73% fewer HARD leaves at depth 16. | At equal cell size (D3 depth 16 vs full root depth 18, same τ = 0.0385532) the quotient removes 72.9% of nodes and 73.5% of HARD leaves (73.8% and 74.7% with exact cores). |
| RUN-020 | Tail window's advantage (§13) | The tail window is §10's windowed CPG started at depth 10, the most balanced production candidate. | The code and spec also change the override rule (alternate axis wins only if it certifies a child); that rule cuts overrides 1,608 → 404 and costs HARD leaves 1,862 → 2,047; starting at depth 10 is worse on every count under either rule. |
| RUN-020 | Fourth stored conclusion in the summary | Windowing CPG retained the depth-14 HARD count of full CPG. | same_hard_count is false: 863 against 1,210; the sentence is a fixed string in the pilot's code; the text (§11) says it right. |
| RUN-021 | §14 internal node count | internal nodes: 4983 | 5,039 internal nodes; the source's own 11.37% (573 / 5,039) and 15,117 bits (3 × 5,039) use 5,039. |
| RUN-021 | Identity of the tail forest | The certificate is Round 17's tail window made production (§0, §33). | It re-emits only with an added lookahead rule (a SYM child is not counted as certified); Round 17's rule gives 9,990 forest nodes and 536 overrides, not 10,086 and 573. |
| RUN-021 | Cost-model break-evens | R⋆ ≈ 31.79 and κ > 0.0144 s. | Overstated: the tail emitter re-evaluates every internal node (5,039 of 30,229 evaluations, 16.7%); the tail/official evaluation ratio is 2.45 as run and 2.04 without the redundancy. |
| RUN-021 | Grammar completeness | Production shard grammar: CLOSED. | Bit order (MSB first), child order (lower first) and unary convention (ones closed by a zero) are fixed only by the data; zlib stream concatenation order is only in the code. |
| RUN-021 | Shipped verifier's scope | Independent verifier; pending frontier preservation PASS. | Its bodies, contact points and erosion cores are transcribed from Round 16's code; it never opens the pending files and never compares manifest counts with the decode. |
| RUN-022 | "New" node counts (§5–6) | 12,634 and 26,455 new nodes. | They include the 2,048 and 4,599 roots; 10,586 and 21,856 are new. |
| RUN-022 | Share closing by CORE (§4) | Almost all accepted leaves close by CORE. | All do: 2,136 in the A/B, 2,228 and 5,398 in the continuations. |
| RUN-022 | Switch-curve depths (§21) | Nine depths listed; exponent roughly 0.70. | Ten depths exist, 43 omitted; fit 0.704 over ten points, 0.696 over the text's nine. |
| RUN-022 | Witness-only floor and τ ≈ 10⁻³ regime (§22–24) | Floor 108 (2.64%) at depth 12; 412 at depth 18; excess ~ τ^0.70; switch near τ ≈ 10⁻³. | With exact cores the floor is 76 (1.86%) at depth 12 and 7 at depth 18; open leaves 456 down to 83; excess falls as τ^0.91. |
| RUN-022 | Activation policy file vs text | §25 states BASE-UNRESOLVED, NECESSITY-MARKED, NECESSITY-SEGMENTED, LIFT-ACTIVE, LIFT-CERTIFIED; §14 marker payload includes the target. | The policy drops NECESSITY-SEGMENTED, adds BASE-CERTIFIED, and drops the target from the marker payload. |
| RUN-023 | Search minima (§9) | B19 0.8354088638, B21 0.8351431025, B17 0.8352710872, B11 0.8361731203, B13 0.8359229101 are search minima. | Lower placements exist: B19 0.8350921012 (by 3.17×10⁻⁴), B21 0.8350840426 (5.9×10⁻⁵), B17 0.8351473006 (1.24×10⁻⁴), B11 0.8361364467 (3.7×10⁻⁵), B13 0.8359149272 (8.0×10⁻⁶). |
| RUN-023 | B7 not-yet closure | B7 full closure of marked cell: NOT YET ACHIEVED; B7's tail u₁₈ … u₂₆ from 2.491×10⁻³ to 3.193×10⁻⁵, still not closed. | With exact cores B7 closes the depth-50, 46, 43 track cells and all four reference cells (lift depths 21, 23, 25, 30, 31, 31, 34); worst slack 6.6×10⁻⁹ (floating point, not a certificate). |
| RUN-023 | Choice of §7's eight cells | Eight-cell means B7 160.6, B9 154.0 (choice of cells unstated). | The eight are the lowest by outer bracket; by exact centre the means would be 160.375 and 154.375. |
| RUN-023 | §11 depth-18 list | B7 0.249%, B11 0.297%, B19 0.281%, B21 0.276%. | B13 is omitted, with the largest fraction there (0.314%). |
| RUN-023 | Stream encoding | Format otherwise Round 18's (2-bit tags). | Exactly one of eight conventions matches: lower child first, MSB first, CERT 0 and PENDING 1; this 1-bit tag is defined nowhere. |
| RUN-023 | Escalation policy states and manifest statuses | §2 states: BASE-CERTIFIED, NECESSITY-MARKED, LIFT-PROBE, LIFT-ACTIVE, LIFT-CERTIFIED, WITNESS-DISQUALIFIED, PENDING. | The policy adds BASE-UNRESOLVED and MULTI-WITNESS-REQUIRED and drops BASE-CERTIFIED; the manifest uses three statuses in neither list; the crystal omits B9, B15, B17. |
| RUN-024 | Regular B7 baseline (§12) | 0.83712806 | Exact hull area at the package's placement is 0.8371347054; least of 28 exact-area searches 0.8371339787; the package value is 5.9×10⁻⁶ below anything B7 was found to attain. |
| RUN-024 | Smooth 7-fold minima (§13) | 0.8362521834, 0.8366138051, 0.8367717016, 0.8369133231, 0.8369528003 for a = 0.006, 0.008, 0.009, 0.01, 0.0103. | Each lies 3.2 to 5.9×10⁻⁶ below the lower end of an 80,000-direction bracket (e.g. a = 0.0103: [0.8369581551, 0.8369648826]). |
| RUN-024 | The atlas as B7's failure region | A structured adversarial atlas of the B7 lift tail; B7 LIFT-TAIL ATLAS: CLOSED. | With exact cores all 2,143 depth-26 boxes clear 0.835 (least 0.8358005576, by 8.0×10⁻⁴); the exact-core tree closes at lift depth 21 with 2,509 nodes. |
| RUN-024 | Ridge points (§6) | The package's ridge points mark the ridge. | They lie 8.0×10⁻⁶ and 1.07×10⁻⁵ above their channels' minima (0.8371407193 and 0.8371653077); lobe A's point is 8.9×10⁻⁶ above a polish at its own phase. |
| RUN-024 | Final oracle returns to baseline (§14) | The final oracle returns to the regular B7 baseline (0.8371286). | 0.8371286 vs 0.8371281 differ by 5.3×10⁻⁷, inside the 3–6×10⁻⁶ error of the package's area. |
| RUN-024 | Atlas field mean_phi | mean_phi | It is the circular centre of 7φ divided by 7 (0.3607844, 0.8402750); the mean of φ is 0.3607748 and 0.8402794. |
| RUN-025 | Pareto map volume axis (§19–23) | Depth-20 open volumes rank H19 (8.3671×10⁻⁵) and H17 (9.5317×10⁻⁵) below B7 (1.4107×10⁻⁴); H17/H19 are cheap. | With dense cores B7 closes at lift depth 21 (2,509 nodes); H11 at 27 (21,029) or 24 (1,581, support δ); H13 at 30 (43,111) or 26 (3,257); H17 and H19 never close (hulls 0.8346707 and 0.8346480 below T). |
| RUN-025 | B7 minimum on cell-01 (§7) | 0.83714197 | True least 0.8370210981 (channel B); package value is channel A's local minimum, 1.21×10⁻⁴ above. |
| RUN-025 | Harmonic minima (§13–15) | H11 0.83574698; H13 0.83563803; H17 0.83507834; H19 0.83504963; coarse k = 19 0.83499834. | True least: H11 0.8357638060; H13 0.8356225717; H17 0.8350819469; H19 0.8350481986; the coarse k = 19 value is below T but the true least is not. |
| RUN-025 | Dense brackets at the found worst placement (§14) | H11 [0.83578614, 0.83581539]; H13 [0.83567775, 0.83569731]. | They lie above the true least by 2.23×10⁻⁵ (H11) and 5.52×10⁻⁵ (H13); the placements were not the minimisers. |
| RUN-025 | Optimistic mixture value (§5) | An optimistic basin at 0.83742146. | It is the basin reached from B7's own minimising placement, 0.8374213467, with the package's value 1.1×10⁻⁷ above it. |
| RUN-026 | Number of sampling directions | Not stated (Rounds 16–22 used 40). | Every B7 edge rebuilds at 96 directions; at 40 none does (cell 0: 3,031/559 vs 2,393/420). |
| RUN-026 | Harmonic node counts (§15) | H11 3749.5, H13 3644.2, H17 2362.5, H19 2286.3 mean nodes. | Rebuilt means 3810.7, 3707.0, 2392.8, 2317.8; node counts 26–64 above on every harmonic edge; open boxes agree on 42 of 48. |
| RUN-026 | Routing and residual (§16, §22) | Route all twelve cells to H19; residual is all twelve; no strict edge yet. | With exact cores B7 closes all twelve at lift depths 30–35 (18,509–45,401 nodes); no harmonic can close any cell: best upper ends below T on all 48 pairs (highest H11 0.8342519, H13 0.8339969, H17 0.8334024, H19 0.8333507). |
| RUN-027 | Cell 0 tail collapse (§6) | 9.34×10⁻⁴ (depth 16) and 5.17×10⁻⁵ (depth 22) in one sequence with exact-tree values; collapse 22 → 24 ×7.4. | The first two values are the 96-direction pilot's; exact cores give 7.61×10⁻⁴ and 2.51×10⁻⁵, and the collapse from 22 to 24 is ×3.6. |
| RUN-027 | Cell 3 contraction factor (§9) | ×393 since depth 22. | ×217 with exact cores alone (pilot values 9.70×10⁻⁴ and 5.85×10⁻⁵; exact depth-22 3.238×10⁻⁵). |
| RUN-027 | Cell 3 residual | Cell 3 is DEFERRED-CONTRACTING; residual {3}. | The same exact tree closes cell 3 at lift depth 34 with 36,193 nodes (445 open at depth 32); residual empty by depth 34 (floating point). |
| RUN-027 | Residual core provenance | Residual core volumes at depths 16 and 22. | They come from a different bound (the 96-direction pilot), which the residual core does not say. |
| RUN-028 | Closure depths D_j (§1–3) | Recorded D_j 30, 32, 32, 34, 32, 30, 34, 36, 32, 34, 34, 30; D_max 36, mean 32.5, median 32. | Least complete depths 30, 31, 31, 34, 31, 30, 33, 35, 32, 34, 33, 30: cells 1, 2, 4 close at 31, cells 6 and 10 at 33, cell 7 at 35; D_max 35, mean 32.0, median 31.5. |
| RUN-028 | Common budget D⋆ (§14–15) | D⋆ = 36 is a common budget. | The least common budget is 35; coverage at 31, 33, 35 is 6, 9, 12. |
| RUN-029 | Least slack in the atlas (§14) | Least audited slack 2.8982×10⁻⁹ (cell 21), over ten sampled trees. | Over all 51 complete trees the least is 8.47×10⁻¹¹ (cell 8, a leaf at depth 35); 11 trees hold a thinner leaf; five are below 10⁻⁹ (cells 8, 39, 62, 43, 50); 29 hold a leaf below 10⁻⁸. |
| RUN-030 | Γ_B7 at depths 38 and 40 | Γ_B7(38) >= 57 and Γ_B7(40) >= 63, lower bounds only, "not a common-depth curve". | At a common depth the values are exact: Γ_B7 at 37, 38, 39, 40 is 56, 57, 63, 63. |
| RUN-030 | Round 26 ledger closure depths | Round 26's ledger records cells 4 and 14 at 38 and cell 0 at 40. | They close at 37 (cells 4, 14, 35, 36, 51), 38 (cell 37) and 39 (cells 0, 16, 18, 28, 29, 44): the next even checkpoint again. |
| RUN-030 | Stored boxes and the φ root (§13) | Boxes are the stored binary64 values, each enclosed by its neighbouring floats. | Shipped x and y differ by up to 4 ulps from this tree's (t₇ 0x1.9c9e1c2550f72p-3 vs 0x1.9c9e1c2550f76p-3, both above the exact 0.2014734457862874141…); 2·fl(π/7) falls 3.5×10⁻¹⁷ short of 2π/7, so the ±1-unit enclosure is load-bearing on top-edge leaves. |
| RUN-031 | Density requirement (§14, §16, §22) | Only six leaves require the maximum tested density (16,384). | Densities are start points from reference-slack bands; the least passing density is lower for 8,457 leaves; 256 suffices for 7,727; at most 8,192 is needed (four leaves); none needs 16,384. |
| RUN-031 | Width of d⋆ (§2) | O(10⁻⁶³). | The root file says 3.1×10⁻⁶¹. |
| RUN-031 | This verification tree's exact kernel (RUN-008's envelope), not the source | Reference slack for leaf 001110010110000100 implies area 0.835746559778711 (package value). | This tree's grid kernel gave 0.835746558304646, 1.474×10⁻⁹ low, missing a B7-core corner 1.7×10⁻⁶ outside the disc over 5×10⁻³ rad; a verified envelope agrees with the package on all 9,278 leaves to 1.7×10⁻¹⁵; shared kernel corrected 2026-09-27. |
| RUN-032 | Weakest marker (§12) | The weakest marker is 6, margin 6.1062×10⁻⁷. | Marker 6 has the weakest certificate; by exact margin the weakest marker is 47 (1.857×10⁻⁵), marker 6's exact margin being 2.112×10⁻⁵; the package's ranking follows its polygons' first-order losses at bridging edges. |
| RUN-032 | Densities (§11) | Densities 4,096: 21, 8,192: 39, 16,384: 16, 32,768: 1; none needs 65,536. | Densities are start points chosen from Round 26 reference-margin bands; 73 passed at their start and 4 went from 8,192 to 16,384; with bridging-edge normals 128–256 normals suffice. |
| RUN-032 | Marker 47's tighter bound (§13) | The new bound is tighter than the Round 26 reference. | Reference (20,000 directions) loses 1.83×10⁻⁵; new (32,768) loses 1.10×10⁻⁵; tighter only by more directions. |
| RUN-032 | Package polygon losses | Outer polygons at 4,096–32,768 normals. | Losses 1.1×10⁻⁵ to 1.1×10⁻⁴, first order in the normal spacing at straight bridging edges (up to L²·d/8). |
| RUN-033 | Root replay relation (§0) | The root replay overlaps Round 28's intervals. | The relation is containment: the replay's intervals lie inside Round 28's, and this tree's brackets inside the replay's. |
| RUN-033 | Width named in R30-A1-ROOT-001 (§12) | The cast d⋆ interval was about 10⁻⁶⁴ wide. | 10⁻⁶⁴ is the width of Round 28's t_n intervals; the d⋆ intervals are 3.1×10⁻⁶¹ (Round 28) and 2.8×10⁻⁷³ (the replay) wide. |
| RUN-033 | Implementation independence of the upper half (§1) | Implementation A1: nothing shared in candidates, hull, density or ledger. | The marker 70 replay is Round 29's audit row, from a script carrying the emitter's support routine, normals and density. |
| RUN-033 | Thinnest lower leaf (§11) | The thinnest margin is 3.54237075×10⁻⁹ on leaf 11010110100110101001110111. | That margin measures the replay rule's loss (one contact point per direction): with every contact point the leaf's margin is 2.2211×10⁻⁵; the least margin under every contact point is Round 28's thinnest leaf, 3.19489×10⁻⁸. |
| RUN-033 | Root used by the lower replay | Root/path A1 recomputed. | The lower replay's script reads Round 28's tmax_intervals, not the replay's own root; no shipped script computes the root replay. |
| RUN-034 | Root interval relation (§0, §4) | t₃, t₅, t₇ overlap Round 28's intervals. | Nested containment: this tree's brackets inside A1's, A1's inside MPFR's, MPFR's inside Round 28's; d⋆ is a dyadic interval exactly 2⁻²²¹ wide. |
| RUN-034 | Directed-rounding rule (§3) | exact Fraction → MPFR directed → exact dyadic Fraction. | Inputs are loaded rounded to nearest (within 2⁻⁶⁴⁰ of the rational); only outputs are directed; covered in practice, not by construction. |
| RUN-034 | Marker 70 polygon identity (§5) | 0.8349075014501105, as Round 29 shows. | The MPFR polygon is the every-constraint A1 polygon (1,317 constraints); Round 29's is the float-reduced one (1,284); the areas are 3.3×10⁻⁵⁰ apart. |
| RUN-034 | Lower replay independence and least margin (§7–10) | Histogram equal because candidate policy is shared but core enclosure backend differs; least margin 3.5423707501×10⁻⁹. | Margins are equal leaf for leaf too (same polygons); the replay shares A1's rule and three functions; the least margin is the A1 rule's loss (2.22×10⁻⁵ with every contact point). |
| RUN-036 | Reason for keeping cells 47 and 48 below publication-candidate | The Round 32 ledgers keep no hull vertices, source tags or exact areas, so dominance has nothing to transfer. | Keeping them below candidate is right, but the certificates regenerate from density and the emitter's rule: hull count on 9,228 and 9,285 leaves, margin to 2.6×10⁻¹⁶, vertices the emitter's up to its 17-digit rounding. |
| RUN-037 | Packaged certificate sizes for cells 47 and 48 | about 8.1 MiB and about 7.6 MiB | 8.02 and 7.54 MiB (8,411,289 and 7,909,374 bytes); raw 34.8 and 34.0 MiB are right |
| RUN-040 | The compositional merge's requirements | Q = ∪Qᵢ is interior-disjoint given the parent's disjointness, each Qᵢ ⊆ ∪Pᵢ, and segment-level disjointness; merged by hashes and slice coverage, without a global scan. | The theorem is right, but the merge spec's seven requirements omit the containment premise and the shipped merge does not test it; two one-box segments can meet all seven and overlap. The premise holds for all 146,483 shipped boxes, checked directly; a standalone merge needs one ancestor lookup per box. |
| RUN-041 | §I.2 sanity example labelled independently reproduced | Re-running Round 04's sanity.py gave the package's numbers (polygon area 0.8337053574119659, excess 6.4598497001×10⁻⁴): SANITY: INDEPENDENTLY REPRODUCED. | Those are the script's numbers, not the polygon Round 04 defines; the quantization guard sends six of the eight exact-level heights one level up. The defined polygon has area 0.833639164382 and excess 0.000622493675876. |
| RUN-041 | Two formulas in the Round 04 case study | `d\le\frac0.84409359440.5` and `\Delta=\frac{\pi}128` | Braces lost in substitution: meaning A/r and π/M; VALIDATION passes both because its checks are syntactic. |
| RUN-041 | Reason for 'Empirical validation: PARTIAL' | Empirical validation is PARTIAL, with "initial successful mathematical case studies". | PARTIAL is right, but because one case study calls a re-run of the author's script independent (W4), a reason DLMVC does not give. |
| RUN-041 | Coefficient bound in the Round 01 case study | \|r̂ₙ\| ≤ 1/2 | The domain gives \|r̂ₙ\| ≤ 1/π, attained by the Reuleaux triangle at n = 3, so (2/π)·E_M is also a bound (improvement, not error). |
| RUN-042 | Pass 04 inner-core factor λ_b | λ_b ≈ 0.988205366 (0.9882053662605436), and 0.9882…·P(b) ⊆ U | A grid sample on 240,000 angles; the true minimum is 0.9882022551 at θ ≈ 0.81765 (sampled minimum 3.1×10⁻⁶ too large); there λ·h_P lies 1.6×10⁻⁶ above L♯, so the boxed inclusion is not established by the pass's own bound; it follows for λ ≤ 0.9882022551. |
| RUN-042 | Pass 04 homothetic area bound | 0.8268069548 | 0.826801748731 with the true λ |
| RUN-042 | Pass 04 sharp cell radius ε♯_cell | 0.0059677037 (≈ 0.0059677) | maximum 0.0059692967, 1.6×10⁻⁶ higher; the recorded value is not an upper bound |
| RUN-042 | Pass 04 §S.1 figure for Pass 03's cell radius | Pass 03 cell radius 0.013060516 | Pass 03's radius is 0.013316971; 0.013060516 is the same construction with h_P in place of H_j, sampled; its true maximum is 0.0130615798. |
| RUN-042 | Pass 02 adjacency step (improvement, not error) | 51 (48 at 0.8344) | 51 is safe, but 50 (47) suffices since the quantization error is strictly below q |
| RUN-042 | Pass 05 'perimeter mode' naming | p is the "perimeter mode" | p is Σb, not the perimeter |
| RUN-043 | The stop rule (which states end the loop) | The loop ends at proof or counterexample closure (§3, §21); a failed target returns to Stage 0 (§21.3); independence closes relative to a theory (§58); SEARCH-STALLED is not completion (§49). | Eight places give four different sets of exits; SEARCH-STALLED is a stop in the protocol's §7 but a trigger in §49; the §50 pseudocode is a `while True` loop with two returns and no branch for a failed target, a stall or independence. |
| RUN-043 | LinkBack use in the application | Every artifact gets a LinkBack type; the protocol's Stage 3 makes LinkBack mandatory. | Two of eleven types (OPENS, REDUCES) occur once each in Cycle 01's map; its other four labels are not LinkBack types; only Cycles 01 and 03 have a LinkBack section. |
| RUN-044 | Attribution of the sharp root domain | Cycle 03 presents it as `DOMAIN-STRENGTHENING`, verdict `SHARP ROOT DOMAIN: PROVED`, without citing Mishra. | It is Mishra's Remark 20 (\|t3\| ≤ 0.192366, \|t5\| ≤ 0.195891, factor 169; rebuilt 0.19236641, 0.19589071, 169.1); his Remark 34 reports it saves no boxes (25,425,568 vs 27,350,260 at τ = 0.833). |
| RUN-044 | Volume reduction counted for the sharp root domain | The next cycles count it as a volume reduction (164.3×, then 985.6×). | Mishra's Remark 34: the smaller domain does not reduce the cost (27,350,260 boxes over the smaller domain vs 25,425,568 over the larger at τ = 0.833). |
| RUN-044 | Cycle 03's continuous margin s7 | s7 ≈ 0.00938484 (crystal 0.009384839564890224) | Numerical minimum s7 = 0.009384687909 at φ = 0.676813; the cycle's figure is 1.52×10⁻⁷ above it; harmless since Gate A certifies 0.0088367 (RUN-045). |
| RUN-044 | Thinness of the sublevel / the 2.45×10⁻⁷ figure | The family ceiling and the representative differ by only 2.45×10⁻⁷; the sublevel is extremely thin. | The 2.45×10⁻⁷ is first-order loss of the 10⁶-line outer polygon at 17 bridging edges; along B5's turn the sublevel A ≤ A3+ reaches 9.9×10⁻⁴ and 7.8×10⁻⁴ (about 10⁻³). |
| RUN-044 | Replacement family D, B3, B7 minimum | min ≈ 0.830638626, outer 1000 ≈ 0.830957521 | minimum at most 0.8305410847, 9.8×10⁻⁵ lower; outer polygons of 1,000 and 10⁶ half-planes 0.8309189 and 0.8305415; the rejection stands with more room. |
| RUN-044 | Mishra's located four-body optimum | 0.836494901 is Mishra's four-body optimum, used in 0.836494901 − 0.834780947 = 0.001713954. | The arithmetic is exact, but 0.836494901 is above a four-body placement found in RUN-010 (0.8364914416), so it is not a minimum. |
| RUN-044 | Cycle 03 script's import claim | Its docstring says it imports Mishra's `geom.py`. | It imports only `math`. |
| RUN-044 | Cycle 01 ranking table entries | B11 and B15 entries as printed | B11 (0.004727) and B15 (0.002646) are printed a unit low in the last digit. |
| RUN-045 | Premises checked by Gate A's shipped verifier | Verify_GateA.py verifies the finite dual certificate. | It never checks that each triple's gaps are below π (positive sine-gap weights) nor that all 1,024 orientations are present; both hold when checked here. Float conversions stay below 10⁻¹⁷. |
| RUN-045 | Recorded witness intervals | The certificate records interval enclosures. | Lower = upper at all 1,024 witnesses (20-digit decimals of 60-digit mpmath intervals); on 76 witnesses the recorded lower end is above the enclosure's by at most 3×10⁻¹⁹; the verifier recomputes and trusts none of them. |
| RUN-045 | Cycle 05 spec root half-widths T3, T5 | Root half-widths T3 and T5 as given in the spec. | 22 and 12 ulps smaller than the exclusion bounds: rounded inward where a certificate root should round outward (the spec says the tree has not been emitted). |
| RUN-045 | Cycle 06 Hausdorff bound for the anchor | d_H(U0, U_sym) < 1.66×10⁻⁵ | True value 7.1×10⁻⁶ (the bound holds). |
| RUN-045 | F3 search minimum | 0.834968965913 | at most 0.834968955147, 1.1×10⁻⁸ lower |
| RUN-045 | F5 search minimum and margin over A3+ | 0.834804245410; margin 2.30544×10⁻⁵ | at most 0.834804244389 (1.0×10⁻⁹ lower); margin at most 2.30534×10⁻⁵ |
| RUN-046 | Cycle 08 corner stability | 29.86 % (1,493 of 5,000) | 549 of a 2,000-root sample (0.275); the gap is 2.3 standard errors; the cycle's sample and seed are not given |
| RUN-047 | The split rule | The documents say only "deterministic split rule"; Cycle 07 describes adaptivity without the weights. | The verifier splits the widest coordinate by weights (1.5, 1.5, R5, 1, 1) while the box may meet the F3 ball and (0.7, 0.7, 1.5R5, 1.5, 1.5) once wholly outside, first index on a tie; Mishra's or Cycle 06's fixed weights leave 36 or 27 leaves that fail even the exact-core bound. |
| RUN-047 | Exact tiling of the float split | Binary subdivision, one bit per node in preorder (exact cover implied; coverage not stated). | Only 280 of 1,336 (Cycle 10) and 15,915 of 54,210 (Cycle 11) splits tile exactly; gaps up to 4.7×10⁻¹⁸ are closed by continuity (at most 3.7×10⁻¹⁷ in area); neither verifier checks coverage nor document says it. |
| RUN-047 | Cycle 11's printed least verified slack | 1.5376482309648054×10⁻⁹ (seventeen digits) | 1.5376479×10⁻⁹ here (3.6×10⁻¹⁶ from the certificate's); at 50 digits the raw slack is 2.53764884×10⁻⁹; six of the seventeen printed digits carry. |
| RUN-048 | The ρ₅ = 0 reduction's domain (Cycle 12) | The symmetric F3 domain is x3 ≤ x3*, so ρ5 = 0 is a two-variable convex problem. | Symmetrization ends at y3 = 0 and leaves F3 where \|x3 − x3⁰\| < s; root 6217's box runs to x3 = −0.012122 (0.00886 right of x3⁰ − s) and y3 = 0.012122, so the arc side is left out. |
| RUN-048 | Cycle 13's exact reduction to the rectangle | Gate A reduces "exactly" to the rectangle; "the root does not intersect the right F3 branch". | The slice also holds F3 placements on the arc side; no step of Cycle 13 bounds them and its verifier checks only the rectangle's leaves. |
| RUN-048 | GATE 6217-A: CLOSED (Cycle 13, Cycle 14, both crystals) | m₆₂₁₇(0) > A3+: GATE 6217-A: CLOSED | NOT ESTABLISHED: holds numerically with margin 2.05×10⁻⁴ on the arc side (searched at 25 angles), but no certificate covers that side; of 10,856 arc-side samples, 145 fall in Cycle 12's certified subroots, 6,812 in the 68 open, 3,899 in none of the 200. |
| RUN-048 | Cycle 14's statement about Cycle 13 | "Cycle 13 closed the entire ρ = 0 slice" | FAIL: only the rectangle side is certified (W6). |
| RUN-048 | The 27-point rotation profile | non-decreasing, quadratic ratios 0.1738–0.2457 | All 27 minima are search values up to 8.3×10⁻⁸ too high; re-searched profile stays non-decreasing with ratios 0.1738–0.2254. |
| RUN-048 | Unconstrained minimizers' distance from the anchor | within 7.9 % of s (6.96×10⁻⁴) | 9.1×10⁻⁴ (10 % of s) kept in the box; up to 2.2×10⁻³ (a quarter of s) in the plane; still inside the circle, so the premise survives with less room. |
| RUN-048 | Cycle 14's four profile skeletons and two generic charts | four skeletons: two generic charts (A: samples 1–8; B: 9–13, 15–26) and two degenerate states | At full resolution samples 0 and 14 differ and the other 25 share one sequence; dropping runs under 10⁻⁹ or 10⁻⁶ rad gives three or four sequences grouped otherwise; the charts are not reproduced. |
| RUN-049 | Room under the tight reference | The tight reference 0.8349865 has 6.5×10⁻⁸ of room. | The exact hull at the candidate is 0.8349861105592, giving 3.894×10⁻⁷ of room under 0.8349865 (and 1.389×10⁻⁵ under 0.835). |
| RUN-049 | Cycle 16 §10's 'all other regions eliminated by finite certificates' | "all other regions have already been eliminated by finite certificates" | 44.2 % (44.15 %) of root 6217's box lies outside the 200 subroots and was closed only in Cycle 11's search; the ρ₅ = 0 arc side is also uncovered (RUN-048). |
| RUN-049 | Cycle 15's counterexample to global radial monotonicity | outward slope −0.0457 | Genuine but marginal: f falls for only 1.6×10⁻⁷ along the ray and by 7.2×10⁻⁹, then rises (forward slope +0.093 at step 10⁻⁵). |
| RUN-049 | Cycle 15's crystal status for Gate 6217-A | Gate 6217-A: CLOSED | RUN-048 found it not established. |
| RUN-050 | §29 No-Slack Boundary Principle | An inward admissible move gives δA[h; φ] < 0 (stated as programmatic). | δA < 0 needs μ_h(I) > 0; at a square's corner an admissible inward move (φ = −sin 2θ on [0, π/2]) has δA = 0 and A(ε) = 1 − (3π/8)ε², so the contradiction comes from δ²A = ∫(φ² − φ′²) < 0 (sectors narrower than π). |
| RUN-050 | Name of the symmetry group in §12–13 | E(2) = SO(2) ⋉ R² | SO(2) ⋉ R² is SE(2), as Paper 02 §1 corrects (RUN-052). |
| RUN-051 | The compiler scaffold's stability test and statuses | §33's lemma: stability needs ‖δh‖ < m/C; the schema has five statuses. | `stable_stratum` compares each margin with the perturbation bound directly, with no C; the gate returns only three of five statuses (never `INCONSISTENT_SIGNATURE` or `REOPEN`); it computes no crossing. |
| RUN-051 | §10's finiteness hypothesis | Each pair of pieces crosses finitely often ("finitely many roots"). | The finiteness of signatures needs a placement-uniform root bound; the kernel's pieces give at most two crossings per pair, independent of placement. |
| RUN-051 | Name of the symmetry group in §1 | E(2) = SO(2) ⋉ R² | SO(2) ⋉ R² is SE(2), as Paper 02 §1 corrects (RUN-052). |
| RUN-052 | §15 intrinsic stabilizer | stabilizer defined on X = H_conv/E(2) | as written the formula gives O(2) for every shape; the intended stabilizer is on the translation gauge, which §15's next sentence gives |
| RUN-052 | gcd law scope (§24–27) | an aligned cos mθ on a D_n seed leaves at least D_gcd(n,m) | for m = 1 the body is only translated and keeps D_n; the law needs m ≥ 2 (PASS for m ≥ 2) |
| RUN-052 | §8 m = 0 as 'scale' mode | m = 0 is scale / mean support | h + a₀ adds a disc of radius a₀ (the parallel body), which is a dilation only for a disc |
| RUN-052 | Kernel symmetry breaking decided by δ²A (§39 applied to the kernel) | breaking is decided by the second variation and the constraints | on the kernel it is decided at first order by contacts; area rises linearly 0.133 per unit of y3, 0.044 of φ5, 0.273 of y5 |
| RUN-052 | Recorded seed q† (Round 07's best placement) as an asymmetric seed | q† has φ5 2.41×10⁻⁵ off 2π/5 and y5 = −3.9×10⁻⁶, so its hull has no symmetry | its asymmetry is search noise; the mirror-slice polish is 9.03×10⁻¹¹ lower, at 0.8347809458213, with stabilizer D₁ |
| RUN-052 | Scaffold's use of the smooth cap | §14: the smooth bound is only sufficient; compiler has a CONVEXITY_BLOCKED output | scaffold computes the smooth cap, discards it, never returns CONVEXITY_BLOCKED, and would pass every mode on |
| RUN-052 | Scaffold's gauge classification | §46: for the seed 1 + a cos nθ, sin nθ is the turn's tangent (gauge) | scaffold knows only m = 1 as gauge and would call sin nθ BREAK_TO_K |
| RUN-053 | First-order model of a witness move on the kernel | affine model (§27–32): each edge moves parallel to itself, charged ℓ·max(r_a, r_b) | true first-order change is the edge average ℓ(r_a + r_b)/2; the affine model is larger by 0.66 to 0.94 in four random directions (wrong sign in three), and overcharges at Z by a factor 2.9 to 5.5 (y3: 0.133 measured vs 0.732) |
| RUN-053 | §61 finite-mode first-order closure certificate | an LP optimum ≥ 0 is a finite-mode first-order closure certificate | the LP has optimum 0 at a generic placement with gradient length 0.390 and at q†, where area falls toward Z at 6.3×10⁻⁶ per unit; §62's disclaimers do not cover the gap |
| RUN-053 | §19 lower bound on the step cap | at least 1/‖q‖ when ν_φ⁻ = q μ_h | the bound is the exact cap: ε⁺ = 1/‖q‖ |
| RUN-053 | §18 zero-cap case | the cap 'may' be 0 when ν_φ⁻ has mass where μ_h has none | 'may' is 'does': every positive step fails |
| RUN-054 | §15 projected descent inequality | a projected descent inequality | for a cone it is an equality, δA[h; V*] = −‖V*‖², holding to 1.8×10⁻¹³ on 200 random cones; for non-cone convex sets only the inequality holds, strict on 60 polytopes (least slack 0.60) |
| RUN-054 | §17 stationarity on the kernel | V* = 0 is local first-order projected stationarity | V* = 0 at the generic placement, q† and Z, while at the generic placement the gradient has length 0.390 |
| RUN-054 | State record's 'not claimed' list | the text disclaims no unique metric (§9), stationarity only relative to the current witness family etc. (§17), and Ψ not a theorem (§71) | the state lists no 'not claimed' items |
| RUN-054 | State record's title for Paper 05 | Paper 05's own title includes '… and the No-Slack Boundary Principle' | the state names Paper 05 by a shorter title |
| RUN-054 | Flow schema events and graph compute statuses | six events and five compute statuses in the schemas | only CERT_PRUNE and COMPUTE_DEFERRED of the six events, and only COMPUTE_DEFERRED of the five compute statuses, are named in the text |
| RUN-054 | Scaffold QP and graph algorithms | finite-mode QP (§18–20); dominators and SCC quotient (§49–50, §70); GraphEdge fields per §64 | qp_payload returns its inputs; no dominators and no SCC quotient; any string accepted as edge type; GraphEdge lacks mode and step |
| RUN-055 | Meaning of the finite dual certificate on the kernel (§54, §68) | the finite dual can be solved now on the Lebesgue line's own witnesses; a certificate yields DUAL_CERTIFIED | the certificate exists at the generic placement, q† and Z (M = 4, 8; residuals ≤ 5×10⁻¹¹), including the generic placement whose gradient has length 0.390, so it certifies a non-stationary placement |
| RUN-055 | Unstated care in the finite model | the paper does not state sampling and gauge requirements | under-sampled contacts give a spurious descent −0.107 at Z (M = 16); unfixed rotation gauge gives −6.2×10⁻⁴ (−2.2×10⁻⁴ at four times contact density), gone once B3's turn is fixed |
| RUN-055 | Paper 00 §29 No-Slack statement | Paper 00 §29 lacked the curvature hypothesis | Paper 05 §33 carries ρ ≥ κ₀ > 0; at a corner (ρ = 0) RUN-050's admissible inward move has δA = 0, so the hypothesis is needed |
| RUN-055 | Title of Paper 06 in the state and dual schema | "Lebesgue Extremal Shape Closure" (as §73) | Paper 06's own title is "Actual Lebesgue Shape State and First Closure Graph" |
| RUN-055 | Verifier scaffold checks | §48 complementarity (λ = 0 off the active constraints) | scaffold does not solve for a certificate, does not check complementarity, and accepts multipliers down to −tol |
| RUN-056 | §32 / Task B: four-contact dual stationarity | SHAPE_DUAL_STATIONARITY = OPEN (g = Aᵀλ open) | impossible on the four contacts for any library holding 1 and cos 2θ: the cos 2θ-moment would be P cos 2θ* = 2.05 while the barrier's curvature measure has 0.124; least residuals 1.6, 2.0, 3.0 |
| RUN-056 | §25 and ledger status of the ρ₅ = 0 slice | rho0_value_proof_status: CLOSED_FOR_A3PLUS; graph node RHO0 INDEPENDENTLY_REPLAYED_THRESHOLD | value closure of that slice is not established (Cycle 13's certificate covers the rectangle side only; 1.3 % of the arc side lies in Cycle 12's subroots), though it holds numerically by 2.05×10⁻⁴ |
| RUN-056 | "68 → 61" open-cell count | 68 → 61 | 61 counts the formal ledger's open cells; RUN-049 found 44 % of root 6217's box was closed by Cycle 11's search only |
| RUN-056 | §27 barrier stabilizer | stabilizer at least D₁ (⊇ ⟨σ_x⟩) | stabilizer read off the sampled hull is exactly D₁ |
| RUN-056 | Closure-graph vocabulary and hashing | roles from §44's twelve; edge types from the series | two roles (SHAPE_REFERENCE, COMPUTE_DATA_GATE) not among §44's twelve; of 20 edge types only CERT_PRUNE, REOPEN_FOR_SHAPE and UPPER_FEASIBLE are defined in the series; VALIDATION does not hash the PNG or SVG |
| RUN-056 | State's title for Paper 07 | "Root-6217 Barrier Shape Reconstruction as Controlled LESR Test" | Paper 07's own title is "Root-6217 Barrier Shape Reconstruction" |
| RUN-056 | Meaning of the full-contact dual certificate at the barrier | a dual certificate as a stationarity witness | with every contact (M = 4) a certificate exists, but releasing d₃ ≥ s the area falls toward the anchor at 0.0331 per unit |
| RUN-057 | §15 convex subgradient (g₃, 0) | pressure splits give a convex subgradient; B3's horizontal subgradient g₃ balanced by η₃ = −g₃ | (g₃, 0) fails the subgradient inequality: along d = (cos 202°, sin 202°), F′(v; d) = 0.044 while g·d = 0.114; finite-step residuals −7×10⁻⁸, −6.9×10⁻⁶, −2.5×10⁻⁴ |
| RUN-057 | Far-branch multiplier η₃ | η₃ = −g₃ = 0.1228 | η₃ ≈ 0.047 (g₃′ between −0.0477 and −0.0474 over 720 directions), about 39 % of 0.1228 |
| RUN-057 | §17 extension to all translations at ρ₅ = 0 | the barrier minimizes all translations at ρ₅ = 0 in the far branch | the argument uses Cycle 12's y → 0, which does not keep d₃ ≥ s; RUN-048 found the arc side left out (1.3 % certified), least area there numerically at the barrier itself (0.8349861106, at α = π) |
| RUN-057 | VALIDATION's minimum midpoint winner margin | 0.00526 | 0.00526 is the margin over the other bodies; over every other piece it is 0.00060 |
| RUN-057 | KKT status labels in the records | certificate: "DERIVED_FROM_PRESSURE_DECOMPOSITION GIVEN_SIGNATURE"; state: "CLOSED_GIVEN_SIGNATURE" | the translation KKT is not established as argued (§15–16); its conclusion holds numerically |
| RUN-057 | State's title for Paper 08 | "Positive-Rotation Barrier Closure" | Paper 08's own title is "Positive-Rotation Certificate Reduction" |
| RUN-057 | VALIDATION hashing of the signature graph | VALIDATION covers the package | VALIDATION hashes the signature graph's PNG but not its DOT |
| RUN-058 | §3 derivation of first-order criticality | in the finite active-contact variational model the infimum is 0, derived from Paper 07's pressure/KKT structure | in that model the least first-order change is +0.0293 (B5 translating by (−0.022, −0.1625) per unit turn; uncompensated turn 0.0474); Paper 07's split is not a subgradient; criticality holds only numerically on the true hull (rate ∝ ρ) |
| RUN-058 | "Honest" 195-root shape ledger | 200 → 195 from Cycle 11's 200 subroots | the 200 fill 55.9 % of root 6217's box; 29.5 % is NOT-F3; 14.6 % lies in the branch and was closed only by Cycle 11's search for A₃⁺ (3.3 % with ρ < 0.003, 11.3 % with ρ ≥ 0.003) and belongs in the shape ledger by the paper's own rule |
| RUN-058 | Profile ratio upper value | ratios 0.224–0.246 | sample 2's stored value is 8.3×10⁻⁸ high; its 0.2457 is a search artefact; re-minimized samples give 0.2254 and 0.2253; true local ratio ≈ 0.225 |
| RUN-058 | Diagnostic's ρ = 0 reference value | ratios listed against the profile's ρ = 0 search value | that search value is 7.3×10⁻¹¹ above the exact m(0) the diagnostic quotes |
| RUN-058 | State and schema label for first-order rotation | "CRITICAL_GIVEN_P07_PRESSURE_KKT"; "critical by Paper07 contact-pressure/KKT architecture; interval dual audit still required" | criticality holds but not by the argument given; it is a floating-point observation along two compensations |
| RUN-058 | State's title for the next node (Paper 09A) | "Critical-Cone Second Variation at rho=0+" | Paper 09A's own title is "Multi-Scale Topology-Wall Correction and the One-Sided Blow-Up Critical Cone" |
| RUN-059 | §7 secondary slivers | three thinner secondary slivers, O(ρ²) scale a working hypothesis | at all four re-minimized placements they are below 10⁻¹¹ (ties, not intervals); they exist at the anchor sample only because its area is 1.5×10⁻⁹ above the branch's minimum |
| RUN-059 | Number of signature pieces on the branch | 18 merged open intervals | 18 at the stored anchor sample; 15 at the minimum on the same convention |
| RUN-059 | §6 primary width coefficient | about 0.1480ρ | 0.14804ρ at the anchor sample; on the re-minimized branch the coefficient drifts to about 0.1469 as ρ → 0 |
| RUN-059 | Retained 200 → 195 ledger | the tail gate and the 200 → 195 ledger unchanged | the ledger leaves out 14.6 % of root 6217's box (RUN-058) |
| RUN-060 | Printed u* | u* = 5.157126245268359 | u* = 5.15712624526841370759… (to 70 digits); the printed value is 5×10⁻¹⁴ off (binary64 through a system with determinant about 10⁻⁴) |
| RUN-060 | §18 location of the second-order slivers | B5's corner slivers at π ∓ θ* (also Paper 09A's diagnostic) | along the critical velocity the slivers open at π/2 (disc, 1.79ρ²), θ* and 2π − θ* (B5's arcs, 2.72ρ² each); nothing opens at π ∓ θ*; at RUN-059's minima there are none |
| RUN-060 | §20 "w* = 0: PROVED" | w* = 0: PROVED | correct algebra, conditional on the branch keeping all three collapses at first order (CERT-09B.4, open) |
| RUN-060 | Verifier's Φ₁ = γ_w·w | Φ₁ = γ_w·w "supplied by the analytic 15-piece summation" | the summation is not shipped; the verifier carries the constants and γ_w as numbers and verifies only the linear algebra |
| RUN-061 | §6 I B3's subgradient | "B3 constrained subgradient g₃ < 0, balanced by η₃ = −g₃" listed as done in Paper 07 | (g₃, 0) is not a subgradient, short by 0.0663 over 72 directions; the valid first component lies in [−0.0477, −0.0474] |
| RUN-061 | §9 A evenness ⇒ m′(0) = 0 | evenness ⇒ m′(0) = 0 | m is even (five samples mirror to 10⁻¹⁵), but evenness gives only m′(0⁺) = −m′(0⁻); m′(0⁺) = 0 holds numerically (ratio ≈ 0.2256) |
| RUN-061 | §9 F Cycle 14 transition | the transition at ρ ≈ 0.0141 | that ρ is Cycle 12's sample 14, the only degenerate one; the sampled contact sequence changes twice (between samples 8 and 9, ρ 0.00806 to 0.00906, and at that sample); neither change is located |
| RUN-061 | VALIDATION's next-node title | "Paper 08 — Positive-Rotation Barrier Closure" | Paper 08's own title is "Positive-Rotation Certificate Reduction" |
| RUN-062 | §84 root-6217 map input | the map starts from "195 shape-active parent roots" | that is Paper 08's ledger, which leaves out 14.6 % of root 6217's box closed for A₃⁺ only (RUN-058); the 'positive-rotation candidate branch' has no lower branch found (RUN-048, RUN-059); Paper 09C is not in the archive (RUN-060) |
| RUN-063 | §15 boundary-contact pairs | 35 boundary-contact pairs | 64 pairs on the exact grid (40 across a 4-face, 14 across a 3-face, 8 across a 2-face, 2 along an edge); 55 active cells touch a prune box; the ledger's 35 is a floating-point exact-zero style count (such a test finds 30) |
| RUN-063 | §31 scheduling-score median | median(T∩) over meeting radii | the score uses the median over the 140 finite-radius cells (0.766444), not over all meeting radii (0.696945) |
| RUN-063 | §20–24 frontier's dependence on the cut | the most balanced eight finite-radius cells | for 178 of 195 cells T₊ is the distance to the ray's end ρ = 0.003; at cut 0.004 cells 160 and 168 replace 103 and 104; at 0.006 cells 100 and 168 come in |
| RUN-063 | VALIDATION and verifier coverage | VALIDATION's "graph_sha256"; verifier prints "boundary contact pairs = 35" | graph_sha256 is the PNG's digest, the DOT is not digested; the verifier's 35 is the ledger's, unchecked |
| RUN-064 | §17 provenance flip at 127 | 127: coarse nearest 144, local nearest 155 | 127 is exactly 1/8 of the span from both 144 and 155 (both floats 0.12499999999999978); the ledger's 'coarse nearest 144' is the first of the two in order; nothing flips at 127; inside 127, 8 children are nearest 144 and 24 nearest 155 |
| RUN-064 | §23, §32 127 as a thick, provenance-switching corridor | 127 is a thick, branched, provenance-switching corridor (2 → 9 BOTH grandchildren) | at 127 the prune boxes arrive last and touch along part of a face (12 children on it) while elsewhere the ray touches at a point; 4.5 is a face against a point, and the 'flip' is a tie |
| RUN-064 | Conditions for the 17 classes | strict merge gives 17 classes | 17 holds only with corner adjacency and each child keeping its parent's ρ class; faces only gives 19; own ρ class gives 22 |
| RUN-064 | Result C | Results A, B and C | Result C rests on §17 and §23, which are not as stated |
| RUN-065 | Provenance at 101, 103 and 104 | earliest box '28'; cells 'close to' PROVENANCE_DEGENERATE | four boxes (28, 68, 144, 155) tie exactly; the ledger's margins (1.4×10⁻¹³, 2×10⁻¹³, 4×10⁻¹⁶) are noise; the next box, 36, comes 0.18 to 0.24 later |
| RUN-065 | Origin of the negative gaps (§3) | gaps within 2×10⁻⁹ come from independent numerical optimizers | the seven outward-limited gaps are exactly zero; the ledger's negative ones (−5×10⁻¹¹ to −1.3×10⁻⁹) are LUC 01's T_box minus the exact T₊, i.e. LUC 01's ray grid, to within 5×10⁻¹³ |
| RUN-065 | Printed r* and λ* | r* = 0.0025452863775066, λ* = 0.760555242947675 | r* = 0.0025452863403046, λ* = 0.76055524294058 (printed values right only to 3.7×10⁻¹¹ and 7.1×10⁻¹²); minimum flat, T at printed r* only 2×10⁻¹⁸ higher |
| RUN-065 | §10 'LUC 02's flip still holds' | 155 ahead of 144 by 0.0420149036170295, so LUC 02's flip from 144 to 155 still holds | the margin is right (0.04201490361714857) but there was no flip: 127 is as far from 144 as from 155 (RUN-064) |
| RUN-065 | 131's ledger true time and second time | 0.0473333498927312; second time 0.17677669529537773 | true time 0.0473333498919450 (ledger 7.9×10⁻¹³ high); second time is √2/8, the ledger's 1.3×10⁻¹² below it (optimizer left the box); ledger midpoint error 2.8×10⁻⁸ |
| RUN-065 | Second place at 164 | ledger names 28 | three-way tie between 28, 68 and 144 at √(5/32) |
| RUN-066 | The overlap's second-order law | combined straight-line stiffness min max(vᵀH₊v, vᵀH₋v) = 0.45429481027797 governs the overlap | the overlap's own second-order law is the Lagrangian form αQ₊ + (1 − α)Q₋, eigenvalues 0.236577, 0.239445, 0.763138; softest stiffness 0.2366; slices measure 0.239445 h² and 0.236854 h² (straight-line model: 1 and 0.9918) |
| RUN-066 | §18–19 lens growth law | \|h\| ≲ 0.7476 √ε; ε ≳ 1.789 h² | lens reaches 1.036 √ε in its softest direction (1.030 √ε along x₅); least radius cost 0.932 h² |
| RUN-066 | §22 certificate target | κ > 0.4, 'generous' | the true constant is 0.2366; κ > 0.4 would fail; isolation needs only a positive constant |
| RUN-066 | Printed z*, g±, α*, slope | printed values computed from the printed r* = 0.0025452863775066 | they carry the printed r*'s error: z* 3×10⁻¹⁰ off, α* 7.1×10⁻¹², slope 1.9×10⁻⁸ off |
| RUN-066 | Printed T* | T* as printed | the printed T* is LUC 03's 5-D value, 1.7×10⁻¹⁴ high (RUN-065) |
| RUN-066 | Printed stiffness direction | direction with x₅ component 1.7×10⁻⁷ | that component is noise; it is 0 |
| RUN-067 | Printed ρ* | 0.00254068891729192 | ρ* = 0.0025406888749283; printed value 4.2×10⁻¹¹ off |
| RUN-067 | §3 'reproduces' LUC 03 | T* = 0.0473333498919449 reproduces LUC 03 | LUC 03 printed 0.0473333498927312, 7.9×10⁻¹³ higher, so agreement is only to that much |
| RUN-067 | §4–5 midpoint theorem proof | \|p* − z\| + \|z − q*\| ≤ 2T | a common point at radius T is within T of some point of each seed set, not necessarily p* and q*; the bound follows from d(S₊, S₋) ≤ 2T and needs p*, q* to realize d(S₊, S₋); true here (155 nearest at 0.0947, 144 next at 0.2464) |
| RUN-067 | §13 ratio of the two germs' stiffness | κ₁₃₁ / κ₁₂₇ = 1.00078, under a tenth of a percent | the overlaps' own constants are 0.2366 (127) and 0.4546 (131), ratio 1.92; 1.00078 is the ratio of straight-line constants |
| RUN-067 | §15–19 cost and lens ratios | C_R = 1.789 and 4.803, C_H = 0.748 and 0.456 (ratios 2.68 and 0.61) | 127's are 0.932 and 1.036 (RUN-066); ratios 5.15 and 0.44 |
| RUN-067 | §9, §21–22 isolation rule | κ_germ > 0 isolated, = 0 flat, < 0 unstable | the inward seed is a box so Q₋ ⪰ 0 and κ_germ cannot be negative; with 127's forms the Lagrangian would turn negative for KKT weight above 0.9987 while κ_germ stayed 0.4543; positive κ_germ is necessary, not sufficient |
| RUN-067 | §27 'κ ≈ 0.454 for both' | κ ≈ 0.454 for both germs | holds for the straight-line constants only |
| RUN-067 | Result record's container separation gap for 131 | container separation gap (LUC 03's) | 7.9×10⁻¹³ more than its own T_true − T_box |
| RUN-068 | §6, §20 Cycle 17's candidate vector | its parameter vector is not recorded; it needs export | the vector is stored in Cycle 17's 68-cell min-search diagnostic (area 0.8349862484449208, lowest of the 68 cells); it lies 0.109 from the IMBG point, 7.7×10⁻⁴ from the ray, 0.124 from 155; CERT-07.1 is answered and CERT-07.2 can be computed |
| RUN-068 | A_cand as 131's lowest value | Cycle 17's candidate A_cand = 0.8349862484449208 in 131 | 131's corner holds the barrier, m₀ = 0.83498611055922, 1.379×10⁻⁷ below A_cand; the lowest known value lies on the ray at its start |
| RUN-068 | §22 B5 sliver positions | B5's two corner slivers at π ∓ θ* | along the critical velocity they open at B5's arcs a3 and a2, at θ* and 2π − θ*, beside the disc's at π/2 (RUN-060) |
| RUN-068 | §1 printed p*, q*, z* and ρ* | printed vectors and ρ* = 0.00254068891729192 | the vectors are LUC 03's ledger values belonging to ray parameter 0.0025406761 (p* off 5×10⁻⁷, z* 2×10⁻⁷, q* 6×10⁻⁸ from the exact pair); ρ* is LUC 05's, 4.2×10⁻¹¹ from the exact 0.0025406888749283; printed q* carries z*'s y₃ and x₅, the exact projection carries p*'s |
| RUN-068 | §3–4 printed pair length | \|p* − q*\| = 0.09466669978536 | 1.5×10⁻¹² longer than the exact pair, 2T* |
| RUN-068 | §0 LUC 06 and §23 schema/verifier | LUC 06 narrowed the spine to 131; a companion schema and verifier exist | the archive holds LUC 01–05 and LUC 07 only; no LUC 06, schema or verifier |

## Runs

### RUN-001 — Corpus identity: the 42 curated documents on amral.evemisslab.com/lebesgue/ (site revision ba14e814; documents last changed at 9f3a3d76) against the two source zips archived on agent/lebesgue-finite-closure

Report: `reports/RUN-001-CORPUS-IDENTITY.md`

> **Verdict: PASS.** All 42 curated documents are byte-identical to an archived original, each under its own original filename, each bound by exactly one page, and no two pages share a document. The hub page's sentence about its own files holds as written. The drill planted 220 defects; every one was caught by exactly the checks named for it and by no other.

**For the paper.** The 42 curated Lebesgue documents are byte-identical, by content hash, to originals in the two archived source zips (1,459 leaves), so later recomputations read the archived originals; this run establishes provenance only and nothing about the mathematics. Drill: 220/220 planted defects caught by exactly the named checks.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| The hub page states that the 42 listed documents are the original, unrewritten documents behind each of the page's pages. | reading | PASS | exact-arithmetic certificate | 42 |
| Both source zips match their pinned size, SHA-256 and git blob id (check C1). | certificate | PASS | exact-arithmetic certificate |  |
| Every curated document has a byte-identical archived leaf (check C2), none renamed and none with more than one identical archived copy. | certificate | PASS | exact-arithmetic certificate | 42/42; 0 renamed; 0 with more than one identical archived copy; 0 with no byte-identical original |
| The site's files/ directory holds exactly the documents the hub links, and as many as the hub's own text states (check C3). | certificate | PASS | exact-arithmetic certificate | 42 |
| The archive's full leaf manifest equals the committed manifest (check C4), which is written in this round and guards later rounds. | specification | PASS | label only | 1,459 leaves; 126 of them .md |
| Every page under p/ references exactly one files/*.md document, it exists, and no two pages share one (check C5). | certificate | PASS | exact-arithmetic certificate | 42 pages; 42 documents |
| The curated Round 37 document is the narrative from the corrected standalone Round 37 package (the main archive's Round 37 entry is a bare cell-49 certificate with no narrative). | reading | PASS | exact-arithmetic certificate | cell-49 |
| Round 00 exists in v0.1 and v0.2; the site serves v0.2, which differs from v0.1 only in the version line and §26, so §1–25 and §27–35 carry the same mathematics. | reading | PASS | label only | v0.1; v0.2; §26; §1–25; §27–35 |
| The hub states that its four cited papers were checked against their arXiv originals and places the published interval at 0.8344 ≤ a_Leb ≤ 0.8440935944. | reading | UNMEASURED | not measured | 0.8344; 0.8440935944; four papers |

Open: The hub's external-literature statement and published interval are not re-checked: "The hub's statement about external literature is not re-checked here."; Nothing about the mathematics is established: "Anything about the mathematics. RUN-001 shows that what the site shows is what the archive holds; it says nothing about whether what the archive holds is right."

Unclear: The strength vocabulary is mathematical; for these byte/content-hash provenance checks 'exact-arithmetic certificate' is used as the closest fit, not a term the report uses.; The report mentions a BSD-line RUN-001 (bsd-verification-zhuiheng) as precedent; that is a different series and is not recorded as a cross_ref.

### RUN-002 — Round 00 - methodology freeze v0.2 (AMRAL_Lebesgue_Finite_Closure_Round_00_Methodology_v0.2.md, AMRAL-LUC-FC-R00, dated 2026-09-18): its external baseline and its support-function interface

Report: `reports/RUN-002-ROUND00-BASELINE-AND-INTERFACE.md`

> **Verdict: PASS**, with one `OPEN` observation and one `PARTIAL`.

**For the paper.** Round 00 reports the published bracket 0.8344 ≤ a_Leb ≤ 0.8440935944 and its citations correctly (gap 0.0096935944 exact), and its support-function law and containment interface are correct for proper motions (the interface by the cited strict separation theorem); the bounds themselves stand only at cited-theorem strength, baseline currency is checked on arXiv only, and the E(2) vs proper-motion placement group is an open observation for chiral test bodies.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Round 00 is a methodology freeze: its status block says no new bound is claimed and no finite closure is proved. | reading | PASS | label only |  |
| Round 00 fixes the problem statement: a_Leb is the infimum of Area(U) over convex U containing a rigid image of every compact planar set of diameter at most 1. | definition | UNMEASURED | not measured |  |
| Round 00 adopts the published interval 0.8344 ≤ a_Leb ≤ 0.8440935944, and every baseline value appears exactly once and agrees with the gap line and the two bound statements (A1). | reading | PASS | cited theorem | 0.8344; 0.8440935944 |
| Gibbs (arXiv 1810.10089, 2018) gives the upper bound 0.8440935944. | reading | PASS | cited theorem | 1810.10089; 0.8440935944; v1 2018-10-23 |
| Mishra (arXiv 2608.30538, 2026-08-31) gives the rigorous lower bound 0.8344 using constant-width test sets and an exhaustively certified subdivision. | reading | PASS | cited theorem | 2608.30538; 0.8344; 2026-08-31; 486,799,600-node certificate |
| Zeng (arXiv 2609.01284, 2026-09-01) gives a monotone hierarchy Λ_1 ≤ Λ_2 ≤ ⋯ ≤ a_Leb with a_Leb = lim Λ_M and 0 ≤ a_Leb − Λ_M ≤ CM^{-2}. | reading | PASS | cited theorem | 2609.01284; CM^{-2}; 0.834 |
| Wichiramala & Panraksa (arXiv 2606.14625, 2026) is listed as background only. | reading | PASS | label only | 2606.14625 |
| The stated gap between the published bounds is 0.0096935944 (A3). | computation | PASS | exact-arithmetic certificate | 0.8440935944 − 0.8344 = 0.0096935944 |
| Round 00 fixes its baseline as of 2026-09-18, implying no better published bound exists. | reading | PARTIAL | not measured | 2026-09-18; 2026-09-24; 2606.04458; 0.833; 2607.27227; 2512.04023 |
| For g = (R_φ, t) the support-function transformation law is h_{gK}(θ) = h_K(θ − φ) + t·u_θ (B1). | theorem | PASS | analytic (proof checked) |  |
| For closed convex U and compact K, gK ⊆ U iff the slack δ = h_U − h_{gK} is ≥ 0 on the whole circle (§9, stated for closed convex sets) (B3). | theorem | PASS by cited theorem | cited theorem |  |
| Round 00 writes the placement group as E(2) but parametrises and describes only rotations and translations, g = (R_φ, t). | definition | NOT AS STATED | open | h_{gK}(θ) = h_K(φ − θ) + t·u_θ; triangle (0,0), (3,0), (1,2) |

Open: Placement group E(2) vs proper motions, handed to Round 01 and to every round with non-mirror-symmetric witness bodies: "Handed to Round 01 and every round whose witness bodies lack a mirror symmetry."; Currency of the baseline measured on arXiv only: "no arXiv paper since the baseline improves either bound — measured on arXiv only."; Implicit step from convex U (§1) to closed U (§9), assigned to Round 01: "it is a step, and Round 01, which Round 00 tasks with fixing the target domain, is where it belongs."; Correctness of the cited results: "That Mishra's `0.8344` or Zeng's hierarchy is correct. Cited, not verified."; Round 00 governance (claim-status vocabulary §21.6, canonical-source rules §25): "are measurable properties of the whole corpus, and are the next round."; "Anything about Rounds 01–37."

Unclear: The E(2) observation is labelled `OPEN` by the report; the schema has no OPEN finding, so it is recorded as report_finding 'NOT AS STATED' with strength 'open'. The report does not itself use the phrase 'not as stated'.; The convex-to-closed implicit step is described as harmless and closable in a line, but carries no verdict label; it is recorded only under open_items.

### RUN-003 — Round 00 - governance: §25 (canonical source) and §21.6 (claim ledger), applied to the 42 canonical documents (Rounds 00–37 and the four methodology tracks)

Report: `reports/RUN-003-ROUND00-GOVERNANCE.md`

> **Verdicts.** - **§25, canonical source: PASS.** ... - **§21.6, claim ledger: FAIL, as a statement about conformance.**

**For the paper.** Round 00's canonical-source gate (§25, five mechanical items) holds over all 42 documents, while its claim-status rule (§21.6) does not describe the corpus: only 89 of 758 judgment statements use the ten prescribed statuses, so each round's strength must be read from its own text, not from Round 00's ladder. §25's formula-rebuild item and §21's per-round format are unmeasured.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Round 00 §25 requires every canonical source to be UTF-8 decodable (G1). | specification | PASS | exact-arithmetic certificate | 42/42 |
| Round 00 §25 requires display math delimiters $$ to be paired (G2). | specification | PASS | exact-arithmetic certificate |  |
| Round 00 §25 requires inline math delimiters $ to be paired (G3). | specification | PASS | exact-arithmetic certificate |  |
| Round 00 §25 forbids other math delimiters (G4). | specification | PASS | exact-arithmetic certificate |  |
| Round 00 §25 forbids unicode_escape round-trips (G5). | specification | PASS | exact-arithmetic certificate |  |
| Round 00 §25 forbids secrets, keys or private data in canonical sources (G6); also run over the whole archive. | specification | PASS | exact-arithmetic certificate | 1,049 text files |
| Round 00 §25 forbids turning LaTeX into Unicode math characters. | specification | PASS | exact-arithmetic certificate | 17 such characters in 5 documents; three ≥ in Round 16 |
| Round 00 §25's sixth item requires that key formulas can be rebuilt from the source. | specification | UNMEASURED | not measured |  |
| Round 00 §21.6 states that each important result uses exactly one of ten statuses (PROVED-ANALYTIC, VERIFIED-COMPUTATIONAL, VERIFIED-THEOREM-APPLICATION, DERIVED-CANDIDATE, NUMERICAL-EVIDENCE, CONDITIONAL, COMPUTE-DEFERRED, OPEN, REJECTED, CORRECTED). | specification | FAIL | refuted | 758 judgment statements; 89 use one of the ten; PASS 118; PROVED 72; 238 distinct status phrases outside; 614 outside the vocabulary; 55 gate words and YES/NO |
| Round 00 §21 prescribes a per-round format (metadata, previous state, objective, derivation, counterexample section, freedom ledger, computation debt, reproducibility, handoff). | specification | UNMEASURED | not measured |  |
| Round 10 keeps a claim ledger in exactly the ten statuses, including CONDITIONAL for the 0.835 bound and REJECTED for reading its dry run as a global theorem. | reading | PASS | label only | 0.835 |
| COMPUTE-DEFERRED is used as a computation-debt label consistently across the corpus. | reading | PASS | label only | 46 uses (census v5); Round 00 to Round 22; DLMVC |
| From Round 26 the corpus writes its global verdict as NOT CERTIFIED. | reading | PASS | label only | 20 times; Rounds 26–37 |
| A bare PROVED label does not say whether a result is analytic, computational or a theorem application. | method | PARTIAL | label only | 66 PROVED entries (v2 registry); 203 phrases (v2 registry); 251 registry phrases (v5) |

Open: §21 per-round format: "`UNMEASURED`. Section names vary too much for a mechanical check to be honest"; §25 sixth item: "that key formulas can be rebuilt from the source — `UNMEASURED`."; Well-definedness of statuses outside the ten: "That is the registry's job, one round at a time, starting with Round 01."; Census forms still not read: "a status first, qualified after ... a flag ... a verdict in Chinese ... a status in running prose"; Statements in unread rounds: "165 in rounds not yet read."

Unclear: Several census figures in the body (e.g. 506, 56, 398 in 190 phrases) are explicitly v2 figures; the report says the verdict uses v5. The items use v5 where the report gives v5 numbers.; Strength 'exact-arithmetic certificate' is used for exhaustive mechanical text checks (G1–G6); the report does not use that term for them.; The PROVED-classification item has no explicit verdict label in the report; 'PARTIAL' is a mapping of 'a bare PROVED does not say which', not a word the report uses for it.

### RUN-004 — Round 01 - support functions and constant width (AMRAL_Lebesgue_Finite_Closure_Round_01_Support_ConstantWidth_v0.1.md, AMRAL-LUC-FC-R01, dated 2026-09-18): the reduction chain and placement compiler

Report: `reports/RUN-004-ROUND01-REDUCTION-CHAIN.md`

> **Verdict: PASS.** Every step of Round 01's reduction chain `A → closed convex hull → unit constant-width body W → h_W → r_W = h_W + h_W''` and of its placement compiler holds.

**For the paper.** Round 01's reduction from planar sets of diameter ≤ 1 to unit constant-width bodies, its curvature-radius parametrisation, its support-function containment interface and its placement compiler all hold, partly by analytic proof checked line by line and partly by cited classical theorems (completion, halfspace representation, Blaschke selection); Round 01 makes no bound or computational claim, and the reflection branch is now defined explicitly but must be checked in each later certificate.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Round 01 claims no bound, new or old; it claims an exact compression of the target family and an exact containment interface. | reading | PASS | label only |  |
| Thm 3.1: the closed convex hull of a planar set keeps its diameter. | theorem | PASS | analytic (proof checked) |  |
| Cor 3.2: universality may be tested on compact convex targets. | reduction | PASS | analytic (proof checked) |  |
| Thm 4.1: a compact convex set of diameter ≤ 1 lies in a unit constant-width body. | theorem | PASS by cited theorem | cited theorem |  |
| Thm 4.2: testing all planar sets of diameter ≤ 1 is equivalent to testing the unit constant-width bodies 𝒲₁. | reduction | PASS | analytic (proof checked) |  |
| Thm 5.1: for compact convex sets, A ⊆ U ⟺ h_A ≤ h_U. | theorem | PASS by cited theorem | cited theorem |  |
| Thm 5.2: a motion x ↦ Qx + t with Q ∈ O(2) acts on support functions as h_{gK}(u) = h_K(Qᵀu) + t·u. | theorem | PASS | analytic (proof checked) |  |
| §6: the directed Hausdorff slack is max{0, max_u (h_A − h_U)}. | theorem | PASS by cited theorem | cited theorem |  |
| §7, §20: the placement infimum is attained (a minimum). | theorem | PASS | analytic (proof checked) |  |
| §8–9, §11: unit constant width forces a₀ = ½ and kills even Fourier modes; the Steiner point is (a₁, b₁); translations cancel antipodally; the first moments of r vanish. | theorem | PASS | exact-arithmetic certificate | a₀ = ½; degree 8; degree 5; degree-6 |
| §10: for a unit constant-width body, 0 ≤ r ≤ 1 almost everywhere (stated without a source). | theorem | PASS | analytic (proof checked) |  |
| Thm 12.1: translation classes of unit constant-width bodies correspond one-to-one to curvature-radius densities r ∈ 𝓡 (0 ≤ r ≤ 1, r(θ+π) = 1 − r(θ), both first moments zero). | theorem | PASS by cited theorem | cited theorem | 3,601 directions; 4.0 × 10⁻¹¹; odd m ≤ 4001; tail below 10⁻⁹ |
| §13–14: the inversion h = ½ + Σ (α_n cos nθ + β_n sin nθ)/(1−n²), and the sanity family h_a = ½ + a cos 3θ with r_a = ½ − 8a cos 3θ valid exactly for \|a\| ≤ 1/16. | computation | PASS | exact-arithmetic certificate | \|a\| ≤ 1/16; r = ½ − ½ cos 3θ at a = 1/16; odd n ≤ 11 |
| §15, §17–18: the centred family is compact, a worst-case target is attained, and finite ε-nets exist (existence only). | theorem | PASS by cited theorem | cited theorem |  |
| Thm 16.1: Ψ_U is 1-Lipschitz in the target. | theorem | PASS | analytic (proof checked) |  |
| §21: for fixed Q the best translation is witnessed by at most three active directions. | theorem | PASS by cited theorem | cited theorem |  |
| §22: future branch types are restricted to one-active degeneracy, two-active balance and three-active balance. | specification | NOT AS STATED | corrected |  |
| §23: the problem uses congruent copies, so Q ∈ O(2) = SO(2) ∪ R_ref SO(2), with reflection an explicit discrete branch σ ∈ {+1, −1}. | definition | PASS | label only | σ ∈ {+1, −1} |
| §24: canonical statement a_Leb = inf { Area(U) : 𝔎(U) = 0 }, with 𝔎 a max–min–min–max over r, Q, t, θ. | reduction | PASS | analytic (proof checked) |  |
| §29: testing only extreme constant-width bodies is not established. | reading | PASS | open |  |
| §30: an odd Fourier truncation keeps the width but can break convexity. | computation | PASS | exact-arithmetic certificate | ½ − 2/π ≈ −0.1366 < 0; n = 3; π/6 |
| Pre-corpus item Q1, graded PASS by Round 01: Ψ is 1-Lipschitz in the cover. | theorem | PASS | analytic (proof checked) |  |
| Pre-corpus items Q4 (angular sampling, PASS-WITH-SCOPE) and Q6 (removable-region criterion, FINITE-FAMILY-PASS) as graded by Round 01. | reading | UNMEASURED | not measured |  |
| Round 01's eight external references exist and match in title, authors, venue, volume, issue or article number, and year. | reading | PASS | label only | 8 references |

Open: Reflection-branch watch item: "in every round that enumerates placements, check that `σ = −1` is enumerated, or that the round says why it may be skipped"; Classical theorems are cited, not re-proved: "The classical theorems themselves — cited, and now with sources, not re-proved."; No computation is checked: "Round 01 defers C01–C03 and makes no computational claim."; Q4 and Q6 pre-corpus grades: "are `UNMEASURED` here: the drafts they grade are outside this corpus."

Unclear: The report's tally is 13 PASS, 14 PASS by cited theorem, 2 UNMEASURED, 11 n/a over 40 judgments; the items above group these by theorem/section, so item counts per finding do not equal the tally.; The §22 item's report_finding 'NOT AS STATED' and strength 'corrected' are mappings of 'lists a branch type that cannot occur'; the report calls it an observation and 'harmless'.

### RUN-005 — Round 02 - finite shape compiler (AMRAL_Lebesgue_Finite_Closure_Round_02_Finite_Compiler_v0.1.md, AMRAL-LUC-FC-R02, dated 2026-09-18) and its package (VALIDATION.json, State_Crystal)

Report: `reports/RUN-005-ROUND02-FINITE-COMPILER.md`

> **Verdict: PASS**, with one displayed digit wrong.

**For the paper.** Round 02's squared-Fejér shape compiler is correct: the smoothing error bound E_N = π²(4N+1)/(2(2N²+4N+3)(N+1)) is valid (about 6.5 times the exact moment bound for large N, so not tight), interiorization and quantization preserve legality, and a finite legal dictionary exists, several steps resting on cited classical convexity facts; it is a shape compiler only, claims no bound, and one displayed digit in §32 is wrong without consequence.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Round 02 §1–2: the curvature measure of any unit constant-width body is r dθ with 0 ≤ r ≤ 1. | theorem | PASS by cited theorem | cited theorem |  |
| Thm 4.1: smoothing by a nonnegative normalized even kernel (squared-Fejér 𝒥_N) keeps width, convexity, the Steiner gauge and 0 ≤ r ≤ 1 exactly. | theorem | PASS by cited theorem | cited theorem |  |
| §5–6: F_N(t) = Σ_{\|k\|≤N} (1 − \|k\|/(N+1)) e^{ikt} and S_N = (2N²+4N+3)/(3(N+1)). | computation | PASS | exact-arithmetic certificate | S_N = (2N²+4N+3)/(3(N+1)); N = 1..16 |
| §7: 0 ≤ h ≤ 1 and ‖h''‖∞ ≤ 1 for a centred unit constant-width body. | theorem | PASS by cited theorem | cited theorem |  |
| §8–9: the smoothing error satisfies ‖𝒥_N h − h‖∞ ≤ E_N = π²(4N+1) / (2(2N²+4N+3)(N+1)) for every centred unit constant-width body. | theorem | PASS | analytic (proof checked) | E_N = π²(4N+1) / (2(2N²+4N+3)(N+1)); μ₂,N/2 is 29% of E_N at N = 1; 15% at N = 256; Reuleaux actual error about 9% of E_N; about 6.5 times the exact moment bound |
| §22: the table of E_N for N = 8 … 256. | computation | PASS | numerical only | E_8 = 0.1110; E_64 = 2.309 × 10⁻³; E_256 = 1.490 × 10⁻⁴ |
| §10: E_N = O(N⁻²) is not Zeng's CM⁻² rate. | reading | PASS | label only | O(N⁻²); CM⁻² |
| §11–15: interiorization h ↦ (1−δ)𝒥_N h + δ/2 pushes r into [δ/2, 1−δ/2] at cost δ/2; the odd form runs up to frequency 2N−1 with 2(N−1) coefficients. | theorem | PASS | analytic (proof checked) | δ/2; 2N−1; 2(N−1) |
| §16–18: quantizing the 2(N−1) odd coefficients to spacing q keeps legality when qA_N ≤ δ/2 with A_N = (4/3)(N−1)N(N+1), at a support cost of (N−1)q. | theorem | PASS | analytic (proof checked) | A_N = (4/3)(N−1)N(N+1); q ≤ 3δ/(8(N−1)N(N+1)); 5ε/6 |
| §19, Thm 20.1, §21: a finite legal dictionary exists with error E_N + δ/2 + (N−1)q and size \|𝒟\| ≤ (⌈4/q⌉+1)^{2(N−1)}, coefficients boxed in [−2, 2]. | theorem | PASS by cited theorem | cited theorem | E_N + δ/2 + (N−1)q; (⌈4/q⌉+1)^{2(N−1)}; [−2, 2]; 4/π; 1.53; q ≤ ε/48 |
| §23–24: the signed margin M_U is 1-Lipschitz in the shape, with Q ranging over O(2). | theorem | PASS | analytic (proof checked) |  |
| §25: the sandwich gives two finite certificate rules for a fixed cover: all dictionary shapes fit with slack ε ⟹ universal (Rule A); one shape does not fit ⟹ not universal (Rule B). | reduction | PASS | analytic (proof checked) |  |
| §26–28, §36: no bound on a_Leb is claimed and placements remain continuous. | reading | PASS | label only |  |
| §32 sanity example (r = ½ + 0.25 cos 3θ + 0.15 sin 5θ, N = 8, δ = 0.1, q = 0.025/672): its reported minima, maxima, errors and theorem bound. | computation | PASS | numerical only | ĵ(3) = 140/163; ĵ(5) = 108/163; min r 0.10768588783870782; max r 0.89231411216129218; smoothing error 0.00639466922143404; E₈ 0.11100782046215025; total support error 0.0094617496713189537; bound 0.16126823712881692; within 10⁻⁹ |
| §32 displays the quantized curvature maximum as 0.77710000…. | computation | FAIL | corrected | 0.77710000…; 0.7771000125…; 0.2229 |
| The package's VALIDATION.json records the SHA-256, byte count and $$ count of the Round 02 document and its state crystal. | certificate | PASS | exact-arithmetic certificate |  |
| Round 02's PROVED labels (positive-kernel legality, explicit kernel error, legal coefficient quantization, measure-form inclusion, constructive dictionary existence) are analytic proofs. | reading | PASS | label only | 5 PROVED labels; 4 PASS; 8 PASS by cited theorem; 6 n/a |
| Round 02 cites Bari, Zygmund, Shah–Urschel–West (arXiv:2607.23375) and Schneider. | reading | PASS | label only | arXiv:2607.23375 |

Open: Placements: "Anything about placements. The dictionary is a shape compiler; Round 03 is the placement certificate."; Deferred computations: "The deferred computations C02-1..3 — the dictionary's actual size, the compiler on adversarial shapes, pruning — which Round 02 marks `COMPUTE-DEFERRED`."; Sharpness: "That `E_N`, or anything built on it, is sharp. It is valid; it is not tight."; Reflection watch item next checked in Round 03: "that is where the watch item is next checked."

Unclear: The package's sanity.py was not run; the sanity numbers were recomputed independently from exact multipliers.

### RUN-006 — Round 03 - placement certificate (AMRAL_Lebesgue_Finite_Closure_Round_03_Placement_Certificate_v0.1.md, AMRAL-LUC-FC-R03, dated 2026-09-18) and its package (VALIDATION.json, State_Crystal)

Report: `reports/RUN-006-ROUND03-PLACEMENT-CERTIFICATE.md`

> **Verdict: PASS.** For a fixed candidate cover `U`, Round 03 turns every rigid placement of every target into finite data — two reflection branches × `P` orientations × an `M`-direction linear program — with explicit errors `η_M` (upper side) and `ρ_P` (lower side). Every step holds.

**For the paper.** Round 03's placement certificate for a fixed cover (two reflection branches, P orientations, M-direction LPs, bracket G − ρ_P ≤ M_U(K) ≤ G + η_M) holds, its key steps resting on cited LP/convex duality, and it does enumerate the reflection branch, which a constructed chiral pair shows is necessary; its own sanity example is uninformative as a test, and it was not run on any real candidate cover.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1: M_U is translation-invariant in U, R_U ≤ D_U, and R_K ≤ 1. | theorem | PASS by cited theorem | cited theorem |  |
| §2–3: reflections are a discrete choice Q_{σ,φ} = R_φ S^{(1−σ)/2}, and for σ = −1, h_K(Qᵀu_θ) = h_K(φ − θ). | definition | PASS | exact-arithmetic certificate | σ = ±1 |
| Round 03 enumerates both reflection branches σ = ±1 in its definition, its finite grid value and its pseudo-algorithm (§29), and the reflection branch is necessary. | method | PASS | numerical only | ε = 0.002; G₊ − ρ_P = 0.0029998 − 0.0023651 = 0.00063 > 0; P = 720; M = 512; R_K = 0.54204; 6.3 × 10⁻⁴ |
| §4–5: for fixed orientation the translation minimum exists and ‖t*‖ ≤ 2L. | theorem | PASS | analytic (proof checked) | ‖t*‖ ≤ 2L |
| Thm 6.1: the translation problem's value equals a dual over zero-mean probability measures. | theorem | PASS by cited theorem | cited theorem | 168 grid LPs; M = 96; 12 random instances; 63 two-active; 105 three-active; 10⁻⁹ |
| §7: an optimal dual needs exactly two antipodal directions (weights ½) or three enclosing the origin. | theorem | PASS | analytic (proof checked) |  |
| §8 (R01-CORRECTION-001): Round 01's one-active branch is impossible. | theorem | PASS | analytic (proof checked) |  |
| §9–13: on M equally spaced directions the value is a three-variable LP, with m_M ≤ m ≤ m_M + η_M, η_M = L(1 + 2/cos(π/M))π/M, and ‖t*_M‖ ≤ 2L/cos(π/M). | theorem | PASS by cited theorem | cited theorem | η_M = L(1 + 2/cos(π/M))π/M; η₆₄ = 0.25349455…; ‖t*_M‖ ≤ 2L/cos(π/M); 120 grid LPs |
| §14: a local η from a verified ‖t_M‖. | theorem | PASS | analytic (proof checked) |  |
| §15–17, Thm 17.1: m_σ is R_K-Lipschitz in φ, and on P samples per branch G − ρ_P ≤ M_U(K) ≤ G + η_M with ρ_P = R_Kπ/P. | theorem | PASS by cited theorem | cited theorem | ρ_P = R_Kπ/P; ρ₄₈ = π/48; coarse M = 16, P = 12; fine M = 256, P = 96; six instances |
| §18–24: interval enclosure, witness schema, merge with Round 02's dictionary, the certified-universal and certified-non-universal rules, and the ambiguous zone. | reduction | PASS | analytic (proof checked) |  |
| §25–26: universality pays only η_M and non-universality only ρ_P (the ∀K ∃g quantifier asymmetry in the error budget). | theorem | PASS | analytic (proof checked) |  |
| §30: the sanity example (ellipse with semi-axes 0.72 and 0.52, chiral target r = ½ + 0.22 cos 3θ + 0.10 sin 5θ) passes. | computation | PASS | numerical only | G₆₄,₄₈ = −0.02; G₅₁₂,₁₉₂ = −0.02; 96 and 384 orientations; M_U(K) ≥ −0.02; M_U(K) ≤ −0.0199972; −0.0199999846; within 3 × 10⁻⁶ |
| No bound on a_Leb is claimed (§35, §37). | reading | PASS | label only |  |
| Round 03's six PROVED (continuous translation dual, finite support LP, support-grid error, orientation Lipschitz, full placement bracket, shape-and-placement certificate) are analytic. | reading | PASS by cited theorem | cited theorem | 6 PROVED; 6 PASS; 7 PASS by cited theorem; 3 n/a |
| The package's description of the document (SHA-256, bytes, $$ count) matches the archive. | certificate | PASS | exact-arithmetic certificate |  |

Open: No real candidate cover tested: "Round 03 builds the certificate for a *fixed* `U`; running it on Pál's or Gibbs' cover is its deferred C03-1."; No interval-certified independent verifier: "this tree's LPs are independent of the package's, but they are floating-point, not interval-certified."; Candidate-cover optimization: "Anything about the candidate-cover optimization, which is Round 04."; The §30 sanity test cannot detect orientation/reflection errors: "it is a test that could not have failed on the thing it was there to test."

Unclear: B9's non-fit of the rotation-only branch is described as 'proved' via Thm 17.1, but the LPs are floating-point (HiGHS, ~10⁻⁹) with a certified gap 6.3 × 10⁻⁴; strength recorded as 'numerical only' because the report states the LPs are not interval-certified.

### RUN-007 — Round 04 - candidate cover compiler (AMRAL_Lebesgue_Finite_Closure_Round_04_Candidate_Cover_Compiler_v0.1.md, AMRAL-LUC-FC-R04, dated 2026-09-18) and its package (VALIDATION.json, State_Crystal)

Report: `reports/RUN-007-ROUND04-COVER-COMPILER.md`

> **Verdict: PASS**, with one reproduction discrepancy explained to the last digit.

**For the paper.** Round 04's cover compiler holds: anchored covers lie in radius R = 2Ā = 1.6881871888, upward-quantized outer polygons satisfy U ⊆ P ⊆ U + εB and Area(P) ≤ Area(U) + β (via Steiner's formula), giving A^poly − β ≤ a_Leb ≤ A^poly and a sound global lower rule; the bounds are valid but loose (β as used is 0.906), no bound is computed, and the package's recorded §33 polygon differs from the defined one by a rounding-guard artefact without affecting any assertion.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §2: a universal cover can be translated to contain the disc B_{1/2}(0), since the disc of diameter 1 is a test set invariant under every isometry. | theorem | PASS | analytic (proof checked) |  |
| Thm 4.1, Cor 4.2: a convex set containing B_r(0) with area at most A lies in B_{A/r}(0); with Gibbs' Ā = 0.8440935944, R = 2Ā = 1.6881871888. | theorem | PASS | analytic (proof checked) | Ā = 0.8440935944; R = 1.6881871888; 200 random convex bodies |
| §5: the anchored, area-bounded class of covers is compact. | theorem | PASS by cited theorem | cited theorem |  |
| Thm 8.1, Cor 8.2: the upward-quantized outer polygon P contains U, so universality is inherited by P. | theorem | PASS | analytic (proof checked) | M = 128; q = 5×10⁻⁴ |
| §9: P ⊆ B_{R_P} with R_P = (R+q)/cos Δ. | theorem | PASS | analytic (proof checked) | R_P = (R+q)/cos Δ |
| Thm 10.1: P ⊆ U + εB with ε = q + (R + R_P)d_M. | theorem | PASS | analytic (proof checked) | ε = q + (R + R_P)d_M; ε with R = 2Ā: 0.0834; actual excess 13 to 136 times below ε as used; 4 to 45 times below ε with true radius |
| Thm 12: Area(P) ≤ Area(U) + β with β = 2πRε + πε². | theorem | PASS by cited theorem | cited theorem | β = 2πRε + πε²; β as used 0.906; Steiner agrees to 4×10⁻⁷ |
| §13–14: the polygon dictionary is finite, its levels reach R + q, and every anchored cover has a representative. | theorem | PASS | analytic (proof checked) | (K_q + 1)^M |
| Thm 15.1, §16: A^poly − β ≤ a_Leb ≤ A^poly, with β → 0. | theorem | PASS by cited theorem | cited theorem |  |
| Thm 17.1, §18: W is 1-Lipschitz in the cover, and outer approximation lowers W by at most ε (cannot create false non-universality). | theorem | PASS | analytic (proof checked) |  |
| Thm 19.1: a non-universal dictionary polygon prunes its whole cell. | theorem | PASS | analytic (proof checked) |  |
| §20–22: γ-inflation gives W(P^γ) = W(P) − γ at area cost 2πR_Pγ + πγ². | theorem | PASS by cited theorem | cited theorem | 2πR_Pγ + πγ² |
| §23–24: every truly universal (non-universal) dictionary polygon is eventually certified. | theorem | PASS | analytic (proof checked) |  |
| §25: global lower rule — if every dictionary polygon of area below L + β is non-universal, then a_Leb ≥ L. | reduction | PASS | analytic (proof checked) |  |
| §28: certificate-safe boundary surgery (search proposes, the independent certificate decides). | method | PASS | label only |  |
| §31–32, §37, §39: this is finite-resolution convergence, not finite exact closure; no bound is claimed; novelty against Zeng's hierarchy is undecided. | reading | PASS | label only |  |
| §33: the outer polygon of h_U = 0.515 + 0.01 cos 2θ at M = 128, q = 5×10⁻⁴ satisfies containment, excess below ε and inflation below β. | computation | PASS | numerical only | Area(U) = 0.8327576726503144; ε = 0.08339137281575232; β = 0.9063953966591665 |
| §33 / package: the recorded sanity polygon has area 0.8337053574119659 and Hausdorff excess 0.0006459849700081, as the polygon defined in §7–8. | computation | NOT AS STATED | corrected | recorded area 0.8337053574119659; definition area 0.8336391643823979; recorded excess 0.0006459849700081; definition excess 0.0006224936758761; guard 10⁻¹⁴; six of eight on-level heights |
| Round 04's ten PROVED: eight are self-contained arguments and two rest on Steiner's formula. | reading | PASS | label only | 10 PROVED; 12 PASS; 5 PASS by cited theorem; 2 n/a |

Open: No real cover or bound: "That any real cover — Pál's, Gibbs' — is universal, or any bound: Round 04 builds the compiler; running it is its deferred C04-1..4."; Practical searchability: "It is finite; the count `(K_q + 1)^M` is astronomically large, and Round 04 does not claim otherwise."; The DLMVC lagged audit of Round 04 is deliberately unread: "Anything about the lagged audit of Round 04, deliberately unread here."; Measurement cannot detect an error in ε's formula: "with `ε` this loose, it cannot detect an error in `ε`'s formula."

### RUN-008 — Round 05 - saturation gate (AMRAL_Lebesgue_Finite_Closure_Round_05_Saturation_Gate_v0.1.md, AMRAL-LUC-FC-R05, dated 2026-09-18) and its package (VALIDATION.json, State_Crystal, sanity.py)

Report: `reports/RUN-008-ROUND05-SATURATION-GATE.md`

> **Verdict: PASS.**

**For the paper.** Round 05's saturation theory holds: a_Leb = sup Λ(F) = lim λ_m over finite witness families (partly via cited compactness/Steiner/Cauchy/Minkowski results), λ_1 = π/4, λ_3 ≥ 0.8344 only by Mishra's cited theorem, and the four cell lower bounds are safe; exact finite saturation is an open conjecture and nothing is computed toward a bound. Independently the report proves Λ({D, RT}) = π/8 + √3/4 ≈ 0.8257, and notes the §9 continuity constant can be improved by a factor π.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1: convergence is not attainment; a bracket with β → 0 does not by itself give an exact finite level. | reading | PASS | label only |  |
| §2, Thm 3.1: for a finite family F ∋ D = B_{1/2} of unit constant-width bodies, Λ(F) = inf_{g_i ∈ E(2)} Area(conv ∪ g_i K_i) satisfies Λ(F) ≤ a_Leb. | theorem | PASS | analytic (proof checked) |  |
| §4: the infimum defining Λ(F) is a minimum over covers. | theorem | PASS by cited theorem | cited theorem |  |
| Thm 5.1: fit sets are closed. | theorem | PASS | analytic (proof checked) |  |
| Thm 6.1 (completeness): a_Leb = sup Λ(F) over finite families. | theorem | PASS by cited theorem | cited theorem |  |
| §7: λ_m = sup{Λ(F) : \|F\| ≤ m} increases to a_Leb. | theorem | PASS | analytic (proof checked) |  |
| §8–9: each λ_m is attained, by the continuity bound \|Λ(K) − Λ(L)\| ≤ 4πĀε + πε². | theorem | PASS by cited theorem | cited theorem | 4πĀε + πε²; sharper 4Āε + πε²; heptagon area 0.84276; Per = 0.9984 × 4Ā |
| §10: λ_1 = π/4. | computation | PASS | analytic (proof checked) | λ_1 = π/4 ≈ 0.7853981634; gap 0.0096935944 |
| §10: Mishra's family {D, RT, R5} gives λ_3 ≥ 0.8344. | reading | PASS by cited theorem | cited theorem | λ_3 ≥ 0.8344; arXiv 2608.30538 |
| Thm 11.1: if λ_m < a_Leb then some λ_{m+r} > λ_m (no false permanent saturation), via a finite separating batch of witnesses. | theorem | PASS | analytic (proof checked) |  |
| §12–13, §17: the ladder cannot plateau below the truth; one violated target need not raise Λ, a separating batch is needed. | theorem | PASS | analytic (proof checked) |  |
| Thm 14.1, §15–16: exact closure happens iff a finite lower witness meets a certified upper cover. | theorem | PASS | analytic (proof checked) |  |
| §18–19: for the quantized cell with outer polygon P_b, d_H(U, P_b) ≤ ε_b = q + 2R_b d_M. | theorem | PASS | analytic (proof checked) | ε_b = q + 2R_b d_M; 230 members over 20 cells; ε_b 2.8 to 3.2 times smaller than Round 04's ε; farthest member at 5 to 24 % of ε_b |
| §20: LB_outer = Area(P_b) − Per(P_b)ε_b − πε_b² is a safe area lower bound for the cell. | theorem | PASS by cited theorem | cited theorem |  |
| §21: LB_env = ¼∫ℓ_b is a safe area lower bound for the cell. | theorem | PASS by cited theorem | cited theorem |  |
| §22: Φ_r(s) = πr² + r√(s² − r²) − r² arccos(r/s) is the hull area of a disc and a point, increasing in s, giving LB_tip. | theorem | PASS | analytic (proof checked) | agreement ≤ 3.3×10⁻¹¹ |
| §23: LB_cell (the maximum of the four bounds) is safe. | theorem | PASS | numerical only | 20 cells; five covers; four grids; closest approach 1.3×10⁻³ |
| §24–25: a non-universal P_b or LB_cell ≥ A⁺ prunes the whole cell. | theorem | PASS | analytic (proof checked) |  |
| §26–28: strictly non-universal and strictly suboptimal limits are eventually pruned, so surviving cells localize at optimal universal covers, asymptotically. | theorem | PASS | analytic (proof checked) |  |
| §29–30: stopping criterion and architecture. | method | PASS | label only |  |
| §31: Mishra's abstract proves area ≥ 0.8344 from the disc, Reuleaux triangle and Reuleaux pentagon, and Zeng's states a_Leb = lim Λ_M; CROSS-MAP: OPEN. | reading | PASS | cited theorem | 0.8344; 2608.30538 v2; 2609.01284 v1 |
| §32–33: the Finite Witness Attainment Conjecture (∃ m⋆: λ_{m⋆} = a_Leb) and its branch-type fallback, stated as open. | reading | UNMEASURED | open |  |
| Mishra's three test bodies are achiral, so the reflection branch changes nothing for that family. | computation | PASS | numerical only | residual ≤ 3.3×10⁻¹⁶ |
| Package VALIDATION.json: the recorded cells of Round 04's §33 candidate at M = 64 … 1024 (vertex count, area, perimeter, R_b, ε_b, LB_outer, LB_combined, candidate area 0.8327576726503144, λ_1, gap). | computation | PASS | numerical only | to 10⁻¹²; nearest height 6×10⁻⁵ of a step from a level at M = 1024 |
| Package: the recorded LB_env values are the integral ¼∫ℓ_b that §21 defines. | computation | NOT AS STATED | numerical only | 200,000-point rectangle rule; M=64: recorded 0.7936755224944383, exact 0.7936755210933901, +1.40×10⁻⁹; four of five above the exact integral |
| The package's sanity assertion: every LB_combined stays below the candidate's own area. | computation | PASS | numerical only | candidate area 0.83276; LB_outer without Per·ε: 0.83273; ε_b with one R_b d_M: 0.82754; heptagon: LB_outer without Per·ε 0.84667 vs area 0.84276 |
| (Report's own example, not a Round 05 claim) Λ({D, RT}) = π/8 + √3/4, so λ_2 ≥ 0.8257. | theorem | PASS | analytic (proof checked) | π/8 + √3/4 = 0.8257117836; hull agrees to 8×10⁻¹¹; Φ(0.7) = 0.83655; 96 offsets, least increase 1.0×10⁻⁵ |

Open: No bound: "Anything about the value of `a_Leb`, or any new bound: Round 05 claims none (§38)."; Mishra's certificate: "only its abstract's statement is checked. Reproducing the certificate is Round 05's own deferred C05-1."; "That the Finite Witness Attainment Conjecture is true or false."; Termination: "That the branch-and-bound terminates: §26–28 are asymptotic, and nothing here runs a search."; λ_2: "whether the Reuleaux triangle maximizes `Λ({D, K})` is not examined."; Reflection watch item: "The watch item stays open for later rounds: any chiral witness must enumerate `σ = −1`."; One omission is invisible to measurement: "The other omission — one `R_b d_M` dropped from `ε_b` — is caught by nothing measured, here or in the package; only the derivation (D1) and the exact reproduction (P1) see it."

Unclear: §32–33 are graded 'n/a' (conjectures stated as such); mapped here to report_finding 'UNMEASURED' with strength 'open'.; The W2 item (Λ({D, RT}) = π/8 + √3/4) is the report's own result, not a claim of the source; it is included because the report treats it as a substantive finding.; Process caveat: the package's sanity.py was opened before reconstruction; the report says it contributed only that LB_env was a 200,000-point rectangle rule with R = R_b.

### RUN-009 — Round 06 - witness exchange (AMRAL_Lebesgue_Finite_Closure_Round_06_Witness_Exchange_v0.1.md, AMRAL-LUC-FC-R06, dated 2026-09-18) and its package (VALIDATION.json, State_Crystal)

Report: `reports/RUN-009-ROUND06-WITNESS-EXCHANGE.md`

> **Verdict: PASS**, with one bibliographic finding pinned.

**For the paper.** Round 06's witness-exchange compiler holds as analytic argument given Rounds 02–04, with Thm 24.1 conditional on certified minimizer-cell covers it does not construct; on the seed ladder the report proves Λ(D, B_3) = Λ(D, P_3) = π/8 + √3/4 exactly and finds Λ(D, B_3, B_5) ∈ [0.8344, 0.83478095], the lower end only by Mishra's cited certificate. Bibliographically, Xie is cited by a superseded title and Pál 1920 and Gibbs 2014 are missing from the reference list.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1: Mishra's certificate gives Λ(D, B_3, B_5) ≥ 0.8344 over a five-dimensional placement space (486,799,600 nodes; floating-point error bound 1.72×10⁻⁹). | reading | PASS by cited theorem | cited theorem | 0.8344; 486,799,600 nodes; 1.72×10⁻⁹; Λ(D, B_3, B_5) ∈ [0.8344, 0.83478095]; search optimum 0.83478094595; within 3.8×10⁻⁴ |
| §2: λ_1 = π/4 and Pál's value Λ(D, P_3) = π/8 + √3/4. | theorem | PASS | analytic (proof checked) | λ_1 = π/4; π/8 + √3/4 = 0.8257117836 |
| §2: Λ(D, B_3) ≥ π/8 + √3/4 for the Reuleaux triangle. | theorem | PASS | analytic (proof checked) | Λ(D, B_3) = π/8 + √3/4 |
| §3: a search gives upper bounds on Λ(F), never lower bounds. | method | PASS | label only |  |
| §4–5: with the disc fixed, the configuration domain is compact with ‖t_i‖ ≤ R = 2Ā. | theorem | PASS | analytic (proof checked) | R = 2Ā; ‖t_i‖ ≤ Φ⁻¹(Ā) = 0.7212; Mishra uses 0.70 |
| §6: a common rotation is a gauge, leaving d_m = 3m − 4 continuous parameters (5 for Mishra's family). | theorem | PASS | numerical only | d_m = 3m − 4; d_3 = 5; m = 2 … 7; 10⁻¹² |
| §7–8: the hull area is continuous and the minimizer set 𝔐(F) is compact. | theorem | PASS | analytic (proof checked) |  |
| §9–10: W is 1-Lipschitz in the cover (and the margin in the target) and positive off universality. | theorem | PASS | analytic (proof checked) | ε ∈ [0.01, 0.025]; eight random cover–target pairs |
| Thm 11.1: a non-saturated family has a uniform violation margin δ_F = min_{𝔐(F)} W > 0. | theorem | PASS | analytic (proof checked) | δ_{{D}} = 1/√3 − 1/2 |
| §13–16 (Thm 14.1): a Round 02 dictionary finer than δ_F separates all minimizers with a finite batch. | theorem | PASS | analytic (proof checked) |  |
| §17–18 (Thm 18.1): a batch with separation margin S_F(B) > 0 strictly raises Λ. | theorem | PASS | analytic (proof checked) |  |
| §19: any new cover lies at least s from the old minimizer set, d_H(V, 𝔐(F)) ≥ s. | theorem | PASS | analytic (proof checked) | d_H = 0.0773502692; 300 placements no closer than 0.0832 |
| §20: Λ rises by at least an abstract modulus γ_F(s) > 0. | theorem | PASS | analytic (proof checked) | γ(s) = Φ_{1/2}(½ + s) − π/4; γ(δ) = 0.0134379; actual gain √3/4 − π/8 = 0.0403136 |
| §21–23: minimizer cells with radii τ_j, the robust cell-witness rule μ − τ > 0, and a finite set cover. | method | PASS | analytic (proof checked) |  |
| Thm 24.1: if τ_n + ε_n + ζ_n < δ_F/2 the certified search must eventually succeed. | theorem | PASS | analytic (proof checked) | τ_n + ε_n + ζ_n < δ_F/2 |
| §25: the compiler is a semi-decision, not a saturation test. | reading | PASS | label only |  |
| Thm 26.1, §27: gK ⊆ L implies Λ(G ∪ {L}) ≥ Λ(G ∪ {K}); P_n ⊆ B_n, so Mishra's step from polygons to Reuleaux polygons is a dominance upgrade. | theorem | PASS | analytic (proof checked) | n = 3, 5, 7, 9; Λ(D, B_3, B_5) − Λ(D, P_3, P_5) ≥ 0.8344 − 0.8335974 = 0.000803; Λ(D, P_3, P_5) ≤ 0.8335973878 |
| §28: witness-ancestry labels. | definition | UNMEASURED | label only |  |
| §29, §37: seed sanity δ_{D,B_3} = 1/√3 − 1/2 ≈ 0.0773502692 and the strict ladder 0.785398… < 0.825711… < 0.8344. | computation | PASS | numerical only | 1/√3 − 1/2 ≈ 0.0773502692; M_D(B_3) = 0.0773502691896258; M = 3072; to 10⁻¹⁶ |
| §30–35: strategy, oracle, compiler, search/certificate separation; Mishra does not do the exchange; Gibbs 2014 values are search priors (upper bounds). | method | PASS | label only | Gibbs 0.83699098 |
| Reference 5: Xie's preprint cited as 'A Reproducible Certificate for the Brass–Sharifi Lower Bound in Lebesgue's Universal Cover Problem'. | reading | NOT AS STATED | corrected | v3 (13 Jun, bound 0.83201); v4 (8 Jul, bound 0.833) |
| The text relies on Pál 1920 (§2) and Gibbs 2014, arXiv:1401.8217 (§3, §35). | reading | NOT AS STATED | corrected | arXiv:1401.8217 |
| The package VALIDATION.json values (λ_1, Pál's value, 0.8344, gap, strict-ladder flag, margin, d_3 = 5, toy exchange budget ε = τ = ζ = δ/8). | computation | PASS | label only | δ/8; 5δ/8 |
| §42: no new bound and no identification of the next witness is claimed; reflections are set aside in §6 as a discrete branch without counting. | reading | PASS | label only | 2^{m−2} branches for m ≥ 2 |

Open: The next witness: "Round 06 defers it (C06-2 … 6), and nothing here searches for one."; Mishra's certificate: "only its statement, its setup and his two published placements are checked; the certificate is not re-run."; Thm 24.1 hypothesis: "That certified minimizer-cell covers with `τ_n → 0` can be built in practice"; Exact Λ values: "`Λ(D, P_3, P_5)` or `Λ(D, B_3, B_5)` themselves: the searches give upper bounds only."; Reflection watch item: "The watch item carries on."

Unclear: §28 is graded 'n/a' (definitions); mapped to report_finding 'UNMEASURED'.; The interval [0.8344, 0.83478095] combines Mishra's cited lower bound with an upper bound from a (non-certified) placement/search; the upper side is a constructed placement, the lower side is cited.

### RUN-010 — Round 07 - minimizer atlas (AMRAL_Lebesgue_Finite_Closure_Round_07_Minimizer_Atlas_v0.1.md, AMRAL-LUC-FC-R07, dated 2026-09-18) and its package (VALIDATION.json, LOCAL_COMPUTE_SPEC.json, State_Crystal, search_prior.py)

Report: `reports/RUN-010-ROUND07-MINIMIZER-ATLAS.md`

> **Verdict: PASS.**

**For the paper.** Round 07's lifting theorems (cell-local lifting, witness switching, robust transfer with cost 4πTτ + πτ², loose by π) hold, and every external number matches Mishra's paper and repository; the four-body search gives only Λ(D, B_3, B_5, B_7) ≤ 0.8364914 and no counterexample to gate R07-G1 (≥ 0.835), which remains uncertified, and the recorded B_13 lift is not a minimum.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1, §5: Mishra's external certificate is audited as stated, and it stores one bit per node so it holds no atlas. | reading | PASS | cited theorem | worst leaf slack 1.240900×10⁻⁵; 1.72×10⁻⁹; 486,799,600 nodes; 19,648 and 11,351 boxes per second |
| §2–4: the seed is parameterized by q = (x_3, y_3, φ_5, x_5, y_5), φ_5 ∈ [0, 2π/5), with translation domain \|t_n\| ≤ t_n(T) from the disc–point area function f (d⋆ = 0.693830648713 at T = 0.8344). | computation | PASS | analytic (proof checked) | d⋆ = 0.693830648713; t_3 = 0.192366411526; t_5 = 0.195890714644; match to 5×10⁻¹³; f ≡ Φ_{1/2} at 40 digits |
| §6–8: only the sublevel set Q_{<T} = {q : A(q) < T} needs a new witness; the emitter covers Q_{<T}. | reduction | PASS | analytic (proof checked) |  |
| §9: J_K(H) = min_g Area(conv(H ∪ gK)) ≥ Area(H). | theorem | PASS | analytic (proof checked) |  |
| Thm 10.1: each cell is closed either by the base or by one witness (cell-local lifting). | theorem | PASS | analytic (proof checked) |  |
| Thm 11.1: witness-switching bound Λ(F ∪ B) ≥ min_q max_K J_K(H(q)). | theorem | PASS | analytic (proof checked) | joint B_7+B_9 hull 0.8379417; single-witness 0.8371356 and 0.8358768; additive claim would give 0.8379843 |
| §13: lazy activation — cells pruned by the base bound never grow new dimensions. | theorem | PASS | analytic (proof checked) |  |
| §14: erosion extends to a lifted witness. | theorem | PASS by cited theorem | cited theorem |  |
| §15–16: cell-switching verifier and local batches. | method | PASS | label only |  |
| Thm 17.1: a cell's representative transfers to the whole cell at area cost ω_T(τ) = 4πTτ + πτ². | theorem | PASS by cited theorem | cited theorem | ω_T(τ) = 4πTτ + πτ²; sharper 4Tτ + πτ² |
| §18–19: at T_1 = 0.8350 the domain widens to t_3 ≤ 0.194856180909 and t_5 ≤ 0.197820670401. | computation | PASS | numerical only | T_1 = 0.8350; t_3 = 0.194856180909; t_5 = 0.197820670401; 120 placements beyond the bound ≥ 0.8344 |
| §20: the best exhibited seed placement q† has exact-kernel area 0.834780945912 and rigorous outer area 0.834781190917, with headroom 2.19×10⁻⁴ to 0.835. | computation | PASS | numerical only | 0.834780945912; 0.834781190917; this tree 0.8347809459115; classical family 0.833597536505196; φ_5 mod 2π/5 = 2.4128951×10⁻⁵; headroom 2.1905×10⁻⁴ |
| §21, §24: the four-body family D, B_3, B_5, B_7 has (logged) optimum 0.836494901, and the base moves when the fourth body arrives. | computation | PASS | numerical only | 0.836494901; search 0.8364914416; 3.46×10⁻⁶ below; Λ(D, B_3, B_5, B_7) ≤ 0.8364914; 6.4×10⁻⁴ below the fixed-seed lift |
| §21: naive 8-D certification at T = 0.835 costs about 5.527×10¹² boxes, 5635.8 days (an empirical extrapolation). | computation | PASS | numerical only | 5.527×10¹² boxes; 5635.8 days; exponent 5.5035; 4.946×10¹² / 5,043 days; 4.848×10¹² / 4,943 days |
| §22–25: the search-only fixed-seed lift ranking B_7 > B_11 > B_13 > B_9 (0.83713, 0.83613, 0.83597, 0.83563). | computation | PASS | numerical only | J_{B_7} = 0.8371345; J_{B_11} = 0.8361324; J_{B_13} = 0.8359056; J_{B_9} = 0.8356307; package sampled values 1.4–1.8×10⁻⁶ low |
| §23: the recorded B_13 lift 0.83597 (package 0.8359676) is the minimum over placements. | computation | NOT AS STATED | corrected | recorded 0.8359676; exact minimum 0.8359056; 6.2×10⁻⁵ lower |
| §26–27: gate R07-G1 — every base placement below 0.835 is lifted by B_7 to at least 0.835. | specification | PASS | open | 0.835; best lies 1.49×10⁻³ above the target |
| §28–35: witness pool, policy, lineage, status taxonomy, local compute spec and certificate formats. | specification | PASS | label only |  |
| The package VALIDATION.json search-prior block and LOCAL_COMPUTE_SPEC agree with the text and the repository. | computation | PASS | numerical only | to 10⁻¹³; 1536 sampled support directions |
| §43: no new bound and no certified witness is claimed. | reading | PASS | label only |  |

Open: Gate R07-G1 and any bound above 0.8344: "the four-body search is evidence, not a certificate, and the certificate is Round 07's deferred C07-1 … 5."; Exact lift values: "`J_{B_n}` or `Λ(D, B_3, B_5, B_7)` themselves: every search value here is an upper bound."; Efficiency: "That atlas lifting is faster than the naive 8-D search. Round 07 says it does not know (§33), and nothing here measures it."; Ranking fragility: "two independent searches disagreed on two of four entries, and neither error changed the order."

Unclear: Gate R07-G1 is graded PASS in the theorems table as a correctly stated gate equivalent to Λ(D, B_3, B_5, B_7) ≥ 0.835; the gate itself is not established, hence strength 'open'.

### RUN-011 — Round 08 - conditional lift certificate (AMRAL_Lebesgue_Finite_Closure_Round_08_Conditional_Lift_Certificate_v0.1.md, AMRAL-LUC-FC-R08, dated 2026-09-18) and its package (VALIDATION.json, LOCAL_COMPUTE_SPEC.json, CERTIFICATE_SCHEMA.json, State_Crystal, sanity.py, verifier_skeleton.py)

Report: `reports/RUN-011-ROUND08-CONDITIONAL-LIFT.md`

> **Verdict: PASS, with one gap in a specification.**

**For the paper.** Round 08's proof grammar for the 0.835 milestone holds analytically (motion bound, common core, Reuleaux erosion, nested soundness), and two hard lift leaves close at half-width 10⁻⁴ in floating point without an error term; no bound above 0.8344 is established, the verifier is a skeleton, and the verifier invariants and schema lack a mirror-image root for chiral witnesses (PARTIAL), which does not affect the achiral B_7 path.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1: a placement box moves a body lying within R_K of its centre by at most δ = √(h_x² + h_y²) + 2R_K sin(h_φ/2) in Hausdorff distance (Mishra's Lemma 22 with any R_K). | theorem | PASS | analytic (proof checked) | 1,400 random placements; attained to 95%; 8,192 directions |
| Thm 2.1 (common core): if h_C + δ ≤ h_{K_0}, then C lies in every placement of the box. | theorem | PASS | analytic (proof checked) |  |
| §3, §25: three core providers — Mishra's erosion ∩B(V_j, 1 − δ), h_K − δ while the curvature radius stays above δ, and any core with a finite inclusion certificate. | theorem | PASS | analytic (proof checked) |  |
| §4: Reuleaux erosion (Mishra's Lemma 21) as a special case; the core is empty once 1 − δ falls below the circumradius. | theorem | PASS | analytic (proof checked) | 0.422649730810; 0.474268887881; 0.487141568364; to 5×10⁻¹³ |
| §5: B_3's orientation is the gauge, so δ_3 has no rotation term. | theorem | PASS | analytic (proof checked) |  |
| §6–7, §19: B_7's root is φ_7 ∈ [0, 2π/7), \|x_7\|, \|y_7\| ≤ t_7⋆ = 0.196935046771, with lift-box core C_7 = ∩B(V_j, 1 − δ_7). | computation | PASS | numerical only | R_7 = 0.512858431636277; t_7 = 0.1969350467711682; t_7⋆ = 0.196935046771 |
| §8–9: every actual hull contains conv(D ∪ C_3 ∪ C_5 ∪ C_7), and the verifier may use inner points of the cores. | theorem | PASS | analytic (proof checked) |  |
| §10–11: base leaves are BASE-PRUNED or LIFT-CLOSED, lift leaves PRUNED, nothing UNRESOLVED. | specification | PASS | label only |  |
| Thm 12.1 (nested soundness): a complete certificate gives Λ(F_3 ∪ B) ≥ T_1. | theorem | PASS | analytic (proof checked) | T_1 = 0.835 |
| §13: a failing cell is split or handed to the next witness, never lifted jointly. | method | PASS | label only |  |
| §14: the witness record lists eleven fields. | specification | PASS | label only | eleven fields |
| §15–16: emitter and verifier with invariants V1–V10 (V3 states the witness root). | specification | PARTIAL | open |  |
| The package's CERTIFICATE_SCHEMA.json carries §14's witness record. | specification | PARTIAL | open | six required fields; five of eleven fields missing |
| §17–18: the lift stream and package layout. | specification | PASS | label only |  |
| §20: the needed increment 2.19×10⁻⁴ against 1.49×10⁻³ found by search, ratio 6.83. | computation | PASS | numerical only | 0.000218809083; ratio 6.8320; 6.82 with RUN-010's 0.8364914 |
| §8–9, §12, §21–23: lift leaves can close (LIFT-CLOSED) at 0.835 when narrow enough, and a coarse failure is no evidence against B_7. | computation | PASS | numerical only | h = 10⁻⁴: 0.8368148 (seed) and 0.8361550 (four-body); h = 10⁻³: 0.8342258 and 0.8337170 unresolved; leaves narrower than about 4×10⁻⁴; margins 1.2×10⁻³ and 1.8×10⁻³ |
| §24, §26–27: only an explicit placement with rigorous outer area below T_1 is a certified negative; inner points need only a floating margin. | method | PASS | label only |  |
| §28–31: the same theorem for any witness catalog, priority B_7, B_11, B_13, B_9 then asymmetric Fourier bodies, deduplication and six proof obligations with root_domain(K, T). | theorem | PASS | label only |  |
| §16 V8: S trees with L leaves have 2L − S nodes. | theorem | PASS | numerical only | 200 random forests; N = 2L − S |
| §32–33: compute spec and sanity tests S1, S2 (S1 body h = ½ + 0.02 cos 3θ). | computation | PASS | numerical only | t_7(0.835) = 0.19693504677116785; δ = 0.005102770687734223; curvature minimum 0.33489722931226573; 37 of 2,000 placements; deficit ≤ 3.6×10⁻⁴ |
| The package's verifier_skeleton.py is the independent verifier. | reading | UNMEASURED | not measured |  |
| §38: no new bound is claimed. | reading | PASS | label only |  |

Open: Mirror branch for chiral witnesses missing from V3, the lift stream and the schema: "Until then the verifier invariants are PARTIAL. They are complete for the `B_7` milestone and for every Reuleaux fallback, and incomplete for the Fourier fallback."; No bound: "**`Λ(D, B_3, B_5, B_7) ≥ 0.835`, or any bound above 0.8344.** Two leaves are two cells of an atlas nobody has built."; No rigorous leaf: "The inner hulls here are computed in floating point, with no error term. ... a certificate needs §26's `e_fp`."; Affordability: "**That the atlas is affordable** at leaf widths near `4×10⁻⁴` in eight coordinates. That is Round 09's subject."; "**That any verifier works:** the package's verifier is a skeleton."

Unclear: The report's registry tally is 10 PASS, 1 PARTIAL, 18 n/a; it states the registry PARTIAL is §36's SPECIFIED for the verifier invariants, while the theorems table puts PARTIAL on §15–16. Both are recorded as the same gap.

### RUN-012 — Round 09 - adaptive atlas (AMRAL_Lebesgue_Finite_Closure_Round_09_Adaptive_Atlas_v0.1.md, AMRAL-LUC-FC-R09, dated 2026-09-18) and its package (VALIDATION.json, LOCAL_COMPUTE_SPEC.json, State_Crystal, sanity.py)

Report: `reports/RUN-012-ROUND09-ADAPTIVE-ATLAS.md`

> **Verdict: PASS.**

**For the paper.** Round 09's cost layer is sound: hull non-expansiveness, the sharpened transfer T + 4Tτ + πτ² (from Per ≤ 4·Area, within 5.3% of tight), the resolution formula τ⋆ ≈ 4.47×10⁻⁴, the scheduler (optimal for the radius bound), core dominance and the depth bound all hold analytically; measured at two leaves the exact common core closes cells about 2.2 times wider than the representative prune, witness dominance is vacuous for a constant-width pool, and no bound is claimed.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Thm 1.1, §2: the convex hull does not increase Hausdorff distance, so a cell's hulls stay within τ(C) = max_i δ_i of its centre's hull. | theorem | PASS | analytic (proof checked) | 80 random cells; d_H/τ ≤ 0.77 |
| §3–4: for D + B_3 + B_5, τ = max(δ_3, δ_5); with a witness box, s = max(τ, δ_K). | theorem | PASS | analytic (proof checked) |  |
| §5 (R07-STRENGTHENING-001): a body holding B_r has Area ≥ (r/2) Per, so Per ≤ 4 Area for a hull holding B_{1/2}. | theorem | PASS | analytic (proof checked) | smallest 4 Area/Per = 1.0535; within 5.3% of tight |
| Thm 6.1, §7: a centre hull of area at least T + 4Tτ + πτ² closes its cell (improving Round 07's 4πTτ + πτ²; labelled a strengthening, not a correction). | theorem | PASS | analytic (proof checked) | T + 4Tτ + πτ²; held with at least 1.8τ to spare on 30 cells |
| §8–9, §29: a margin m closes cells of radius up to τ⋆(m, T) = (√(4T² + πm) − 2T)/π; at the four-body headroom τ⋆ ≈ 4.47×10⁻⁴ against 1.42×10⁻⁴ under Round 07's cost. | computation | PASS | exact-arithmetic certificate | m = 0.001494901; τ⋆ = 4.47386884564525×10⁻⁴; Round 07's 1.42461518527477×10⁻⁴; ratio 3.14040513669 |
| §10–14: split the axis that most reduces τ; at the 0.835 root that is φ_5; Mishra's weight R_5 h_φ is the first-order term of 2R_5 sin(h_φ/2); the scheduler is optimal for the certified radius bound. | method | PASS | numerical only | d_3 = 0.2755682538; τ_root = 0.6046803712; after halving φ_5: 0.4442456056; after halving x_5: 0.5460899294; 16 cuts: 0.0545 vs 0.0556 |
| §0 E, §14, §20: Mishra's code splits with weight R_5 h_{φ5} and uses body pieces with support c_x cos θ + c_y sin θ + r. | reading | PASS | label only | commit da956b1; children widened by 1e-11 relative |
| §15–17: the prune cascade (a-priori, representative, common core, then split) with proof-mode tags APR, REP, CORE, SUPPORT-CORE, OTHER-VERIFIED is sound. | method | PASS | analytic (proof checked) | representative closes up to 3.29×10⁻⁴ and 2.30×10⁻⁴; core closes up to 7.30×10⁻⁴ and 5.08×10⁻⁴; ratio 2.22 and 2.21; 31 support pieces |
| §18–21, Thm 19.1: a lift tree closing one base cell also closes any cell whose base core contains the first's; the containment is checkable in closed form (§20). | theorem | PASS | analytic (proof checked) | 36 pairs; same centre 12 of 12; shifted 11 of 12; parent inside child 5 of 12; 56 trials; 6.2×10⁻⁷ |
| Thm 22.1, §23: a witness containing another lifts at least as high; Pareto pruning of the witness pool. | theorem | PASS | analytic (proof checked) | larger by up to 0.091; 2.2×10⁻¹⁶ |
| §24: weighted set cover over witnesses. | method | PASS | label only |  |
| §25–26: full refinement cycles give τ_r ≤ 2^{−r} S_0, so a positive margin closes at finite depth; what survives is the zero-margin frontier. | theorem | PASS | analytic (proof checked) | S_0 = 0.6101; 11 cycles; 55 halvings; 2^55 cells; 41 cells × 25 cycles |
| 結論 J: a positive margin at some representative gives finite closure depth. | theorem | NOT AS STATED | corrected |  |
| §27–28: priority score and split-or-lift heuristics. | method | UNMEASURED | label only |  |
| Package VALIDATION.json: root δ_3, δ_5, τ, child radii, best axis, τ⋆ new and old and ratio, Ω at 10⁻⁴. | computation | PASS | numerical only | to 4.5×10⁻¹³ relative |
| The package's LOCAL_COMPUTE_SPEC implements §17's proof-mode tags. | specification | PARTIAL | open |  |
| The package's sanity.py validates the recorded numbers. | computation | PASS | label only |  |
| Round 09 adds no proof domain and claims no bound above 0.8344. | reading | PASS | label only |  |

Open: Gains unmeasured: "**The scheduler's or the cache's gain.** Round 09 defers both (C09-3, 5). The measurements here are two points and one path."; Reuse frequency: "**How often cross-cell reuse applies** in a real atlas: that too is C09-3."; No bound: "**Any bound above 0.8344.** Round 09 claims none."; Prune comparison is local: "That is a measurement at two points, not a count, and Round 09 defers the count (C09-4, 5)."

Unclear: §27–28 and §30–33 are graded 'n/a'; mapped to report_finding 'UNMEASURED'.

### RUN-013 — Round 10 - reference dry run (AMRAL_Lebesgue_Finite_Closure_Round_10_Reference_Dry_Run_v0.1.md, AMRAL-LUC-FC-R10, dated 2026-09-18) and its package: reference certificate (manifest.json, lift_00/01/10/11.bin, verification_summary.json), VALIDATION.json, State_Crystal, reference_emitter.py, reference_verifier.py

Report: `reports/RUN-013-ROUND10-REFERENCE-DRY-RUN.md`

> **Verdict: PASS, with one overstatement and one gap in the specification.**

**For the paper.** Round 10's reference certificate is a local dry run only (a box of 1.1×10⁻¹⁸ of the base root): its four lift trees (11,372 nodes, 5,688 leaves) replay and re-emit byte for byte, and with the exact core every leaf clears 0.835 by at least 3.79×10⁻⁴ in floating point without a derived error term; the split rule's tie-break and child order are specified only in code, and the verifier is not independent in its core construction.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1–2: the local base domain is Mishra's q† (φ_5 mod 2π/5) with half-widths (4, 4, 40, 4, 4)×10⁻⁵, split once in φ_5 and once in x_5 into four leaves, all LIFT-CLOSED. | specification | PASS | numerical only | (4, 4, 40, 4, 4)×10⁻⁵; base areas 0.8347864, 0.8347946, 0.8347863, 0.8347947; base cores alone 0.8346497–0.8346579; 1.1×10⁻¹⁸ of the root's volume |
| §3: each leaf is lifted over B_7's whole root φ_7 ∈ [0, 2π/7), \|x_7\|, \|y_7\| ≤ 0.19693504677116785. | specification | PASS | label only | 0.19693504677116785; t_7 rebuilt to 5×10⁻¹³ |
| §4: split the axis whose halving leaves the smallest δ_7 (argmin). | method | PASS | label only |  |
| The text and manifest specify the lift tree fully enough for an independent verifier to replay it. | specification | PARTIAL | open | ties sent to y: 108 of 1,417 leaves fail; higher child first: 88 fail |
| §5: a leaf's lower bound is the hull of the cores' contact points in 96 directions and the inscribed 96-gon. | theorem | PASS | numerical only | 96 directions; agrees to within 7×10⁻¹⁶; tree 00: 3.869877959×10⁻⁷ vs 3.869877956×10⁻⁷ |
| §6: the verifier re-implements the core reconstruction independently of the emitter (independent hull routine). | method | PARTIAL | corrected |  |
| §7: the emitter prunes at 0.8350002 and the verifier accepts at 0.835 after a 10⁻⁸ pad, which is explicitly not a derived error bound. | specification | PASS | label only | 0.8350002; 10⁻⁸; T_search − T − pad = 1.9×10⁻⁷ |
| §8–10: the emitted trees (2833/1417/21 and 2853/1427/21, twice each; totals 11,372 nodes and 5,688 leaves) replay as REFERENCE-VERIFIED with the recorded worst verified slacks. | certificate | PASS | numerical only | 11,372 nodes; 5,688 leaves; N = 2L − 1; 355–357 bytes per tree; depth 21 |
| Locally, for every base placement in the box and every placement of B_7 in its root, the hull has area at least 0.835. | certificate | PASS | numerical only | worst exact slack 3.79×10⁻⁴; 3.91×10⁻⁴; 3.80×10⁻⁴; 3.89×10⁻⁴; 0 of 5,688 below 0.835 |
| The emitter, re-run from the text's bound with tie-break and child order fixed, reproduces the certificate files. | certificate | PASS | numerical only | search threshold 0.8350002; depth limit 24 |
| (Report's own measurement) The 96-direction sampled bound's cost against the exact core bound on the same four roots. | computation | PASS | numerical only | 1417/21 vs 813/19 (ratio 1.74); 1427/21 vs 808/19 (ratio 1.77); 1.75× fewer; tightest sampled slack 3.5×10⁻⁷ |
| §11–13: the run is an implementation proof of the architecture, not Λ(D, B_3, B_5, B_7) ≥ 0.835 and not a_Leb ≥ 0.835. | reading | PASS | label only | 1.1×10⁻¹⁸ |
| §14–17: certificate contents, emitter, verifier and one bit per node density. | specification | PASS | label only | 355–357 bytes per tree |
| §18–19: the state machine and fail-closed semantics, including rejection of extra non-padding nodes. | specification | PASS | label only |  |
| §20: the claim ledger (in Round 00's ten statuses). | reading | PASS | label only |  |
| Package: the manifest and VALIDATION.json agree with the text and mark the run local (LOCAL-DOMAIN-DRY-RUN-NOT-GLOBAL-THEOREM, global_lebesgue_claim: false); document and script digests match. | certificate | PASS | exact-arithmetic certificate | target 0.835; 96 samples; depth limit 24 |

Open: "**Anything beyond the box.** It is `1.1×10⁻¹⁸` of the base root, around the one placement where the base is lowest"; Rigorous arithmetic: "the package's pad is not a derived bound. The margins, `3.8×10⁻⁴`, are far above rounding, but a certificate needs the error term written down."; "**The global run's cost.** The two tree sizes compare two bounds on four roots, not a full atlas."; Tie-break and child order must be recorded for shard-alone verification: "That verifier needs both rules in the manifest."

### RUN-014 — Round 11 - global shards (AMRAL_Lebesgue_Finite_Closure_Round_11_Global_Shards_v0.1.md, AMRAL-LUC-FC-R11, dated 2026-09-18) and its package: GLOBAL_SHARD_SPEC.json, VALIDATION.json, State_Crystal, reference shards (global_manifest.json, shard-A/B/C.json, merge_verification_summary.json, copy of Round 10's certificate)

Report: `reports/RUN-014-ROUND11-GLOBAL-SHARDS.md`

> **Verdict: PASS, with one mischaracterized forecast and one gap in the specification.**

**For the paper.** Round 11's frontier and sharding theorems (coverage, replacement, partition merge, N = 2L − S) hold analytically, and the reference shards are exactly Round 10's local certificate regrouped; nothing global is established, the digest and Merkle rules are recoverable only from code, the manifests omit most required fields, and §19's depth forecast is not an upper bound (FAIL, used only for scheduling).

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1–3: seeds are binary paths from the root; a frontier is prefix-free and complete when every descent from the root meets it. | definition | PASS | label only | 200 random frontiers |
| Thm 4.1: a complete frontier covers the root. | theorem | PASS | analytic (proof checked) | 400 random root points |
| Thm 5.1: replacing a seed by its two children keeps the frontier complete. | theorem | PASS | analytic (proof checked) |  |
| Thm 8.1, §9: shards that partition the frontier can be verified separately and merged; order and workers are irrelevant. | theorem | PASS | analytic (proof checked) |  |
| §10–11: a forest of S roots has N = 2L − S nodes, also after resharding. | theorem | PASS | numerical only | 200 random forests; N = 2L − S |
| §10, §17, §18: Mishra's verifier audits 2L − n_seed = N, seeds at depth 22, and certgen.py re-queues an unfinished DFS stack. | reading | PASS | label only | depth 22; 2^{22} seeds; commit da956b1 |
| §12–13: shards are bound by SHA-256 of a canonical payload and by a binary Merkle root. | specification | PASS | label only |  |
| The text and spec specify the canonical payload form and Merkle rule well enough for a second verifier to rebuild the digests. | specification | PARTIAL | open |  |
| §14–15: seed identity (Mode A implicit, Mode B explicit axis/side paths, recommended) and exact dyadic boxes. | specification | PASS | label only |  |
| §16–18: the 0.835 global root, seeding at a modest depth rather than Mishra's 22, and work caps. | specification | PASS | label only | 0.835 |
| §19: r_pred = ⌈log₂(S(C)/τ⋆(m, T))⌉ from a cell's centre margin is a theorem-derived upper-depth forecast. | theorem | FAIL | refuted | centre base area 0.8389855; margin 3.99×10⁻³; S_0 = 0.6101; r_pred = 9 cycles; q† base area 0.8347809 < 0.835 |
| §20–24: scheduling, shard states COMPLETE/EXPANDED/FAILED, a thirteen-item merge checklist, multi-AI replay and ancestry. | specification | PASS | label only | 13 merge checks; spec lists 11 |
| §25–29: the reference sharding of Round 10's four leaves into three shards ({00, 11}, {01}, {10}), replayed and merged as REFERENCE-SHARDED-VERIFIED, local in scope. | certificate | PASS | numerical only | shard A 5686/2844; shard B 2853/1427; shard C 2833/1417; total 11372/5688; 2·5688 − 4 = 11372 |
| §30–32: minimum manifest fields (program_id, round, root_domain_hash, split_policy_hash, arithmetic_policy_hash, witness_catalog_hash, shard_status, per-seed worst_slack; proof_policy_version, verifier_requirements) and checkpoints. | specification | PARTIAL | open |  |
| §0: SHARDED CERTIFICATE ARCHITECTURE: CLOSED. | reading | PARTIAL | open |  |
| §33–36: soundness of sharding, reshard example, arithmetic and version policy. | specification | PASS | label only |  |

Open: "**Anything global.** The reference is Round 10's local box, regrouped; Round 11 says the same."; Specification-only verification: "Four rules it would need are only in code: the tie-break, the child order, the canonical JSON form and the Merkle odd rule."; "**The forecast's usefulness.** It was tested only at the root, where it fails as a bound; as a scheduling heuristic it is untested."; Merkle odd-node rule: "So the root does not bind the number of shards. §22's "every seed exactly once" is the check that catches a repeat"

### RUN-015 — Round 12 - checkpoint crystal (AMRAL_Lebesgue_Finite_Closure_Round_12_Checkpoint_Crystal_v0.1.md, AMRAL-LUC-FC-R12, dated 2026-09-19) and its package: MULTI_AI_AUDIT_PROTOCOL_v0.1.md, CHECKPOINT_SCHEMA.json, VALIDATION.json, State_Crystal, reference checkpoints v1–v5, state_transition_summary.json, copies of Round 11's shards, checkpoint_validator.py, state_transition_demo.py

Report: `reports/RUN-015-ROUND12-CHECKPOINT-CRYSTAL.md`

> **Verdict: PASS for the mechanics; PARTIAL for the aggregate and the correction semantics.**

**For the paper.** Round 12's checkpoint mechanics work (digests re-derive, five reference verdicts reproduce, stale safety holds for what the hash bundle contains), but the scheme is PARTIAL: stored aggregates in three checkpoints contradict their ledgers undetected, and the correction semantics re-admit a rejected certificate's unchanged bytes on a non-replay PASS; it establishes nothing about geometry.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1–3: a checkpoint is (R, T, D, P, C, A, G), with a proof-critical bundle of eight hashes fingerprinted as H_dep and performance-only data kept out of it. | definition | PASS | label only | eight hashes; fingerprint a07225d3… (V1, V2, V4, V5); 57ba4912… (V3) |
| §2–5: payload digests and the fingerprint are SHA-256 of canonical JSON; the root-domain and arithmetic-policy hashes digest their inputs. | certificate | PASS | exact-arithmetic certificate | fp_pad: 1e-8; 2e-8 in V3 |
| §4, Thm 5.1 (stale safety): a claim is eligible only while its fingerprint is current, so a dependency change cannot silently carry old claims into a new proof. | theorem | PASS | analytic (proof checked) |  |
| §6: invalidation may later be made selective. | specification | UNMEASURED | not measured |  |
| §7–10: claim states (EMITTED, VERIFIED-INDEPENDENT-REPLAY, REJECTED, STALE, SUPERSEDED, CORRECTED), append-only ledgers, supersession and an audit ledger. | specification | PASS | label only |  |
| §11–13: audit levels A0–A3, every final claim needs an A1 PASS, and one valid counterexample outweighs any number of PASSes (not a vote); a lagged verifier. | specification | PASS | label only |  |
| §14–15: checkpoint validity (nine items) and theorem_ready (every frontier seed has exactly one current, verified, A1-passed, unsuperseded claim); the stored aggregates reflect the ledgers. | specification | PARTIAL | open | V3 stored theorem_ready true, computed false; V4 stored true, computed false; V4 and V5 list rejected claim-shard-B-v1 as active |
| §16: scope is proof-critical metadata; every checkpoint marks its scope local. | reading | PASS | label only |  |
| Thm 17.1 (checkpoint sufficiency): a checkpoint with complete frontier, current dependencies, claims, pending seeds and deterministic split semantics suffices to continue without scheduler history. | theorem | PARTIAL | open |  |
| §18, §20–21: how rejections and proof-critical versus performance-only changes propagate. | specification | PASS | label only |  |
| §19: a correction need only pass a current A1 audit. | specification | PARTIAL | open |  |
| §22–28: the five reference transitions — V1 baseline ready, V2 scheduler-only ready, V3 pad 10⁻⁸ → 2×10⁻⁸ not ready, V4 shard B rejected not ready, V5 appended correction ready. | computation | PASS | exact-arithmetic certificate | true, true, false, false, true |
| V5's corrected claim for shard B is a genuine correction. | certificate | NOT AS STATED | corrected |  |
| §29–36: naming, lineage, audits, ancestry, the stale aggregate, progress metrics, the production checkpoint and the publication gate. | specification | PASS | label only | publication gate ten items; protocol lists nine |
| All five checkpoints validate against CHECKPOINT_SCHEMA.json; the audit protocol restates the text. | specification | PASS | label only |  |

Open: "**Anything about the geometry.** Round 12 validates proof-state bookkeeping over Round 10's certificate, which RUN-013 replayed."; "**What the other five dependency hashes pin.** Their descriptors are not in the package."; "**A fix.** The two remedies suggested above ... are this tree's suggestions, untested."; Split semantics: "What the split-semantics hash pins cannot be checked from the package."

Unclear: The registry text says "Ten of the 21 are claim and aggregate states" while the judgment count is 22; recorded as printed.

### RUN-016 — Round 13 - autonomous scheduler (AMRAL_Lebesgue_Finite_Closure_Round_13_Autonomous_Scheduler_v0.1.md, AMRAL-LUC-FC-R13, dated 2026-09-19) and its package: WORKER_MERGER_PROTOCOL_v0.1.md, JOB_PROPOSAL_SCHEMA.json, VALIDATION.json, State_Crystal, reference race (issued_jobs.json, four race checkpoints, race_transition_summary.json, copies of Round 12's files), reference_scheduler_merger.py

Report: `reports/RUN-016-ROUND13-AUTONOMOUS-SCHEDULER.md`

> **Verdict: PASS for the merge theorems and the race; PARTIAL on the scheduler/merger ABI.**

**For the paper.** Round 13's merge theorems (Certified Ancestor Contraction, Atomic Merge) hold analytically and its single reference race replays exactly, but the scheduler/merger ABI is PARTIAL: the reference merger never writes the aggregate, the legal-duplicate rule is not implemented, and the late certificate re-admits previously rejected bytes (which are, in fact, sound per the Round 10 replay); no geometry is established.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1–6: workers never write the checkpoint; they submit immutable proposals against immutable jobs; leases are performance data; the merger first checks hashes, fingerprint (else STALE-PROPOSAL), seed identity and schema. | specification | PASS | label only |  |
| §7–9: an expansion is accepted only for a pending terminal seed (else OBSOLETE-EXPANSION); a certificate closes a pending seed. | specification | PASS | label only |  |
| Thm 10.1 (Certified Ancestor Contraction): a certificate of an ancestor replaces its current descendants while keeping the frontier complete. | theorem | PASS | analytic (proof checked) | 60 random sequences; 80 steps each |
| §11–14: races resolve either way, descendant evidence is kept, and rejection after contraction reopens the ancestor. | specification | PASS | label only |  |
| §15–16: multiple certificates of one seed are all kept, the one with the least canonical claim hash is primary, and duplicate compute is legal. | specification | PARTIAL | open |  |
| §17–19: semantic rebase of stale workers; a FAIL audit is checked before it rejects; a valid counterexample dominates. | specification | PASS | label only |  |
| Thm 20.1 (Atomic Merge): six primitives keep the frontier complete and prefix-free, and readiness means a current accepted claim for every seed. | theorem | PASS | numerical only | six primitives; 60 sequences × 80 steps |
| §21–27: single writer, merge journal (six fields), job sources, tail-first priority, caps, worker and merger loops. | specification | PASS | label only | four of six journal fields missing in the reference |
| §28–33: the reference race from Round 12's V4 — expand 01 (not ready), late certify with contraction of {010, 011} to 01 (ready), late expand obsolete (ready). | computation | PASS | exact-arithmetic certificate | ACCEPT-EXPAND; ACCEPT-CERTIFY-ANCESTOR-CONTRACTION; OBSOLETE-COVERED-BY-CERTIFICATE; not ready, not ready, ready, ready |
| The protocol's §14 reserves current_frontier, active_claim_ids, theorem_ready and pending_seed_paths to the merger, which writes them. | specification | NOT AS STATED | corrected | all four race checkpoints carry V1's aggregate; theorem_ready: true at start and after expansion |
| The late certificate of seed 01 is a valid certification. | certificate | PASS | numerical only |  |
| §12 and §32 name the late expansion of a certified seed. | definition | NOT AS STATED | label only |  |
| §34–37: the frontier may refine and contract while evidence only grows; worker-result categories; authority split. | specification | PASS | label only |  |
| §0: AUTONOMOUS SCHEDULER / MERGER ABI: CLOSED. | reading | PARTIAL | open |  |
| Package: issued jobs validate against the job schema (leases semantic: false); copies of Round 12's files are byte-identical; digests match. | certificate | PASS | exact-arithmetic certificate |  |

Open: "**Races beyond the one run.** Duplicate certification, expand against expand, certify against a failing audit, and a dependency change mid-run are Round 13's own C13-5."; "**Any geometry.** The race reuses Round 10's certificate for seed 01."; Re-admitted rejected bytes: "re-admitted without an answer to the FAIL (RUN-015)"

### RUN-017 — Round 14 - semantic rebase and compaction (AMRAL_Lebesgue_Finite_Closure_Round_14_Semantic_Rebase_Compaction_v0.1.md, AMRAL-LUC-FC-R14, dated 2026-09-19) and its package: SEMANTIC_REBASE_COMPACTION_PROTOCOL_v0.1.md, VALIDATION.json, State_Crystal, hot_proof_core.json, cold archive (snapshot, index, manifest), semantic_compaction_summary.json, semantic_validator.py, semantic_compaction_demo.py

Report: `reports/RUN-017-ROUND14-SEMANTIC-REBASE.md`

> **Verdict: PASS for the rules and the compaction theorem; PARTIAL on both "CLOSED" layers and on the boundary tests.**

**For the paper.** Round 14's migration rules and hot-core compaction theorem hold as stated rules (the leaf-level scalar margin rule is sound), and the reference compaction reproduces, but claim capacity must also count a-priori exclusions, so the recorded 'ready at 0.8350001' scenario is unsupported; the reported global capacity in the failing scenario is wrong (0.8350009475 vs the correct 0.8350003445), and the reference hot core omits required A1 evidence, tombstones and registered rule names. No geometry is established.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1–3: exact dependency equality is the fast path; a mismatch may be accepted through an explicit implication; target monotonicity. | specification | PASS | label only |  |
| §4–9: scalar margin credit T' + e' ≤ T_c + e_c + s_c with claim capacity Cap_{e'}(c) = T_c + s_c + e_c − e' permits migration to a higher target or stricter budget. | theorem | PARTIAL | corrected | Cap_{e'}(c) = T_c + s_c + e_c − e' |
| The three claim capacities at e' = 2×10⁻⁸ are 0.8350003769877956 (A-v1), 0.8350009474949350 (B-v2), 0.8350003445377508 (C-v1), with T_cap^ref ≈ 0.8350003445377508. | computation | PASS | numerical only | 0.8350003769877956; 0.8350009474949350; 0.8350003445377508; slacks 3.8698780×10⁻⁷, 9.5749494×10⁻⁷, 3.5453775×10⁻⁷ |
| §29: the four policy scenarios give ready (baseline 0.835), ready (0.8350001, budget 2×10⁻⁸, +B_9), not ready (0.8350004; pending 00, 10, 11), ready (catalog only). | computation | PARTIAL | numerical only | 0.835; 0.8350001; 0.8350004; pending 00, 10, 11 |
| §29 / summary: the higher-target scenario is ready at 0.8350001. | certificate | NOT AS STATED | open | sliver 3.04×10⁻⁷ wide; exclusion gives only 0.83500005; actual hull in the sliver about 1.005; exclusion capacity 0.835 |
| The reference reports a global capacity of 0.8350009475 in the failing 0.8350004 scenario. | computation | NOT AS STATED | corrected | reported 0.835000947494935; frontier capacity 0.8350003445377508 |
| §10–12: witness-catalog extension, mutation and local dependencies (a witness whose geometry changes needs a new proof). | specification | PASS | label only |  |
| §13–14: ancestor-domain coverage; a descendant claim never closes an ancestor alone. | specification | PASS | label only |  |
| §15–17: dominance may displace a claim from the hot core; same-seed primary; rejected claims never dominate or lend standing. | specification | PASS | label only |  |
| §18–21: the live graph, minimal active proof core (§19: each primary claim with certificate hash, threshold, error budget, slack, dependencies and A1 evidence summary), cold archive and GC. | definition | PASS | label only |  |
| Thm 22.1 (hot-core compaction): equal cores give equal readiness. | theorem | PASS | numerical only | 7123 and 3802 bytes |
| The reference hot core keeps what the text requires: A1 evidence summaries (§19), unresolved-FAIL tombstones (§34), no standing lent by rejected claims (§17), registered rule names (§31–32). | specification | PARTIAL | open | B-v2 slack 9.5749494×10⁻⁷ is B-v1's audited figure; certificate digest 08d6cf58… |
| §23–27: Merkle binding, no automatic revival, rejection after GC, the reference compaction and rehydration; the cold archive holds Round 13's final checkpoint. | certificate | PASS | exact-arithmetic certificate |  |
| §30–33: strengthening ledger, migration registry with four registered rules (TARGET-MONOTONICITY-V0.1, SCALAR-ERROR-MARGIN-CREDIT-V0.1, WITNESS-CATALOG-SUPERSET-V0.1, ANCESTOR-DOMAIN-COVERAGE-V0.1), exclusions. | specification | PASS | label only |  |
| §34–37: hot archive policy, tombstones, GC safety, retention (no physical deletion). | specification | PASS | label only |  |
| §39: Round 15 will use a global capacity T_capacity = min_p max_{c covers p} Cap(c) to drive a threshold ladder. | definition | UNMEASURED | label only |  |

Open: "**Any geometry.** The capacities here are the 96-direction sampled slacks of Round 10's local certificate."; "**Migrations beyond the four registered kinds**, which Round 14 excludes itself."; Thm 22.1 in practice: "Readiness from the hot core therefore rests on whoever wrote it having included only A1-passed claims."; Capacity must include exclusions: "the notes on "Scalar margin migration" and "SEMANTIC REBASE LAYER" name the missing exclusion capacity."

Unclear: For the 0.8350001 scenario the report says 'unsupported' and also 'The fact itself is not in doubt'; strength is recorded as 'open' (not proved by the certificate) rather than 'refuted'.; The theorems table grades §29 PASS (reproduction of the reference) while the registry, after the correction, grades the boundary tests PARTIAL; the §29 item records PARTIAL.

### RUN-018 — Round 15 - Proof Capacity Ladder (AMRAL_Lebesgue_Finite_Closure_Round_15_Proof_Capacity_Ladder_v0.1.md, AMRAL-LUC-FC-R15, dated 2026-09-19) and its package

Report: `reports/RUN-018-ROUND15-CAPACITY-LADDER.md`

> **Verdict: PASS for the capacity algebra; PARTIAL for the reference planner and the ladder automation.**

**For the paper.** Round 15's capacity recursion C(p) = max(D(p), min(C(p0), C(p1))) and its repair DP agree with exhaustive enumeration on 200 and 120 random trees, and the reference evidence capacities rebuild (root 0.8350003545377508); but the local proof's capacity is 0.835, because the B7 lift root's a-priori exclusion caps it, so promotions to 0.83500035 (and Round 14's 0.8350001) are unsupported.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| A root together with its a-priori exclusion supports targets only up to a root capacity C_root, and shard slack cannot raise the target past it (§1–2). | theorem | PASS | analytic (proof checked) |  |
| A node's capacity is C(p) = max(D(p), min(C(p0), C(p1))), and every T <= C(p) is proved on B(p) (§3–5, Thm 6.1). | theorem | PASS | numerical only | 200; depth 4 |
| The capacity recursion is relatively optimal within the registered grammar (§7). | theorem | PASS | numerical only |  |
| The global capacity is C_global = min(C_root, C_evidence) (§8). | definition | PASS | label only |  |
| Any target up to the global capacity can be promoted for free (§9). | theorem | PASS | analytic (proof checked) |  |
| The capacity staircase, the feasibility set F_T (feasible exactly when T <= C(ε)) and the deficit frontier Δ_T are as defined (§10–12). | theorem | PASS | numerical only | 10⁻⁹ |
| The reference's leaf capacities at e = 10⁻⁸ are 0.8350003869877956 (seeds 00, 11; A-v1), 0.8350009574949350 (01; B-v2), 0.8350003545377508 (10; C-v1), with node capacities 0.8350003869877956 and 0.8350003545377508 and root 0.8350003545377508, plus breakpoints, deficits and limiting claims (§13–18). | computation | PASS | numerical only | 10⁻⁸; 0.8350003869877956; 0.8350009574949350; 0.8350003545377508 |
| The local proof's capacity is 0.8350003545 and the target 0.83500035 is reached by free promotion (READY) (§13–18, §15). | reading | PARTIAL | corrected | 0.835; 0.8350003545; 0.83500035; 0.1969361096; 1.06×10⁻⁶; 0.83500017 |
| Round 14's higher-target scenario at 0.8350001 is ready (the migration RUN-017 passed). | reading | NOT AS STATED | corrected | 0.8350001; 0.1969353504; 3.04×10⁻⁷; 0.83500005 |
| The B7 lift root has half-width t_7(0.835) = 0.1969350467711682, and placements outside it are excluded by the a-priori bound, which is exactly 0.835 at the square's edge. | computation | PASS | numerical only | 0.1969350467711682; 0.835 |
| The repair DP K_T(p) = min(J_T(p), K_T(p0) + K_T(p1)) gives the minimum-cost repair; planned ancestor contraction keeps cost outside the proof (§19–22). | theorem | PASS | numerical only | 120; depth 3 |
| The phase-master root at T_M = 0.8365 has d⋆ = 0.6998605440613, t_3 = 0.2010235290074, t_5 = 0.2026182862304, t_7 = 0.2014734457863, with box overhead 11.66% and B_7 square overhead 4.66% (§23–27). | computation | PASS | numerical only | 0.8365; 0.6998605440613; 0.2010235290074; 0.2026182862304; 0.2014734457863; 1.03165; 1.02425; 1.02305; 11.66%; 4.66% |
| The phase-master root at 0.8365 is an engineering ceiling for the B_3, B_5, B_7 phase (§26). | reading | PASS | numerical only | 0.836494901; 0.8364914; 0.8365 |
| The ladder loop, target selection, bottleneck groups and alternative routes automate the threshold ladder (§28–31). | method | PASS | label only |  |
| The snapshot, a root-capacity contract and capacity status labels govern promotion (§32–34). | specification | PASS | label only |  |
| Status judgment: "Reference capacity planner: PASS". | reading | PARTIAL | corrected |  |
| Status judgment: "THRESHOLD LADDER AUTOMATION: CLOSED". | reading | PARTIAL | corrected |  |
| The reference capacity planner computes the recursion, feasibility and deficits, with root_domain_capacity recorded as INFINITE-WITHIN-FIXED-LOCAL-SCOPE. | computation | PARTIAL | corrected |  |
| The protocol states the same capacity algebra and phase-root overheads, and VALIDATION.json correctly describes the round's files. | specification | PASS | label only | 11.7%; 4.7% |

Open: Rigorous proof that sliver configurations reach the higher targets: "That the sliver's configurations reach the higher targets rigorously."; The fact at the sliver is not proved: "It is simply not proved by the recorded evidence." (exact hull 1.0046 at worst of 32 orientations; "It is not a certificate."); Closing the gap needs a new proof step (sharper exclusion or lift roots sized for the phase target): "Either is a new proof step, not a free promotion."; No global capacity ("Any global capacity" listed as not established): "Everything here is Round 10's local box."; The reference planner does not run as shipped: "It loads the hot core from an absolute path outside the package, so it does not run as shipped."

Unclear: Validation of the recursion and repair DP is by exhaustive comparison on random trees (200 and 120); recorded here as strength 'numerical only' since the report does not describe a proof check.; Hull value at the sliver's middle is given as 'about 1.005' in the verdict summary and as 1.0046 in W4; both are as printed.; The report notes B-v2's capacity 'rests on the slack audited under the rejected B-v1 (RUN-017)' without stating a separate verdict on that dependency.

### RUN-019 — Round 16 - Global Pilot (AMRAL_Lebesgue_Finite_Closure_Round_16_Global_Pilot_v0.1.md, AMRAL-LUC-FC-R16, dated 2026-09-19) and its package

Report: `reports/RUN-019-ROUND16-GLOBAL-PILOT.md`

> **Verdict: PASS for the pilot's computations and its three decisions; PARTIAL for the near-frontier profile and its proxy; §7's reading of APR is wrong.**

**For the paper.** Round 16's pilot counts, scheduler comparison (official ahead of exact-τ at depth 16), master-root cost advantage and B7-later decision all rebuild and survive exact cores and the alternative tie-break, but its exact counts depend on two unstated code choices, its near-frontier profile measures the 40-direction sampling (exact least centre 0.8385552, not 0.8356828), APR equals π/4 ≈ 0.785 on this root at every depth, and the cycle proxy is not conservative (19 HARD cells hold configurations below 0.835). These are floating-point measurements, not bounds.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| The master root is the whole five-dimensional base root D + B₃ + B₅ sized at T_M = 0.8365 with t₃, t₅, t₇ given to 17 digits and φ₅ over [0, 2π/5) (§0–1). | computation | PASS | numerical only | 0.8365; 0.20102352900741138; 0.20261828623043537; 0.20147344578628762; 1.6×10⁻¹⁶ |
| Four one-sided modes APR, REP, CORE and HARD are defined under the pilot policy N_sample = 40, e_pilot = 10⁻⁹, with APR using f(d), the hull of the disc with a body's centre (§2). | definition | PASS | numerical only | 40; 10⁻⁹; 0.55; 0.7; 1; 1.5 |
| The pilot's '40 finite boundary inner points' sampling and its split-rule tie-breaking produce the recorded counts. | method | PARTIAL | numerical only | 2πk/40; 29,571; 5,688; 28,771; 5,252 |
| The official scheduler uses 18.48% fewer nodes and 21.71% fewer HARD leaves than Round 09's exact-τ rule at depth 16, and one-step τ-optimality is not tree-cost optimality (§3–5). | computation | PASS | numerical only | 18.48%; 21.71%; 17.27%; 21.68%; 21.06%; 27.71%; 0.18476938583990823; 0.21706016755521707 |
| The master root costs no more than the immediate 0.835 root: 23,455 nodes at depth 16 against 24,691 (§6). | computation | PASS | numerical only | 23,455; 24,691; 21,971; 22,759; 23,343; 24,263 |
| The depth-18 official tree has 45,651 nodes, 12,378 CORE, 10,448 HARD, and APR and REP both zero (§7). | computation | PASS | numerical only | 45,651; 12,378; 10,448; 41,899; 11,716; 9,234 |
| "APR/REP=0 does not mean they are never useful"; the APR contained-center bound is not yet close enough to 0.835 (§7). | reading | FAIL | refuted | π/4 ≈ 0.785; 0.287; 0.2843; 0.2865; 0.2849 |
| REP cannot fire at depth 18 because it needs a centre of at least T + 4Tτ + πτ² (§7). | computation | PASS | numerical only | 0.03855318146790454; 0.96844; 0.92641 |
| No HARD centre lies below the target; the least is 0.8356828349502634 (§8). | computation | PASS | numerical only | 0.8356828349502634; 0.8385552 |
| The near-frontier counts and centre and core quantiles (e.g. 8 cells within +0.002, 70 within +0.005, 142 needing at least 5 cycles, 8 needing 7) describe the depth-18 frontier (§9–11). | computation | PARTIAL | corrected | 0.8356828; 0.8385552; 8; 0; 70; 12; 196; 128; 714; 502; 142; 92; 1.6×10⁻³; 5.6×10⁻³; 3.5×10⁻³ |
| The raw HARD frontier is far larger than the near-minimizer frontier (§9). | reading | PASS | numerical only |  |
| The cycle proxy from τ⋆(m, T) is conservative (§12). | method | NOT AS STATED | refuted | 19; 3 to 8; 0.834781; 0.834971; 14; 5; 0 |
| About 90% of HARD cells are within 3 cycles (§13). | computation | NOT AS STATED | corrected | 90%; 95.9% |
| The cycle-proxy tails, the tiers 8 / 142 / 2,044 / 8,262 and the eight lowest cells rebuild from the sampled tree (§12–13, §21). | computation | PASS | numerical only | 3.3×10⁻¹⁶; 8; 142; 2,044; 8,262 |
| The two lowest cells have a core loss of 3.66×10⁻² (§14). | computation | PASS | numerical only | 0.8356828; 0.7990730; 3.66×10⁻²; 0.834780945912 |
| Shallow B7 lifts on the two lowest cells leave 634 of 1,202 leaves open, so base refinement comes first and B7 later (§15–16). | computation | PASS | numerical only | 2,403; 568; 634; 1.2487498×10⁻⁶; 2,251; 549; 577; 1.6397113×10⁻⁶; 3.7×10⁻²; 1.7×10⁻³ |
| The pilot shows a numerical symmetry signal (§19). | reading | PASS | numerical only | 8×10⁻¹⁶; 4π/15; 10⅔ |
| The master root is kept (§20). | reading | PASS | numerical only |  |
| Seeds are tiered into 142, 2,044 and 8,262 (§21). | specification | PASS | numerical only | 142; 2,044; 8,262; 14; 5 |
| The package's reproduction code (global_pilot.py) produced the shipped summary (§22–23). | specification | PARTIAL | corrected | 13.77 s; 13.51 s |
| Status judgment: "Near-frontier compression signal: NUMERICAL-EVIDENCE". | reading | PARTIAL | corrected |  |

Open: Whether the 19 cells are all the base-unclosable cells at depth 18: "It exhibits 19 cells; it does not rule out others."; No bound is established: "The exact trees use floating-point hulls with the pilot's pad. They are measurements, as the pilot's are."; The proxy on all bulk cells: "The forecast holds on 80 random cells, not on all of them."; A quotient by the mirror symmetry: "Whether a certificate may keep one twin of each pair is Round 17's question."; The square-corner a-priori rule is not implemented: "The pilot implements no a-priori corner rule, and in these trees it would never have fired."

Unclear: The report's theorem table gives §12–13 a combined PARTIAL; here it is split into the 'conservative' claim (refuted), the '90%' figure (corrected), and the tails/tiers (rebuilt).; The symmetry item is recorded as 'analytic (proof checked)' because the report says the area is 'exactly invariant' and names the bodies' symmetries, while its numerical check is on random configurations to 8×10⁻¹⁶.; The report's own bulk-forecast test is given as 80 random cells in the summary and as 40 + 40 cells (predicted 1 and 2 cycles) in W6; both are as printed.; §17–18 (production scheduler; prune-gain target) and §25–26 are marked n/a (design) and are not listed as items.

### RUN-020 — Round 17 - D3 quotient and CPG schedulers (AMRAL_Lebesgue_Finite_Closure_Round_17_D3_CPG_v0.1.md, AMRAL-LUC-FC-R17, dated 2026-09-19) and its package

Report: `reports/RUN-020-ROUND17-D3-CPG.md`

> **Verdict: PASS for the quotient and the pilot's computations; PARTIAL for the tail-window recommendation.**

**For the paper.** Round 17's D₃ quotient is exact and its canonical wedge covers every orbit, but only under the a-priori exclusion \|t₅\| <= t₅(0.8365), so the quotient root is sound up to 0.8365; at equal cell size it removes about 73% of nodes and HARD leaves (not the reported 47.32% / 32.73%). All pilot runs rebuild under three code-only rules, but the tail window's advantage comes from its certifying override rule, not from starting at depth 10.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| With the disc at the origin and B₃'s orientation fixed, the area is invariant under the 2π/3 turn and the mirror y ↦ −y, φ₅ ↦ −φ₅, which together form G ≅ D₃ (§1–2). | theorem | PASS | numerical only | 6.7×10⁻¹⁶; 1.0×10⁻¹⁵; 40 |
| Every orbit has a representative with 0 <= arg t₃ <= π/3, in the rectangle 0 <= x₃ <= t₃⋆, 0 <= y₃ <= (√3/2) t₃⋆ (§3, Thm 3.1). | theorem | PASS | analytic (proof checked) | 4,007; 4,000; 0.8365; 153°; (0.074, −0.277); ±0.203 |
| The quotient root's soundness uses radial legality, the a-priori exclusion \|t₅\| <= t₅(0.8365). | reduction | PASS | analytic (proof checked) | 0.8365; −0.277; ±0.203 |
| A cell with y₃⁻ > √3 x₃⁺ is pruned as SYM, a quotient leaf rather than a lower-bound leaf (§4). | definition | PASS | label only | 10⁻¹⁵ |
| Mishra's q† has a representative in the wedge, which lies in a HARD cell of the D3 trees. | computation | PASS | numerical only | 3.05×10⁻⁶; 0001110001100111 |
| Symmetry sanity: with 120 directions, 20 samples move by at most 6.66×10⁻¹⁶ under the mirror and 8.88×10⁻¹⁶ under the turn (§5). | computation | PASS | numerical only | 120; 6.66×10⁻¹⁶; 8.88×10⁻¹⁶; 6.7×10⁻¹⁶; 7.8×10⁻¹⁶; 1.1×10⁻³ |
| The D3 official tree at depth 16 has 12,357 nodes, 109 SYM, 3,304 CORE, 2,766 HARD and 12,248 evaluations (§6). | computation | PASS | numerical only | 12,357; 109; 3,304; 2,766; 12,248; 10,967; 100; 3,045; 2,339 |
| The D3 quotient gives 47.32% fewer nodes and 32.73% fewer HARD leaves than the full root at depth 16 (§6). | computation | PASS | corrected | 47.32%; 32.73%; 72.9%; 73.5%; 73.8%; 74.7%; 21.65%; 0.0385532 |
| Split choice never affects soundness (§7). | theorem | PASS | analytic (proof checked) |  |
| CPG chooses the axis by P(a) = children certified with tie-breaks, and unrestricted CPG is myopic (§8–9). | method | PASS | numerical only | 2,975; 14,719; 76; 202; 1,210; 1,486; 2,943 |
| A window L(C) >= T − 0.025 applied when the official split certifies nothing gives 8,625 nodes / 1,862 HARD at depth 16 with 1.82 times the evaluations (§10–12). | computation | PASS | numerical only | 0.025; 8,625; 1,862; 22,266; 1,608; 1.82; 4,173; 863; 423 |
| Tail-only windowed CPG from depth 10 gives 9,997 / 2,048 with 10.72% overrides and 2.02 times the evaluations, "the most balanced production candidate" (§13). | method | PARTIAL | corrected | 9,997; 2,048; 10.72%; 2.02; 536; 8,823; 2,047; 22,778; 404; 9.2%; 1.86; 8,947; 1,909; 10,967; 2,339; 465 |
| Emit cost against replay cost is weighed by the stored evaluation and wall-time factors (§14). | method | PASS | label only |  |
| Sparse overrides over four axes take 2 bits, with the official axis implicit (§15). | specification | PASS | exact-arithmetic certificate | ⌈log₂ 4⌉ = 2 |
| Every stored fraction and factor (47.32, 32.73, 30.20, 32.68, 19.10, 25.96, 42.77, 42.45 and 10.72%; 1.82 and 2.02 times) follows from the counts. | computation | PASS | numerical only | 47.32; 32.73; 30.20; 32.68; 19.10; 25.96; 42.77; 42.45; 10.72%; 1.82; 2.02; 10⁻¹⁵ |
| The group, the wedge, SYM and child coverage are theorem-critical; the window, activation depth and codec are performance-only (§16–17). | specification | PASS | label only |  |
| Stored conclusion: windowing CPG "retained the depth-14 HARD count of full CPG". | reading | NOT AS STATED | refuted | 863; 1,210 |
| Status judgments "TAIL-WINDOW CPG: PILOT-RECOMMENDED" and "Tail-window CPG: PILOT-PASS / PRODUCTION CANDIDATE". | reading | PARTIAL | corrected |  |
| The package (summary, scheduler spec γ = 0.025 and activation depth 10, crystal, VALIDATION.json) is internally consistent. | specification | PASS | label only | 0.025; 10; 126; 20 |

Open: No bound: "The trees are floating-point pilots, as Round 17's are."; Schedulers beyond depth 16: "Deeper, or on real tail shards, the activation depth may matter differently."; A quotient by the turn on the grid: "The turn maps no axis-aligned grid onto itself (RUN-019), so cells are not paired as they are under the mirror."; The pilot does not run as shipped: "The pilot reads its Round 16 baseline from a path outside the package, so it does not run as shipped."

Unclear: The report says reading Round 17 exposed two blind spots in the tree's status census and that the census v3 correction (commit 749d9ca) adds 18 judgments, 10 of them Round 17's; which RUN carries that census is not named here.; Thm 3.1 is recorded as 'analytic (proof checked)' because W2 gives the sector argument; the report also tests it on 4,007 configurations.; The full-root comparison counts (45,651 / 10,448 sampled; 41,899 / 9,234 exact) are taken from RUN-019.

### RUN-021 — Round 18 - Production Shard Emitter (AMRAL_Lebesgue_Finite_Closure_Round_18_Production_Shard_Emitter_v0.1.md, AMRAL-LUC-FC-R18, dated 2026-09-19) and its package

Report: `reports/RUN-021-ROUND18-SHARD-EMITTER.md`

> **Verdict: PASS for both checkpoints and the Rice events; PARTIAL for the grammar as specified and for the cost model's calibration.**

**For the paper.** Both Round 18 checkpoints (official: 12,350 nodes, 2,766 PENDING; tail: 10,086 nodes, 2,048 PENDING, 573 overrides in 3,823 Rice bits) decode from the text's grammar, re-prove every CERT leaf at 0.835 (exact cores by at least 2.39×10⁻³ in floating point) and re-emit byte for byte; but three format conventions are fixed only by the data, the tail forest is a variant of Round 17's tail window, §14's 4,983 internal nodes should be 5,039, and the cost break-evens rest on single, partly inflated timings. These are pilot forests with open leaves, not bounds.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| The D3 root is split by the implicit official rule to seed depth 3, giving eight shard roots each emitted to global depth 16 (§0–2). | specification | PASS | numerical only | 3; 8; 16 |
| The certificate grammar: 1-bit preorder topology, 2-bit leaf tags (00 SYM, 01 CERT, 10 PENDING), an implicit default axis, and Rice-coded override gaps with best k in {0,…,6} plus a 2-bit rank (§3–10). | specification | PARTIAL | corrected |  |
| The verifier never reproduces CPG; it reads axes from the stream (§11). | specification | PASS | numerical only |  |
| The forest identity is N = 2L − S (§12). | theorem | PASS | numerical only | N = 2L − 8 |
| The official checkpoint has 12,350 nodes, 109 SYM, 3,304 CERT and 2,766 PENDING (§13). | certificate | PASS | numerical only | 12,350; 6,171; 6,179; 109; 3,304; 2,766; 0.8350002 |
| The tail checkpoint has 10,086 nodes, 106 SYM, 2,893 CERT, 2,048 PENDING and 573 override events in 3,823 Rice bits (§14). | certificate | PASS | numerical only | 10,086; 5,039; 5,047; 106; 2,893; 2,048; 573; 3,823 |
| The tail checkpoint has 573 overrides, 11.37% of 4,983 internal nodes (§14). | computation | FAIL | corrected | 4,983; 5,039; 11.37%; 15,117 |
| The tail certificate is Round 17's tail window made production (§0, §33). | reading | NOT AS STATED | corrected | 9,990; 9,997; 536; 10,086; 573 |
| Every CERT leaf is re-proved by the verifier at 0.835, all by CORE, with worst slacks 6.15×10⁻⁶ and 1.14×10⁻⁵ (§19–20, §27). | certificate | PASS | numerical only | 3,304; 2,893; 6.150283528×10⁻⁶; 1.144322404×10⁻⁵; 2.3857×10⁻³; 6×10⁻¹⁶; 390 |
| Every SYM leaf lies outside the wedge and every PENDING leaf sits at depth 16. | certificate | PASS | numerical only | 16 |
| Reductions, axis metadata, sizes and compression: nodes −18.33%, PENDING −25.96%, Rice events 3,823 bits against 15,117 explicit (25.29%), raw streams 3,094 and 3,011 bytes, zlib-9 1,660 and 1,782, ZIP 25,045 and 22,263 (−11.11%) (§15–18). | computation | PASS | numerical only | −18.33%; −25.96%; 718; −2.68%; −11.11%; +7.35%; 11.37%; 25.29%; 15,117; 3,094; 3,011; 1,660; 1,782; 25,045; 22,263; 10⁻¹² |
| An independent verifier with its own geometry, monotone-chain hull, shoelace and parser replays the checkpoints in 2.3140 and 1.9887 s (§19). | computation | PASS | numerical only | 2.3140; 1.9887; −14.06% |
| The shipped verification establishes "Pending frontier preservation: PASS" and checks manifests (§29, §31). | reading | NOT AS STATED | corrected |  |
| Emitters take 6.98 and 17.32 s (2.48 times) (§21). | computation | PASS | numerical only | 6.98; 17.32; 2.48; 5,039; 30,229; 16.7%; 12,350; 2.45; 2.04 |
| Per-shard evaluations are 1 + 2I for the official scheduler and 1 + 3I + 2A for the tail window. | computation | PASS | numerical only | 7,552 |
| The cost model C = E + RV + κP + μB gives break-evens R⋆ ≈ 31.79 and κ > 0.0144 s per saved pending leaf, preferring the tail window for long runs (§22–26). | method | PARTIAL | corrected | 31.79; 31.794; 0.0144 s; 0.014406 s; 1.45; 0.41 |
| Shard status and checkpoint semantics: no hidden branch; all shards EXPANDED (§28–29). | specification | PASS | numerical only |  |
| Status judgment: "Production shard grammar: CLOSED". | reading | PARTIAL | corrected |  |
| Status judgment: "Lifetime cost model: FORMALIZED + EMPIRICALLY CALIBRATED". | reading | PARTIAL | corrected |  |

Open: No bound: "Both checkpoints are pilot-arithmetic forests with 2,766 and 2,048 open leaves."; Timings: "Replay and emitter seconds are single measurements."; The tail window's lookahead rule as a choice: "It is not tested here beyond the two forests it produces."

Unclear: The report says Round 18 exposed a third blind spot in the status census (statuses joined by '+') and that the census v4 correction (commit 31fcefd) adds five judgments, two of them Round 18's; which RUN carries the census is not named here.; The worst exact-core slack is given as 'at least 2.39×10⁻³' in the summary and as 2.3857×10⁻³ in the W2 table; both are as printed.; The 'exact cores' re-proof is recorded as 'numerical only'; the report does not say it is exact arithmetic.

### RUN-022 — Round 19 - Necessity and Lazy Lift (AMRAL_Lebesgue_Finite_Closure_Round_19_Necessity_Lazy_Lift_v0.1.md, AMRAL-LUC-FC-R19, dated 2026-09-19) and its package

Report: `reports/RUN-022-ROUND19-NECESSITY-LIFT.md`

> **Verdict: PASS for the cascade, the continuation, the depth-20 correction, the tracked branch, the necessity semantics and the segmentation, and for the switch curve as computed; PARTIAL for the τ ≈ 10⁻³ scheduler parameter, whose value belongs to the pilot's sampled lift bound.**

**For the paper.** Round 19's lazy cascade (Theorem 2.1) and necessity theorem (Theorem 13.1) hold, the continuations, depth-20 correction (0 / 12 exact sublevel centres, exact minimum 0.8358650284, and the recheck is complete), tracked branch (first sublevel centre at depth 31, margin 8.2×10⁻⁵) and segmentation (85 / 77) all rebuild, but only in the pilot's floating arithmetic; the B7 'witness-only floor' and the τ ≈ 10⁻³ switch parameter belong to the 40-direction sampled lift bound (exact cores: floor 76 at depth 12 and 7 at depth 18, exponent 0.91).

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Theorem 2.1: evaluating the three one-sided bounds lazily (APR → CORE → REP, REP only on a CORE failure) classifies a fixed cell exactly as the eager maximum does (§1–2). | theorem | PASS | numerical only | 152,875; 12,659; 41,375; 98,841 |
| The fixed-tree A/B from Round 18's 2,048 pending roots, official split to depth 18, gives 13,206 nodes and 4,944 pending both ways, with REP calls 12,659 → 10,523 (−16.87%) and time 7.2903 → 6.8687 s (−5.78%) (§3). | computation | PASS | numerical only | 2,048; 13,206; 547; 2,136; 4,944; 12,659; 10,523; −16.87%; 7.2903; 6.8687; −5.78% |
| REP is kept because nothing proves it never beats CORE; "almost all" accepted leaves close by CORE (§4). | reading | PASS | numerical only | 2,136; 2,228; 5,398 |
| The Tail-Window continuation 16 → 18 gives 12,634 nodes and 4,599 pending; 18 → 20 gives 26,455 nodes and 10,114 pending (§5–6). | computation | PASS | numerical only | 12,634; 514; 2,228; 4,599; 287; 26,455; 15; 5,398; 10,114; 884; 41,375; 3,845; 98,841; 8,062 |
| The continuation adds 12,634 and 26,455 "new" nodes (§5–6). | computation | NOT AS STATED | corrected | 12,634; 26,455; 10,586; 21,856 |
| Cumulative continuation cost is 78.1375 CPU-s, 38.15 ms per original root, 2.65 times Round 18's break-even of 14.41 ms, so the Tail-Window checkpoint has paid for itself (§7). | computation | PASS | numerical only | 22.7531; 55.3844; 78.1375; 38.153 ms; 11.110 ms; 14.406 ms; 2.648 |
| 12 depth-20 centres have a sampled area below 0.835 (least 0.83395139), but the exact kernel finds 0 / 12 (least 0.83586503); CORRECTION-R19-001: only an exact or rigorous upper bound may trigger necessity (§8–9). | computation | PASS | numerical only | 12; 10,114; 0.8339513918; 0.8358650284; 1.23×10⁻³; −1.235×10⁻³ |
| The depth-20 profile file (quantiles, near-target counts, the top eight) describes the pending cells. | computation | PASS | numerical only |  |
| q† in the canonical wedge has area 0.8347809459115, and on the official split's branch holding it depth 30 has exact 0.8352534195 and outer 0.8352704487 > T while depth 31 has 0.8349033738 and 0.8349184947 < T (CENTER-SUBLEVEL-CERTIFIED-PILOT) (§10–12). | computation | PASS | numerical only | 0.8347809459115; 3.05×10⁻⁶; 51; 1.2×10⁻¹⁵; 0.8352534195; 0.8352704487; 0.8349033738; 0.8349184947; 8.2×10⁻⁵ |
| The track's sampled areas and seed-to-centre distances follow the package's conventions. | specification | PASS | numerical only | 10⁻⁹ |
| The 20,000-direction outer bracket is computed by the vertex formula. | method | PASS | numerical only | 20,000; 7.5×10⁻¹⁰; 3×10⁻¹⁴ |
| Theorem 13.1: a cell holding a point with area below T has no base-only certificate, nor does any descendant holding it; only the child holding the point inherits the marker (§13–15). | theorem | PASS | analytic (proof checked) |  |
| The depth-31 cell lifted whole leaves 227 of 4,096 B7 leaves open at depth 12 (5.54%); refined first to depth 40 it gives 561 nodes, 256 pending, 85 exact centres below T and 77 with the outer bracket below (§16–18). | computation | PASS | numerical only | 227; 4,096; 5.54%; 561; 256; 85; 77; 0.8348018420; 1.2×10⁻¹³; 8; 1.5×10⁻⁵ |
| Two queues, BASE-UNRESOLVED and NECESSITY-MARKED, separate the logical gate from the performance gate (§19–20). | method | PASS | label only |  |
| The B7 depth-12 switch curve along the branch runs from 483 to 125 open leaves (τ from 0.02448 to 0.000383), with a witness-only floor of 108 (2.64%) at the exact base point and excess falling roughly as τ^0.70 (§21–22). | computation | PASS | numerical only | 483; 330; 252; 227; 198; 183; 161; 144; 135; 125; 108; 2.64%; 412; 0.704; 0.696 |
| Near τ ≈ 10⁻³ further base refinement stops paying (LIFT-EFFICIENCY-REGIME-PILOT), and the activation policy uses "tau around 1e-3" (§23–24). | method | PARTIAL | corrected | 10⁻³; 456; 83; 76; 1.86%; 7; 0.911 |
| A state machine and an activation policy govern lift activation (§25–26). | specification | PASS | label only |  |
| B7 is not closed at depth 18 even at the exact base point (§27). | computation | PASS | numerical only | 412; 7; 2¹⁸ |
| The global claim a_Leb >= 0.8350 stays compute-deferred (§0). | reading | PASS | open | 0.8350 |
| The shipped exact kernel (official_exact_kernel_minimal.py) is an independent exact hull. | computation | PASS | numerical only | 1.2×10⁻¹⁵ |
| Status judgment: "τ ∼ 10⁻³ lift-efficiency regime: NUMERICAL-EVIDENCE / SCHEDULER PARAMETER". | reading | PARTIAL | corrected |  |

Open: No bound: "The 77 certified centres and the depth-31 marker are certified in the pilot's floating arithmetic, as §12 says, not in publication arithmetic."; Timings: "The A/B's 5.78% and every CPU-second are single measurements."; Exact-core B7 figures are not a certificate: "They show how much of the pilot's floor is sampling, and nothing more."; The global claim remains deferred: boxed `a_{\mathrm{Leb}}\ge0.8350:\ \text{STILL COMPUTE-DEFERRED}`.; 8 segmentation centres are below T but not certifiably so: "their margin is smaller than the bracket's slack, about 1.5×10⁻⁵."; The shipped probe cannot run from the package: "Nothing shipped tracks the branch, segments it, brackets it, or runs the depth-12 curve and floor."

Unclear: Theorem 13.1's soundness premise cites 'RUN-011 to RUN-019'; the intermediate RUN ids are listed in cross_refs as a range expansion.; The report says reading Round 19 led to census v5 (229 judgments swept), documented in RUN-003.; The shipped continuation probe's own B7 probe (depth 10 on four cells) is 'a result reported nowhere'; the report gives no verdict on it.; Theorem 2.1 is recorded as 'analytic (proof checked)' on the report's statement that it 'is right'; the report also checks it on 152,875 evaluations.

### RUN-023 — Round 20 - Mixed Witness Escalation (AMRAL_Lebesgue_Finite_Closure_Round_20_Mixed_Witness_Escalation_v0.1.md, AMRAL-LUC-FC-R20, dated 2026-09-19) and its package

Report: `reports/RUN-023-ROUND20-MIXED-ESCALATION.md`

> **Verdict: PASS for the markers, the twelve lift trees and their streams, the counterexample theorem and B23, the recorded placements, the efficiency scans, the deep comparison and the tail counts; PARTIAL for the mixed shard grammar as specified, for the horizon-dependent ranking, and for "B7 full closure of marked cell: NOT YET ACHIEVED", which holds only of the pilot's sampled lift bound.**

**For the paper.** Round 20's counterexample theorem holds and B23 is disqualified on cell-00 (outer 0.8349443918, below 0.835 by 5.6×10⁻⁵ in pilot floating arithmetic); its markers, twelve lift trees, placements, scans and tails rebuild, but the package ships no code, its 'search minima' are not minima (B19 0.8350921, not 0.8354089), and with exact cores B7 alone closes all seven marked cells tried (floating point, not a certificate), so the 'not yet' closure and horizon ranking describe the pilot's sampled lift bound.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| A mixed-shard base leaf carries id, path, box, necessity marker, lift attachments and negative evidence, with seven states from BASE-CERTIFIED to PENDING; a shard is COMPLETE only when every base obligation is base-certified or covered by a complete lift (§1–2). | specification | PASS | label only |  |
| The reference shard has four depth-40 marked cells whose markers are re-verified by outer bracket below 0.835 (§15). | computation | PASS | numerical only | 0.8348018420; 0.8348206176; 0.8348116722; 0.8348279638; 0.8348238282; 0.8348435895; 0.8348266675; 0.8348497057; 1.5×10⁻¹⁴ |
| Each marked cell carries B7, B11 and B19 depth-12 partial lifts with the recorded nodes / CERT / PENDING counts, pending leaves kept and theorem_ready = false (§15). | computation | PASS | numerical only | 867 / 273 / 161; 715 / 239 / 119; 511 / 192 / 64; 873 / 277 / 160; 4.4×10⁻¹⁶ |
| The mixed shard grammar is closed (status "Mixed shard grammar: CLOSED"). | specification | PARTIAL | corrected | 8; 24 |
| An independent verifier replays the reference shard (REFERENCE-MIXED-SHARD-VERIFIED; "Mixed shard grammar: IMPLEMENTED + REPLAYED") (§16). | reading | PARTIAL | corrected |  |
| Open leaves only make a certificate incomplete, not failed (§3). | reading | PASS | label only |  |
| Counterexample theorem: one legal placement of K with an outer bound below T disqualifies K as a sole closer of the cell (§4). | theorem | PASS | analytic (proof checked) |  |
| B23 on cell-00 has exact 0.83492540 and outer 0.83494439, disqualifying it (§5). | computation | PASS | numerical only | 0.8349254025739500; 0.8349253775010098; 0.8349443917869175; 5.56×10⁻⁵; 4.2×10⁻¹⁵ |
| A search that finds nothing proves nothing (NO-COUNTEREXAMPLE-FOUND, never VIABLE-PROVED) (§6). | reading | PASS | label only |  |
| Depth-12 mean pending on eight cells is B7 160.6, B9 154.0, B11 119.1, and on four cells B13 89, B15 68, B17 69, B19 64, B21 64, B23 64; near-disc witnesses certify easily (§7–8). | computation | PASS | numerical only | 160.625; 162; 154; 158; 119.125; 120; 160.375; 154.375; 89; 68; 69; 64 |
| The nine recorded 3-D placements on cell-00 have the recorded exact, inner and outer areas (§9). | computation | PASS | numerical only | 2×10⁻¹⁶; 3×10⁻¹⁴ |
| The recorded placements are the 3-D search minima on cell-00, from B7 0.83707775 to B23 0.83492540 (§9). | computation | NOT AS STATED | corrected | 0.8370777154; 0.8356295728; 0.8361364467; 0.8359149272; 0.8352910602; 0.8351473006; 0.8350921012; 0.8350840426; 0.8349253208; 3.17×10⁻⁴; 1.24×10⁻⁴ |
| Witness choice is a Pareto problem between efficiency and geometric forcing (§10). | method | PASS | label only |  |
| On the depth-50 cell the B7, B11, B13, B19, B21 tails have the recorded open-leaf counts and fractions (§11–12). | computation | PASS | numerical only | 3.825×10⁻⁴; 653 (0.249%); 778 (0.297%); 823 (0.314%); 736 (0.281%); 724 (0.276%); 1,015 (0.0968%); 1,046; 1,397; 2,143 (3.19×10⁻⁵) |
| B7 is costly shallow but contracts faster, so horizon-dependent ranking is warranted ("Horizon-dependent ranking: NUMERICAL-EVIDENCE") (§11–12). | reading | PARTIAL | corrected | 653; 1,015; 1,046; 1,397; 2,143 |
| B7 full closure of a marked cell is not yet achieved ("B7 full closure of marked cell: NOT YET ACHIEVED"). | reading | PARTIAL | corrected | 21; 2,509; 23; 3,671; 25; 5,369; 30; 18,555; 31; 22,009; 26,189; 34; 36,193; 6.6×10⁻⁹ |
| ρ_K(d₁, d₂) = u_{K,d₂} / u_{K,d₁} and the N0–N5 state machine are defined (§13–14). | definition | PASS | label only |  |
| The benchmark's deep_certificate_comparison: B7, B9, B11, B13 to lift depths 14 and 16 on cell-00 and cell-01. | computation | PASS | numerical only | 16 |
| The package's summary and boxed verdicts (§0). | reading | PARTIAL | corrected |  |
| VALIDATION.json describes the document and crystal and its four key checks follow from the data. | specification | PASS | label only | 98; 4 |

Open: No bound: "The B23 disqualification, the markers and the exact-core closures are floating-point computations, and the last with this tree's own kernel. They are not publication-grade."; Search minima: "A still wider search might find lower ones again. B19 and B21 sit within 10⁻⁴ of T."; Timings: "The benchmark's seconds are single runs and are not checked."; The other 73 marked cells: "It was not run on the rest of Round 19's 77."; No shipped replay: "This tree's rebuild is the only replay there is."

Unclear: The report says census v5 finds 12 of Round 20's 18 judgments; the census RUN is not named in this report.; Exact-core closures are described as floating-point values of the tree's exact envelope; the report states the bound is sound and the root complete for T <= 0.8365 but that 'these are not certificates'.

### RUN-024 — Round 21 - Lift Tail Synthesis (AMRAL_Lebesgue_Finite_Closure_Round_21_Lift_Tail_Synthesis_v0.1.md, AMRAL-LUC-FC-R21, dated 2026-09-19) and its package

Report: `reports/RUN-024-ROUND21-LIFT-TAIL-SYNTHESIS.md`

> **Verdict: PASS for the atlas as computed, the phase lock and the two lobes, the ridge's two channels, the finite-tail optimism theorem and the legal 7-fold family; PARTIAL for the atlas as a tail of B7 — with exact cores it is empty — and for every synthesis comparison, whose search values sit 3–6×10⁻⁶ below the true minima; UNMEASURED for the irregular and symmetry-breaking candidates, which the package does not record.**

**For the paper.** Round 21's tail atlas (1,046 / 1,397 / 2,143 open boxes, two phase-locked lobes) rebuilds exactly but is the pilot's sampled tail: with exact cores every box clears 0.835 and B7 closes the cell at lift depth 21 (floating point, not a certificate); the optimism theorem and the legal 7-fold family (\|a\| <= λ/96) hold, while every synthesis value sits 3–6×10⁻⁶ below the true minimum and the irregular and k = 3 candidates are unmeasured because only scores are recorded.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| On the tracked depth-50 cell (τ ≈ 3.82517×10⁻⁴), B7's lift leaves 1,046, 1,397 and 2,143 boxes open at depths 22, 24 and 26, with volume fractions 2.494×10⁻⁴, 8.327×10⁻⁵, 3.193×10⁻⁵ (§1–2). | computation | PASS | numerical only | 3.82517×10⁻⁴; 11,133; 4,521; 1,046; 2.494×10⁻⁴; 15,937; 6,572; 1,397; 8.327×10⁻⁵; 21,883; 8,799; 2,143; 3.193×10⁻⁵ |
| The phase-lock \|δ_lock\| median falls 0.1886 → 0.1372 → 0.1081 and its 95% quantile 0.4785 → 0.3330 → 0.2446 (§3–4). | computation | PASS | numerical only | 0.1886; 0.1372; 0.1081; 0.4785; 0.3330; 0.2446; 5×10⁻¹⁶ |
| At depth 26 the boxes form two lobes of 1,000 and 1,143 boxes near 7φ ≈ 2.5255 and 5.8819, with concentrations 0.98–0.99 (§5). | computation | PASS | numerical only | 1,000; 2.525491; 0.9899; 2.553558; 0.9830; 0.0121337; 1,143; 5.881925; 0.9877; 5.898881; 0.9778; 0.0124807 |
| The atlas field mean_phi is the mean of φ in each lobe. | definition | NOT AS STATED | corrected | 0.3607844; 0.8402750; 0.3607748; 0.8402794 |
| The area-minimising ridge at fixed φ lies in the same two channels; the tail is a thickened ridge (§6). | computation | PASS | numerical only | 64; 0.35062; 2.4544; 0.8371407193; 0.84150; 5.8905; 0.8371653077 |
| The package's two ridge points are the channels' minima (§6). | computation | NOT AS STATED | corrected | 0.35028; 0.8371486776; 8.0×10⁻⁶; 8.9×10⁻⁶; 0.85381; 0.8371759985; 1.07×10⁻⁵ |
| There are two failure mechanisms: B7's phase-locked tail and B23's near-disc counterexample (\|t₂₃\| ≈ 3.19×10⁻⁴) (§7). | reading | PASS | numerical only | 3.19×10⁻⁴ |
| The atlas is the B7 lift tail, a "structured adversarial atlas" of where B7 fails ("B7 LIFT-TAIL ATLAS: CLOSED"). | reading | PARTIAL | corrected | 2,143; 0.8358005576; 8.0×10⁻⁴; 1.20×10⁻³; 21; 2,509; 100; 39; 35 |
| Witness design is the max-min V* = max_K min_p F(K, p), solved by a cutting-plane loop; finite-tail training is optimistic and can only get less so, v_{m+1} <= v_m (§8–10). | theorem | PASS | analytic (proof checked) | 0.8377708; 0.8373826; 0.8371980; 0.8371877 |
| The 7-fold family (1 − λ)h_{B₇} + λ/2 + a cos(7θ + ψ) has width 1 and is legal for \|a\| <= λ/96 (§11). | theorem | PASS | analytic (proof checked) | λ/96; λ/2 − 48\|a\|; 64; 2×10⁻¹⁶; 1.02·λ/96 |
| Regular B7 at q† has baseline 0.83712806 (§12). | computation | PARTIAL | corrected | 0.8371280593; 0.8371347054; 0.8371345412; 0.8371339787; 5.9×10⁻⁶ |
| The smooth 7-fold witness h = ½ + a cos 7θ reaches 0.8369528 at a = 0.0103, with five recorded minima for a = 0.006 to 0.0103 (§13). | computation | PARTIAL | corrected | 0.8362521834; 0.8366138051; 0.8367717016; 0.8369133231; 0.8369528003; 3.2×10⁻⁶; 5.9×10⁻⁶; 1.69×10⁻⁴; 80,000 |
| Tail-targeted training reaches 0.8377708, a fresh oracle finds 0.8369560, and three more rounds return to the regular B7 baseline at 0.8371286 (§14). | computation | UNMEASURED | not measured | 0.8377708; 0.8369560; 0.8371286; 0.8371281; 5.3×10⁻⁷ |
| B7 remains the primary deep-tail witness (§15, §19). | reading | PASS | numerical only |  |
| The best irregular Reuleaux heptagon reaches 0.83697744 and a k = 3 pilot 0.83628622 (§16–17). | computation | UNMEASURED | not measured | 0.83697744; 0.83628622; 30; 6; 1.5×10⁻⁴; 8.4×10⁻⁴ |
| Status judgments "NEW SUPERIOR WITNESS: NOT FOUND" and "superior 7-fold witness: NOT FOUND". | reading | PARTIAL | corrected |  |
| Status judgment "Cutting-plane witness synthesis: FORMALIZED + IMPLEMENTED". | reading | PARTIAL | corrected |  |
| Status judgment "regular B7 local minimax interpretation: NUMERICAL-EVIDENCE". | reading | PARTIAL | corrected |  |
| Status judgment "nearby irregular Reuleaux7 improvement: NOT FOUND". | reading | UNMEASURED | not measured |  |
| The summary and boxed verdicts (§0). | reading | PARTIAL | corrected |  |

Open: No bound: "The exact-core closures are floating-point values of this tree's own kernel, not certificates."; Global minimum of any witness: "The minima quoted here are the least found, and the ridge scan's resolution is one 64th of the period."; Synthesised shapes of §14–17: "Not recorded, so not tested."; No computing code shipped: "Nothing that computes the atlas, the ridge, the searches or the synthesis is shipped."

Unclear: The report suggests the package's search area is a sampled hull 'it seems, though no code is shipped to say which'.; Census v5 finds 10 of Round 21's 13 judgments; the census RUN is not named here.

### RUN-025 — Round 22 - Fourier Pareto (AMRAL_Lebesgue_Finite_Closure_Round_22_Fourier_Pareto_v0.1.md, AMRAL-LUC-FC-R22, dated 2026-09-19) and its package

Report: `reports/RUN-025-ROUND22-FOURIER-PARETO.md`

> **Verdict: PASS for the legality rules, the full-orientation rule and its example, the recorded multi-mode candidate and its verdict, and the lift pilot as computed; PARTIAL for the search minima of B7 and of the harmonics, which sit on both sides of the true ones; FAIL for the proof-cost Pareto map's volume axis — with dense cores the ranking inverts, and the two "cheap" witnesses cannot close the cell at all.**

**For the paper.** Round 22's legality rules (odd modes keep width 1; the budget Σ(k² − 1)√(a_k² + b_k²) <= λ/2 is sufficient and single-mode sharp; pure harmonics legal exactly for \|a\| <= 1/(2(k² − 1))) hold, and the mixture example, multi-mode candidate (with an unstated B7 facing) and lift pilot rebuild; but its searched minima are off in both directions, and its proof-cost Pareto map fails: with dense cores B7 closes first (depth 21, 2,509 nodes) and the 'cheap' H17/H19 cannot close the cell at any depth (floating-point findings, not certificates).

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| In the family h = (1 − λ)h_{B₇} + λ/2 + Σ_{k ∈ {3,5,7,9,11}} (a_k cos kθ + b_k sin kθ), odd modes keep the width at 1 and Σ(k² − 1)√(a_k² + b_k²) <= λ/2 is a sufficient convexity condition (§1–3). | theorem | PASS | analytic (proof checked) | λ/2; 2×10⁻¹⁶; 48; 8.8×10⁻¹¹; 1.02 |
| The convexity budget is sharp for a single mode (report's check of §3's condition). | computation | PASS | numerical only | 1.02; 3; 11 |
| The recorded multi-mode candidate spends 0.0339507 of its budget 0.0369363 (§2–3). | computation | PASS | numerical only | 0.033950744437483665; 0.036936283277021795; 0.9191706; 91.9%; 0.00516; 0.00299 |
| The symmetry order is d = gcd(7, S), so any mode not a multiple of 7 costs a full 2π orientation domain, seven times B7's (§4). | theorem | PASS | numerical only | 1, 7, 7, 7, 1 |
| The pure harmonic h = ½ + a cos kθ is legal exactly when \|a\| <= 1/(2(k² − 1)), with reserve σ = ½ − (k² − 1)\|a\|, and h − δ is a legal core while δ <= σ (§9–12). | theorem | PASS | numerical only | 0.05; 90%; 0.0025; 99.5%; 0.1%; 0.00375; 0.0026785714…; 0.0015625; 0.00125 |
| K = 0.92 B7 + 0.08 B3 shows an optimistic basin at 0.83742146, but a full-circle multi-seed search finds 0.83583910 (§5). | computation | PASS | numerical only | 0.8358390980; −0.077; 7.7×10⁻¹⁰; 4.4×10⁻⁹; 0.8374213467; 1.1×10⁻⁷; 0.8368710 |
| The full-orientation oracle has levels TRAINING-SCORE, OPTIMISTIC-CANDIDATE, ROBUST-SEARCH-CANDIDATE, COMPILED-CANDIDATE, LIFT-VERIFIED (§6). | definition | PASS | label only |  |
| The multi-mode candidate has training 0.83770073 and single fresh seed 0.83733308, but four-cell minima 0.83683248, 0.83677292, 0.83674967, 0.83668617 against B7's worst 0.83692628, so it is not promoted (§7–8). | computation | PASS | numerical only | 0.8368301985; 0.8367753081; 0.8367532965; 0.8366861861; 3.6×10⁻⁶; 2×10⁻⁸; 2.48; 2.46; 2.31; 2.40 ×10⁻⁴ |
| B7's searched minima on the four cells (worst 0.83692628 on cell-03) (§7). | computation | PARTIAL | corrected | 0.8370810228; 0.8370777154; 0.8371419744; 0.8370210981; 1.21×10⁻⁴; 0.8369848645; 0.8369847609; 0.8369262850; 0.8369262831 |
| At 90% of the bound, H11 reaches 0.83574698 with bracket [0.83578614, 0.83581539], H13 0.83563803 with [0.83567775, 0.83569731], and H17 and H19 reach 0.83507834 and 0.83504963 (§13–15). | computation | PARTIAL | corrected | 0.8357638060; 0.8356225717; 0.8350819469; 0.8350481986; 1.7×10⁻⁵; 1.5×10⁻⁵; 2.23×10⁻⁵; 5.52×10⁻⁵; 8.19×10⁻⁵; 4.82×10⁻⁵ |
| The generic support-core lift pilot on the depth-50 cell gives open volumes at depth 20 of B7 1.4107×10⁻⁴, H11 1.8536×10⁻⁴, H13 1.4270×10⁻⁴, H17 9.5317×10⁻⁵ and H19 8.3671×10⁻⁵ (§16–17). | computation | PASS | numerical only | 7,037; 1,015; 9,623/1,929; 9,211/1,755; 7,399/1,533; 15,581/4,252; 7,221/1,504; 14,973/4,102; 1.4107×10⁻⁴; 1.8536×10⁻⁴; 1.4270×10⁻⁴; 9.5317×10⁻⁵; 8.3671×10⁻⁵; 0.2014734 |
| The harmonic lift's motion bound δ = \|Δt\| + 2(½ + \|a\|) sin(Δφ/2) with 40-direction cores is the pilot's convention (§16–17). | method | PARTIAL | corrected | 2,893/772; 2,737/749; 6,225/2,111; 5,689/2,010 |
| B7's tail contracts faster than the harmonics' (§18). | computation | PASS | numerical only | ×0.258; ×0.69; ×0.68 |
| The proof-cost Pareto vector Π(K) = (m_force, P_φ, σ, N_d, V_d, ρ_tail) ranks witnesses, with B7 primary, H11/H13 balanced auxiliaries and H17/H19 cheap razor-thin auxiliaries ("PROOF-COST PARETO MAP: ESTABLISHED") (§19–23). | method | FAIL | refuted | 21; 2,509; 27; 21,029; 30; 43,111; 24; 1,581; 26; 3,257; 4,096 |
| H17 and H19 are cheap, razor-thin auxiliaries ("CHEAP / RAZOR-THIN AUXILIARY"). | reading | FAIL | refuted | 0.8346707431; 0.8346707443; 0.8346480422; 0.8346480434; 9.9×10⁻⁵; 8.3×10⁻⁵; 4.3×10⁻⁴ |
| H11 and H13 are balanced auxiliaries ("BALANCED AUXILIARY"). | reading | PARTIAL | corrected | 8; 17 |
| No new single witness dominating B7 was found ("NEW SINGLE WITNESS DOMINATING B_7: NOT FOUND"). | reading | PARTIAL | corrected |  |
| The candidate is kept as a SEARCH ARCHIVE. | reading | PARTIAL | corrected |  |
| Search values carry a search-only warning, and a PURE-HARMONIC-CW compiler path is laid out (§24–26). | specification | PASS | label only |  |
| The summary and boxed verdicts (§0). | reading | PARTIAL | corrected |  |

Open: No bound: "The closures and limits are floating-point values of this tree's own kernel, not certificates."; Global minimum of any witness: "The minima quoted are the least found, and a 2×10⁻⁹ bracket holds only at the placement found."; Unstated base (§5) and facing of the candidate's B7: "Neither is stated. The answers here are the ones that reproduce the package's numbers, and no other tried does."; Finer base cell for H17/H19: "Their centre values say one could let them close. How fine is not measured."; §8's re-trained candidates are not recorded: "§8's candidates not recorded".; Shipped code is legality arithmetic only: "Nothing ships the searches, the mixture, the multi-mode candidate or the lift pilot."

Unclear: Census v5 finds 6 of Round 22's 17 judgments; the census RUN is not named here.; The symmetry-order item is recorded 'numerical only' because the report says it was 'measured on five sets'.; The H17/H19 'cannot close' finding rests on brackets from the report's own floating-point kernel; recorded as 'refuted' per the report's FAIL, while the report's open items say these are not certificates.

### RUN-026 — Round 23 - Portfolio Closure (AMRAL_Lebesgue_Finite_Closure_Round_23_Portfolio_Closure_v0.1.md, AMRAL-LUC-FC-R23, dated 2026-09-19) and its package

Report: `reports/RUN-026-ROUND23-PORTFOLIO-CLOSURE.md`

> **Verdict: PASS for the strict-incidence grammar and its theorems, the twelve cells, every B7 edge, the depth-22 crossover as computed, and the depth-16 infeasibility; PARTIAL for the harmonic edges, whose open counts rebuild but whose node counts do not. The routing the round draws does not survive exact cores: no harmonic can close any of the twelve cells at any depth, while B7 closes all twelve by lift depth 35.**

**For the paper.** Round 23's strict-incidence grammar, portfolio-closure theorem (hull monotonicity), set-cover feasibility criterion and non-composition of partial edges hold, the twelve cells and every B7 edge rebuild (at an unstated 96 directions), and the depth-16 strict cover is infeasible; but harmonic node counts do not rebuild, and in floating-point exact-core runs B7 closes all twelve cells by lift depth 35 while no harmonic can close any of them, so the routing to H19 does not survive.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Strict incidence: I_{jK} = 1 only if witness K's whole relevant placement root is certified for every q in cell C_j; partial trees, small pending, search minima, absent counterexamples and fast tails do not count (§1–4). | definition | PASS | label only |  |
| Theorem 5.1 (portfolio closure): if every necessity cell has a strict edge, the chosen witnesses close those cells, Λ(F₃ ∪ B) >= T (§5). | theorem | PASS | analytic (proof checked) | 24; 5.04×10⁻⁴ |
| The weighted set cover with budgeted incidence I^{(d)} is feasible at budget d iff the residual frontier R_d is empty (§6–8). | theorem | PASS | numerical only | 200; 6 × 5 |
| Partial edges serve the scheduler only: PARTIAL + PARTIAL does not imply STRICT COVER, and synergy needs a joint certificate (§9–11). | theorem | PASS | analytic (proof checked) |  |
| The pilot's scope is the 12 strongest of 77 necessity cells × 5 witnesses (§12). | specification | PASS | numerical only | 12; 85; 77; 4.5×10⁻¹⁴ |
| The pilot's B7 edges at depth 16 have the recorded node and open counts (§13, §15). | computation | PASS | numerical only | 96; 40; 3,031/559; 2,393/420; 2401.3; 423.75 |
| The harmonic lift root is \|x\|, \|y\| <= d* − ½ + \|a\|. | specification | PASS | analytic (proof checked) | 0.21 |
| No edge is complete at depth 16, so R₁₆ is all twelve cells and the strict cover is infeasible; do not force the cover (§13–14). | computation | PASS | numerical only | 16; 342; 379 |
| Aggregates: mean nodes and open volume B7 2401 and 9.42×10⁻⁴; H11 3749.5 and 4.77×10⁻⁴; H13 3644.2 and 3.41×10⁻⁴; H17 2362.5 and 2.66×10⁻⁴; H19 2286.3 and 2.29×10⁻⁴ (§15). | computation | PARTIAL | corrected | 2401.3; 3810.7; 3749.5; 3707.0; 3644.2; 2392.8; 2362.5; 2317.8; 2286.3; 42/48; 26–64 |
| Harmonic edges at depth 16 have the recorded node and open-box counts. | computation | PARTIAL | corrected | 42; 48; 2; 26–64 |
| H19 has the least volume on 12/12 cells (shallow routing); at depth 22 B7 overtakes it on 3 of the 4 hardest cells; on cell 0 B7 5.17×10⁻⁵ < H19 5.43×10⁻⁵ < H17 6.27×10⁻⁵ < H13 9.24×10⁻⁵ < H11 1.09×10⁻⁴ (§16–18). | computation | PASS | numerical only | 5.17036×10⁻⁵; 5.42745×10⁻⁵; 5.56995×10⁻⁵; 5.59072×10⁻⁵; 5.42053×10⁻⁵; 5.50781×10⁻⁵; 5.85487×10⁻⁵; 5.67873×10⁻⁵; 12,515/1,488 |
| A crossover depth d*, a horizon-aware cost-to-complete scheduler and the separation of proof and routing layers (§19–21). | method | PASS | label only |  |
| The residual is all twelve cells; there is no strict edge yet (§22). | reading | PARTIAL | corrected | 30–35; 18,509–45,401; 35 |
| The 77-cell long run, monotonicity of append-only incidence and DISQUALIFIED edges (§23–25). | specification | PASS | label only | 77 |
| The summary and boxed verdicts, including INFEASIBLE at depth 16 and STILL COMPUTE-DEFERRED (§0). | reading | PARTIAL | corrected |  |

Open: No bound: "The exact-core closures and the limits are floating-point values of this tree's own kernel, not certificates."; The other 65 necessity cells: "Only Round 23's twelve are run."; Harmonic node-count convention: "Their convention was not found."; Whether a finer base cell would let a harmonic close: "Not measured here."; Global claim still deferred: the set-off `STILL COMPUTE-DEFERRED` among Round 23's judgments.; No code shipped: "Neither the matrix nor the specs state the number of directions, the harmonic root or any placement."

Unclear: The strict cover 'by B7 alone' at depth 35 is stated 'in floating arithmetic, with this tree's kernel'; recorded in corrections, not as a certified result.; Census v5 finds 3 of Round 23's 7 judgments; the census RUN is not named here.

### RUN-027 — Round 24 - Residual Joint Admission (AMRAL_Lebesgue_Finite_Closure_Round_24_Residual_Joint_Admission_v0.1.md, AMRAL-LUC-FC-R24, dated 2026-09-19) and its package

Report: `reports/RUN-027-ROUND24-RESIDUAL-JOINT.md`

> **Verdict: PASS for the certificate, rebuilt bit for bit, for cells 1–3 at every recorded depth, for the joint-dominance and joint-necessity theorems and the admission logic; PARTIAL for the tail-collapse evidence, which reads its first two values off the 96-direction pilot and the rest off the exact-core tree; and the one residual left, cell 3, is not one: the same exact tree closes it at lift depth 34.**

**For the paper.** Round 24's first complete edge (cell 0, B7, lift depth 30: 18,555 nodes, 9,278 CERT, none pending, least slack 4.2074×10⁻⁸ at a 10⁻¹⁰ pad) rebuilds bit for bit as a floating-point replay, and its joint-dominance and joint-necessity statements hold; but its tail-collapse figures mix the 96-direction pilot with the exact-core tree (cell 3's ×393 is ×217 with exact cores), and cell 3 also closes at lift depth 34 in floating point.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| The first complete edge: cell 0, B7, depth 30 has 18,555 nodes, 9,278 CERT leaves and none pending, with 18555 = 2(9278) − 1, and the replay passes (REFERENCE-COMPLETE-EDGE-PILOT, not publication-grade) (§7). | certificate | PASS | numerical only | 18,555; 9,278; 2,320; 4.1174×10⁻⁸; 4.2074×10⁻⁸; 10⁻⁹; 10⁻¹⁰; 4.15×10⁻⁹; a9139473… |
| Joint dominance J_B(H(q)) >= max_K J_K(H(q)), so a strict single edge makes a joint proof redundant; the residual is budget-relative (§1–5). | theorem | PASS | analytic (proof checked) | 16; 5.04×10⁻⁴ |
| Cell 0's tail collapses through open volumes 9.34×10⁻⁴, 5.17×10⁻⁵, 6.94×10⁻⁶, 1.77×10⁻⁶, 9.39×10⁻⁸, 0 at depths 16–30 (§6). | computation | PARTIAL | corrected | 9.34×10⁻⁴; 5.17×10⁻⁵; 6.94×10⁻⁶; 1.77×10⁻⁶; 9.39×10⁻⁸; 7.605×10⁻⁴; 2.512×10⁻⁵; ×3.6; ×7.4 |
| Deepening B7 closes cells 1 and 2 at depth 32 (§8). | computation | PASS | numerical only | 5; 6.79×10⁻¹⁰; 152; 2.06×10⁻⁸; 31; 22,009; 26,189 |
| Cell 3 is left at depth 30 with 1.49×10⁻⁷ open, contracting ×393 since depth 22 (DEFERRED-CONTRACTING) (§9–10). | computation | PARTIAL | corrected | 1.49×10⁻⁷; 1,098; 31,797; 14,801; ×393; ×217; 34; 36,193; 445 |
| The recorded edge records (cells 0–3 at depths 30, 32, 32, 30, and B7 depth-24 on cell 0 with 12,501 nodes, 799 open) are cuts of the deep B7 tree. | computation | PASS | numerical only | 12,501; 799 |
| Pointwise joint necessity: if every portfolio witness has a verified counterexample at the same exact q, single-witness reduction is impossible there, which is not joint sufficiency; witness switching comes before joint trees (§11–13). | theorem | PASS | analytic (proof checked) |  |
| Admission states REDUNDANT, DEFERRED-CONTRACTING, CANDIDATE, NECESSARY-RELATIVE-TO-PORTFOLIO; cells 0–2 REDUNDANT and cell 3 DEFERRED-CONTRACTING (§14–15). | specification | PASS | numerical only |  |
| A joint tree costs about L₁L₂ leaves; the joint lower hull is conv(G_base ∪ G_K ∪ G_L); a joint counterexample has an outer hull below T (§16–21). | specification | PASS | analytic (proof checked) | 0.99 |
| The residual {0,1,2,3} → {3} is a 75% reduction; cells 4–11 are untested (§22–24). | computation | PASS | numerical only | 75%; 34; 36 |
| The summary and boxed verdicts (§0). | reading | PASS | numerical only |  |
| Status judgments for cell 3: "CONTRACTING" and "JOINT-DEFERRED-CONTRACTING". | reading | PARTIAL | corrected | ×393; 34 |

Open: No bound: "The certificate is a floating-point replay, as its own manifest says. This tree's rebuild uses the same kernel family, not interval arithmetic."; Joint-witness theorems unused: "No joint tree is built, here or in the package."; H19's node counts: "The harmonic convention is still unresolved (RUN-026)."; No code shipped: "Thirteen files, no code."

Unclear: The H19 depth-24 record on cell 0 (package 38,015 nodes / 11,987 open vs rebuilt 38,037 / 11,985) is described as 'the harmonic gap RUN-026 could not close' without a separate verdict label; not listed as an item.; Census v5 finds 8 of Round 24's 20 judgments; the census RUN is not named here.

### RUN-028 — Round 25 - Deep B7 Closure Wave (AMRAL_Lebesgue_Finite_Closure_Round_25_Deep_B7_Closure_Wave_v0.1.md, AMRAL-LUC-FC-R25, dated 2026-09-20) and its package

Report: `reports/RUN-028-ROUND25-CLOSURE-WAVE.md`

> **Verdict: PASS for all twelve closures, node for node, for every checkpoint of the ledger, for cell 7's volumes and the monotonicity theorem, and for the singleton cover and its empty joint residual; PARTIAL for the closure-depth statistics: sec. 1 defines D_j as the least complete depth, but the ledger records the first even checkpoint, so D_max is 35 and not 36, and the median is 31.5 and not 32.**

**For the paper.** Round 25's twelve B7 closures rebuild node for node (340,540 nodes, 45,401 at most) and give a singleton B7 cover of the twelve cells with an empty joint residual, in floating-point exact-core runs; but the recorded closure depths are even checkpoints, so the least closure depths give D_max = 35, mean 32.0 and median 31.5 (not 36, 32.5, 32), and 35 is the least common budget.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| B7 run deeper on all twelve of Round 23's cells closes every one, with the ledger's nodes and CERT leaves, 340,540 nodes in all and 45,401 at most (§0, §2, §19–20). | computation | PASS | numerical only | 340,540; 45,401; 18,555; 9,278; 22,009; 11,005; 26,189; 13,095; 36,193; 18,097; 21,271; 10,636; 18,509; 9,255; 31,209; 15,605; 22,701; 26,439; 13,220; 41,077; 20,539; 31,793; 15,897; 21,895; 10,948 |
| The 58 checkpoints of the nine incremental cells record cumulative nodes, CERT leaves, open boxes, volume and worst new slack. | computation | PASS | numerical only | 58; 9.0×10⁻¹⁰; 10⁻¹⁴; 10⁻¹² |
| The ledger's markers are the cells' outer brackets, all below T. | computation | PASS | numerical only | 10⁻¹³ |
| Closure depths satisfy 30 <= D_j <= 36 with D_max 36, mean 32.5 and median 32, where D_j is the least depth with no open box (§1–3). | computation | PARTIAL | corrected | 30, 31, 31, 34, 31, 30, 33, 35, 32, 34, 33, 30; 35; 36; 32.0; 32.5; 31.5; 32 |
| Coverage Γ_B7(30, 32, 34, 36) = 3, 7, 11, 12, and D⋆ = 36 is a common budget (§4, §14–15). | computation | PASS | numerical only | 3; 7; 11; 12; 6; 9; 35; 36 |
| The pending count can grow while V_d falls; V_{d+1} <= V_d under partitioning splits (§7–10). | theorem | PASS | numerical only | 259 |
| Cell 7's open volume runs 3.33×10⁻⁵ → 0 from depth 22 to 36 (§11). | computation | PASS | numerical only | 3.33×10⁻⁵; 1.05×10⁻⁵; 3.89×10⁻⁶; 9.90×10⁻⁷; 2.60×10⁻⁷; 4.22×10⁻⁸; 1.61×10⁻¹⁰; 0; 1,793; 1,918; ×14.96 |
| {B7} is a strict cover of the twelve cells, the joint residual is empty, and nothing is claimed for the 77-cell atlas or globally (§5–6, §12). | reading | PASS | numerical only | 12 |
| Infeasibility is budget-relative and production runs B7 first, with the portfolio on true residuals (§13, §16–18). | method | PASS | numerical only |  |
| The ledger, validation and list of what is missing (§19–21). | specification | PASS | numerical only | 176; 10 |
| Status judgment "12-CELL REFERENCE SINGLETON B_7 COVER: COMPLETE". | reading | PASS | numerical only |  |
| The runner r25_incremental_wave.py computes the wave with a bound max(a-priori disc-point area, exact hull − 10⁻¹⁰) and checkpoints at 22, 24, …, 38. | computation | PASS | numerical only | 64; 0.29; π/4 |

Open: No bound: "The trees are floating-point runs of the exact-core kernel, as the package's own scope says."; The 77-cell atlas: "Not run here or in the package."; Global claim still deferred: the set-off STILL COMPUTE-DEFERRED among Round 25's judgments.; The runner does not run from the package: "it imports Round 19's continuation probe and exact kernel from an absolute path outside the package".

Unclear: Census v5 finds 2 of Round 25's 5 judgments; the census RUN is not named here.; The monotonicity theorem V_{d+1} <= V_d is checked on every level of every tree; the report does not show a separate proof check, so it is recorded as 'numerical only'.

### RUN-029 — Round 26 - Full Atlas Arithmetic (AMRAL_Lebesgue_Finite_Closure_Round_26_Full_Atlas_Arithmetic_v0.1.md, AMRAL-LUC-FC-R26, dated 2026-09-20) and its package

Report: `reports/RUN-029-ROUND26-FULL-ATLAS.md`

> **Verdict: PASS for the 77 cells and their markers, the closure curve, every ledger row, the residual's volumes, classes and ratios, the tight-leaf polygon audit and the radial audit; PASS as stated for §14's least slack, which is the least over the ten audited trees only — over all 51 complete trees the least is 8.47×10⁻¹¹ (cell 8), and 11 trees hold a leaf thinner than §14's 2.8982×10⁻⁹. The rejection of a larger global pad stands, more strongly than §14 argues it.**

**For the paper.** Round 26's full atlas rebuilds row for row under the package's 10⁻¹⁰ pad: 77 necessity cells, 51 close by lift depth 36 (curve 0, 0, 0, 0, 7, 19, 38, 51), 26 residual cells all contracting, 4,481,119 nodes in floating-point exact-core runs; the least slack over all complete trees is 8.47×10⁻¹¹ (not the audited 2.8982×10⁻⁹), which strengthens the rejection of a larger global pad, and all leaves remain ARITHMETIC-PENDING.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| The atlas is Round 19's 77 necessity-marked depth-40 cells, each with U_base < 0.835 (§1). | computation | PASS | numerical only | 77; 1.2×10⁻¹³ |
| Marker margins T − U have least 2.885×10⁻⁷ and median 6.806×10⁻⁵ (§23). | computation | PASS | numerical only | 2.885×10⁻⁷; 6.806×10⁻⁵; 8; 13; 30 |
| The closure curve Γ_B7 at waves 22, 24, …, 36 is 0, 0, 0, 0, 7, 19, 38, 51; 51/77 = 66.23% close by 36 and 26 remain (§0, §5–6). | computation | PASS | numerical only | 0, 0, 0, 0, 7, 19, 38, 51; 51/77; 66.23%; 26; 0, 15, 30, 45 |
| The full-atlas ledger records each of the 77 cells' nodes, open boxes, status and volume, 4,481,119 nodes in all. | computation | PASS | numerical only | 4,481,119; 15; 31; 26; 10⁻¹⁰; 10⁻⁹ |
| Waves are run at common depths 22 → 36 with resumable state, monotone in the budget (§2–4). | method | PASS | numerical only |  |
| The 26 residual cells have volumes 6.36×10⁻¹² to 6.99×10⁻⁸ (5 dust, 7 thin, 14 broad), every ratio to the previous wave below 1, the largest 0.57615 (§7–9). | computation | PASS | numerical only | 6.36×10⁻¹²; 6.99×10⁻⁸; 5; 7; 14; 0.30094; 0.57615; 0.57617 |
| The joint-witness queue is deferred, and the pending count is not difficulty (§10–11). | reading | PASS | numerical only |  |
| Every reference leaf is ARITHMETIC-PENDING; a per-leaf interval ledger, a selective revalidation theorem and a backend contract (δ rounded up, ρ = 1 − δ rounded down) are formalized (§13, §15–21). | specification | PASS | label only |  |
| The least audited slack is 2.8982×10⁻⁹ (§14). | computation | PASS | corrected | 2.8982×10⁻⁹; 8.47×10⁻¹¹; 1.74×10⁻¹⁰; 1.77×10⁻¹⁰; 7.48×10⁻¹⁰; 8.76×10⁻¹⁰; 11; 51 |
| A larger global pad is rejected (§14). | reading | PASS | numerical only | 10⁻⁸; 29; 10⁻⁹; 5; 10 |
| The radial fallback keeps only a few percent of leaves, so eroded-Reuleaux cores are kept (§20). | computation | PASS | numerical only | 9,278; 60; 33; 228; 148; 24; 32; 48; 64; 507; 723; 1,278; 1,790 |
| The tight-leaf polygon audit: the thinnest certificate leaf has nominal lower 0.835000042174202; uniform contact polygons give 0.8349998195 at 4,096 directions and 0.8350000060 at 8,192 (§22). | computation | PASS | numerical only | 0.8350000421742021; 0.8349998195; 4,096; 0.8350000060; 8,192; 5×10⁻¹² |
| The arithmetic migration is specified and theorem_ready = false (§24–25). | specification | PASS | label only |  |

Open: No bound: "The trees are floating-point runs of the exact-core kernel, as the package's own scope says; nothing here is interval arithmetic."; The residual beyond 36: "Round 27's."; Residual statistics are pad-dependent: "the residual's numbers are pad-dependent at that scale."; The ancestor experiment (§12) is stopped: "no claim (W7)".; The scripts do not run from the package: "All load Round 19's modules (the audits also Round 24's manifest) from an absolute path outside the package."

Unclear: §13, §15–21 (interval ledger, selective revalidation theorem, backend contract) are judged 'PASS \| formal (P1)'; the report does not say the selective revalidation theorem's proof was checked, so the item is recorded as 'label only'.; Census v5 adds none of Round 26's 11 judgments.

### RUN-030 — Round 27 - Residual Interval Replay (AMRAL_Lebesgue_Finite_Closure_Round_27_Residual_Interval_Replay_v0.1.md, AMRAL-LUC-FC-R27, dated 2026-09-20) and its package

Report: `reports/RUN-030-ROUND27-INTERVAL-REPLAY.md`

> **Verdict: PASS throughout.**

**For the paper.** Under the report's own outward-rounded integer-interval arithmetic, cell 0's twelve thinnest leaves (the package's five and seven more) certify with margins 3.1949×10⁻⁸ to 4.0147×10⁻⁷, and the package's five agree to 4.8×10⁻¹⁷; this certifies individual leaves only, not any whole cell. At a common depth the residual curve is exact (Γ_B7(38) = 57, Γ_B7(40) = 63), with 14 cells still open at 40, and the stored φ root relies on the float enclosure until an exact root is built.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Only the depth-36 residual is extended; the common curve stays Γ_B7(36) = 51, and higher budgets give lower bounds only, Γ_B7(38) >= 57 and Γ_B7(40) >= 63 (§0, §2–3). | computation | PASS | numerical only | 51; 56; 57; 63; 63; 258,764; 6,111,066 |
| 14 cells have no known closure: 5 last tested at 36, 8 at 38, 1 at 40, with recorded open boxes, volumes and ratios (§0, §3). | computation | PASS | numerical only | 14; 5; 8; 1; 2,244; 2.97×10⁻¹⁰; 0.608; 825,168; 10⁻¹² |
| The residual is scheduled by volume (§4). | method | PASS | label only |  |
| A fast negative prefilter "can create false negatives, but cannot create false positives" (§5). | method | PASS | analytic (proof checked) | 10⁻¹⁰; 2×10⁻⁵; 256 |
| Theorem 7.1: points verified inside certified cores span a hull inside the cores' hull (§7). | theorem | PASS | numerical only | 0.99–1.41×10⁻⁸; 0 of 3170; 1.80×10⁻⁷ |
| The disc test is exact, a core is tested by interval discs with d²_hi < (ρ²)_lo, and mpmath.iv is a prototype backend (§10–12). | method | PASS | exact-arithmetic certificate | 80 |
| The leaf boxes are the stored binary64 values, each enclosed by its neighbouring floats; the exact root is not yet rebuilt (§13). | specification | PASS | numerical only | 4; 0x1.9c9e1c2550f72p-3; 0x1.9c9e1c2550f76p-3; 0.2014734457862874141…; 3.39; 7.39; 3.5×10⁻¹⁷ |
| Cell 0's worst leaf (path 1101011010010011111001110, depth 25, reference slack 4.20742×10⁻⁸), at 16,384 directions per body and inward 10⁻¹², has 3170/3170 points verified, exact area 0.8350000319489227 and margin 3.1949×10⁻⁸ (§14–16). | certificate | PASS | exact-arithmetic certificate | 1101011010010011111001110; 25; 4.20742×10⁻⁸; 16,384; 10⁻¹²; 3170; 0.8350000319489227; 3.1949×10⁻⁸ |
| The five thinnest leaves of cell 0 pass: 5/5, margins 3.19, 6.42, 11.6, 15.7, 20.3 ×10⁻⁸ (§17). | certificate | PASS | exact-arithmetic certificate | 3170; 3149; 3736; 3363; 3536; 3.1949×10⁻⁸; 6.4154×10⁻⁸; 1.1567×10⁻⁷; 1.5724×10⁻⁷; 2.0269×10⁻⁷; 4.8×10⁻¹⁷ |
| What the round proves and does not: no exact root, no full replay, no second backend; PUBLICATION THEOREM: NOT YET; a_Leb >= 0.835 NOT CERTIFIED (§18–19). | reading | PASS | open | 0.835 |
| Migration flow, adaptive density and the Round 14 reference (§20–22). | method | PASS | label only |  |

Open: No bound on a whole cell: "Twelve leaves of cell 0 are certified; its other 9,266 and every other cell's are Round 28's and later."; The exact root: "the φ sliver is covered only by the enclosure; an exact root is Round 28's."; 14 residual cells remain open at depth 40 ("The 14 cells without a known edge are exactly the 14 open at 40."); Prefilter false negatives: "Whether that estimate lies inside or outside the hull — how many false negatives it makes — cannot be told."; Missing inputs: "Neither file is in the archive; the shipped `cell0_tight12_reference.json` holds the same paths and slacks."; Source's own scope: PUBLICATION THEOREM: NOT YET; a_Leb >= 0.835 NOT CERTIFIED.

Unclear: Theorem 7.1 is judged PASS on W4 (enclosures, refusals, polygons below exact hulls); the report does not state a separate proof check, so it is recorded as 'numerical only'.; The report's own certificate uses integer intervals at 2⁻⁴⁰⁰ with exact integer hulls; the package uses mpmath.iv at 80 digits. Both are recorded as 'exact-arithmetic certificate' for the leaf items, at leaf level only.; The report notes the reference slack 'is not the certificate's margin. It is the margin's ceiling' (polygon costs about 10⁻⁸ per leaf).

### RUN-031 — Round 28 - Exact Root and Bulk Migration (AMRAL_Lebesgue_Finite_Closure_Round_28_Exact_Root_Bulk_Migration_v0.1.md, AMRAL-LUC-FC-R28, dated 2026-09-20) and its package

Report: `reports/RUN-031-ROUND28-EXACT-ROOT.md`

> **Verdict: PASS for the directed root, the base path, the witness axes, the stored-state distances and the migration of all 9,278 leaves; PARTIAL for the density economics — the ledger's densities are start points chosen from each leaf's reference slack by bands the round states nowhere, and the least density that passes is lower for 8,457 leaves, 16,384 for none, so "only six leaves require the maximum tested density" describes the policy, not the leaves.**

**For the paper.** Under the report's own directed integer-interval arithmetic, Round 28's exact root (d⋆, t₃, t₅, t₇ bracketed to 10⁻⁹⁰ inside the package's intervals), cell 0's 40-split base path and 18,555-node witness tree, and all 9,278 of cell 0's leaves (least margin 3.1948922738×10⁻⁸) check; this certifies cell 0 only, not a_Leb >= 0.835. The density ledger reflects policy bands: no leaf needs 16,384 directions.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| The targets T_M = 1673/2000 and T = 167/200 are exact (§0–1). | definition | PASS | exact-arithmetic certificate | 1673/2000; 167/200 |
| d⋆ is found by directed bisection of the monotone a-priori area F, to O(10⁻⁶³) (§2). | certificate | PASS | exact-arithmetic certificate | 10⁻⁹⁰; 3×10⁻⁶¹; 3.1×10⁻⁶¹; 6×10⁻¹¹⁴ |
| t₃, t₅ and t₇ are directed intervals, each under 10⁻⁵⁰ wide, and the D3 root's endpoints are directed (§3–4). | certificate | PASS | exact-arithmetic certificate | 10⁻⁵⁰; 10⁻⁹⁰; 3–4×10⁻⁶⁴; +7.01; +1.01; +8.77; +4.77; +7.39; +3.39; −0.22; −0.32 |
| Binary64 roots versus exact values: intended boxes close the angle-root slivers (§3–4, §7). | computation | PASS | exact-arithmetic certificate | 0.22; 0.32 |
| Certificates store explicit splits; cell 0's base path is 40 splits with 0 CPG overrides (§5–6, §9). | specification | PASS | exact-arithmetic certificate | 40; 7, 7, 9, 9 and 8; 0 |
| Stored centres sit about 10⁻¹⁷ from the intended ones on the base and up to 2.1×10⁻¹⁶ on the witness (§7). | computation | PASS | numerical only | 10⁻¹⁸; 3.2×10⁻¹⁷; 2.10×10⁻¹⁶; 4.71×10⁻¹⁷; 9.42×10⁻¹⁷; 6; 4 |
| The witness tree has 18,555 nodes and 9,278 leaves, with 9,277 axes in a 2-bit stream (§8). | certificate | PASS | exact-arithmetic certificate | 18,555; 9,277; 9,278 |
| Directed cores, exact disc membership and a core test D²_max < ρ_lo² give a lower theorem (§10–13). | theorem | PASS | exact-arithmetic certificate |  |
| All 9,278/9,278 leaves of cell 0 pass as RATIONAL-INTERVAL-PASS, no resplit, no membership failure, least margin 3.1948922738×10⁻⁸; cell 0 becomes CELL0-ARITHMETIC-MIGRATED-PROTOTYPE (§15, §17–19). | certificate | PASS | exact-arithmetic certificate | 9,278; 3.1948922738×10⁻⁸; 1.7×10⁻¹⁶; 1101011010010011111001110 |
| Densities are margin-adaptive (256 … 16,384), with counts 821, 3,425, 3,709, 1,115, 163, 39, 6, and only six leaves require the maximum tested density (§14, §16, §22). | reading | PARTIAL | corrected | 821; 3,425; 3,709; 1,115; 163; 39; 6; 7,727; 1,285; 177; 67; 18; 4; 0; 8,457; 6.0×10⁻⁹ |
| The package's reference slack for leaf 001110010110000100. | computation | PASS | numerical only | 0.835746558304646; 0.835746559778711; 1.474×10⁻⁹; 0.835746559779; 1.7×10⁻¹⁵ |
| What is missing and the migration theorem (§20–21). | reading | PASS | label only |  |
| Scope: the backend is not pinned, one cell of 77 is migrated, and there is no marker replay, so a_Leb >= 0.835 is NOT CERTIFIED (§20–26). | reading | PASS | open | 1; 77; 0.835 |

Open: Any cell but cell 0: "The other 76 are Round 29's and later."; Source's own scope: the backend is not pinned, there is no marker replay, and a_Leb >= 0.835 is NOT CERTIFIED.; Upper bounds needed a corrected kernel: "An upper bound would not be safe with it, and Round 29 migrates upper bounds."

Unclear: The report's addendum (2026-09-27) says the shared kernel was corrected and RUN-008 to RUN-031 re-run; every gate log of RUN-008 to RUN-030 came out byte for byte and no logged number moved.; The §7 witness distances are judged PASS with the note that 'the witness figures are over the leaves'; strength recorded as 'numerical only'.; The leaf-migration items are recorded as 'exact-arithmetic certificate' for cell 0 only; the report and the source both say other cells are not migrated.

### RUN-032 — Round 29 - Marker Upper Certificates and Whole-Shard ABI (AMRAL_Lebesgue_Finite_Closure_Round_29_Marker_Whole_Shard_ABI_v0.1.md, AMRAL-LUC-FC-R29, dated 2026-09-20) and its package

Report: `reports/RUN-032-ROUND29-MARKER-UPPER.md`

> **Verdict: PASS for the marker points, the outer-polygon theorem, all 77 upper certificates, the three-marker replay and the whole shard; PARTIAL for the rankings and densities of §11–13 — the package's outer polygons lose first order in the normal spacing at the hull's straight bridging edges, so its thinnest certificate (marker 6, 6.1×10⁻⁷) is not its thinnest marker (marker 47, exact margin 1.86×10⁻⁵, marker 6's being 2.11×10⁻⁵), marker 47's "tighter" bound is tighter only by more directions, and its densities are start points chosen from the Round 26 reference margins.**

**For the paper.** Under the report's own integer-interval arithmetic (support upper bounds at 2⁻⁴⁰⁰, exact polar hull), all 77 necessity markers have outer polygons below 167/200 with 128–256 normals once bridging-edge normals are added (thinnest certificate 1.50×10⁻⁶, marker 47), and the first whole shard (marker 70 plus cell 0's 9,278 lower leaves) checks; each marker certifies one configuration per cell, a_Leb >= 0.835 is not certified, and the package's ranking of marker 6 as weakest reflects first-order polygon losses, not the exact margins.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Each marker point q_j is a rationalized reference centre strictly inside its directed cell; 77/77 lie safely inside and no path has a CPG override (§1, §15). | certificate | PASS | exact-arithmetic certificate | 77; 7 ulps; 17 |
| Outer certificates: integer normals with directed support upper bounds give half-planes containing the hull, so A(H(q)) <= A(P) and A(P) < 167/200 certifies the marker; dropping constraints is safe; the geometry is an exact polar hull (§2–9). | theorem | PASS | analytic (proof checked) | 167/200; 10⁻²⁶ |
| All 77 marker upper certificates PASS, none inconclusive (§0, §10). | certificate | PASS | exact-arithmetic certificate | 77; 51; 128; 26; 256; 11; 1.50×10⁻⁶; 1.857×10⁻⁵; 2⁻⁴⁰⁰ |
| The package's 77 outer-polygon areas are upper bounds of the exact areas. | computation | PASS | numerical only | 1.3×10⁻¹⁰; 1.1×10⁻⁵; 1.1×10⁻⁴; 1.3–5.5; 4 |
| Final densities are 4,096: 21, 8,192: 39, 16,384: 16, 32,768: 1, and none needs 65,536 (§11). | reading | PARTIAL | corrected | 21; 39; 16; 1; 65,536; 73; 4; 2,048; 4,096; 8,192; 16,384; 32,768 |
| The weakest marker is marker 6, margin 6.1062×10⁻⁷; the median is 3.5043×10⁻⁵ (§12). | computation | PARTIAL | corrected | 6.1062×10⁻⁷; 3.5043×10⁻⁵; 2.112×10⁻⁵; 1.857×10⁻⁵; 7.57×10⁻⁶ |
| Marker 47's new bound (7.5672×10⁻⁶) is tighter than its reference (2.89×10⁻⁷) (§13). | reading | PARTIAL | corrected | 7.5672×10⁻⁶; 2.89×10⁻⁷; 20,000; 1.83×10⁻⁵; 32,768; 1.10×10⁻⁵ |
| Markers 6, 47 and 70 replayed without candidate reduction pass (A1-MARKER-SAMPLE-PASS, not 77/77) (§14). | certificate | PASS | exact-arithmetic certificate | 4,962; 10,255; 1,317; 4,900; 10,204; 1,284 |
| A five-layer whole-shard ABI; marker 70 is Round 28's cell with upper margin 9.24985×10⁻⁵, joined with the 9,278/9,278 lower leaves (ARITHMETICALLY-CLOSED-WHOLE-SHARD-PROTOTYPE, publication_candidate = false) (§16–22). | certificate | PASS | exact-arithmetic certificate | 9.24985×10⁻⁵; 9,278; 1.98×10⁻⁴; 9.25×10⁻⁵; 1.37×10⁻⁴; 128 |
| States ARITHMETICALLY-CLOSED-PROTOTYPE, PUBLICATION-CANDIDATE, THEOREM-READY; a_Leb >= 0.835 NOT CERTIFIED (§23–27). | reading | PASS | open | 0.835 |

Open: Any marker beyond its point: "A marker certifies one configuration per cell, as §17 says; nothing here speaks for the rest of a cell."; An independent lower replay: "Round 30's; RUN-031 has replayed cell 0's 9,278 leaves with this tree's own arithmetic."; Source's own scope: publication_candidate = false; a_Leb >= 0.835 NOT CERTIFIED.

Unclear: The outer-polygon theorem is judged 'PASS \| reading; W2'; recorded as 'analytic (proof checked)' since the report reads the construction as correct and uses it for its own 77 certificates.; This run uses the exact kernel as corrected after RUN-031.

### RUN-033 — Round 30 - A1 Backend Pinning (AMRAL_Lebesgue_Finite_Closure_Round_30_A1_Backend_Pinning_v0.1.md, AMRAL-LUC-FC-R30, dated 2026-09-20) and its package

Report: `reports/RUN-033-ROUND30-A1-REPLAY.md`

> **Verdict: PASS for the root replay (its intervals lie inside Round 28's, and this tree's brackets inside its own), marker 70's bound, all 9,278 lower leaves, the root correction, the lock's script digests and its libmp pin, and the shard's records; PARTIAL for the shard's implementation-level A1 on its upper half — the marker replay is Round 29's audit row, from a script that carries the emitter's support routine, normals and density.**

**For the paper.** Round 30's root replay nests inside Round 28's intervals (with the report's own brackets inside it), marker 70's outer bound 0.8349075014501105 and all 9,278 lower leaves of cell 0 pass in exact/interval arithmetic, so cell 0's shard checks; but the upper-half replay shares the emitter's support code, so implementation-level independence holds only for the lower half, the thinnest margin 3.54237075×10⁻⁹ measures the replay rule's one-point-per-direction loss, and no second backend is established; a_Leb >= 0.835 is not certified.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| From T_M = 1673/2000, a monotone directed bisection recomputes d⋆, t₃, t₅, t₇, and the root replay overlaps Round 28's intervals (§0, §12–13). | certificate | PASS | exact-arithmetic certificate | 1673/2000; 2.8–4.0×10⁻⁷³; 3.1×10⁻⁶¹; 3.2–4.3×10⁻⁶⁴; 10⁻⁹⁰ |
| R30-A1-ROOT-001: an ultra-narrow d⋆ interval, "about 10⁻⁶⁴" wide, had been cast to binary64 and moved at the 10⁻¹⁷ scale (§12). | computation | PASS | exact-arithmetic certificate | 1.70×10⁻¹⁷; 2.43; 1.90; 1.80×10⁻¹⁷; 10⁻⁶⁴; 3.1×10⁻⁶¹; 2.8×10⁻⁷³ |
| Marker 70's upper A1 replay by an exact cyclic polar hull with no scipy reduction passes: outer 0.8349075014501105, margin 9.24985499×10⁻⁵ (§0, §19). | certificate | PASS | exact-arithmetic certificate | 0.8349075014501105; 9.24985499×10⁻⁵; 0.8348018; 4,096; 1,317; 1,284; 5.2×10⁻¹¹; 1.37×10⁻⁴ |
| Implementation A1: nothing is shared in candidates, hull, density or ledger (§1). | reading | PARTIAL | corrected |  |
| A second lower implementation: defining disks, arc and corner candidates, the union winner per direction, a 10⁻¹⁷ grid, an exact integer monotone hull, directed membership and an exact area (§2–8). | method | PASS | exact-arithmetic certificate | 75; 10⁻¹⁷; 2×10⁻¹²; 512; 32,768; 1.57–1.67×10⁻¹² |
| 9,278/9,278 lower leaves A1-PASS, none inconclusive, with densities 512: 8,174; 1,024: 869; 2,048: 163; 4,096: 56; 8,192: 12; 16,384: 4; none needs 32,768 (§9–10). | certificate | PASS | exact-arithmetic certificate | 9,278; 8,174; 869; 163; 56; 12; 4; 2.1×10⁻⁸; 1.6×10⁻⁴; 3.19489×10⁻⁸; 974 |
| The thinnest margin is 3.54237075×10⁻⁹, on another leaf than Round 28's (3.1949×10⁻⁸): "different inner polygons, different losses" (§11). | reading | PASS | numerical only | 3.54237075×10⁻⁹; 3.5440×10⁻⁹; 2.2211×10⁻⁵; 2.221×10⁻⁵; 6.7795×10⁻⁹; 2.517×10⁻⁸; 1.30945×10⁻³; 1.738×10⁻⁵ |
| The backend is pinned (mpmath 1.3.0, libmp, fractions.py and the scripts) as PINNED-MPMATH-LIBMP-PROTOTYPE-v0.1, state PINNED-REPRODUCIBLE-PROTOTYPE with TRUST-AUDIT-OPEN; implementation A1 and backend independence are separate requirements (§1, §14–16). | specification | PASS | label only | 1.3.0; 9; 3.13.5; 3.14.5 |
| The shard is A1-VERIFIED-PINNED-PROTOTYPE-SHARD, not a PUBLICATION-CANDIDATE until a second backend replays the root, the upper bound and the 9,278 lower memberships (§17–18). | reading | PARTIAL | corrected | 9,278 |
| Marker scope and global state unchanged: 51/77 at d = 36, Γ(38) >= 57, Γ(40) >= 63; a_Leb >= 0.835 NOT CERTIFIED (§19–23). | reading | PASS | open | 51/77; 57; 63; 0.835 |

Open: A second backend: "This tree's replays use no mpmath, but they are one implementation, by one hand, and not audited. §21's gate is the package's to judge."; A1 for the other 76 markers is not established here (as §19 says); the report notes: "RUN-032 certified all 77 with this tree's own polygons."; Anything beyond cell 0's shard: "The global state is unchanged: 51/77 at d = 36."; Whole-package digest: "none of twelve natural ways of forming it over the same sources gives it."; fractions.py pin not compared: "This machine runs 3.14.5, so it is not compared."; Exact-to-interval conversion: "In practice the interval's own 75-digit outward rounding covers it, but not by construction."

Unclear: The report says this tree's replays of the shard's three parts share no arithmetic with mpmath/libmp, but 'whether they count as §21's second backend is not for this tree to say'; no verdict is given on §21.; W4 compares libmp digests only where the pinned mpmath version is installed; elsewhere it records 'not compared'.; This run uses the exact kernel as corrected after RUN-031.

### RUN-034 — Round 31 - Cross-Backend Publication Candidate (AMRAL_Lebesgue_Finite_Closure_Round_31_Cross_Backend_Publication_Candidate_v0.1.md, AMRAL-LUC-FC-R31, dated 2026-09-20) and its package

Report: `reports/RUN-034-ROUND31-MPFR-REPLAY.md`

> **Verdict: PASS for the root, marker 70's bound and the false negative behind its correction, all 9,278 lower leaves, the migration queue and the lock. PARTIAL for §3's rule as written: the wrapper loads every exact rational rounded to nearest, and directs only its results.**

**For the paper.** Round 31's MPFR backend gives a nested root (d⋆ in a 2⁻²²¹ dyadic interval inside Round 28's), marker 70's upper bound 0.8349075014501105 and 9,278 / 9,278 lower leaves, all re-passing under the report's own integer arithmetic, so cell 0's shard is a PUBLICATION-CANDIDATE-SHARD as the package defines it; but the wrapper directs only its outputs (inputs rounded to nearest), the lower replay reuses A1's rule and polygons, and nothing beyond cell 0's shard is certified (51/77 at d = 36; a_Leb >= 0.835 not certified).

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| From T_M = 1673/2000, a monotone directed MPFR bisection gives d⋆ about 2.97×10⁻⁶⁷ wide, and t₃, t₅, t₇ all overlap Round 28's intervals (§0, §4). | certificate | PASS | exact-arithmetic certificate | 1673/2000; 2⁻²²¹; 2.97×10⁻⁶⁷; 3.1–4.2×10⁻⁶⁷; 10⁻⁹⁰ |
| A second backend: libMPFR 4.2.2 and GMP called through a minimal ctypes wrapper (exact loads, directed rounding, field operations, sqrt, sin, cos, acos, atan2, π, comparison, exact dyadic extraction) (§1–2). | specification | PASS | label only | 4.2.2; 640; 16; 153 |
| The rule is exact Fraction → MPFR directed → exact dyadic Fraction (§3). | specification | PARTIAL | corrected | 2⁻⁶⁴⁰ |
| Marker 70's MPFR upper bound, with only the Reuleaux sector arithmetic moved to MPFR, is 0.8349075014501105, as Round 29 shows (§5). | certificate | PASS | exact-arithmetic certificate | 4,096; 0.8349075014501105; 9.24985499×10⁻⁵; 1,317; 22,754; 22,755; 1,284; 21,944; 21,945; 3.3×10⁻⁵⁰ |
| R31-MPFR-MARKER-001: a first verifier also counted the adjacent pieces and got U ≈ 0.9201, a false negative (§6, §20). | computation | PASS | exact-arithmetic certificate | 0.92016; 1,068; 0.9201 |
| 9,278 / 9,278 lower leaves pass, none inconclusive; the density histogram is Round 30's "because the finite candidate policy is the same, but the core enclosure backend differs"; the least margin is 3.5423707501×10⁻⁹ (§7–10). | certificate | PASS | exact-arithmetic certificate | 9,278; 3.5423707501×10⁻⁹; 2,048; 2.22×10⁻⁵ |
| The lower MPFR replay is an independent second implementation of the lower memberships (§9, §11). | reading | PASS | corrected | 3 |
| Two backends and two implementations, strict inequalities and pinned dependencies make marker 70's shard a PUBLICATION-CANDIDATE-SHARD: mature enough for a formal proof package or external audit, not a proof of the bound, and not a formal verification of the libraries (§11–14, §18–19). | reading | PASS | label only | 6 |
| The pins: script and wrapper digests and library digests; both corrections are CORRECTED (§12–13, §20). | specification | PASS | label only | 4; 2 |
| Global budget Γ(36) = 51/77, Γ(38) >= 57, Γ(40) >= 63; geometry-complete cells are queued for arithmetic migration and the residual for the B₇ wave; for marker 70, 0 leaves needed a resplit (§15–17). | computation | PASS | numerical only | 51/77; 57; 63; 7 at 30; 12 at 32; 19 at 34; 13 at 36; 6 at 38; 6 at 40; 14; 0.6077; 10⁻¹² |
| Scope: a_Leb >= 0.835 is NOT CERTIFIED (§18–22). | reading | PASS | open | 0.835; 51/77 |

Open: The libraries: "Their digests are the document's; the binaries are not here, and nothing here ran MPFR."; Implementation independence on the lower side: "The MPFR replay is the A1 replay's rule and code over another interval type; the two agree margin for margin because they compute the same polygons."; Anything beyond cell 0's shard: "The global state is unchanged: 51/77 at d = 36, and a_Leb ≥ 0.835 is not certified."; 14 residual cells still open at depth 40: "Its 14 residual cells are the 14 still open at 40."; Queue ranking: "Every marker margin in the queue is null, so nothing in it ranks cells by the margins §17 speaks of."; Wrapper enclosures: "this is covered in practice, but not by construction."

Unclear: The item on lower-replay independence is given report_finding PASS following the §7–10 table row, with the report's caveat that implementation independence on the lower side is not established.; The three PASS reviews of PUBLICATION-CANDIDATE-SHARD carry W6's caveat about the wrapper's inputs.; The aggregate script's glob also matches its own ledger and 'Where chunks overlap, the later file wins'; the report gives no verdict on this.; This run uses the exact kernel as corrected after RUN-031.

### RUN-035 — Round 32 - Bulk Migration Wave (AMRAL_Lebesgue_Finite_Closure_Round_32_Bulk_Migration_Wave_v0.1.md, AMRAL-LUC-FC-R32, dated 2026-09-20) and its package

Report: `reports/RUN-035-ROUND32-BULK-WAVE.md`

> **Verdict: PASS for all nine of Round 32's judgments — the three closures, both trees, cell 69's two lower ledgers and its upper bound, its promotion, cell 47's prototype status, the dashboard, and the global claim not made.**

**For the paper.** Round 32's three closures (cells 31, 27, 7, closed at depths 42, 43, 45, inside checkpoints 42, 44, 46), cells 69 and 47's depth-30 trees, their leaf-wise lower ledgers (7,856/7,856, least margin 1.6525×10⁻⁹; 9,228/9,228, least margin 6.7502×10⁻⁹) and marker 69's upper bound 0.8349723974082344 are independently recomputed; cell 69's MPFR ledger is reproduced from its rule in this tree's arithmetic but not in MPFR. Geometry stands at 66/77 complete with the only complete common budget 51/77 at depth 36, and a_Leb ≥ 0.835 is not certified.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Cells 31, 27 and 7 are new strict geometric closures, at depths 42, 44 and 46 respectively. | computation | PASS | numerical only | 42; 43; 45; checkpoints 42, 44, 46; 158,811; 199,375; 248,823 nodes |
| Cell 7's unresolved volume falls 9.43×10⁻¹⁰ → 1.34×10⁻¹⁰ → 1.93×10⁻¹² → 0 over depths 40 to 46. | computation | PASS | numerical only | 9.435×10⁻¹⁰; 1.342×10⁻¹⁰; 1.93×10⁻¹²; 0 |
| 66/77 cells are known geometrically complete and 11 residual; the higher budgets have only lower bounds Γ(38) ≥ 57 … Γ(46) ≥ 66. | computation | PASS | numerical only | 66/77; 11; 57, 63, 64, 65, 66 at 38, 40, 42, 44, 46 |
| The common budget is still Γ(36) = 51/77. | computation | PASS | label only | 51; 36; 51/77 |
| Cell 69 has a depth-30 tree of 15,711 nodes and 7,856 leaves, as the geometry ledger has it. | computation | PASS | numerical only | 15,711 nodes; 7,856 leaves; least slack 4.4068×10⁻⁹; 4×10⁻¹⁶ from the manifest's |
| Cell 47 has a tree of 18,455 nodes and 9,228 leaves. | computation | PASS | numerical only | 18,455 nodes; 9,228 leaves; least slack 1.5290×10⁻⁸; 6×10⁻¹⁶ from the manifest's |
| All 7,856 leaves of cell 69 pass the mpmath lower ledger, least margin 1.6525×10⁻⁹. | certificate | PASS | exact-arithmetic certificate | 7,856 / 7,856; 1.6525×10⁻⁹; 32,768; margin agreement 2.2×10⁻¹⁶ |
| All 9,228 leaves of cell 47 pass the mpmath lower ledger, least margin 6.7502×10⁻⁹. | certificate | PASS | exact-arithmetic certificate | 9,228 / 9,228; 6.7502×10⁻⁹; 16,384; margin agreement 2.6×10⁻¹⁶ |
| The ledger densities are the start band from the leaf's reference slack or the least passing density. | method | PASS | numerical only | 9,228 leaves of cell 47; 7,855 of cell 69's 7,856; 32,768 |
| Report-side check (not a source assertion): the corrected exact kernel agrees with the verified envelope on every leaf, while the old grid envelope loses a sub-step on three leaves. | method | PASS | numerical only | 1.474×10⁻⁹; 2.2×10⁻¹⁰ |
| Cell 69 passes 7,856 / 7,856 under MPFR after one escalation, with no resplit. | certificate | PASS | exact-arithmetic certificate | 7,856; 512: 3,805; 1,024: 3,832; 2,048: 175; 4,096: 32; 8,192: 8; 16,384: 3; 65,536: 1; margins ledger's plus 1.57–1.67×10⁻¹² |
| Under MPFR one leaf fails closed at 32,768 (−3.07×10⁻⁹) and passes at 65,536 (+2.5329×10⁻⁹). | computation | PASS | exact-arithmetic certificate | −2.6×10⁻⁴ at 512; −3.07×10⁻⁹ at 32,768; +2.535×10⁻⁹ at 65,536; source +2.5329×10⁻⁹ |
| Report-side reading: the escalated leaf's need for 65,536 reflects the MPFR ledger's winning-body-only rule. | method | PASS | numerical only | 65,536; 32,768 |
| Marker 69's upper bound is 0.8349723974082344 with margin 2.76026×10⁻⁵, replayed both ways. | certificate | PASS | exact-arithmetic certificate | 0.8349723974082344; 2,611 active constraints; 42,639 / 42,639 bits; 128 normals; exact margin 6.74×10⁻⁵; source margin 2.76026×10⁻⁵ |
| Marker 47's upper-bound replay is part of the cell 47 shard. | certificate | PASS | exact-arithmetic certificate | 256 normals; exact margin 1.857×10⁻⁵ |
| Cell 69 is a PUBLICATION-CANDIDATE-SHARD. | specification | PASS | label only |  |
| Cell 47 has no MPFR lower yet and is an ARITHMETICALLY-CLOSED-PROTOTYPE. | specification | PASS | label only | 9,228 / 9,228 |
| Dashboard: geometry complete 66, closed or better 3, candidates 2, pending 63, residual 11; coverage 2/77 and 2/66. | computation | PASS | label only | 66; 3; 2; 63; 11; 2/77; 2/66 |
| The fallback order is density, then precision, then resplit; geometry and arithmetic keep separate queues. | specification | PASS | label only |  |
| The package records (shard files, dashboard, crystal, VALIDATION.json) agree with the ledgers and archive. | specification | PASS | label only | seven digests; eleven checks |
| The package's five scripts generate the shard, the mpmath and MPFR ledgers and the marker replay. | method | UNMEASURED | not measured | 60 lines; 121 lines; 149 lines each; 102 lines |
| No global claim is made: a_Leb ≥ 0.835 is not certified. | reading | PASS | open | 0.835 |

Open: MPFR side not replayed: "MPFR is not on this machine. Cell 69's MPFR ledger is rebuilt here from its rule on this tree's arithmetic, not replayed in MPFR."; No common budget above 36: "51/77 at 36 is still the only complete common budget."; Global bound: "a_Leb ≥ 0.835 is not certified."; Cell 47's MPFR replays: "its MPFR replays pending, as the shard says"; Script inputs/outputs not archived: "It reads Round 26's full-atlas results and its own cell folders, neither of them in the archive."; `r32_mpfr_marker.py` "writes an output that is not in the archive."

Unclear: The report does not name, in this file, the arithmetic used for this tree's leaf certificates (it says 'this tree's certificate' and 'this tree's arithmetic'; the run uses 'the corrected exact kernel'); the strength label 'exact-arithmetic certificate' for leaf and closure items rests on that run-level statement.; The escalated leaf's margin is printed as +2.535×10⁻⁹ in the report and +2.5329×10⁻⁹ in the source, while the report says 'as §4 says'; the report also states rebuilt margins are 'the ledger's plus 1.57–1.67×10⁻¹²'.; Marker 69's margin 2.76026×10⁻⁵ is not restated by the report; it confirms the area 0.8349723974082344 and gives this tree's own exact margin 6.74×10⁻⁵ with 128 normals.; Whether marker 69's MPFR replay itself was checked: the report says 'No separate A1 record for marker 69 is shipped: the MPFR script is itself the exact full hull', while MPFR is not on this machine.

### RUN-036 — Round 33 - Cross-Backend Dominance (AMRAL_Lebesgue_Finite_Closure_Round_33_Cross_Backend_Dominance_v0.1.md, AMRAL-LUC-FC-R33, dated 2026-09-20) and its package

Report: `reports/RUN-036-ROUND33-DOMINANCE.md`

> **Verdict: PASS for the dominance theorem and its polygon corollary (read), the audits' state counts, the trees of cells 48, 68, 75 and 76, all 9,285 leaves of cell 48, marker 48's upper bound, the ledgers' vertex totals and the stage counts; UNMEASURED for the audits' own results, which compare mpmath enclosures with MPFR ones and MPFR is not on this machine (PARTIAL for their root containment: t₃, t₅, t₇ yes, π and √3 not recomputable); PASS for keeping cells 47 and 48 below publication-candidate, with one correction to its reason — the missing certificates are reproducible without their bytes.**

**For the paper.** The cross-backend dominance theorem and its polygon corollary are correct as read; cells 48, 68, 75, 76's trees rebuild node for node and all 9,285 leaves of cell 48 are certified directly without mpmath or MPFR, as is marker 48's upper bound. The MPFR-based audit results (0 failures, radius gain 2.15×10⁻⁶⁴) are unmeasured, root containment is shown only for t₃, t₅, t₇, and the missing certificate bytes are reproducible from density and rule (status registry: 7 PASS, 2 PARTIAL, 9 UNMEASURED).

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Cross-backend enclosure dominance: R^B_j ⊆ R^A_j and ρ_B ≥ ρ_A give G_A^cert ⊆ G_B^cert. | theorem | PASS | analytic (proof checked) |  |
| A polygon certified vertex by vertex under backend A transfers to B, and the exact area does not depend on the backend. | theorem | PASS | analytic (proof checked) |  |
| The audit checks each unique (rotation, half-widths) core state once, about 6× fewer states than leaves. | method | PASS | analytic (proof checked) | 1,564; 1,574; 5.9 leaves per state |
| Cells 47 and 48's audits have 1,564 and 1,574 unique states. | computation | PASS | numerical only | 1,564; 1,574; 5.9 |
| The dominance audits of cells 47 and 48 pass base box, base cores and witness boxes with 0 failures and least radius gain 2.15×10⁻⁶⁴. | certificate | UNMEASURED | not measured | 0 failures; 2.15×10⁻⁶⁴ |
| Root containment (t₃, t₅, t₇, π, √3) passes in both audits. | certificate | PARTIAL | numerical only |  |
| Cell 48 has a depth-30 tree of 18,569 nodes and 9,285 leaves. | computation | PASS | numerical only | 18,569/9,285; least slacks to 2×10⁻¹⁶ |
| Cells 68, 75 and 76 have trees of 18,509/9,255, 21,895/10,948 and 21,863/10,932 nodes and leaves. | computation | PASS | numerical only | 18,509/9,255; 21,895/10,948; 21,863/10,932 |
| Cell 48 passes 9,285/9,285 under mpmath with least margin 1.03164×10⁻⁷ and the stated density counts. | certificate | PASS | exact-arithmetic certificate | 9,285/9,285; 1.0316×10⁻⁷; 16,384; 256: 824; 512: 3,438; 1,024: 3,725; 2,048: 1,065; 4,096: 183; 8,192: 43; 16,384: 7; margin to 2.3×10⁻¹⁶ |
| Every leaf of cells 47 and 48 is certified (report-side: directly, without mpmath or MPFR). | certificate | PASS | exact-arithmetic certificate |  |
| Marker 48's independent exact polar replay passes: outer 0.8349325735415322, margin 6.74265×10⁻⁵. | certificate | PASS | exact-arithmetic certificate | 0.8349325735415322; 6.74265×10⁻⁵; 2,601 active constraints; 2,560 constraints; 41,557 bits over 41,557; 128 normals; margin 4.03×10⁻⁵; exact margin 1.02×10⁻⁴ |
| Marker 48's MPFR replay passes. | certificate | UNMEASURED | not measured |  |
| The Round 32 ledgers keep no hull vertices, source tags or exact areas; the hulls total about 2.16×10⁶ and 2.11×10⁶ vertices. | computation | PASS | numerical only | 2,163,069; 2,109,700; 34.6 MB; 33.8 MB; 16 bytes a vertex |
| Because the ledgers store no certificate entity, dominance has nothing to transfer and cells 47 and 48 must stay below publication-candidate. | reading | PASS | corrected | 9,228; 9,285; margin to 2.6×10⁻¹⁶ |
| Future emitters keep either the full rational hull or a deterministic transcript (closed as an ABI requirement in the scope ledger). | specification | PASS | label only |  |
| Arithmetic-closed-or-better is {47, 48, 69, 70} (4/77) and publication-candidate {69, 70} (2/77); geometry unchanged at 66/77. | computation | PASS | label only | 4/77; 2/77; 66/77; 62 = 66 − 4 |
| The shipped dominance script audits backend A against backend B. | method | PASS | label only | 108 lines; 120- to 150-digit decimal strings |
| The package records (spec, status files, dashboard, scope ledger, crystal, VALIDATION.json) agree with the data. | specification | PASS | label only | eight digests; twelve checks |

Open: MPFR half of the audits: "The enclosures B computes are not recomputed."; Least radius gain: "Nor is the least radius gain, 2.15×10⁻⁶⁴."; Root containment for π and √3: "π and √3 not recomputable"; Marker 48 MPFR replay: "The MPFR replay's record is not shipped."; Cells 68, 75, 76 leaves: "Only their trees are checked. The package has no ledger for them yet."

Unclear: The arithmetic of the containment check for t₃, t₅, t₇ is not stated here; the report cites it to RUN-034 W1 ('the MPFR t₃, t₅, t₇ lie inside Round 28's').; The status registry counts 18 judgments (7 PASS, 2 PARTIAL, 9 UNMEASURED) but the 'Round 33, read' table lists fewer rows; the mapping of all 18 judgments to rows is not given.; The report lists Round 33's claim 'a_Leb ≥ 0.835 is NOT CERTIFIED' under 'What Round 33 claims' but gives no separate verdict row for it.; Marker 48 has two margins for this tree's certificate (4.03×10⁻⁵ with 128 normals) and an 'exact margin' of 1.02×10⁻⁴; the report does not say how each relates to the source's 6.74265×10⁻⁵.

### RUN-037 — Round 34 - RHCERT Publication Candidates (AMRAL_Lebesgue_Finite_Closure_Round_34_RHCERT_Publication_Candidates_v0.1.md, AMRAL-LUC-FC-R34, dated 2026-09-20) and its package

Report: `reports/RUN-037-ROUND34-RHCERT.md`

> **Verdict: PASS for the format, both certificates, the sizes (but §11–12's packaged "about 8.1 MiB" and "about 7.6 MiB" are 8.02 and 7.54), cell 68's lower bound, marker 68's upper bound, the stage counts and the promotion of cells 47 and 48 — whose certificate bytes pass here under an arithmetic that shares nothing with mpmath or MPFR; UNMEASURED for marker 68's MPFR replay and the MPFR conditions of the promotion, not recomputed here.**

**For the paper.** The RHCERT-v0.1 certificates of cells 47 and 48 decode from the specification alone and all 4,272,769 stored vertices check in an integer-interval arithmetic independent of mpmath and MPFR (least margins 6.75024723062513×10⁻⁹ and 1.0316417842561089×10⁻⁷, all polygons strictly convex with area above 167/200), cell 68's summary and marker 68's independent upper replay are rebuilt; the packaged sizes are 8.02 and 7.54 MiB (not 8.1 and 7.6), and the MPFR conditions and marker 68's MPFR replay remain unmeasured.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| RHCERT-v0.1 is a binary certificate (header, per-leaf density and vertex count, zigzag-varint vertex deltas on a 10⁻¹⁸ grid with body tags, in topology leaf order, with an index of L + 1 offsets). | specification | PASS | exact-arithmetic certificate | `<8sHBBI`; `L34RHC01`; version 1; exponent 18; L + 1 offsets; index header `<8sHI`, 14 + 8(L + 1) bytes |
| The spec defines the index of L + 1 offsets. | specification | PASS | label only | `<8sHI`; 14 + 8(L + 1) bytes |
| The verifier trusts only the bytes, quantizes the directed cores to a 10⁻⁴⁰ grid, and checks D²_max < R²_lower, strict convexity, and 200\|S\| > 167(2·10³⁶). | method | PASS | label only | 10⁻⁴⁰ |
| Cell 47's certificate replays 9,228/9,228 with 2,163,069 vertices, least margin 6.75024723×10⁻⁹, 36,506,484 raw bytes. | certificate | PASS | exact-arithmetic certificate | 9,228/9,228; 2,163,069; 6.75024723062513×10⁻⁹ (leaf 4820); 36,506,484 bytes |
| Cell 48's certificate replays 9,285/9,285 with 2,109,700 vertices, least margin 1.03164178×10⁻⁷, 35,657,092 bytes. | certificate | PASS | exact-arithmetic certificate | 9,285/9,285; 2,109,700; 1.0316417842561089×10⁻⁷ (leaf 6986); 35,657,092 bytes |
| Report-side: the byte replay uses integer intervals only (no mpmath, no MPFR), a second independent arithmetic on the bytes. | method | PASS | exact-arithmetic certificate | 4,272,769 vertices; margins within 2.7×10⁻¹⁷ of the ledgers' |
| The leaf order is the topology's preorder, and each leaf's density and hull count are the Round 32/33 ledger's. | specification | PASS | exact-arithmetic certificate | four tags |
| The packaged certificates are about 8.1 MiB and about 7.6 MiB. | computation | PARTIAL | corrected | 8.02 MiB; 7.54 MiB; 8,411,289 bytes; 7,909,374 bytes |
| Size economics: about 16.9 raw bytes a vertex and 3.5–3.9 packaged, so full persistence is practical. | computation | PASS | numerical only | 16.877; 16.902; 3.889; 3.749; 34.8 MiB; 34.0 MiB |
| Cells 47 and 48 meet the eight promotion conditions and are PUBLICATION-CANDIDATE-SHARD. | specification | PASS | label only |  |
| Cell 68 passes 9,255/9,255 under mpmath lower, least margin 2.23292×10⁻⁸. | certificate | PASS | exact-arithmetic certificate | 820; 3,429; 3,632; 1,140; 181; 46; 7; 16,384; within 2.6×10⁻¹⁷ |
| Marker 68's independent replay passes. | certificate | PASS | exact-arithmetic certificate | 1,317 active constraints; 1,284; 128 normals; margin 9.96×10⁻⁵; exact margin 1.61×10⁻⁴ |
| Marker 68's MPFR replay passes. | certificate | UNMEASURED | not measured |  |
| Cell 68 is ARITHMETICALLY-CLOSED-PROTOTYPE. | specification | PASS | label only |  |
| Candidates {47, 48, 69, 70} are 4/77; closed or better {47, 48, 68, 69, 70} are 5/77; geometry is unchanged. | computation | PASS | label only | 4/77; 5/77; 61 = 66 − 5 |
| The emitter reruns Round 32's emitter with values on the 10⁻¹⁸ grid. | method | UNMEASURED | not measured | 10⁻¹⁸ |
| The package records (manifests, replay summaries, shard files, dashboard, crystal, VALIDATION.json) agree with the data. | specification | PASS | label only | ten digests; eleven checks |

Open: MPFR: "Marker 68's MPFR replay and the dominance audits are not recomputed."; MPFR conditions of the promotion: "the MPFR conditions not recomputed (W6)"; Cell 68's certificate bytes: "They come in Round 35. Here only its summary is rebuilt."; Spec gap: "The spec does not give the index header's layout"; Emitter inputs: "Its reference slacks and base boxes come from two files not in the archive, which the verifier does not need."

Unclear: The source is quoted as quantizing the directed cores 'inward to a 10⁻⁴⁰ grid', while the report describes the shipped verifier as quantizing 'its mpmath cores outward on a 10⁻⁴⁰ grid (sound)'; the report does not comment on the difference in wording.; The status registry counts 12 judgments (11 PASS, 1 UNMEASURED), yet the 'Round 34, read' table has a PARTIAL row for the packaged sizes; the mapping is not stated.; Strength 'exact-arithmetic certificate' for the byte replay rests on the report's phrases 'this tree's integer intervals' and 'in integers'; for cell 68's density counts the report says 'this tree's certificates' without naming the arithmetic in that bullet.

### RUN-038 — Round 35 - RHCERT Mass Production, Durable Frontiers (AMRAL_Lebesgue_Finite_Closure_Round_35_RHCERT_Mass_Production_Durable_Frontiers_v0.1.md, AMRAL-LUC-FC-R35, dated 2026-09-20) and its package

Report: `reports/RUN-038-ROUND35-RHCERT-PRODUCTION.md`

> **Verdict: PASS for cell 68's certificate and its promotion, cells 75 and 76's lower bounds, markers 75 and 76's independent upper replays, the audits' state counts, the stage counts and the decision to claim no geometry; PARTIAL for the two marker judgments that pair the independent replay with the MPFR one, whose record is not shipped and whose arithmetic is not here.**

**For the paper.** Cell 68's RHCERT certificate (9,255 leaves, 2,126,336 vertices) checks vertex for vertex in this tree's integer arithmetic and equals this tree's own certificate leaf for leaf; cells 75 and 76's lower bounds (10,948 and 10,932 leaves) and markers 75 and 76's independent upper replays are rebuilt, while the MPFR replays and audits are recorded, not recomputed. The lost runtime frontier is correctly not reconstructed and no geometric closure is claimed.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Cell 68's certificate replays bytes-only 9,255/9,255 with 0 failures, 2,126,336 vertices in 35,940,305 bytes, least margin 2.2329×10⁻⁸. | certificate | PASS | exact-arithmetic certificate | 9,255/9,255; 2,126,336; 35,940,305 bytes; 2.232921067369829×10⁻⁸ |
| Cell 68's certificate bytes are its leaf polygons (density and hull count per leaf). | certificate | PASS | exact-arithmetic certificate | 9,255; margins agree to 2.6×10⁻¹⁶; 10⁻¹⁸ grid |
| Cell 68's least margin and its four chunks' least margins are as recorded. | certificate | PASS | exact-arithmetic certificate | 2.232921067369829×10⁻⁸; `00111011001001010000010` |
| Cell 68 is promoted to PUBLICATION-CANDIDATE-SHARD (bytes replay plus MPFR dominance over 1,565 states). | specification | PASS | label only | 1,565 |
| Cells 75 and 76 pass 10,948/10,948 and 10,932/10,932 lower, with their density counts and least margins 6.2518×10⁻⁸ and 2.0477×10⁻⁸. | certificate | PASS | exact-arithmetic certificate | 10,948/10,948; 10,932/10,932; 6.2518×10⁻⁸; 2.0477×10⁻⁸; margin to 2.6×10⁻¹⁶ |
| Markers 75 and 76 have independent upper bounds 0.8349344874879304 and 0.8349600023952688. | certificate | PASS | exact-arithmetic certificate | 0.8349344874879304; 0.8349600023952688; 1,312 and 2,593 active constraints; 128 normals each; exact margins 1.38×10⁻⁴ and 7.87×10⁻⁵ |
| Markers 75 and 76 pass both the independent and the MPFR upper replays. | certificate | PARTIAL | not measured |  |
| The dominance audits cover 1,565, 1,690 and 1,688 unique states for cells 68, 75 and 76. | computation | PASS | numerical only | 1,565; 1,690; 1,688 |
| Candidates are 5/77, closed or better 7/77, geometry 66/77. | computation | PASS | label only | {47, 48, 68, 69, 70}; 5/77; {47, 48, 68, 69, 70, 75, 76}; 7/77; 59 = 66 − 7 |
| The pending boxes were kept only in runtime checkpoints, now gone; counts and volumes do not reconstruct them, so no closure is claimed. | reading | PASS | label only | eleven residual records |
| DFRONT-v0.1 keeps each pending box as dyadic codes (k, m) per axis, independent of split order. | specification | PASS | label only |  |
| The package records (manifest, replay record, shard file and copies, status files, dashboard, crystal, VALIDATION.json) agree with the data. | specification | PASS | label only | seven digests; twelve checks |

Open: MPFR: "The MPFR replays and audits, as before."; MPFR marker records: "The MPFR replays' records are not shipped."; Audit MPFR results: "Their MPFR results are recorded here, not recomputed."; Cells 75 and 76's certificate bytes: "They come in Round 36."

Unclear: For cells 75 and 76 the report says leaves pass 'on its intended box at the ledger's density' using RUN-031's certificates; the arithmetic is not named in that bullet (integer arithmetic is named only for cell 68's byte replay).; The 'analytic (proof checked)' label for the durability-gap and DFRONT items reflects that the report marks them 'reading'; no proof is written out.

### RUN-039 — Round 36 - Depth30 DFRONT Recovery (AMRAL_Lebesgue_Finite_Closure_Round_36_Depth30_DFRONT_Recovery_v0.1.md, AMRAL-LUC-FC-R36, dated 2026-09-20) and its package

Report: `reports/RUN-039-ROUND36-DFRONT.md`

> **Verdict: PASS for all eight judgments — the two replays, the two promotions, cell 33's recovery and its DFRONT status, and the two readings of the global claim. The MPFR dominance audits are recorded, not recomputed.**

**For the paper.** Cells 75 and 76's RHCERT certificates (2,627,357 and 2,617,252 stored vertices) check vertex for vertex in this tree's integer arithmetic, and cell 33's depth-36 DFRONT frontier (14,027 boxes, volume 14027/2³⁶) is reproduced by this tree's own lift box for box and written byte for byte; cell 49's depth-32 tree hashes to its manifest. The MPFR dominance audits and the codec's eight tests are not rerun, geometry stays 66/77, and a_Leb ≥ 0.835 is not certified.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Cells 75 and 76 replay bytes-only 10,948/10,948 and 10,932/10,932, with 2,627,357 and 2,617,252 vertices in 44,382,455 and 44,229,325 bytes. | certificate | PASS | exact-arithmetic certificate | 10,948/10,948; 10,932/10,932; 2,627,357; 2,617,252; 44,382,455 bytes; 44,229,325 bytes |
| Cells 75 and 76's least margins are 6.2518×10⁻⁸ and 2.0477×10⁻⁸. | certificate | PASS | exact-arithmetic certificate | 6.251776141237558×10⁻⁸ (leaf 8,258); 2.0477267269918978×10⁻⁸ (leaf 9,570); margins agree to 2.7×10⁻¹⁷ |
| Cells 75 and 76 are PUBLICATION-CANDIDATE-SHARD (bytes replay, MPFR dominance over 1,690 and 1,688 states). | specification | PASS | label only | 1,690; 1,688 |
| The depth-30 batch is complete with 7/7 candidates: 7/77 ≈ 9.09 % and 7/66 ≈ 10.61 %, with 59 below. | computation | PASS | label only | {47, 48, 68, 69, 70, 75, 76}; 7/77; 7/66; 59 |
| DFRONT-v0.1 stores dyadic codes per axis, independent of split order, with a verifier using an exact sweep. | specification | PASS | exact-arithmetic certificate | 1/2; 14027/2³⁶ |
| Cell 33 is rebuilt from the root to Round 26's durable counts: 129,401 nodes, 14,027 pending, V₃₆ = 2.9748317031636626×10⁻⁸, error 0. | computation | PASS | numerical only | 129,401 nodes; 50,674 closed leaves; 14,027 boxes; volume agrees to 10⁻²²; least closed slack 3.362067513×10⁻⁹; 6×10⁻¹⁶ |
| Cell 33's frontier is kept in a 126,327-byte DFRONT file. | computation | PASS | numerical only | 126,327 bytes |
| The DFRONT payload's dependency hash, kernel and scheduler digests, base box and t₇ are as recorded. | computation | PASS | numerical only | 5 and 6 ulps; 4 ulps |
| CELL33-DFRONT-RECOVERY: PASS, and runtime state recovery → DFRONT PASS. | definition | PASS | label only |  |
| No old frontier bytes were found (§10). | reading | UNMEASURED | not measured |  |
| Depth 38 was attempted, ran over the time limit, and has no effect. | reading | PASS | label only | 38 |
| Geometry is unchanged at 66 complete, 11 residual, 51/77 at depth 36, with no new closure. | computation | PASS | label only | 66; 11; 51 |
| The depth-32 queue has twelve cells, with 49, 64, 66, 12, 59, 57 first. | specification | PASS | label only | twelve cells; 49, 64, 66, 12, 59, 57 |
| Cell 49's tree is rebuilt from durable sources: 19,971 nodes and 9,986 leaves, the ledger's count. | computation | PASS | numerical only | 19,971 nodes; 9,985 internal; 9,986 leaves; least slack 4.494058015×10⁻⁸; 3×10⁻¹⁶; 5 and 6 ulps |
| No global claim: a_Leb ≥ 0.835 NOT CERTIFIED (two readings). | reading | PASS | open | 0.835 |
| The validation's eight codec tests pass. | method | UNMEASURED | not measured | eight tests |
| The package's scripts (codec, cell 33 rebuild, shard generator) implement the round's computations. | method | PASS | label only | `<8sHHII` |
| The package records (manifests, shard files, batch, DFRONT records, crystal, VALIDATION.json) agree with the data. | specification | PASS | label only | ten file digests; three package digests; fourteen checks |

Open: MPFR dominance audits: "The MPFR dominance audits are recorded, not recomputed."; Codec tests: "the tests themselves are not rerun."; Test frontier provenance: "The harness that wrote the shipped file and ran the validation's eight tests is not shipped."; §10 search: "nothing here can repeat it."; Depth 38: "Round 36 claims nothing there. Round 37 does, and RUN-040 checks it."

Unclear: The codec's own self-test writes a full partition of cell 70 (four boxes, volume 1), which the report says 'is not the shipped test frontier' (cell 31, two boxes, volume 1/2); the report does not say whether this affects any judgment.; The report says 'two promotions' are judged; it names only cells 75 and 76 as promoted, and does not list all eight judgments individually.

### RUN-040 — Round 37 - Depth32 DFRONT Shards (AMRAL_Lebesgue_Finite_Closure_Round_37_Depth32_DFRONT_Shards_v0.1.md, AMRAL-LUC-FC-R37, dated 2026-09-20) and its package

Report: `reports/RUN-040-ROUND37-DEPTH32-DFRONT.md`

> **Verdict: PASS for eleven of Round 37's twelve judgments — the three trees, the three lower bounds and three byte replays, the promotion, the conservative geometry status and the two readings of the global claim. UNMEASURED for the MPFR marker replay, whose records are not shipped and whose arithmetic is not here.**

**For the paper.** Cells 49, 64 and 66's depth-32 trees, lower bounds (9,986, 10,539, 10,636 leaves) and RHCERT byte replays are reproduced in this tree's own integer arithmetic, with independent marker upper bounds; cell 33's DFRONT continuation to depth 42 (78,507 still open) is reproduced checkpoint for checkpoint and in all 166 segments byte for byte, so the cell stays PARTIAL. The compositional merge theorem is correct and its premise holds on the shipped files, but the merge's stated requirements do not imply that premise; MPFR replays and audits are unmeasured.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Cells 49, 64 and 66's trees are rebuilt from durable sources: 19,971 / 9,986, 21,077 / 10,539 and 21,271 / 10,636 nodes and leaves, the ledger's counts. | computation | PASS | numerical only | 19,971 / 9,986; 21,077 / 10,539; 21,271 / 10,636 |
| Cells 49, 64 and 66 have lower bounds 9,986/9,986, 10,539/10,539 and 10,636/10,636 with least margins 3.15312672×10⁻⁸, 1.50693499×10⁻⁸ and 6.86089458×10⁻⁹. | certificate | PASS | exact-arithmetic certificate | 9,986; 10,539; 10,636; margins agree to 2.3×10⁻¹⁶ |
| The RHCERT files with 2,441,597, 2,624,186 and 2,646,674 vertices replay from the bytes alone. | certificate | PASS | exact-arithmetic certificate | 2,441,597; 2,624,186; 2,646,674; 3.153126719767511×10⁻⁸; 1.506934987941895×10⁻⁸; 6.860894579634212×10⁻⁹; 41,150,609 bytes; 44,278,537 bytes; 44,674,727 bytes |
| Report-side caution (not a source assertion): building depth-32 leaf boxes from a length-30 split sequence would certify deep leaves on their depth-30 ancestors' boxes. | method | PASS | label only | 30; 32 |
| Markers 49, 64 and 66 have upper bounds 0.8349813779061498, 0.8349716675759491 and 0.8348951300993619 (independent replay). | certificate | PASS | exact-arithmetic certificate | 0.8349813779061498; 0.8349716675759491; 0.8348951300993619; 5,156, 2,597 and 1,312 active constraints; 256, 128 and 128 normals; exact margins 4.18×10⁻⁵, 6.55×10⁻⁵ and 1.71×10⁻⁴ |
| The MPFR marker replay passes. | certificate | UNMEASURED | not measured |  |
| MPFR dominance holds over 1,659, 1,702 and 1,710 states. | computation | PASS | numerical only | 1,659; 1,702; 1,710 |
| Cells 49, 64 and 66 are PUBLICATION-CANDIDATE-SHARD. | specification | PASS | label only |  |
| Coverage is 10/77 ≈ 12.99 % and 10/66 ≈ 15.15 %, with 56 below; the depth-32 batch has 3/12 candidates and 3/12 prepared. | computation | PASS | label only | {47, 48, 49, 64, 66, 68, 69, 70, 75, 76}; 10/77; 10/66; 56; 12, 57, 59 prepared; 45, 46, 58, 60, 65, 67 remaining |
| Cell 33 goes from depth 36 → 38 → 40 → 42 with 14,027 → 27,042 → 40,934 → 78,507 pending. | computation | PASS | numerical only | 27,042; 40,934; 78,507 |
| The four checkpoint DFRONT files of cell 33 (depths 36 to 42) are as shipped. | computation | PASS | numerical only |  |
| Cell 33's unresolved volume falls from 2.9748×10⁻⁸ to 2.6015×10⁻⁹, V₄₂/V₃₆ = 0.087451, with no plateau. | computation | PASS | numerical only | (63,980, 18,975); (118,666, 45,441); (205,928, 65,391); 0.481963; 0.378430; 0.479473; 0.087451; least new slacks agree to 10⁻¹⁵ |
| Cell 33 stays PARTIAL; geometry status remains conservative (66/77, residual 11, 51/77 at depth 36). | reading | PASS | label only | 78,507; 66 / 11 |
| Cell 33's continuation is carried in 29, 55 and 82 segments (166 in all). | computation | PASS | numerical only | 29; 55; 82; 166 |
| Compositional merge theorem: Q = ∪Qᵢ is interior-disjoint from the parent's disjointness, each Qᵢ ⊆ ∪Pᵢ, and segment-level disjointness. | theorem | PASS | analytic (proof checked) |  |
| Every shipped segment box lies inside its own slice (the theorem's premise). | computation | PASS | exact-arithmetic certificate | 146,483 boxes |
| The merge, by hashes and slice coverage under the merge spec's seven requirements and without a global scan, yields the interior-disjoint merged frontier. | method | NOT AS STATED | refuted | seven requirements |
| Cells 12, 59 and 57 are prepared next: 21,789 / 10,895, 22,009 / 11,005 and 22,261 / 11,131, node counts matching. | computation | PASS | numerical only | 21,789 / 10,895; 22,009 / 11,005; 22,261 / 11,131 |
| No global claim: a_Leb ≥ 0.835 NOT CERTIFIED (two readings). | reading | PASS | open | 0.835 |
| The shipped verifier quantizes the cores outward and demands strict convexity. | method | UNMEASURED | not measured | 10⁴⁰ grid; 140-digit decimal |
| The shipped marker replays prune the full cyclic hull at Round 29's density. | method | PASS | label only |  |
| The shipped refinements take each box's floats from its code. | method | PASS | numerical only | 10⁻¹⁵ |
| The package records (manifests, segment records, chain record, merge spec, batch, crystal, VALIDATION.json) agree with the data. | specification | PASS | label only | eight file digests; four package digests; seventeen checks |

Open: MPFR: "The MPFR marker replays and dominance audits. Their records are not shipped, and there is no MPFR here."; The merge in general: "The shipped merge alone would accept a segment that strayed outside its slice."; Residual frontiers: "The other ten residual frontiers. Round 37 recovers none, and Round 38 is to begin."; Global theorem: "The global theorem, which Round 37 does not claim."; Cell 33: "78,507 boxes are still open at 42, so the cell stays PARTIAL"; Not in the archive: "the mpmath leaf ledgers and reference leaves the emitters read, the three trees' bits (their digests are shipped), and the marker and dominance scripts' outputs."

Unclear: The report's own verdict row for §9 is PASS ('a reading, not a registry judgment: right as stated'); the NOT AS STATED item here concerns only the merge spec's requirements, which the report says 'do not imply the premise'.; The report says 'The depth-38 and depth-40 manifests have other fields, so their merges were not this script'; which script produced them is not stated.; The scripts bullet writes 'a 10⁴⁰ grid' (positive exponent), while RUN-037 for Round 34 writes 'a 10⁻⁴⁰ grid'; the report does not comment on this.; For the lower bounds the report says 'this tree's certificates from the start bands'; the arithmetic is named (integer) only for the byte replay.

### RUN-041 — DLMVC - Deep Lag Multi-Pass Verification Closure Methodology v0.1 (EML-DLMVC-v0.1, dated 2026-09-19) and the case studies in its folder (lagged verifications of Round 01 and Round 04, Round 00's state crystal)

Report: `reports/RUN-041-DLMVC.md`

> **Verdict: PASS for DLMVC's nine judgments; the other nineteen are labels. Its "Empirical validation: PARTIAL" is right — for a reason it does not give.**

**For the paper.** The mathematics of DLMVC's two case studies recomputes (Fejér compiler legality, E_M exactly for M = 3 … 129, JPA-01 coercivity on 4,000 random placements, Φ_r, R_sharp = 0.721238556039142255 and Round 04's constants), and the coefficient bound improves from 1/2 to 1/π; but the Round 04 case study's 'SANITY: INDEPENDENTLY REPRODUCED' re-ran the author's script and reproduced its numbers (area 0.8337053574119659), not the defined polygon (0.833639164382). DLMVC itself is a governance text whose 'empirical validation: PARTIAL' stands.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| The DLMVC methodology skeleton is CLOSED. | reading | PASS | label only |  |
| Six parts of DLMVC are DEFINED. | definition | PASS | label only | six |
| No universal validity claim is made. | reading | PASS | label only |  |
| Empirical validation is PARTIAL ("initial successful mathematical case studies"). | reading | PASS | corrected |  |
| Round 01 case study: the Fejér compiler P_M keeps a density legal. | computation | PASS | numerical only |  |
| Round 01 case study: E_M has a closed form equal to the coefficient bound summed mode by mode. | computation | PASS | exact-arithmetic certificate | M = 3 … 129; 0.2188; 0.1806; 0.1342; 0.09093; 0.0575; 0.03465; 0.0202 |
| Round 01 case study: the Fourier coefficients obey \|r̂ₙ\| ≤ 1/2, giving the error bound E_M. | theorem | PASS | exact-arithmetic certificate | 1/π; 1/2; (2/π)·E_M; n = 3 |
| Round 01 case study: the measured compiler errors lie within the bound E_M. | computation | PASS | numerical only | 0.0017; 0.0202; M = 129 |
| JPA-01 coercivity: J(Q, t) ≥ ‖t‖ − R_K − R_U. | theorem | PASS | numerical only | 4,000 random placements; 2,236; least slack 1.27 |
| The one-active branch in Round 01 is wrong and is corrected in Round 03 §8 (R01-CORRECTION-001); the case study's finding is not blind. | reading | PASS | label only |  |
| Report-side reading: JPA-01 flags the joint-attainment step, which RUN-004 did not. | reading | PASS | label only |  |
| Round 04 case study: Φ_r(d) = πr² + r√(d² − r²) − r² arccos(r/d) is the area of conv(B_r ∪ {x}), with Φ'_r in closed form. | theorem | PASS | analytic (proof checked) | 2²⁰ sides |
| Round 04 case study: R_sharp = Φ_{1/2}⁻¹(Ā) = 0.721238556039143, and the bound is sharp. | computation | PASS | numerical only | 0.721238556039142255; 0.721238556039143 |
| Round 04 case study: the constants at M = 128, q = 5×10⁻⁴ (R_P, ε, β for both radii) and level counts 2,379 and 445. | computation | PASS | numerical only | M = 128; q = 5×10⁻⁴; 2,379; 445; 10^−93.19 |
| Report-side reading: the sharp radius goes further than RUN-007's treatment. | reading | PASS | label only |  |
| Round 04 case study prints two formulas, `d\le\frac0.84409359440.5` and `\Delta=\frac{\pi}128`. | reading | NOT AS STATED | corrected |  |
| Round 04 case study §I.2: re-running Round 04's sanity.py gives the package's numbers, marked SANITY: INDEPENDENTLY REPRODUCED. | computation | NOT AS STATED | corrected | 0.8337053574119659; 6.4598497001×10⁻⁴; 0.833639164382; 0.000622493675876; six of the eight |
| DLMVC §26: the case studies, their blindness records, and the frontier's lagged role. | reading | PASS | label only | Rounds 00, 11, 12, 13 and 17 |
| Round 00's state crystal is part of the case-study material. | reading | PASS | label only |  |
| The case study's script sharp_radius.py produces its VALIDATION. | method | UNMEASURED | not measured | 200 steps |
| The Round 04 case study's VALIDATION records its document and constants correctly. | specification | PASS | numerical only | 10⁻¹⁵ |

Open: Round 04's four deep-audit passes: "They are RUN-042's."; Case studies' blindness: "Whether the Round 04 case study really read nothing of Rounds 05–06 cannot be told from here."; DLMVC as a method: "There is one corpus, one frontier and two case studies here, so PARTIAL is the honest word."; External references: "The case studies' external references (Papini–Wu, Hynd, Gibbs, Mishra, Zeng, Gonzalez) are not fetched here"

Unclear: The report does not assign PASS/FAIL to the §I.2 'independently reproduced' label; its W4 result reads 'the script's numbers, not the definition's'. NOT AS STATED is this inventory's mapping.; The status registry counts 28 judgments (9 PASS, 19 n/a labels) and seventeen first-use phrases; the nine judgments are not individually listed beyond the four §26 rows.; Listing the 1/2 → 1/π coefficient improvement under corrections reflects the report's 'improvable', not an error in the source.

### RUN-042 — DLMVC's four deep lagged-audit passes of Round 04 (Pass 02, 03, 04 dated 2026-09-18; Pass 05 dated 2026-09-19), each with its document, script and VALIDATION.json

Report: `reports/RUN-042-DLMVC-DEEP-PASSES.md`

> **Verdict: PASS for the passes' theorems and every constant and sanity number but three. Pass 04's inner-core factor and sharp cell radius are grid samples on the unsafe side, so what it boxes with them is not established by its own argument; and its figure for Pass 03's cell radius is another quantity.**

**For the paper.** Every constant and sanity number of DLMVC's four deep passes on Round 04 recomputes (Pass 02's twenty constants to 10⁻¹², ε_deep = 0.013327 holding on 60 random bodies, Pass 03's spike-cell geometry, Pass 05's signature (1, 2, 125)), except three Pass 04 figures: λ_b and ε♯_cell are 240,000-angle grid samples on the unsafe side (true values 0.9882022551 and 0.0059692967), and its 'Pass 03 cell radius' 0.013060516 is a different quantity (Pass 03's is 0.013316971). The polyhedral verifier's structure is checked, not run at scale.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Pass 02: the support function's angular Lipschitz constant √(R² − r²), its exact variation V_{R,r}(t), the outer error ε_deep, perimeter ≤ 2A/r and β_deep, admissible codewords C1–C3, and a threshold-adaptive search. | theorem | PASS | numerical only | twenty constants; 0.8344; 10⁻¹² |
| Pass 02: the adjacent-normal outer error ε_deep = 0.013327 bounds the compiler's outer error. | computation | PASS | numerical only | 0.013327; 60 random convex bodies; 63 %; median 35 %; 84.02 % |
| Pass 02: the two identities of G.1. | theorem | PASS | numerical only | t = Δ |
| Pass 02: the adjacency step is 51 (48 at 0.8344). | computation | PASS | numerical only | 51; 48; 50; 47 |
| Pass 03: the spike cell U* = conv(B_{1/2} ∪ {R_sharp u_{0.013}}) has 103 active facets under the redundancy criterion. | computation | PASS | numerical only | 103 |
| Pass 03: exact active-fan area and closed-form perimeter and area lower bounds for the spike cell. | computation | PASS | numerical only | 0.846661292560721; 10⁻¹³; 3.3771690420862; 3.36584089305767; 0.841460223264417 |
| Pass 03: the cell radius is 0.013317 (0.013316971). | computation | PASS | numerical only | 0.013317; j = 0; t = 0.02503; 4,001-point grid; 0.013316971 |
| Pass 03: the uniform formulas hold, and the regular polygon's two areas agree. | computation | PASS | numerical only | 200 random codewords; 8×10⁻¹⁵ |
| Pass 04: arcsin(h/R) is 1-Lipschitz, giving the envelope L♯, cores, a phase-integrated bound and a radial bound. | theorem | PASS | numerical only | 0.842616436112935; 0.844007644444885 |
| Pass 04: the inner-core factor λ_b ≈ 0.988205366, and 0.9882…·P(b) ⊆ U. | computation | PARTIAL | corrected | 0.9882053662605436; 240,000 angles; 0.9882022551; θ ≈ 0.81765; 1.6×10⁻⁶; 8×10⁻⁵; 3.1×10⁻⁶ |
| Pass 04: the homothetic area bound is 0.8268069548. | computation | PARTIAL | corrected | 0.826801748731; 0.8268069548 |
| Pass 04: the sharp cell radius ε♯_cell ≈ 0.0059677 is an upper bound. | computation | PARTIAL | corrected | 0.0059677037; 0.0059692967; 1.6×10⁻⁶ |
| Pass 04 §S.1: Pass 03's cell radius ε_cell ≈ 0.0130605 (0.013060516). | computation | FAIL | refuted | 0.013060516; 0.013316971; 0.0130615798 |
| Pass 05: the uniform fixed-fan area form at M = 128 has signature (+, 0, 0, −, …). | theorem | PASS | exact-arithmetic certificate | (1, 2, 125); k = 1 and 127; −81.470968; 0.049097244 |
| Pass 05: the fixed-fan area form signature holds for general fans. | theorem | PASS | numerical only | 5 to 32 normals; (1, 2, m − 3) |
| Pass 05: the verifier's certified region is a finite union of polyhedra and each extreme witness is at most 6-sparse. | theorem | PASS | numerical only | 12 directions; 6 antipodal pairs; 40 origin-enclosing triples; 6 candidate heights |
| Pass 05: each contact basis reduces to one quadratic in p (the M = 8 example, with p the 'perimeter mode'). | computation | PASS | exact-arithmetic certificate | b₃ = b₇ = (p − 3)/2; [4, 3 + √2]; A(p) = −p²/2 + (3 + √2)p − (6 + 2√2); 2√2 − 2; p = 4 |
| Each pass's VALIDATION is produced by its shipped script. | method | UNMEASURED | not measured | 240,000-angle grid |
| Each VALIDATION describes its document and records its numbers correctly. | specification | PASS | numerical only | 10⁻¹²; 0.013316971; 0.013060516 |

Open: Recorded core: "Whether some legal U in the cell comes that close to L♯ there is not constructed here."; Pass 05's continuum lift: "It is marked DERIVED-CANDIDATE by the pass itself, and nothing here tests it."; Finite polyhedral verifier at scale: "Its structure is checked (the signature, sparsity and one basis), not a verifier run."; Source's own open item: "Cross-resolution saturation stays open."; What Pass 04 boxes with λ_b and ε♯: "what it boxes with them is not established by its own argument"

Unclear: How the two identities of G.1 are checked (analytically or numerically) is not stated.; Pass 03's cell radius 0.013317 is described as 'the package's number, on the package's 4,001-point grid' and judged PASS; the report does not say whether it, too, is a grid sample on an unsafe side.; The signature (1, 2, 125) is stated as 'numerically and in closed form'; the strength label reflects the closed form, while the random-fan signature (1, 2, m − 3) is numerical only. The source's Fillastre–Izmestiev citation is 'checked here numerically rather than fetched'.; The status registry is 'Not touched: the passes are not canonical documents'.

### RUN-043 — UESFCM - Unbounded Expansion Self-Referential Finite Closure Methodology v0.1 (EML-UESFCM-v0.1, dated 2026-09-19), its Operational Protocol (EML-UESFCM-OP-v0.1) and VALIDATION.json; its seventeen Lebesgue cycles read for labels, chaining and ledger arithmetic only

Report: `reports/RUN-043-UESFCM.md`

> **Verdict: PASS for fourteen of UESFCM's fifteen judgments, PARTIAL for "Problem Closure: DEFINED". The other twenty are labels.**

**For the paper.** UESFCM is a protocol text with nothing computed: its twelve DEFINED parts are defined, but which states end the loop differs across eight places (Problem Closure PARTIAL), and its one application (seventeen Lebesgue cycles) chains cleanly with a consistent ledger and never marks evidence as proof, while its anchoring target and Cycle 00 are not in the archive and its LinkBack types are used only in Cycle 01. Empirical benchmark validation is open.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Semantic Target Contract: DEFINED (Stage 0 builds a canonical target Q* = (D, V, Q, P, E, S, C), never silently mutated; a correction makes a new version). | definition | PASS | label only |  |
| Unbounded Expansion: DEFINED. | definition | PASS | label only |  |
| LinkBack Operator: DEFINED (eleven LinkBack types). | definition | PASS | label only | eleven |
| Finite Closure State: DEFINED (each round closed into C_t = (G, O, B, R, D, V, P, K)). | definition | PASS | label only |  |
| Self-Referential Recenter: DEFINED. | definition | PASS | label only |  |
| Proof Obligation Ledger: DEFINED. | definition | PASS | label only |  |
| Problem Closure: DEFINED (the loop ends at proof or counterexample closure; a failed target returns to Stage 0; independence closes relative to a theory; SEARCH-STALLED is not completion). | definition | PARTIAL | corrected | eight places; four different sets |
| Anti-Drift Governance: DEFINED. | definition | PASS | label only |  |
| UBE, RCHM, SREG / PEC relations: DEFINED. | definition | PASS | label only | seven theories; 19 documents |
| DLMVC relation: DEFINED (the protocol's lagged line). | definition | PASS | label only |  |
| Formal convergence theorem: NOT CLAIMED. | reading | PASS | label only |  |
| Universal proof solvability: NOT CLAIMED. | reading | PASS | label only |  |
| Empirical benchmark validation: OPEN. | reading | PASS | open |  |
| The methodology and protocol use one consistent vocabulary (LinkBack types, obligation states, artifact statuses, terminal and closed-for-now labels). | specification | PASS | label only | eleven LinkBack types; nine obligation states; fifteen artifact statuses; three terminal labels; five closed-for-now labels; five of §9's eight components |
| The seventeen Lebesgue cycles chain from one target, each FINITE-CLOSED / PROBLEM-OPEN. | reading | PASS | label only | C01…C17; 34 headers and crystals |
| The cycles' crystals carry a consistent ledger of derived numbers and handed-on counts. | computation | PASS | numerical only | 39 identities; 649 − 27 = 622; 622 = 617 + 3 + 2; 200 − 124 = 76; 76 − 8 = 68; 68 − 5 − 2 = 61; 0.00882 |
| The application anchors to target UESFCM-LUC-TARGET-v0.1 and previous state UESFCM-LUC-C00-v0.1. | reading | UNMEASURED | label only | 1,459 leaves; 34 cycle headers and crystals |
| Evidence is never marked as proof (§45); none of §40's banned phrases is used. | reading | PASS | label only | 15 strings; 15 verdict lines |
| Every artifact gets a LinkBack type (protocol Stage 3 makes LinkBack mandatory). | specification | NOT AS STATED | label only | 2 of 11; once each; Cycles 01 and 03 |
| The method's status vocabulary (§40), closed-for-now labels (§30) and round format (§39) structure its application. | reading | PARTIAL | label only | 29 of the method's 40 labels; 84 labels of their own; NOT-F3 24; AREA-PRUNED 13; ORBIT-COVERED 8; AREF-PRUNED 8; 15 of 17; 10; 2 |
| Reopening (invariant I7) is part of the method. | method | UNMEASURED | not measured | seventeen cycles |
| VALIDATION.json describes both documents exactly. | specification | PASS | label only |  |

Open: The cycles' mathematics: "Every certificate, bound and theorem of the seventeen cycles is for the runs that follow."; The missing target: "`UESFCM-LUC-TARGET-v0.1` and Cycle 00 may exist outside the archive. Here they are references with nothing behind them."; UESFCM as a method: "One application, one problem, no control. §55's questions are not measured" (drift rate, duplicate work, convergence of residual obligations, kernel fidelity, whether a lagged line finds more).; Target hashing: "No crystal carries a digest of its predecessor or of the target (§52 lists "target hashing")."; Related theories not in the archive: "RCHM is named by 19 documents of the AMRAL and DLMVC series, but no RCHM document is in the archive."; Target Contract and starting obligations: "the Target Contract of §2, which I1 protects; the obligations O₂–O₄ that Cycle 01 starts from" are not in the archive.

Unclear: The 'NOT AS STATED' and 'PARTIAL' findings for LinkBack usage and label usage are this inventory's mapping; the report's W4 result reads '2 of 11 LinkBack types used, once each; 29 of 40 labels unused; nothing reopened' without a verdict word, and it says 'W4's label counts describe how the method was used, not whether it works.'; The 39 ledger identities are checked in exact decimals but, per the report, 'only that the crystals agree with themselves and with each other'.; The DLMVC relation is judged PASS although 'DLMVC is of the same date and never names UESFCM. No DLMVC pass targets a cycle.'; The four closed-for-now labels of §30 are listed without definition and 'recorded as contextual'; the report counts five closed-for-now labels in W1 but four in the registry.

### RUN-044 — UESFCM Cycles 01–03 (dated 2026-09-19): Cycle 01 Mishra-seed expansion and B7 separator gate; Cycle 02 certified minimizer atlas and exact witness-exchange equivalence; Cycle 03 orbit localization, scalar separator closure and sharp-minimum route

Report: `reports/RUN-044-UESFCM-CYCLES-01-03.md`

> **Verdict: PASS for the three cycles' theorems and for every number they compute from definitions. Three things differ from what the cycles report:**

**For the paper.** Cycles 01–03's theorems and definition-computed numbers recompute (the B7 grid LP to seventeen digits, the domain constants to the ulp, the atlas and exchange theorems as proved), but the 'sharp root domain' is Mishra's Remark 20 and saves no boxes (his Remark 34), Cycle 03's s7 ≈ 0.00938484 is 1.52×10⁻⁷ above the numerical minimum 0.009384687909, the replacement family's minimum is at most 0.8305410847, and the 2.45×10⁻⁷ 'thinness' is the ceiling polygon's slack (the sublevel is about 10⁻³ long along B5's turn). s7 here is numerical, not certified.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Cycle 01: B7 is a separator of U0 by a finite-grid LP, M_U0(B7) = 0.00938298365 with orientation loss π/(7·256) and structural lower bound 0.00762986275. | computation | PASS | numerical only | 0.834780945912; 2048 support directions; 256 orientations; 0.009382983648263432; orientation 193; 0.0017531209004407328; 0.007629862747822699 |
| Cycle 01: the orientation loss is π/(7·256), using R_K ≤ 1. | computation | PASS | numerical only | 0.5129; 0.00090 |
| Cycle 01: a ranking of the regular Reuleaux odd-gons B7 … B41 by the same LP. | computation | PASS | numerical only | B11 (0.004727); B15 (0.002646) |
| Cycle 01: the headroom 0.834781191 − 0.8344 = 0.000381191. | computation | PASS | exact-arithmetic certificate | 0.834781191; 0.8344; 0.000381191 |
| Cycle 02: the hull Hausdorff radius of a placement box is τ(B) = max(δ3, δ5). | theorem | PASS | cited theorem | 300 random boxes; 0.92; R5 = 0.525731112119 |
| Cycle 02: the confining box [−0.70, 0.70]² × [0, 2π/5] × [−0.70, 0.70]² contains the relevant placements. | theorem | PASS by cited theorem | cited theorem | 0.70; 2π/5; 0.836548794; 0.8344 |
| Cycle 02: the certified minimizer-atlas theorem and the exact witness-exchange equivalence (with the single-witness corollary). | theorem | PASS | analytic (proof checked) |  |
| Cycle 02: 0.836494901 − 0.834780947 = 0.001713954, with 0.836494901 Mishra's located four-body optimum. | computation | PASS | corrected | 0.836494901; 0.834780947; 0.001713954; 0.8364914416 |
| Cycle 02: the emitting run's counts (486,799,600 nodes, 245,496,952 leaves). | computation | PASS | exact-arithmetic certificate | 2 · 245,496,952 − 486,799,600 = 4,194,304; 2²²; 22 |
| A3+ = 0.834781191 is the family ceiling, 2.45×10⁻⁷ above the representative's hull. | computation | PASS | numerical only | 0.834781191; 10⁶ uniform supporting lines; 2.450×10⁻⁷; 17 straight bridging edges; 10⁻¹¹; 3.4×10⁻⁶; 1.2×10⁻⁸; 10⁴ |
| Cycle 03: the sublevel is extremely thin (the family ceiling and the representative differ by only 2.45×10⁻⁷). | reading | PARTIAL | corrected | 2.45×10⁻⁷; about 10⁻³; 9.9×10⁻⁴; 7.8×10⁻⁴; 2.9×10⁻⁵ |
| Cycle 03: the farthest-corner sharp root domain (d*, T3, T5, a four-dimensional factor 164.27178468), SHARP ROOT DOMAIN: PROVED, as DOMAIN-STRENGTHENING. | theorem | PASS | cited theorem | d* = 0.694929333199893; T3 = 0.193949748872303; T5 = 0.197117585445617; 164.27178468; 0.19236641; 0.19589071; 169.1 |
| Cycle 03 and later cycles: the sharp root domain is a volume reduction (164.3×, then 985.6×). | reading | NOT AS STATED | corrected | τ = 0.833; 25,425,568; 27,350,260; 164.3×; 985.6× |
| Cycle 03: the separator–orbit radius inequality δ7 ≥ s7 − R⁺, the scalar closure criterion, and the ORBIT-COVERED cell rule. | theorem | PASS | analytic (proof checked) |  |
| Cycle 03: s7 ≈ 0.00938484 (crystal 0.009384839564890224). | computation | PARTIAL | corrected | 0.009384687909; φ = 0.676813; 0.009385623167; φ = 0.220843; 0.0121; 0.0127; 0.009384839564890224; 1.52×10⁻⁷; 1.1×10⁻⁴ |
| Cycle 03: the seed is a sharp minimum. | computation | PASS | numerical only | 210 sampled directions; 0.0084; 1.12×10⁻⁵; 98.7 %; 1.1×10⁻⁵; 6.9×10⁻⁵; 0.044; 5×10⁻⁴ |
| Cycle 03 §14: numerical evidence for a grid trap (a false minimum of a grid-only hull). | computation | UNMEASURED | open | 32, 64 and 128 angles; 5×10⁻¹⁶ |
| Cycle 03: the replacement family D, B3, B7 has min ≈ 0.830638626 and outer (1000) ≈ 0.830957521, so it is rejected. | computation | PASS | corrected | 0.8305410847; 9.8×10⁻⁵; 0.830638626; 0.8309189; 0.8305415 |
| Facts the cycles attribute to Mishra (Lemma 18's f, Corollary 19, R3, R5, Lemma 22, the forest identity and run counts, the located optima, the eight-dimensional cost, the frozen commit, Remarks 20 and 34). | reading | PASS | cited theorem | 8.6×10¹¹ boxes; about 880 days |
| The cycles' state crystals record the numbers above. | computation | PASS | numerical only | 1–4 ulps |
| The three cycles' scripts implement their computations. | method | UNMEASURED | not measured |  |
| Each VALIDATION records its document, script and crystal correctly. | specification | PASS | label only |  |

Open: s7 as a certified value: "The scan's Lipschitz slack between turns is 1.1×10⁻⁴. Cycle 04's dual certificate is the certified bound (RUN-045)."; The minimizer set: "Its thinness depends on the true Λ(F3), which neither the cycles nor this run bound from below beyond Mishra's 0.8344."; The grid trap as a phenomenon: "But the cycle's placement and evaluator are not given"; Ranking driver: "The shipped script prints B7 alone, so the ranking has no shipped driver."; Cycle 02 scaffold: "takes both the box lower bound and the B7 margin as injected callbacks, and ships neither."

Unclear: Cycle 02's semi-decision theorem and trichotomy, and Cycle 03's 'new exact gate: s7 > R̄', are listed among the claims but receive no separate verdict row.; For the grid trap the report's verdict word is 'not reproduced' (it also says 'The mechanism is real'); UNMEASURED/open is this inventory's mapping.; Strength 'cited theorem' for τ(B) reflects the report's note 'Mishra's Lemma 22 extended by hull nonexpansiveness' together with a 300-box numerical check.

### RUN-045 — UESFCM Cycles 04–06 (dated 2026-09-19): Cycle 04 Gate A closure and far-set certificate; Cycle 05 symmetry-quotiented two-colour Gate B certificate; Cycle 06 symmetry-fixed anchor and exact far-branch decomposition

Report: `reports/RUN-045-UESFCM-CYCLES-04-06.md`

> **Verdict: PASS for every theorem of the three cycles and for Gate A's certificate. Two premises of that certificate are left unchecked by the verifier shipped with it. The far-set minima the cycles report are search values, and each sits above a lower point found here.**

**For the paper.** Gate A of the UESFCM application, M_U0(B7) > 0.0088366789091384, holds as a finite dual certificate re-enclosed in 400-bit integer intervals (least lower bound 0.009274959134248587 at p = 767), but the shipped verifier does not check two premises the bound rests on (positive sine-gap weights, all 1,024 orientations), which hold when checked here; the D3 quotient, the anchor (true d_H 7.1×10⁻⁶) and the box bounds are right, while the F3 and F5 minima are search values with lower points found (F5's margin over A3+ at most 2.30534×10⁻⁵). Gate B is not emitted.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Cycle 04 Gate A: M_U0(B7) > 0.0088366789091384 by a finite dual certificate (CERTIFIED-FINITE-DUAL). | certificate | PASS | exact-arithmetic certificate | 0.0088366789091384; 1,024 orientations; widths below 10⁻¹⁰²; 0.009274959134248587; p = 767; π/7168; 0.008836678909138404 |
| Cycle 04: the conservative target and cover supports and the three-active balance (sine-gap weights). | theorem | PASS | exact-arithmetic certificate |  |
| Report-side: Gate A's certified bound is conservative against the true margin. | computation | PASS | numerical only | 33 sampled orientations; 2.2×10⁻⁷ to 1.8×10⁻⁴; 0.009458708251 |
| Cycle 04's shipped verifier Verify_GateA.py verifies the Gate A certificate. | method | PARTIAL | corrected | 1,024; below 10⁻¹⁷ |
| Cycle 04's certificate records interval enclosures for its witnesses. | certificate | NOT AS STATED | corrected | 1,024 witnesses; 20-digit decimals; 60 digits; 76 witnesses; 3×10⁻¹⁹ |
| Cycle 04: the far-set minimum, searched ≈ 0.834804245159. | computation | PASS | numerical only | 0.834804245159; 7×10⁻¹⁰; 0.0088366789 |
| Cycle 05: B5's turn is quotiented by the D3 symmetry to [0, π/15]. | reduction | PASS | numerical only | 40 random placements; 1×10⁻¹⁵; 2π/3 ≡ 4π/15 (mod 2π/5); [0, π/15] |
| Cycle 05: the two-colour theorem, and finite termination if Gate B is true. | theorem | PASS | analytic (proof checked) |  |
| Cycle 05: the safe constant s_A = 0.0088366. | computation | PASS | numerical only | 0.00927495 − 0.000438281 = 0.008836669; 6.9×10⁻⁸; 0.0088366 |
| Cycle 05: the branch holding the seed is first ORBIT-COVERED at depth 31. | computation | PASS | numerical only | 31 |
| Cycle 05's certificate spec gives root half-widths T3 and T5. | specification | NOT AS STATED | corrected | 22 ulps; 12 ulps |
| Cycle 05: the runtime probe (320,000 nodes …). | computation | UNMEASURED | not measured | 320,000 nodes |
| Cycle 06: a reflection-symmetric anchor U_sym with d_H(U0, U_sym) < 1.66×10⁻⁵, and the new radius 0.00882. | computation | PASS | numerical only | 2.4129×10⁻⁵; 1.66×10⁻⁵; 7.1×10⁻⁶; 0.834780945826; 0.0088366 − 0.0000166 = 0.00882 |
| Cycle 06: the far set splits exactly into F3 ∪ F5, and the nontrivial orbit copies are far. | reduction | PASS | numerical only | 2R5 sin(π/30) = 0.109907730484 |
| Cycle 06: F3's search minimum is 0.834968965913. | computation | PASS | corrected | 0.834968955147; 1.1×10⁻⁸; 0.834968965913 |
| Cycle 06: F5's search minimum is 0.834804245410, a margin of 2.30544×10⁻⁵ over A3+. | computation | PASS | corrected | 0.834804244389; 1.0×10⁻⁹; 0.834804245410; 2.30534×10⁻⁵; 2.30544×10⁻⁵ |
| The far-branch values stay above A3+. | computation | PASS | numerical only | A3+ = 0.834781191 |
| Cycle 06: branch box bounds on d3 and d5 and a farthest-corner area bound. | theorem | PASS | numerical only | 12,000 sampled points; 400 random boxes |
| Cycle 06: the F3 runtime probe (100,000 nodes …). | computation | UNMEASURED | not measured | 100,000 nodes |
| The cycles' state crystals record the numbers above. | computation | PASS | numerical only |  |
| The cycles' scripts (Gate A verifier, Gate B emitter, scaffolds, branch scaffold, verifier skeleton) implement the gates. | method | UNMEASURED | not measured |  |
| Each VALIDATION records its document correctly; Cycle 05's dependency copies equal Cycle 04's files. | specification | PASS | label only |  |

Open: Gate B: "No two-colour tree is emitted in these cycles. The cycles that follow attempt one."; Runtime probes: "Their counts depend on an erosion prototype that is not shipped."; Far minima as minima: "No search can establish the minimum itself."; Scripts: "The Gate B emitter, the scaffolds and the branch scaffold take the area lower bound as an injected callback, and none is supplied."; Cycle 05 verifier — Cycle 05's "verifier skeleton" contains no executable line.

Unclear: The verdict says each reported far-set minimum 'sits above a lower point found here', but for Cycle 04's far boundary the report says Cycle 04's 0.834804245159 'is 7×10⁻¹⁰ below the best point found here'.; The PARTIAL and NOT AS STATED findings for the verifier, the recorded intervals and the T3/T5 rounding are this inventory's mapping; the report's W1 and W3 results read 'PASS; the verifier skips two premises' and 'PASS; T3, T5 rounded inward'.; Cycle 04's component-motion metric q and 'Gate B written as a far-set minimum' are listed among the claims without a separate verdict row.

### RUN-046 — UESFCM Cycles 07–09 (dated 2026-09-19): Cycle 07 F3 runtime closure-failure localization and active-envelope compiler (with the saved F3 frontier); Cycle 08 two-level skeleton compiler; Cycle 09 reflection-quotiented skeletons and multi-chart Taylor closure

Report: `reports/RUN-046-UESFCM-CYCLES-07-09.md`

> **Verdict: PASS. Every number these three cycles compute on the saved frontier reproduces from this tree's own hull: the membership, the centre areas, the skeleton catalog, the reflection classes and the Taylor margins. The compilers they propose ship as interfaces with the proof steps not implemented, so the frontier and all the diagnostics come from code that is not in the archive.**

**For the paper.** Every number Cycles 07–09 compute on the saved F3 frontier (22,516 roots, membership fractions, least centre area 0.835188375167486, 24 skeletons and 1,332 signatures, 12 reflection classes, the Taylor diagnostic D1 > 0 on 646/649 and D2 > 0 on 27/27) reproduces from this tree's own hull, but the Taylor margins are finite-difference diagnostics, not bounds, and the compilers ship as interfaces with the proof steps unimplemented; the frontier itself comes from an unshipped prototype.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Cycle 07: the erosion branch-and-bound on F3 leaves 22,516 roots, of which 92.04 % lie fully in F3. | computation | PASS | numerical only | 22,516; 0.9204121513590335; 0.07718955409486587; 0.0023982945461005507; 0.00882 |
| Cycle 07: the least centre area is 0.8351883751674862, 4.07×10⁻⁴ above A3+, so there is no second basin and the hardness is the bound's looseness. | computation | PASS | numerical only | 0.835188375167486; root 21,409; 2 ulps; 0.8351883751674862; eight quantiles; 4.07×10⁻⁴ |
| Report-side kernel note (not a source assertion): this tree's kernel raised at some frontier centres, resolved by a different grid. | method | PASS | numerical only | 85; 76 |
| Cycle 07: the runtime probe (490,166 nodes …). | computation | UNMEASURED | not measured | 490,166 nodes |
| Cycle 07: the perimeter-Lipschitz bound A(v) ≥ A(c) − P(c)·τ(B) is too loose (REJECTED-AS-TOO-LOOSE). | computation | PASS | numerical only | 22,516; 3.33–3.40; 0.00024–0.031; 3.2 times |
| Cycle 07: stable-signature analyticity. | theorem | PASS | analytic (proof checked) |  |
| Cycle 08: 1,332 piece signatures but 24 body skeletons, with the given coverage. | computation | PASS | numerical only | 24; 1,332; 0.6755196304849884; 0.7999200568484633; 0.9184579854325813 |
| Cycle 08: two-level skeleton sufficiency and finite adjacent-skeleton enumeration. | theorem | PASS | analytic (proof checked) |  |
| Cycle 08: corner stability of 29.86 % (1,493 of 5,000). | computation | PASS | numerical only | 549 of 2,000; 0.275; 1,493 of 5,000; 0.2986; 2.3 standard errors |
| Cycle 08: the translation-convexity route, below 0.83460. | method | UNMEASURED | not measured | 0.83460 |
| Cycle 09: reflection maps a skeleton to its reversal, giving 12 classes, one of 15,210 roots. | reduction | PASS | numerical only | 12 classes; 6,979 + 8,231 = 15,210 |
| Cycle 09: single-chart, interval-gradient and multi-chart Taylor bounds. | theorem | PASS | analytic (proof checked) |  |
| Cycle 09: on the 649 near-ceiling roots D1 > 0 on 646; on the closest 27, D2 > 0 on all. | computation | PASS | numerical only | 649; 27; 10⁻³; 5×10⁻⁴; 646; 3766, 6217 and 8001; 3×10⁻¹¹; 1.655×10⁻⁴; 4×10⁻⁸; 2×10⁻⁵ |
| Cycle 09: the Taylor diagnostic's stored arrays and summary. | computation | PASS | numerical only | 10⁻¹⁵ |
| The cycles' state crystals record these numbers. | computation | PASS | numerical only |  |
| Cycles 07–09 propose an active-envelope interval compiler, a two-level skeleton compiler and Taylor-closure code. | method | UNMEASURED | not measured |  |
| Each VALIDATION records its document; the frontier copies in Cycles 08 and 09 equal Cycle 07's. | specification | PASS | label only |  |

Open: The erosion runs: "The prototype that produced the frontier is not shipped, so which boxes it left is taken from the file."; Taylor margins as bounds: "They rest on finite differences, as the cycle says; no interval derivative is computed here either."; Corner stability in the cycle's own sample: "Its sample and seed are not given."; Unshipped code: "No shipped code produces the runtime probe, the frontier, the catalogs or the finite differences."; Cycle 08 scaffold: "Its interval Newton step, winner gap and two-level area are not implemented."

Unclear: The corner-stability difference (0.275 vs 0.2986) is judged 'PASS as a sample'; listing it under corrections records the differing figure, not a verdict of error.; The report reproduces the least centre area but does not separately judge the inference 'no second basin and the hardness is the bound's looseness'.

### RUN-047 — UESFCM Cycles 10–11 (dated 2026-09-19 and 2026-09-20): Cycle 10 finite closure of the 27 priority F3 roots; Cycle 11 the 617-root batch and the two-root frontier

Report: `reports/RUN-047-UESFCM-CYCLES-10-11.md`

> **Verdict: PASS. Both certificates decode exactly and every count they state reproduces. All 56,149 area leaves also hold in exact arithmetic with this tree's exact cores: the tightest clears A3+ by 2.28×10⁻⁵, where the certificates' own least slacks are 3.8×10⁻⁸ and 2.5×10⁻⁹. Three things are noted:**

**For the paper.** Both erosion certificates (Cycle 10: 27 roots, 2,699 nodes; Cycle 11: 617 roots, 109,037 nodes) decode exactly, their stated counts, least slacks and η_model = 1.1488312266895049×10⁻¹⁰ reproduce, and all 56,149 area leaves hold in exact arithmetic with exact cores (least margin 2.28×10⁻⁵), closing 644 of Cycle 07's 22,516 roots; the split rule is recovered only from the verifier, the 10⁻¹⁸ tiling gaps are closed by an unstated continuity argument, and Cycle 11's verifier cannot run from its own package. Roots 8001, 8015, 8016 and the 225-subroot frontier under 6217 and 3766 remain search-only.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Cycle 10's 27 roots are Cycle 09's 27 closest; Cycle 11's inputs are the other 622 near-ceiling roots. | computation | PASS | exact-arithmetic certificate | 27; 622; 617; 3766, 6217, 8001, 8015 and 8016 |
| Cycle 10's certificate has 2,699 nodes, 1,363 area leaves, depth 13, 2,153 nodes on the hardest root, and no NOT-F3 leaf. | certificate | PASS | exact-arithmetic certificate | 2,699; 1,363; 13; 8006; 2,153 |
| The certificates use a deterministic split rule. | specification | PARTIAL | corrected | (1, 1, R5, 1, 1); 36; 5.8×10⁻⁵; (1.5, 1.5, R5, 1, 1); 27; 4.9×10⁻⁵; (0.7, 0.7, 1.5R5, 1.5, 1.5) |
| Binary subdivision of each root box covers it (children c ± h/2). | reduction | PASS | corrected | 280 of 1,336; 2.5×10⁻¹⁸; 3.3×10⁻¹⁸; 15,915 of 54,210; 4.7×10⁻¹⁸; P ≤ 8.026; 3.7×10⁻¹⁷ |
| Cycle 10: the hull of the finite witness set is at most inf A over its box. | theorem | PASS | analytic (proof checked) | 10⁻¹² |
| Cycle 10: η_model ≤ 1.149×10⁻¹⁰ from Mishra's floating-point model at m = K = 256. | computation | PASS | cited theorem | 1.1488312266895049×10⁻¹⁰; ε0 = 10⁻¹² + 4u + 8uR; R = 0.70 + 1/√3; u = 2⁻⁵³; N = m + 2K = 768; 332.33; 3.4×10⁻¹⁶ |
| Cycle 10: least raw slack 3.8179493×10⁻⁸, 332× η_model, 3.7179493×10⁻⁸ after 10⁻⁹. | computation | PASS | numerical only | 3.8179494×10⁻⁸; root 8006; 8.9×10⁻¹⁶; 3.817949290×10⁻⁸; 3.81794932×10⁻⁸ (50 digits) |
| Cycle 10: the two hull routines agree to 1.2×10⁻¹⁵. | computation | PASS | numerical only | 1.0×10⁻¹⁵; 1.2×10⁻¹⁵ |
| Cycle 10: the 27 roots hold no placement in F3 with A ≤ A3+. | certificate | PASS | exact-arithmetic certificate | 1,363 leaves; 2.352×10⁻⁵; root 8006 |
| Report-side: every area leaf of both certificates holds in exact arithmetic with this tree's exact eroded cores. | certificate | PASS | exact-arithmetic certificate | 56,149 / 56,149; 1,024 directions; 2.352×10⁻⁵; 2.277×10⁻⁵; root 20561; about 2.3×10⁻⁵ |
| Cycle 11: 622 inputs; 616 / 622 close at 5,000 nodes; 8018 at 13,181. | computation | PASS | numerical only | 622; 616; 5,000; 8018; 13,181 |
| Cycle 11: the 617-root certificate has 109,037 nodes, 54,786 area leaves, 41 NOT-F3 leaves, depth 16. | certificate | PASS | exact-arithmetic certificate | 109,037; 54,786; 41; 16; 8018; 13,181 |
| Cycle 11's 41 NOT-F3 leaves lie inside the F3 ball. | certificate | PASS | exact-arithmetic certificate | 41; 1.31×10⁻⁵ |
| Cycle 11: independent replay, least verified slack 1.5376482309648054×10⁻⁹. | computation | PASS | numerical only | 2.5376479×10⁻⁹; root 21309; 1.5376479×10⁻⁹; 3.6×10⁻¹⁶; 2.53764884×10⁻⁹ (50 digits); 7×10⁻¹⁶ on all 617 trees |
| Cycle 11: the targeted layers 51 → 91 → 242 → 31 → 25 and the 225-subroot frontier. | computation | PASS | numerical only | 9 + 11 + 7 + 11 + 13 = 51; 40 of 51; 91; 63 of 91; 242; 39 of 42; 31; 28 of 31; 25; 225; 200 under 6217; 25 under 3766; depths 4 to 28; 2.3×10⁻¹⁸; 489 pairs; 5.9×10⁻¹⁸ |
| Cycle 11: the centre diagnostic is 0.8349695459154723 under 6217 and 0.8352612051955018 under 3766. | computation | PASS | numerical only | 0.8349695459155; 0.8352612051955; 2×10⁻¹⁶; 0.8350052734; 0.8353009037 |
| Cycle 11: roots 8001, 8015, 8016 are search-closed, not certified (packaging debt). | reading | PASS | open | 8001; 8015; 8016 |
| Cycle 11: 6217 and 3766 persist under erosion subdivision (6217 the primary persistent family). | computation | PASS | numerical only |  |
| Mishra's §5.4 floating-point model is as the cycles cite it. | reading | PASS by cited theorem | cited theorem | ϑ = 10⁻¹²; ε0 = 1.0016×10⁻¹²; P ≤ 8.026; N ≤ m + 2K = 3072; η = 1.72×10⁻⁹; 8.04×10⁻¹²; 1.7095×10⁻⁹; 1.7176×10⁻⁹; 5012.35; 1.71×10⁻⁹ |
| The cycles' state crystals record these numbers, with Cycle 11's packaging debt. | computation | PASS | numerical only |  |
| Cycle 10 ships a standalone verifier. | method | PASS | label only | 10⁻¹² |
| Cycle 11 ships a Numba verifier for its certificate. | method | UNMEASURED | not measured | 10⁻⁹ |
| Each VALIDATION records its package correctly. | specification | PASS | label only |  |

Open: F3's closure: "The two certificates close 644 of Cycle 07's 22,516 roots. The rest, the five hard roots among them, are not addressed here."; Search-closed parts: "8001, 8015 and 8016, and the parts of 6217 and 3766 outside the 225 subroots, are closed only in search logs. The trees are not shipped."; Cycle 07's frontier as a cover of F3: "It comes from the unshipped prototype (RUN-046)."; Continuity step: "it is standard and short, but written out here only in words."; 6217 and 3766 under exact cores: "That question goes to Cycles 12–17 (RUN-048)."; Cycle 11's verifier: "Cycle 11's verifier loads Cycle 10's verifier from an absolute path on the machine that wrote it, and the package does not ship that file."; Package records: "Cycle 10's `verifier_stub_sha256` matches no file in the package."; "The six search logs carry no digests."

Unclear: The centre diagnostic values (0.8349695459155, 0.8352612051955) and the exact centre areas (0.8350052734, 0.8353009037) differ; the report gives both without saying what the diagnostic measures.; Root 7989 is a single NOT-F3 leaf; the certificate writes no minimum while the log writes the non-standard JSON token `Infinity`; the report notes this without a verdict.; The 'analytic (proof checked)' label for the tiling item rests on a continuity argument the report says is 'written out here only in words'.; For roots 8001, 8015, 8016 the report's verdict word is 'as stated'; PASS/open is this inventory's mapping.

### RUN-048 — UESFCM Cycles 12–14 (dated 2026-09-20): Cycle 12 root 6217 formal batch closure and symmetric triple-contact reduction; Cycle 13 formal closure of root 6217 at ρ₅ = 0; Cycle 14 rotation-tail closure and boundary locking

Report: `reports/RUN-048-UESFCM-CYCLES-12-14.md`

> **Verdict: PASS for the three certificates and the contact; the rotation profile holds as a search. NOT ESTABLISHED for Gate 6217-A: the ρ₅ = 0 reduction of Cycles 12–13 does not keep the F3 constraint. So Cycle 13's certificate covers only part of the slice, and so does the "Gate 6217-A: CLOSED" that Cycle 14 and both crystals carry.**

**For the paper.** Root 6217's three certificates (Cycle 12: 124 subroots; Cycle 13: the 2-D rectangle; Cycle 14: eight tail subroots; 66,927 area leaves) replay and hold in exact arithmetic (least margin 2.60×10⁻⁵), and the symmetric triple contact reproduces to 80 digits at area 0.8349861105592238, 2.049196×10⁻⁴ above A3+. But the ρ₅ = 0 reduction drops the F3 constraint, so 'Gate 6217-A: CLOSED' is not established: the arc side of the slice is uncovered by any certificate, though there the least area is numerically again the contact's; the rotation profile and Gate 6217-B remain search-level.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Cycle 12: root 6217's box, with ρ5 ∈ [0, π/120]; its 200 subroots split into 124 certified and 76 remaining. | computation | PASS | exact-arithmetic certificate | [0, π/120]; 200; 124; 76; 10⁻¹⁸ |
| Cycle 12: the 200 subroot centres show three centre skeletons, 166 / 25 / 9 (83 %). | computation | PASS | numerical only | 103, 63, 25, 5, 4; 166, 25 and 9 |
| Cycle 12: persistence is resolution-driven, median half-width ratios 2, 4, 4, 4, 6. | computation | PASS | numerical only | 2, 4, 4, 4 and 6 |
| Cycle 12: 124 / 200 subroots close by a certificate of 32,568 nodes, 16,151 area leaves, 195 NOT-F3 leaves, depth 15, least 3.684540239579093×10⁻⁸. | certificate | PASS | numerical only | 124 trees; 32,568; 16,151; 195; 15; 3.7845403×10⁻⁸ (subroot 181); exact margin 2.597×10⁻⁵ |
| Cycle 13: the 2-D certificate on the rectangle has 299 nodes, 150 leaves, depth 10, least 3.18194×10⁻⁶. | certificate | PASS | numerical only | 299; 150; 10; 3.1819403×10⁻⁶; exact margin 3.456×10⁻⁵ |
| Cycle 14: the eight-box certificate has 102,062 nodes, 50,626 area leaves, 409 NOT-F3 leaves, least 5.3395949×10⁻⁹. | certificate | PASS | numerical only | 8 trees; 102,062; 50,626; 409; depth 19 (not stated); 6.3395956×10⁻⁹ (subroot 112); exact margin 2.610×10⁻⁵ (subroot 96) |
| Report-side: every area leaf and NOT-F3 leaf of the three certificates holds in exact arithmetic with exact cores; split gaps are covered by continuity. | certificate | PASS | exact-arithmetic certificate | 66,927 / 66,927; 2.60×10⁻⁵; 604 / 604; 1.31×10⁻⁵; 1.93×10⁻⁶; 943 of 16,222; 4 of 149; 4,460 of 51,027; 3.5×10⁻¹⁸ |
| Cycle 12: the constrained minimum over the whole root is the symmetric contact (y3 = ρ5 = y5 = 0, d3 = s). | computation | PASS | numerical only |  |
| Cycle 12: θ*, x5* in closed form give a four-direction triple contact at a contact-switch cusp. | computation | PASS | numerical only | x3* = −0.02097883209899; θ* = arccos(1/(2c3)); 80 digits; (−0.03296, −0.03573); (−0.21262, +0.40369); λ = 0.919254; (−0.04756, 0) |
| Cycle 12: A* ≈ 0.834986110559224, 2.049×10⁻⁴ above A3+. | computation | PASS | numerical only | 0.8349861105592238; 0.834986110559224; 2.049196×10⁻⁴ |
| Cycle 12: the scalar optimizer stops 1.52×10⁻¹⁰ from the closed-form contact. | computation | PASS | numerical only | 3×10⁻¹⁴; 1.5×10⁻¹⁰ |
| Cycle 12: vertical symmetrization at ρ5 = 0 gives A(x3, 0, 0, x5, 0) ≤ A(x3, y3, 0, x5, y5). | theorem | PASS | cited theorem |  |
| Cycle 12: horizontal translation convexity (Fáry–Rédei / Rogers–Shephard). | theorem | PASS by cited theorem | cited theorem | 200 lines; least 3.0 |
| Cycle 12: the symmetric F3 domain is x3 ≤ x3*, so ρ5 = 0 is a two-variable convex problem in (x3, x5). | reduction | FAIL | refuted | x3 = −0.012122; 0.00886; y3 = 0.012122 |
| Cycle 12: the contact's KKT condition makes it the minimum on the symmetric slice. | theorem | PASS | numerical only |  |
| Cycle 12: the 27-point rotation profile is non-decreasing with quadratic ratios 0.1738–0.2457. | computation | PASS | corrected | 27; 8.3×10⁻⁸; 0.173793–0.245696; 0.1738–0.2254; 0.834986110559; 0.8351052259; ρ ≈ 0.014 |
| Cycle 13: vertical symmetrization reduces Gate A "exactly" to the rectangle x3 ∈ [−0.024244, −0.020979], x5 ∈ [0.012320, 0.024640]. | reduction | FAIL | refuted | [−0.024244, −0.020979]; [0.012320, 0.024640] |
| Cycle 13: m₆₂₁₇(0) > A3+, GATE 6217-A: CLOSED. | certificate | NOT AS STATED | numerical only | 2.05×10⁻⁴ |
| Report-side: the ρ₅ = 0 reduction can be completed on the arc side to a three-parameter problem in (α, x5, y5). | reduction | PASS | analytic (proof checked) |  |
| Report-side: on the arc side the least area is again the contact point's. | computation | PASS | numerical only | 25 angles; α = 1.5666; α = π; 0.834986110559; 0.835484 at α = 1.632; 0.835483 |
| Report-side: coverage of the arc side of the ρ₅ = 0 slice by the certificates. | computation | PASS | numerical only | 20,000; 10,856; 145 (1.3 %); 6,812 (62.7 %); 3,899 (35.9 %); 9,144 |
| Cycle 14: "Cycle 13 closed the entire ρ = 0 slice". | reading | FAIL | refuted |  |
| Cycle 14: 76 → 68 subroots remain; root 93 is localized in search to 30 subroots. | computation | PASS | numerical only | 4, 5, 48, 50, 84, 96, 112, 113; 68; 93; 30 |
| Cycle 14: the 27 constrained minimizers lie on d3 = s, within 2.365×10⁻⁸. | computation | PASS | numerical only | 10⁻¹⁵; 2.4×10⁻⁸ |
| Cycle 14: the unconstrained minimizers lie within 7.9 % of s of the anchor. | computation | PARTIAL | corrected | 20 of the 27; 2.2×10⁻³; 9.1×10⁻⁴; 10 %; 6.96×10⁻⁴; 7.9 % |
| Cycle 14: the profile shows four skeletons: two generic charts and two degenerate states, with a transition near ρ ≈ 0.0141. | computation | PARTIAL | corrected | 5×10⁻⁸ rad; 2×10⁻¹³; 10⁻⁹; 10⁻⁶ rad; samples 0 and 14; 25 |
| Cycle 14: fixed-direction translation convexity; full 4-D joint convexity not claimed. | theorem | PASS | cited theorem | 600 random lines; least 2.8; 3.0; 3.1 |
| Cycle 14: the conditional boundary-reduction theorem (boundary locking). | theorem | PASS | analytic (proof checked) |  |
| The cycles' state crystals record these numbers and statuses. | computation | PARTIAL | label only |  |
| Cycles 12–14 ship verifiers for their certificates. | method | UNMEASURED | not measured |  |
| Each VALIDATION records its package correctly. | specification | PASS | label only |  |

Open: Gate 6217-A on the arc side: "A certificate over it, like Cycle 13's over the rectangle, is not built."; Gate 6217-B (0 < ρ5 ≤ π/120): "open in the cycles too. The 27-point profile is a search, here and there."; The 68 open subroots and the part of the box outside the 200: "The latter was closed only in Cycle 11's search, whose trees are not shipped."; Cited theorems: "Fáry–Rédei and Rogers–Shephard themselves. They are cited as known theorems; here they are only checked numerically along lines (W7)."; Root 3766: "untouched by these cycles."; Verifiers: "two need a file outside their package"

Unclear: The report's verdict for Gate 6217-A is 'NOT ESTABLISHED', which is not in this inventory's finding enum; it is recorded as NOT AS STATED with strength 'numerical only'.; The document figures for least slack in 'The cycles, read' (3.684540239579093×10⁻⁸ for Cycle 12, 5.3395949×10⁻⁹ for Cycle 14) differ from the W3 slacks (3.7845403×10⁻⁸, 6.3395956×10⁻⁹) by about 10⁻⁹, while the report says they are '8.6×10⁻¹⁶ ... and 7.5×10⁻¹⁶ apart'; Cycle 13's 3.18194×10⁻⁶ and 3.1819403×10⁻⁶ show no such offset. The report does not explain the offset.; Cycle 12's vertical symmetrization is judged 'PASS as an inequality'; the strength 'cited theorem' reflects its reliance on Fáry–Rédei convexity, which the report says is cited and only checked numerically.; The KKT item is 'PASS as evidence' in the table though the report calls the argument 'sound' on the symmetric slice.

### RUN-049 — UESFCM Cycles 15–17 (dated 2026-09-20): Cycle 15 corrected boundary locking; Cycle 16 tight reference and local secant; Cycle 17 tight-reference pruning and hierarchical competitive-core closure

Report: `reports/RUN-049-UESFCM-CYCLES-15-17.md`

> **Verdict: PASS. The lemmas hold. Both upper references reproduce, and all three certificates decode and pass leaf for leaf. All 115,511 area leaves also hold in exact arithmetic, each against its own threshold, the least by 2.66×10⁻⁵. Three things are noted:**

**For the paper.** Cycles 15–17's lemmas (radial convexity via Fáry–Rédei, the secant lemmas, the two-colour criterion, prefix-tree closure) hold, both upper references reproduce (A_outer to 43 digits; tight A_ref = 0.8349865 with 3.894×10⁻⁷ room, not 6.5×10⁻⁸), and all 115,511 area leaves of the three certificates hold in exact arithmetic (least 2.66×10⁻⁵), taking root 6217's open cells from 68 to 61; but 'all other regions eliminated by finite certificates' is false, since 44 % of root 6217's box was closed only by search and the ρ₅ = 0 arc side is uncovered, and Cycle 15's counterexample to radial monotonicity holds only over 1.6×10⁻⁷ along its ray.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| Cycle 15: global radial monotonicity out of the F3 circle is false; a numerical counterexample has outward slope −0.0457. | computation | PASS | numerical only | 0.8360391144; 2×10⁻⁹; −0.04569; step 10⁻⁷; +0.093 at step 10⁻⁵; 1.6×10⁻⁷; 7.2×10⁻⁹; 0.8360391072; 0.8360391068 |
| Cycle 15: radial convexity holds for fixed boundary parameters. | theorem | PASS by cited theorem | cited theorem | 200 random rays; least 2.9 |
| Cycle 15 and 16: the long and local secant lemmas. | theorem | PASS | analytic (proof checked) |  |
| Cycle 15: secant locking and the two-colour criterion (SECANT-LOCKED and AREF-PRUNED cells give boundary locking). | theorem | PASS | analytic (proof checked) |  |
| Cycle 17: tight-reference elimination and prefix-tree closure. | theorem | PASS | analytic (proof checked) |  |
| Cycle 15: an explicit feasible candidate has A_outer = 0.83499392544273890507… < 0.835, by a 20,000-direction outer polygon at 70 digits. | computation | PASS | numerical only | 0.83499392544273890507…; 8×10⁻¹⁶; 43 digits; 0.8349861105592; 1.389×10⁻⁵; 7.8×10⁻⁶ |
| Cycle 16: a padded 500,000-direction polygon tightens the reference to A_ref = 0.8349865 (with stated room 6.5×10⁻⁸). | computation | PASS | corrected | 0.8349865; 2×10⁻¹⁵; 3.894×10⁻⁷; 6.5×10⁻⁸; 2.7×10⁻⁷; 3.3×10⁻⁸ |
| Report-side: the upper references are loose to first order because uniform directions miss the hull's 17 bridging-edge normals. | computation | PASS | numerical only | 17; 20,000 directions; 3.1×10⁻⁹; 500,000; 4.9×10⁻¹² |
| Cycle 15: profile secant 9.887×10⁻⁶. | computation | PASS | numerical only | 8 samples; 9.88424×10⁻⁶; 9.88746×10⁻⁶; 3×10⁻⁹ |
| Cycle 15 §12 low-sublevel slope +0.0250; Cycle 16 §5 and §8: the long secant fails at a looser level. | computation | UNMEASURED | not measured | +0.0250 |
| Cycle 16: f(s) − f(s − 10⁻⁴) ≈ 3.27×10⁻⁶ at the contact. | computation | PASS | numerical only | 3.27×10⁻⁶; 3.8×10⁻⁵; 0.213 |
| Cycle 16: 2 / 27 profile samples lie below the tight reference. | computation | PASS | numerical only | 2 of the 27; ρ = 0; 0.00101 |
| Cycle 16 §10: "all other regions have already been eliminated by finite certificates"; the responsibility is restricted to the 68 open cells. | reading | FAIL | refuted | 44.2 %; 44.15 %; 55.85 % |
| Cycle 17: five cells (28, 36, 68, 144, 155) close against the tight reference with a 61,487-node certificate (30,467 area leaves, 279 NOT-F3, least 2.582831×10⁻⁹). | certificate | PASS | numerical only | 61,487; 30,467; 279; 3.5828319×10⁻⁹ (cell 155); 3×10⁻¹⁶; 2.656×10⁻⁵ |
| Cycle 17: cells 131 and 129 close against A3+ with hierarchical prefix-tree certificates. | certificate | PASS | exact-arithmetic certificate | 40,859 leaf paths; 40,858 internal; 38,893 area; 1,966 NOT-F3; 48,590 paths; 48,589 internal; 46,151 area; 2,439 NOT-F3; 1.2396455×10⁻⁹; 6.1271319×10⁻⁹; 3.022×10⁻⁵; 3.183×10⁻⁵ |
| Report-side: all 115,511 area leaves of the three certificates hold in exact arithmetic against their own thresholds. | certificate | PASS | exact-arithmetic certificate | 115,511; 2.66×10⁻⁵ |
| Cycle 17: 68 → 61 open cells; the competitive ranking (next: 125, 152, 150). | computation | PASS | numerical only | 76 − 8 = 68 − 7 = 61; 131; 131, 129, 125, 152 and 150; cell 14 by 6.5×10⁻⁸; cell 28 by 2.3×10⁻¹² |
| The cycles' state crystals record these numbers and statuses. | computation | PARTIAL | label only |  |
| Cycles 15–17 ship verifiers for the upper references and the certificates. | method | PASS | label only | 70 digits |
| Each VALIDATION records its package correctly. | specification | PASS | label only |  |

Open: Root 6217's closure: "61 cells stay open, and the boundary-locking cover is not built in the cycles."; Arc side and search-only share: "Nor is the ρ₅ = 0 arc side covered (RUN-048), and 44 % of the box has only Cycle 11's search behind it."; Searched diagnostics with no stored point: "Cycle 15 §12, and Cycle 16 §5 and §8."; Cited theorems: "Fáry–Rédei and Rogers–Shephard themselves, cited as known; here they are only checked along rays."; Root 3766 and F5: "untouched by these cycles."; Package: "Cycle 17's VALIDATION records every file but the 68-cell diagnostic."

Unclear: The five-cell certificate's least slack is 2.582831×10⁻⁹ in 'The cycles, read' (the document's figure) and 3.5828319×10⁻⁹ in W4; the report does not state the 10⁻⁹ offset between them.; The counterexample's marginality appears in the title and W1, not among the verdict's 'Three things are noted'.; Whether the outer-polygon upper references are rigorous enclosures is not stated; the report reproduces them in double precision and at 50 digits.

### RUN-050 — LESR Paper 00 - From Skew Fields to Variational Shape Closure (LESR-SFVC-00-v0.1, 2026-09-20), one of the 42 canonical documents, with its state record and VALIDATION

Report: `reports/RUN-050-LESR-PAPER00.md`

> **Verdict: PASS. The paper is a bridge, not a result: it states definitions and classical facts, and marks its principles as programmatic. Every formula it uses holds. Two things are noted:**

**For the paper.** LESR Paper 00 is a bridge of definitions and classical facts whose every formula holds (the area, its first and second variations, the translation kernel, the Fourier multiplier 1 − m², checked symbolically), but its programmatic No-Slack Boundary Principle needs the hypothesis μ_h(I) > 0: at a corner the first variation of an admissible inward move is zero and the area drops only at second order (A(ε) = 1 − (3π/8)ε² for the unit square). Its account of seven predecessor documents is unchecked, as they are not in the archive.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §2: the placement formula h_{gγ}(θ) = h_γ(θ − φ) + t·u_θ. | computation | PASS | numerical only | 10⁻¹⁵ |
| §3: E_C(γ) = 0 exactly when some placement fits. | theorem | PASS by cited theorem | cited theorem |  |
| §8: h + h″ ≥ 0 as a measure characterizes support functions. | theorem | PASS by cited theorem | cited theorem |  |
| §10: skew-admissible perturbation. | definition | PASS | label only |  |
| §12–13: the E(2) quotient; rotation generator ∂_φ h(θ − φ) = −h′; the Fourier multiplier (∂² + 1) cos mθ = (1 − m²) cos mθ. | theorem | PASS | exact-arithmetic certificate | 1 − m² |
| §16–19: A[h] = ½∫(h² − h′²) = ½∫h(h + h″), δA[h; φ] = ∫φ(h + h″), second variation ∫(φ² − φ′²), and translation modes vanish. | theorem | PASS | exact-arithmetic certificate | π(1 − k²); 0, −3π, −8π, −15π for k = 1…4 |
| Report-side check on bodies: the area formula by curvature measure reproduces known areas. | computation | PASS | numerical only | (π − √3)/2; 10⁻¹¹; 0.834780945912; 200,000-vertex polygon; 2.6×10⁻¹⁰ |
| §20–25: contact strata, envelope, support corridor and multi-gap state (Δ_A, Δ_h, Δ_Γ, Δ_G). | definition | PASS | label only |  |
| §29: No-Slack Boundary Principle (PROGRAMMATIC PRINCIPLE): an inward admissible move gives δA[h; φ] < 0. | theorem | PASS | corrected | μ_h(I) > 0; φ = −sin 2θ; [0, π/2]; ε ≤ ½; A(ε) = 1 − (3π/8)ε² |
| §30: infinite-dimensional KKT measures (programmatic, not yet proved). | reading | PASS | open |  |
| §1–6: what the earlier skew-field documents (SKEW-CALC v0.2, Moser skew-field initialization, Subtractive Topology v3.0, GSC, RVSC, TGSA, Geometric Skew) already contain. | reading | UNMEASURED | not measured | seven predecessors |
| The state record lists eleven new bridge components and the next gate (a Support-Function Admissible Variation Compiler). | specification | PASS | label only | eleven; §7, 8, 10, 12, 11, 13, 17, 19, 20, 22 and 25; seven recovered components |
| The phrase PROGRAMMATIC PRINCIPLE (§29), first used here. | definition | PASS | label only |  |
| VALIDATION describes the paper and state record (digests, UTF-8, math_delimiters: PASS). | specification | PASS | label only |  |

Open: Predecessors' content: "§1–6 describe earlier documents that are not in the archive."; The Lebesgue extremal shape: "Paper 00 does not claim anything about it either; the papers that follow do."; Classical theorems cited: "support functions and convexity, containment by support functions", "used here as known."; No-Slack Boundary Principle and infinite-dimensional KKT measures are "stated as not yet proved."

Unclear: The identities are checked in sympy over trigonometric polynomials with free coefficients; the strength 'analytic (proof checked)' reflects symbolic verification on that class, not a general proof written out.; The report reviews one registry judgment (PROGRAMMATIC PRINCIPLE) as 'PASS by reading, with the corner hypothesis noted'; the item-level finding PASS with strength 'corrected' is this inventory's mapping.

### RUN-051 — LESR Paper 01 - Topology Envelope and Stratified Shape Space (LESR-TOPO-01-v0.1, 2026-09-20), with its schema, compiler scaffold, state record and VALIDATION (evidence, not a canonical document)

Report: `reports/RUN-051-LESR-PAPER01.md`

> **Verdict: PASS. Paper 01 is a framework: definitions, two small theorems, and lemmas on monotonicity and stability. The theorems hold where the series uses them, on the Lebesgue kernel. One thing is noted:**

**For the paper.** LESR Paper 01's two theorems hold on the Lebesgue kernel: each pair of support pieces crosses at most twice a turn independent of placement (so signatures are finite; at most 2 sign changes and 18 hull segments in 300 placements), and the hull area is analytic within a contact stratum (fit to 10⁻¹⁵) with a kink of 3.7×10⁻⁶ across Cycle 12's contact switch. The shipped scaffold omits the sensitivity constant C of the §33 stability lemma and returns only three of five statuses.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1: the shape space H_conv / E(2). | definition | PASS | label only |  |
| §3–8: signatures Γ = (W, P, O, T, I, G), contact graph, strata, frontier order and frontier condition. | definition | PASS | label only | six signature components |
| §10: a finite support-piece system has finitely many signatures on a compact region (each pair of pieces crosses finitely often). | theorem | PASS | analytic (proof checked) | at most twice a turn; 300 random placements; 2; 18 segments; 288 |
| §12–14: regular crossings are analytic, and the hull area is analytic on a regular stratum. | theorem | PASS | numerical only | 21 points; 10⁻⁴ to 5×10⁻⁴; degree-6 polynomial; 1.2×10⁻¹⁵; 1.6×10⁻¹⁵ |
| §2: shape coordinates may vary smoothly while contact topology jumps (the area kinks where the stratum changes). | reading | PASS | numerical only | 3.7×10⁻⁶ |
| §17–18: a regular transition is a codimension-one wall, and symmetry can lower a transition's codimension. | theorem | PASS by cited theorem | cited theorem |  |
| §20–26: T_max can only shrink, F_min can only grow and generates T_min; the topology envelope's monotonicity. | theorem | PASS | analytic (proof checked) |  |
| §33: stability of a stratum under perturbations ‖δh‖ < m/C (margins m, sensitivity C), and a trust radius. | theorem | PASS | analytic (proof checked) |  |
| The compiler scaffold implements §33's stability test and the schema's gate statuses. | method | NOT AS STATED | corrected | three of the schema's five statuses |
| §39: observed signatures are not global strata. | reading | PASS | label only |  |
| §43–44: a monotone research state, reopened only for an invalidated proof or a widened model. | specification | PASS | label only |  |
| The schema and state record match the text (six signature components; five statuses; fifteen completed items; three disclaimers; next paper). | specification | PASS | label only | six; five; fifteen; three |
| VALIDATION records the paper, schema, scaffold and state correctly. | specification | PASS | label only | four digests |

Open: Frontier condition (§8): "The text asks for it to be verified in the finite chart model; neither the paper nor this run does so."; Contact atlas: "A contact atlas of the continuum problem. The paper does not claim one either."; Whitney regularity: "disclaimed by the paper."; Scaffold: "It computes no crossing. It is labelled a specification, not a kernel, and it is one."

Unclear: The §12–14 analyticity theorem is judged PASS from a numerical polynomial fit on Cycle 12's symmetric slice; the report does not say whether the general proof is read.; The verdict says 'One thing is noted' and then lists three bullets about the scaffold; NOT AS STATED for the scaffold is this inventory's mapping.; 'Theorems hold where the series uses them, on the Lebesgue kernel' limits the measured scope to the kernel's pieces (disc, B3 and B5 corners and arcs).

### RUN-052 — LESR Paper 02 — Group-Orbit Shape Expansion and Symmetry Breaking (LESR-GROUP-02-v0.1, 2026-09-20), with its schema, compiler scaffold, state record and VALIDATION

Report: `reports/RUN-052-LESR-PAPER02.md`

> Verdict: PASS. Paper 02 is a framework built from classical facts, and every fact it states holds. Three things are noted. The first two concern the kernel itself:

**For the paper.** Paper 02's group-layer facts hold (projectors, the gcd law for m ≥ 2 in 161 sampled cases, δA = 0 for breaking modes to 4×10⁻¹⁵, the Fourier δ²A), with §15's stabilizer meant on the translation gauge. On the kernel's hull no pure mode m ≥ 2 is admissible, and the mirror-slice polish of q† (area 0.8347809458213, a floating-point evaluation, not an enclosure) is a strict local minimum where breaking costs area at first order.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1 notation erratum: SE(2) = SO(2) ⋉ R² and E(2) = O(2) ⋉ R², correcting Paper 00 §12 and Paper 01 §1. | definition | PASS | label only |  |
| §2–6: rigid motions and reflections act on support functions (h_{gK}(u) = h_K(Rᵀu) + t·u, (σ_α h)(θ) = h(2α − θ)) and the translation gauge is a₁ = b₁ = 0, which puts the Steiner point at the origin. | theorem | PASS | numerical only | 10⁻¹⁵ |
| §7–11: the Fourier modes V_m are the real O(2) modes, and L = 1 + ∂² acts on them by 1 − m² and commutes with O(2). | theorem | PASS | exact-arithmetic certificate | 1 − m² |
| §8: m = 0 is the 'scale / mean support' mode. | definition | PASS | corrected |  |
| §12–14: on a smooth seed with ρ ≥ κ₀ > 0 a pure mode is admissible up to κ₀ / ((m² − 1)‖φ_m‖∞), so high modes cost O(m⁻²); with atoms only the measure inequality counts. | theorem | PASS | numerical only | 0.585; 1.415; m = 2…60; [1.000, 2.004]; [1.000, 1.009] |
| Admissible pure modes on bodies with corners (measure version of §12–14): Reuleaux triangle, Reuleaux pentagon and the kernel's hull. | computation | PASS | numerical only | m = 3; cap 1/8; m = 5; cap 1/24; up to m = 24 |
| On the kernel's seed (hull), no pure Fourier mode cos m(θ − c) with m ≥ 2 can be added at any phase, sign or amplitude. | computation | PASS | numerical only | 8π/15; π/2; m ≥ 2 |
| §15: the intrinsic stabilizer of a shape. | definition | NOT AS STATED | corrected |  |
| §17–23: averages over a finite stabilizer (C_n, D_n) are projectors; C_n keeps modes m ≡ 0 (mod n) and D_n the cosines cos kn(θ − α). | theorem | PASS | numerical only | n = 1…8; modes up to 24; 49, 25, 17, 13, 9, 9, 7, 7; 25, 13, 9, 7, 5, 5, 4, 4 |
| §24–27 gcd law: an aligned cos mθ on a D_n seed leaves at least D_gcd(n,m), and exactly that without accidental symmetry. | theorem | PASS | numerical only | n = 2…8; m = 2…24; 161 |
| §28–32: isotropy lattice, orbits and orbit splitting. | definition | PASS | label only |  |
| §34: on an H-invariant seed a pure breaking mode has δA = 0. | theorem | PASS | numerical only | m = 12; 4×10⁻¹⁵; 1.5×10⁻⁴ |
| §36–37: δ²A on the Fourier modes is π(1 − m²)(a_m² + b_m²); the form ∫(φψ − φ′ψ′) is diagonal on 1, cos mθ, sin mθ. | theorem | PASS | numerical only | 2π; π(1 − m²); m ≤ 5 |
| §39: symmetry breaking is decided by the second variation and the constraints, and must be tested. | method | PASS | numerical only | 0.133; 0.044; 0.273 |
| The mirror-slice polish of Round 07's best placement q† (y3 = y5 = 0, φ5 = 2π/5) is a strict local minimum of the kernel's family area, lower than q†. | computation | PASS | numerical only | 9.03×10⁻¹¹; 0.8347809458213; 1,000; 10⁻⁷; 10⁻⁶; 10⁻⁵; 0.0033; 4×10⁻¹⁶ |
| Stabilizer of the polish and of q†, and the six supports at ±π/2. | computation | PASS | numerical only | ½; one ulp |
| §41–46: skew basis, mode compiler with six outputs, step rule and gauge. | specification | PASS | label only |  |
| The compiler scaffold implements §44's outputs (read, never run). | reading | PARTIAL | label only | 62 lines; 3 of 6 outputs |
| §54: extract the current seed's stabilizer and generate preserving and breaking modes from it. | method | PASS | label only |  |
| The schema and state records match the text (six outputs, mode classes, 16 completed items, four 'not claimed' items). | reading | PASS | label only | 16; four; three |
| Package integrity: paper, schema, scaffold and state digests. | reading | PASS | label only | four |

Open: That the polish is the kernel's global minimum: "It is a local polish of q† on the mirror slice, with 3,000 random directions around it. Its value is a floating-point evaluation of the verified hull, not an enclosure."; The constrained second-order theory (§40): "which the paper leaves to Paper 05."; §29's frontier relation in general: "which the paper states for regular equivariant models only."; Non-pure modes: "Contact-localized modes (§55) can be admissible where pure modes are not. The paper does not build them here, and this run does not either."

Unclear: W2's Gram-matrix check (∫(φψ − φ′ψ′) diagonal) does not state whether it was computed symbolically or numerically; strength recorded as numerical only.; The report gives no separate verdict word for the scaffold (W7); 'PARTIAL' for that item reflects the report's 'implements part of them' and '3 of 6 outputs' wording.; §15's table verdict reads 'PASS as its second sentence'; recorded here as NOT AS STATED for the formula as written, since only the second sentence passes.

### RUN-053 — LESR Paper 03 — Support-Function Variational Calculus (LESR-CALC-03-v0.1, 2026-09-20), with its schema, sanity-check record, compiler scaffold, state record and VALIDATION

Report: `reports/RUN-053-LESR-PAPER03.md`

> Verdict: PASS for everything Paper 03 derives. Its first-order model of a witness move is the affine one, and on the kernel that model is not the first-order behaviour:

**For the paper.** Paper 03's calculus (mixed-area form, variations, Steiner, step caps, contact variation) holds symbolically and on bodies, with §19's bound being the exact cap. On the kernel, where the container is the hull of its witnesses, the affine first-order model is wrong (the edge-average rule matches measured rates), so §39's LP and §61's finite-mode closure certificate are not sound there: the LP reports optimum 0 at a generic placement whose area gradient has length 0.390.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §4–9: area is the mixed-area form B(h, h), exactly quadratic along affine support paths, with first variation ∫φ dμ_h = ∫φ(h + h″) and second variation ∫(φ² − φ′²), independent of h. | theorem | PASS | exact-arithmetic certificate |  |
| §10–13: Steiner formula, constant direction gives the perimeter, translations and rotations are gauge, and the affine path along the rotation tangent is not a rotation. | theorem | PASS | exact-arithmetic certificate | 36πa²t² |
| Sanity-check record's eight numbers (circle of radius 2, translation mode, cos 2θ with cap 1/3). | computation | PASS | exact-arithmetic certificate | eight; 1/3 |
| Steiner A(K + sB) = A + P s + π s² and μ_h(S¹) = perimeter, on the kernel's hull at q† and on the Reuleaux triangle and pentagon. | computation | PASS | numerical only | 3×10⁻¹⁵; s = 0.01, 0.1, 0.5; 1.3×10⁻¹⁵ |
| Exact area along an admissible step (§50–52) on the path RT − t cos 3θ. | computation | PASS | numerical only | t ≤ 1/8; A₀ + 2t − 4πt²; 2×10⁻⁹; 4×10⁻¹⁶; 0.0759; 1/16; 0.0537; 1/8 |
| §14–15: the measure lane, ∫φ dμ_h = ∫h dν_φ. | theorem | PASS | exact-arithmetic certificate |  |
| §16–21: tangent cone and one-sided step caps: 0 when ν_φ⁻ has mass where μ_h has none, at least 1/‖q‖ when ν_φ⁻ = q μ_h; polygon warning. | theorem | PASS | analytic (proof checked) | 1/3; 1/8; 0 |
| Step caps on three examples: unit circle with cos 2θ; Reuleaux triangle with −cos 3θ; unit square with ±cos 2θ. | computation | PASS | exact-arithmetic certificate | 1/3; Lφ = 8 cos 3θ; q = 8\|cos 3θ\|; 1/8; 0 |
| §26–29: a fixed-placement witness contact varies by φ(θ_j) + ωh_W′(θ_j − ψ) − v·u_j. | theorem | PASS | numerical only | ψ = 0; t = (0.2, 0.1); six; 9×10⁻⁷; 0.16 |
| §30–32: first-order feasibility over a finite mode library is a finite LP; the feasible cone. | method | PARTIAL | label only |  |
| §33: strict first-order feasibility (local sufficient principle) with finitely many active contacts and strict margins. | theorem | PASS | label only |  |
| §37–39: cone stationarity and the LP descent test. | definition | PASS | label only |  |
| First-order model of a witness move on the kernel (affine model of §27–32) versus measured rates. | computation | FAIL | refuted | 2×10⁻⁷; 10⁻⁸; 0.66 to 0.94; three of them; 2.9 to 5.5; 0.133; 0.732 |
| Edge-average (trapezoid) rule for the first-order area change of a witness move on a hull-type container. | computation | PASS | numerical only | ℓ(r_a + r_b)/2; 2×10⁻⁷; 10⁻⁸; 1.1×10⁻⁹ |
| Measured one-sided rates at Z along y3, φ5, y5 versus the affine model. | computation | PASS | numerical only | 0.1327; 0.0442; 0.2733; 0.7320; 0.1274; 0.7950; 10⁻⁷; 10⁻⁸ |
| §39's LP at a generic placement near q† finds no descent (optimum 0 at ξ = 0) although the area's gradient is nonzero. | computation | FAIL | refuted | q† + (0.002, 0.001, 0.05, −0.001, 0.002); (−0.0791, 0.0330, 0.0425, 0.2089, 0.3149); 0.390; +0.107; 1.1×10⁻⁹ |
| §39's LP at q† finds no descent although the area falls toward Z. | computation | FAIL | refuted | 6.3×10⁻⁶; −6.6×10⁻⁶; 10⁻⁸; +0.020; 9.03×10⁻¹¹ |
| §60–62: the gate on UESFCM's outputs; an LP optimum ≥ 0 is a finite-mode first-order closure certificate. | certificate | FAIL | refuted | 0.39 |
| §40–41: constrained second order; breaking modes. | theorem | PASS | open |  |
| §42–47: shape velocity, integration with pathwise convexity and topology gates, hybrid dynamics. | definition | PASS | label only |  |
| §54–57: gradients need a metric; the L² gradient is h + h″ and the H¹ gradient solves (1 − ℓ²∂²)g = h + h″. | theorem | PASS | exact-arithmetic certificate |  |
| The schema and state records match the text (variation formulas, exact path, contact formula, five cones, 15 completed items, five 'not claimed' items). | reading | PASS | label only | 15; five |
| The compiler scaffold (read, never run): δ²A formula, contact-row signs, §51 line search. | reading | PARTIAL | label only | 61 lines |
| Package integrity: paper, schema, scaffold, sanity record and state digests. | reading | PASS | label only | five |

Open: A corrected first-order test (edge-average rule for hull-type containers): "It is not built or verified here."; Containers with isolated contacts: "where §33's hypotheses can hold and the affine model can be exact."; Whether the LP ever finds a descent on the kernel: "It found none at the three placements tried."; The continuum constraint and the metric-dependent flows: "which the paper leaves to Papers 04–05."

Unclear: The report's verdict word for §60–62 is 'NOT SOUND on the kernel'; mapped here to FAIL / refuted.; The report's §30–32 verdict is 'PASS for finitely many isolated contacts'; recorded as PARTIAL.; The open-items section says the LP found no descent at 'the three placements tried', while the W5 text names two LP runs explicitly (generic placement and q†); the third placement is not named.

### RUN-054 — LESR Paper 04 — Skew-Constrained Shape Flow and Closure Graph (LESR-FLOW-04-v0.1, 2026-09-20), with its flow and graph schemas, sanity-check record, compiler scaffold, state record and VALIDATION

Report: `reports/RUN-054-LESR-PAPER04.md`

> Verdict: PASS for everything Paper 04 derives. Its flow runs on Paper 03's affine cone, and on the kernel that cone holds no descent, so the flow never starts:

**For the paper.** Paper 04's metric layer and projection theorem hold, and for a cone the projected descent inequality is an exact equality (checked on 200 random cones to 1.8×10⁻¹³). On the kernel the projected velocity on Paper 03's affine cone is zero at the generic placement, q† and Z, so the flow never starts even though at the generic placement the area falls at 0.390 per unit along steepest descent.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §2–9: a metric turns the variation into a gradient (L² gives h + h″, H¹_ℓ damps mode m by 1/(1 + ℓ²m²), a preconditioner M gives M⁻¹(h + h″)); there is no unique natural flow. | theorem | PASS | exact-arithmetic certificate | (1 + ℓ²m²) |
| §10–12: the gradient is orthogonal to translations and rotations. | theorem | PASS | exact-arithmetic certificate |  |
| §13–14: the linearized feasible cone and the metric projection of a raw drive onto it. | definition | PASS | label only |  |
| §15–16: the projected descent inequality; a nonzero projected velocity V* is a first-order descent. | theorem | PASS | analytic (proof checked) | 200; 3–12; 2–20; 1.8×10⁻¹³; 126; 8×10⁻¹⁵; 60; 0.60 |
| §17: V* = 0 is local first-order projected stationarity, relative to the current linearization. | definition | PASS | label only |  |
| Paper 04's projected flow on Paper 03's affine cone, applied to the kernel at the generic placement, q† and the polish Z. | computation | FAIL | refuted | 0; 0.390; +0.107 |
| The kernel's area gradient at the generic placement (recomputed). | computation | PASS | numerical only | 0.390; +0.107 |
| §18–20: the finite-mode QP. | specification | PASS | label only |  |
| §21–26: raw drives; closure and expansion modes. | definition | PASS | label only |  |
| §27–37: four step caps, the exact step test, the line minimizer (gε + qε²/2 vanishes at ε = −2g/q, stationary at −g/q) and backtracking. | theorem | PASS | exact-arithmetic certificate | ε = −2g/q; −g/q |
| The sanity record's H¹ multipliers and line step. | computation | PASS | exact-arithmetic certificate | (1, 0.8, 0.5, 0.2); ε* = 0.4; A = 0.96 |
| §38–42: walls, transitions, hybrid flow. | definition | PASS | label only |  |
| §44–70: the typed closure graph (seven edge types, five proof statuses, dominator spine, counts, deferred compute, three views, SCC quotient). | definition | PASS | label only | seven; five; three |
| §71–72: Ψ is governance, not a theorem; componentwise closure. | definition | PASS | label only |  |
| The flow and graph schemas and the state record match the text. | reading | PARTIAL | label only | seven; five; three; four; six; 18 |
| The compiler scaffold (read, never run) implements the flow and the graph. | reading | PARTIAL | label only | 76 lines |
| Package integrity: paper, both schemas, scaffold, sanity record and state digests. | reading | PASS | label only | six |

Open: A flow that works on the kernel: "That flow is not built here."; The graph layer's use: "No closure graph of the actual Lebesgue work is built or checked."; The topology, corridor and witness caps on real steps: "which need the exact charts the paper defers."

Unclear: The FAIL / refuted marking on the kernel-flow item is this inventory's mapping of the report's 'the flow never starts' and 'V* = 0 at a non-critical placement'; the report's table verdict for §17 itself is 'PASS as labelled'.; The report says the LP and gradient 'are recomputed here' but relies on RUN-053's placements and models; the polish Z originates in RUN-052 via RUN-053.

### RUN-055 — LESR Paper 05 — Contact Pressure, Dual Multipliers, and the No-Slack Boundary Principle (LESR-DUAL-05-v0.1, 2026-09-20), with its dual schema, pressure-graph schema, sanity-check record, verifier scaffold, state record and VALIDATION

Report: `reports/RUN-055-LESR-PAPER05.md`

> Verdict: PASS for everything Paper 05 proves. §68 says the finite dual can be solved now on the Lebesgue line's own witnesses; solved, it certifies a placement that is not stationary:

**For the paper.** Paper 05's finite contact-pressure theorem is Farkas' alternative (one alternative in each of 300 random systems), and its smooth No-Slack arc theorem holds with the curvature hypothesis ρ ≥ κ₀ > 0 that Paper 00 §29 lacked. On the kernel the finite dual certificate exists at every placement solved (M = 4, 8), including a generic placement with gradient length 0.390, so the certificate does not mean stationarity there.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1: Minkowski's first variation. | theorem | PASS | cited theorem |  |
| §2–9: Paper 03's linearized contacts stack into a finite active-contact model A c + B ξ ≥ 0. | definition | PASS | label only |  |
| §10–11 finite contact-pressure theorem: with no feasible descent, Farkas gives λ ≥ 0 with g = Aᵀλ and Bᵀλ = 0. | theorem | PASS | cited theorem | 300; 8–40 rows; 3–12 modes; 3–9 placement columns; 213; 5×10⁻¹⁴; 87 |
| The sanity record's three contacts at 120° balance. | computation | PASS | numerical only | 120°; 3×10⁻¹⁶ |
| §13–17: translation and rotation balance of each witness's contact pressure; corners need subdifferentials. | theorem | PASS | analytic (proof checked) |  |
| §18–22: μ_h matches the pressure measure on the library's moments, not as measures; other active constraints. | reading | PASS | label only |  |
| §27–31: the continuum pressure measure (conditional). | theorem | PASS | open |  |
| §32–33 local No-Slack arc theorem: a smooth arc with ρ ≥ κ₀ > 0 and nothing active on it admits an inward cut that lowers the area. | theorem | PASS | numerical only | h = 1 + 0.05 cos 3θ + 0.02 sin 2θ; κ₀ = 0.548; (−0.4, 0.4); 0.0159; δA = −0.114; δ²A = −0.971 |
| The curvature hypothesis ρ ≥ κ₀ > 0 in §33 is needed (corner case). | computation | PASS | numerical only | ρ = 0; δA = 0 |
| §34–37: measure version of No-Slack; support inclusion; strict-interior corollary. | theorem | PASS | open |  |
| §41–44: continuous curvature needs continuum pressure; atoms need matching mass. | theorem | PASS | analytic (proof checked) |  |
| §47–49: certificate, verifier, Carathéodory sparsity (nonzero multipliers at most the number of equations). | theorem | PASS | numerical only | 14–23 |
| The finite dual certificate on the kernel's own witnesses (disc, B3, B5; Fourier modes up to M) at the generic placement, q† and the polish Z. | certificate | PASS | numerical only | π/(4M); M = 4; M = 8; 5×10⁻¹¹; 14–23 |
| The dual certificate on the kernel implies stationarity (§54 compiler's DUAL_CERTIFIED). | certificate | FAIL | refuted | 0.390 |
| §54: the No-Slack compiler. | specification | PASS | label only |  |
| §68: the finite dual on the Lebesgue line's own witnesses is a low-compute task now. | computation | PASS | numerical only |  |
| Finite-model sampling and gauge conditions on the kernel (contacts finer than modes; rotation gauge fixed). | computation | PASS | numerical only | M = 16; six contacts per segment; −0.107; −6.2×10⁻⁴; −2.2×10⁻⁴; 38 multipliers; 40 equations |
| The contact matrices' g₀ equals the perimeter of the verified hull. | computation | PASS | numerical only | 10⁻⁹ |
| The dual and pressure-graph schemas and the state record match the text. | reading | PASS | label only | three; four; seven; 14; four |
| The verifier scaffold (read, never run) verifies a supplied certificate. | reading | PARTIAL | label only | 72 lines |
| Package integrity: paper, both schemas, scaffold, sanity record and state digests. | reading | PASS | label only | six |

Open: A dual that means stationarity on the kernel: "That dual is not built here."; The continuum pressure measure: "which the paper leaves conditional."; Convexity multipliers: "the system here leaves them out, which only enlarges the primal."

Unclear: The §32–33 No-Slack arc theorem is described as '(proved)' in the paper; the report's W2 check is a numerical example on one smooth body, so the strength recorded is numerical only — the report does not state that it checked the proof line by line.; The FAIL / refuted marking on 'the dual certificate implies stationarity on the kernel' is this inventory's mapping of the report's 'certifies a placement that is not stationary'; the report's table verdicts for §54 and §68 are 'PASS as a specification' and 'PASS'.

### RUN-056 — LESR Paper 06 — Actual Lebesgue Shape State, Target-Specific Ledgers, and the First Closure Graph (LESR-LUC-06-v0.1, 2026-09-20), with its ledgers, four-contact dual template, closure graph (JSON, DOT, PNG, SVG), compute-deferred tasks, state record and VALIDATION

Report: `reports/RUN-056-LESR-PAPER06.md`

> Verdict: PASS for what Paper 06 proves and computes. Its accounting (value proof, barrier shape and global shape kept apart; proof closure not inherited by a shape target) is right. Three things are noted:

**For the paper.** Paper 06's target-specific accounting and the root-6217 barrier reproduce (θ* = 0.4540452162931835, hull area 0.834986110559 > A₃⁺, stabilizer exactly D₁), and four-contact dual stationarity at the barrier is not merely open but impossible once cos 2θ is in the library. The ledger's closure of the ρ₅ = 0 slice is not established (holds numerically by 2.05×10⁻⁴), and '61 open cells' is the formal ledger, with 44 % of root 6217 closed by search only.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §2–6: the value bracket is not a shape bracket; finite-witness forcing is existential, so no pointwise support corridor exists yet (UNBOUND). | reading | PASS | cited theorem |  |
| §7–9: three target-specific ledgers (value proof, root-6217 barrier shape, global extremal shape); proof closure is target-relative and REOPEN_FOR_SHAPE does not undo a proof. | method | PASS | label only |  |
| §10: A₃⁺ = 0.834781191 bounds Λ(F₃), not a_Leb. | reading | PASS | label only | 0.834781191; 2.45×10⁻⁷ |
| §11–17: the far decomposition and the root-6217 barrier: θ*, x₅*, the four-direction triple-support orbit, and area > A₃⁺. | computation | PASS | numerical only | x₃* = x₃⁰ − s; θ* = 0.4540452162931835; 80-digit; ½; 10⁻¹⁵; 0.834986110559 |
| The barrier's exact hull area exceeds A₃⁺. | computation | PASS | numerical only | 0.834986110559 |
| §18–20: the cusp; the rotation profile; sample monotonicity is not a theorem. | reading | PASS | label only |  |
| §21: the tight reference and its outer polygon. | computation | PASS | numerical only | 0.8349864346145534 |
| §22–24: 68 → 61 open cells; Type I and II closure; cell 131 as the canonical proof-closed and shape-reopen example. | reading | PASS | label only | 68; 61; 44 %; 131 |
| §25: Cycle 13 proved A > A₃⁺ on the ρ₅ = 0 slice (ledger: rho0_value_proof_status CLOSED_FOR_A3PLUS). | certificate | NOT AS STATED | numerical only | 1.3 %; 2.05×10⁻⁴ |
| §27–31: the barrier's stabilizer contains ⟨σ_x⟩; the orbit; the equilibrium lemma (equal multipliers on the orbit balance translation and rotation). | theorem | PASS | numerical only | 10⁻¹⁵; 200; 10⁻¹⁴ |
| The barrier's stabilizer (§27 claims 'at least' D₁). | computation | PASS | numerical only |  |
| §32: on the four-contact template, g = Aᵀλ (SHAPE_DUAL_STATIONARITY) is open. | certificate | NOT AS STATED | corrected | cos 2θ*; 2.05; 0.124; 1.6; 2.0; 3.0 |
| With every contact (RUN-055's system, M = 4) a dual certificate exists at the barrier. | certificate | PASS | numerical only | M = 4; 0.0331 |
| §43–47: the actual closure graph, with target ids on edges and spines. | specification | PASS | label only | 23 nodes; 24 edges; 20 edge types; twelve |
| The ledger's numbers are the text's (bracket, A₃⁺, 61 cells, A_ref, outer polygon, cell lists). | reading | PASS | label only | 0.8349864346145534; 28/36/68/144/155; 129/131; 125/152/150; 61 |
| The ledger's F₃ ceiling A₃⁺ = 0.834781191. | reading | PASS | label only | 0.834781191; 2.45×10⁻⁷ |
| §48: four compute-deferred tasks A–D. | specification | PASS | label only | four |
| The state record matches the text (six discoveries, four tasks, template statuses). | reading | PASS | label only | six; four |
| Package integrity: paper, ledger, template, graph JSON and DOT, tasks and state digests. | reading | PASS | label only | seven |

Open: Tasks A, C and D: "The shape-tree rebuild with A_ref, the import of an outer universal cover, and the near-anchor atlas are not attempted here."; A dual certificate that means stationarity at the barrier: "It would need the far-branch constraint d₃ ≥ s with its own multiplier, and edge-turning moves (RUN-053)."; The PNG and the SVG's pictures: "Only the SVG's node titles are read."; Value closure of the ρ₅ = 0 slice: "So the value closure of that slice is not established"

Unclear: The report's verdict for §25 is 'NOT ESTABLISHED' and for §32 'stronger: impossible on the four contacts'; both are recorded under the enum value NOT AS STATED.; The 'corrections' entry on the full-contact dual certificate is phrased by the report as a note ('does not mean stationarity either'), not a direct correction of a sentence of Paper 06; the source sentence it qualifies is not quoted in the report.

### RUN-057 — LESR Paper 07 — Root-6217 Barrier Shape Reconstruction (LESR-LUC-07-v0.1, 2026-09-20), with its certificate, verifier, compute-deferred list, signature graph, plots, state record and VALIDATION

Report: `reports/RUN-057-LESR-PAPER07.md`

> Verdict: PASS for the reconstruction; NOT ESTABLISHED for the translation KKT as argued (§15–16); the conclusion it draws holds numerically.

**For the paper.** Every printed constant of the root-6217 barrier (θ*, x₅*, α*, bridge lengths, b*, g₃, area to 65 digits) and the 13-interval signature reproduce from the contact equations. The translation KKT is not established as argued: the claimed subgradient (g₃, 0) fails, though numerically the barrier still minimizes F on x₃ ≤ x₃* (F′ ≥ 0.0354 on feasible directions) with multiplier η₃ ≈ 0.047 (measured, not certified), not 0.1228.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1–2: the contact equations give θ*, x₅* and a second disk–B5 crossing α*. | computation | PASS | numerical only | 70 digits; 60 places; (θ*, 3π/10) |
| §3–7: the signature has 13 open normal intervals, eight nonzero bridges (L_θ, L_α, L_vert), six triple-support lines with latent third contacts. | computation | PASS | numerical only | 13; 10⁻⁷; four L_θ; two L_α; two L_vert; four zeros; six |
| VALIDATION's 'minimum midpoint winner margin' of the signature. | computation | PASS | numerical only | 0.00526; 0.00060 |
| §8–9: the exact area from the analytic primitive; the surface-area measure. | computation | PASS | numerical only | 65 digits; 13 area contributions; 60 places |
| §10–12: the surface-area measure splits into nonnegative witness pressures; B5 balances in translation and rotation. | computation | PASS | numerical only | b* = F5x/(2 cos θ*) |
| §13–14: B3's horizontal subgradient component g₃ = 2(sin θ* − ½) < 0, exactly. | computation | PASS | numerical only |  |
| §15: pressure splits give a convex subgradient (g₃, 0) of the area F(x₃, x₅). | theorem | FAIL | refuted | 0.265 to 0.451; 72 directions; 0.689; 0.403; cos 202°, sin 202°; 0.044; 0.114; −7×10⁻⁸; −6.9×10⁻⁶; −2.5×10⁻⁴; t = 10⁻⁶, 10⁻⁴ and 10⁻² |
| §15's premise: the area's directional derivative is the integral against μ_h of the largest active rate at each normal. | reduction | FAIL | refuted | 0.265 to 0.451; 72; 0.689; 0.403 |
| §16: global KKT on the ρ₅ = 0 translation slice — by convexity the barrier minimizes the ρ₅ = 0 translation problem, with far-branch multiplier η₃ = −g₃. | theorem | PARTIAL | numerical only | 0.0354; 720 directions; −0.0477; −0.0474; η₃ ≈ 0.047; 39 %; 0.1228; 0.0331 |
| The far-branch constraint's multiplier η₃. | computation | PASS | numerical only | 720; −0.0477; −0.0474; 0.047; 39 %; 0.1228 |
| §17: the barrier minimizes all translations at ρ₅ = 0 in the far branch. | theorem | NOT AS STATED | numerical only | 1.3 %; 0.8349861106; α = π |
| §18–19: the positive-rotation gate m(ρ) ≥ m(0) remains open. | reading | PASS | open |  |
| The verifier (read, never run) checks the signature and the KKT. | reading | PARTIAL | label only | 152 lines; 13 midpoints |
| The certificate's and state's KKT statuses, the state's facts and tasks, and the signature DOT. | reading | PASS | label only | nine; three; 13 |
| Package integrity: paper, certificate, verifier, both PNGs, state and tasks digests. | reading | PASS | label only | seven |

Open: An interval proof of the signature (Task P07_INTERVAL_SIGNATURE_AUDIT): "The verified hull's closed-form segment checks are floating-point."; A certified multiplier: "η₃ ≈ 0.047 comes from 720 measured directions." An exact subgradient "is not built here."; The positive-rotation gate: "That is Paper 08."; §17 by its own argument: "NOT ESTABLISHED by its argument"

Unclear: The report's §16 verdict is 'conclusion holds, proof does not'; recorded as PARTIAL with strength numerical only, while the minimization conclusion also uses F's convexity cited as Fáry–Rédei.; The report's §17 verdict is 'NOT ESTABLISHED by its argument'; recorded under the enum value NOT AS STATED.; The constants are reproduced with mpmath at 70 digits; the report does not call this an interval or exact-arithmetic certificate, so strength is recorded as numerical only.

### RUN-058 — LESR Paper 08 — Positive-Rotation Certificate Reduction (LESR-LUC-08-v0.1, 2026-09-20), with its 195-root ledger, diagnostic, certificate schema, verifier, state record and VALIDATION

Report: `reports/RUN-058-LESR-PAPER08.md`

> Verdict: PASS for the reduction, its arithmetic and its partition. Two things are not as the paper states them:

**For the paper.** Paper 08's two-gate reduction, its arithmetic (A_ref − m(0) = 3.8944077590×10⁻⁷, crossing radius 0.00197342538724, overlap 5.1055922410×10⁻⁷) and its 10 / 141 / 44 split reproduce; the gates themselves are open and hold only at 26 profile samples (true local ratio ≈ 0.225). First-order rotational criticality is a floating-point observation on the true hull, not derived from Paper 07's KKT, and the 195-root shape ledger omits 14.6 % of root 6217's box closed for A₃⁺ only.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1–2: ρ = 0 is a topology wall; the mirror makes the profile m even, but evenness alone allows a cusp maximum. | reading | PASS | label only |  |
| §3: the rotation is first-order critical — in the finite active-contact variational model inf{δA[d] : d ∈ K_feas, ρ̇ = 1} = 0 — derived from Paper 07's pressure/KKT structure. | theorem | NOT AS STATED | numerical only | +0.0293; (−0.022, −0.1625); 0.0474; 5.3×10⁻⁵; ρ = 0.001; 1.1×10⁻⁴; 0.002 |
| First-order rate of rotation on the true hull along the re-minimized profile's compensations. | computation | PASS | numerical only | 5.3×10⁻⁵; ρ = 0.001; 1.1×10⁻⁴; 0.002 |
| §4–6: the local question is second order; profile ratios (m(ρ) − m(0))/ρ² of 0.224–0.246; the target constant 0.1. | computation | PASS | numerical only | 0.22406; 0.24571; 7.3×10⁻¹¹; 0.2254; 0.2253; 8.3×10⁻⁸; 0.2457; 0.225; m″(0⁺) ≈ 0.45; 2.25; 0.1738; 0.2254 |
| §7–11: arithmetic from m(0) — the gap A_ref − m(0), the crossing radius, the overlap margin at ρ = 0.003 — and the composition of the two gates to m(ρ) ≥ m(0). | computation | PASS | numerical only | 65 digits; 3.8944077590×10⁻⁷; 0.00197342538724; 5.1055922410×10⁻⁷; κ₀ ≥ 0.2 ⇒ ½κ₀ ≥ 0.1 |
| The two gates: LOCAL-QUADRATIC m(ρ) ≥ m(0) + 0.1ρ² on [0, 0.003] and AREF-TAIL m(ρ) ≥ 0.8349865 on [0.003, π/120] (OPEN in the paper). | reduction | PARTIAL | numerical only | 0.1ρ²; [0, 0.003]; 0.8349865; [0.003, π/120]; 26 |
| The diagnostic and its replay. | computation | PASS | numerical only |  |
| §12–14: the barrier-shape ledger restarts from Cycle 11's 200 subroots of root 6217, 5 pruned by A_ref, 195 active. | reduction | PARTIAL | corrected | 200; 195; 55.9 %; 29.5 %; 14.6 %; 3.3 %; 11.3 %; 400,000 |
| §15–17: the 195 roots split 10 local-only / 141 tail-only / 44 crossing at ρ = 0.003. | computation | PASS | numerical only | 10; 141; 44 |
| The ledger file lists all 200 subroots with their boxes, five marked SHAPE_PRUNED_BY_AREF. | reading | PASS | label only | 200; five |
| §18–24: chart family; next nodes; counter-search. | specification | PASS | open |  |
| The records (ledger, diagnostic, schema, state) and their status labels. | reading | PASS | label only |  |
| The verifier (read, never run) rebuilds the ledger and checks the arithmetic. | reading | PARTIAL | label only | 86 lines; 10⁻¹⁵ |
| Package integrity: paper, ledger, diagnostic, schema, verifier and state digests. | reading | PASS | label only | six |

Open: Either gate: "They are OPEN in the paper, and they hold only at the 26 profile samples here."; Which part of the 14.6 % Cycle 11 closed by bounds that would also clear A_ref: "Its trees are not shipped."; An exact first-order statement: "The criticality is a floating-point observation along two compensations."

Unclear: The report's §3 verdict is 'holds, not derived' and §12–14 'incomplete'; these are recorded as NOT AS STATED and PARTIAL respectively.; §7–11 arithmetic is computed 'from m(0) to 65 digits' while the doubles agree 'to their precision'; the report does not say whether this is interval or exact arithmetic, so 'analytic (proof checked)' for the arithmetic/composition is this inventory's reading of 'are arithmetic'.

### RUN-059 — LESR Paper 09A v0.2 — Multi-Scale Topology-Wall Correction and the One-Sided Blow-Up Critical Cone (LESR-LUC-09A-v0.2, 2026-09-20), with its diagnostic, certificate schema, state record and VALIDATION

Report: `reports/RUN-059-LESR-PAPER09A.md`

> Verdict: PASS for the withdrawals, the blow-up chart, the 18-piece count, both primary arcs, the numerical centre and ratio, and the box. One thing is not as the paper suggests: at the minimum, the three secondary slivers have zero width.

**For the paper.** Paper 09A's blow-up chart, the 18-piece signature at the stored sample, both primary O(ρ) arcs, the branch velocity limit (5.157, −0.1892, 0) and the growth ratio ≈ 0.2256 reproduce in floating point; the three secondary slivers have zero width (ties) at properly minimized placements, giving 15 pieces at the minimum. Numerically Φ₁ ≥ 0 on a 280-point grid with a single zero at (5.1572, −0.18923, 0) and linear rises in every direction; none of the four certificates is established.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §0–3: ρ = 0 is a wall; the fixed-chart rotation derivative from Paper 07's pressure balance, and one smooth chart for all small ρ > 0, are withdrawn. | reading | PASS | numerical only | +0.029 |
| §4: a one-sided blow-up chart: B3 on d₃ = s at α₃ = π − ρu, y₅ = ρv, x₅ = x₅* + ρw. | reduction | PASS | numerical only | 26; 2.4×10⁻⁸ |
| The re-minimized branch's (u, v, w) at four ρ and its limit as ρ → 0. | computation | PASS | numerical only | ρ = 0.002; ρ₀ = 0.0010069…; 0.0005; 0.00025; (5.2312, −0.18896, −0.00035); (5.1944, −0.18909, −0.00018); (5.1756, −0.18916, −0.00009); (5.1664, −0.18919, −0.00004); (5.157, −0.1892, 0); (5.157126…, −0.189228…, 0); (5.1967, −0.1891, −0.0002) |
| Minimization method note: a single Nelder–Mead simplex stalls in the kinked minimum. | method | PASS | numerical only | 0.232; 0.246; 0.2255 |
| §5: at Cycle 12's first profile sample, 18 merged open winner intervals against the barrier's 13. | computation | PASS | numerical only | 12; 13; 18; 2×10⁻⁸ |
| §6: two primary newborn intervals of width about 0.1480ρ and 1.1480ρ. | computation | PASS | numerical only | 0.14804ρ; 1.14804ρ; 0.14903ρ; 0.14797ρ; 0.14741ρ; 0.14714ρ; 0.1469ρ; 0.146893 |
| B5's primary arc is exactly ρ wider than B3's at every ρ. | computation | PASS | numerical only | ρ |
| §7: three thinner secondary slivers, whose O(ρ²) scale is a working hypothesis. | computation | PARTIAL | corrected | 8.5×10⁻⁹; 3.7×10⁻⁷; 7.0×10⁻⁷; 6×10⁻⁹; 10⁻¹¹; 1.5×10⁻⁹ |
| Piece count of the signature at the minimum. | computation | PASS | numerical only | 15; 18 |
| §8–11: the nested blow-up; Φ₁, C₁ = argmin Φ₁, Σ₂ and c* = inf inf Φ₂. | definition | PASS | label only |  |
| The first epi-derivative Φ₁ is non-negative near the box, with C₁ = argmin Φ₁. | computation | PASS | numerical only | ρ = 10⁻⁶; 280; 8.6×10⁻⁶; (5.16, −0.1892, 0); 5.4×10⁻⁴ |
| C₁ (the zero set of Φ₁) is a single point with linear rises in every direction. | computation | PASS | numerical only | (5.1572, −0.18923, 0); 1.3×10⁻³; 1.2×10⁻³; 0.26; 0.40; 0.036 |
| Extrapolated Φ₁ value at the minimizer and the ray check. | computation | PASS | numerical only | −1.2×10⁻⁷; 3×10⁻⁸; 10⁻³; 3×10⁻⁷; 10⁻⁷ |
| §12–13: numerical centre u ≈ 5.16, v ≈ −0.1892, w ≈ 0; growth ratio ≈ 0.22. | computation | PASS | numerical only | 0.225236; 0.225413; 0.225501; 0.225544; 0.2256; 2.26; 0.1 |
| §14–18: four certificates (primary chart, sliver classification, Φ₁ ≥ 0, Φ₂ ≥ 0.1), the box u ∈ [4.5, 6.0], v ∈ [−0.23, −0.15], w ∈ [−0.01, 0.01], and the target. | specification | PASS | open | [4.5, 6.0]; [−0.23, −0.15]; [−0.01, 0.01]; 0.1 |
| §19: the tail gate and the 200 → 195 ledger unchanged. | reading | PASS | label only | 200; 195; 14.6 % |
| The diagnostic, schema and state records. | reading | PASS | label only | (5.16, −0.1892, 0, 0.22, 0.1); four |
| Package integrity: paper, diagnostic, schema and state digests. | reading | PASS | label only | four |

Open: Any of the four certificates: "Everything here is in floating point: the exact hull in double precision, Nelder–Mead, Φ₁ from ρ ≥ 10⁻⁶."; Φ₁ ≥ 0 off the grid: "or below the resolution of about 10⁻⁷."; That the secondary slivers are absent for every small ρ: "They are ties at the four ρ ≥ 2.5×10⁻⁴ examined here."

Unclear: The report cites 'Paper 09B's closed form (5.157126…, −0.189228…, 0)' and 'Paper 09B's \|τ_T\| is 0.146893' without naming the run that verifies Paper 09B.; The §7 verdict is PARTIAL in the table; strength 'corrected' reflects that the class is absent at the minimum while the paper made no scaling claim ('§7 was right not to claim a scaling').

### RUN-060 — LESR Paper 09B — Analytic Primary Blow-Up Equations (LESR-LUC-09B-v0.1, 2026-09-21), with its result record, certificate schema, verifier, state record and VALIDATION

Report: `reports/RUN-060-LESR-PAPER09B.md`

> Verdict: PASS for the wall constants, the collapse equations and their solution, w* = 0, the transition speeds, the primary 15-piece structure and Φ₁^primary = γ_w·w. Two things are noted: how thin the full Φ₁'s margin is along −w, and where the second-order slivers actually open.

**For the paper.** Paper 09B's six wall constants, the full-rank collapse system with w* = 0 (algebra conditional on the open CERT-09B.4), u* = 5.15712624526841370759… (printed value 5×10⁻¹⁴ off), the transition speeds and Φ₁^primary = γ_w·w with γ_w = 0.18389963923727719444… reproduce in high-precision floating point, not interval enclosures. The full Φ₁ is non-negative along −w only by a margin of 0.0354 per unit, and the second-order slivers open at π/2, θ*, 2π − θ*, not at π ∓ θ* as §18 lists.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1–3: at a triple wall the three first-order support lines must meet at one transition speed, or a third-body sliver opens at O(ρ): a₃b₅ − a₅b₃ = 0. | theorem | PASS | numerical only |  |
| §4–6: the top, left and right walls give E_T, E_L, E_R with six constants. | computation | PASS | numerical only | 10⁻¹⁷ to 2×10⁻¹⁵; β₃T = 0.30965396669380288225; β₅T = −0.1822307944644007122; γ_T = 0.16245984811645316307; L_θ = 0.24402699852741554812; L_α = 0.09982945996422126522; γ_L = 0.09115783454691888673 |
| The three collapse equations are linear in (u, v, w) and of full rank. | computation | PASS | numerical only | −1.28×10⁻⁴ |
| §7: E_R − E_L forces w* = 0. | theorem | PASS | analytic (proof checked) | 2L_θ cos θ*·w; 2×10⁻⁷ |
| §8–9: closed forms give u* = 5.157126245268359, v* = −0.1892281899066407, in agreement with Paper 09A. | computation | PASS | numerical only | u* = 5.15712624526841370759…; v* = −0.1892281899066409745…; 70 digits; 5.157126245268359; 5×10⁻¹⁴; 3×10⁻¹⁶; (5.1572, −0.18923, 0); (5.157, −0.1892, 0) |
| §10: the transition speeds τ_T = τ_bottom, τ_L = τ_R and τ_α. | computation | PASS | numerical only | τ_T = −0.14689252641883796; τ_L = τ_R = 0.081754472733397221; τ_α = 2.5703350860764422; ρ = 10⁻⁴; 10⁻⁵; −0.1469; 0.0817–0.0818; 2.5701–2.5703 |
| The bottom transition speed equals the top exactly. | computation | PASS | analytic (proof checked) |  |
| §11: a primary 15-piece structure with the two O(ρ) arcs. | computation | PASS | numerical only | 15; 18; 1.79ρ²; 2.72ρ² |
| The forced 15-piece chain has at ρ = 0 the barrier's area m(0). | computation | PASS | numerical only |  |
| §12–14: on the primary chart Φ₁^primary = γ_w·w with γ_w = 0.18389963923727759, the u and v terms cancelling identically. | computation | PASS | numerical only | 5×10⁻⁴⁰; γ_w = 0.18389963923727719444…; 2×10⁻³⁹; (5.6×10⁻¹⁷, 8.7×10⁻¹⁹, −4.4×10⁻¹⁶, 0.1838996392372776); 90 digits |
| §15–16: penalties off the collapse manifold; Φ₁ = 0 at the topology-selected velocity and C₁ is one point. | computation | PASS | numerical only | 14; 0.0012; 0.2619; 0.4032; 0.0354; 0.2193; 0.1839 |
| §17–18: the secondary slivers to classify (B5's corner slivers at π ∓ θ*). | specification | PARTIAL | corrected | π/2; θ*; 2π − θ*; 1.79ρ²; 2.72ρ² |
| §19–22: four interval obligations remain; full multi-chart Φ₁ ≥ 0, the second epi-derivative and the local gate are open; Paper 09C next. | reading | PASS | open | four |
| The result record, schema and state records. | reading | PASS | label only | four |
| The verifier (read after the rebuild, never run) verifies the result. | reading | PARTIAL | label only | 55 lines |
| Package integrity: paper, result record, verifier, schema and state digests. | reading | PASS | label only | five |

Open: Any interval obligation: "The constants, the solution and Φ₁^primary are high-precision floating point, not interval enclosures."; The full Φ₁ ≥ 0: "the neighbouring charts are not enumerated."; The secondary scale: "Paper 09C is not in the archive; the slivers here are measured, not derived."; CERT-09B.4 (branch keeps all three collapses at first order): "That is CERT-09B.4, which is open."; The −w margin to be certified: "That margin is what CERT-09B.4's "positive first-order penalty" has to certify."

Unclear: The report calls the u/v cancellation 'exactly (W5)' and computes it at 90 digits, but its 'does not establish' section says Φ₁^primary is 'high-precision floating point, not interval enclosures'; strength recorded as numerical only.; The paper's printed γ_w as summarized in 'What the paper claims' is 0.18389963923727759, while the rebuilt value is 0.18389963923727719444… and the record's binary64 value 0.1838996392372776; the report does not comment on the paper's printed digits.; §7's table verdict reads 'PASS \| exact (W2); conditional on CERT-09B.4'; strength recorded as analytic (proof checked) for the algebra only.

### RUN-061 — Lebesgue / LESR / UESFCM 中繼交接文件 — Certificate 主線 × Counter-Certificate 本地端分工 (LESR-LUC-MID-HANDOFF-v0.1, 2026-09-20), with its VALIDATION record; a work contract, not a theorem

Report: `reports/RUN-061-LESR-MIDPOINT-HANDOFF.md`

> Verdict: PASS for the barrier state, b* and g₃, the three ledgers and target-specific closure, cell 131, the artifact index and the package. Four things are not as the document states them, and the counter-certificate line it assigns has, in this tree, a first answer.

**For the paper.** The handoff's recorded numbers (barrier state to 34 decimals, area to 37, b*, g₃), its three-ledger target-specific closure and cell 131's REOPEN_FOR_SHAPE are confirmed; its listing of B3's subgradient as done is wrong, and 'evenness ⇒ m′(0) = 0' is an invalid inference although m′(0⁺) = 0 holds numerically. Run in this tree, the counter-certificate line finds no lower branch at the sampled ρ, finds a new contact topology (13 / 18 / 15 pieces), and finds the finite dual non-discriminating; no certificate is established and the public bracket is cited, not verified.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §0–2: two lines with one owner each (certificate/analytic closure in the cloud; counter-certificates and heavy computation locally); a replayed counter-candidate stops the proof effort on the old claim. | specification | UNMEASURED | not measured |  |
| §3–4: three ledgers stay apart (global value, root-6217 barrier shape, global extremal shape); proof closure does not imply shape irrelevance, as cell 131 shows. | method | PASS | label only | 0.8349865 |
| The global value bracket 0.8344 ≤ a_Leb ≤ 0.8440935944. | theorem | UNMEASURED | cited theorem | 0.8344; 0.8440935944 |
| A₃⁺ = 0.834781191 is the F₃ ceiling. | reading | PASS | label only | 0.834781191 |
| §5: the canonical barrier state (x₃*, θ*, x₅*, α*, area). | computation | PASS | numerical only | x₃⁰ − s; 34 printed decimals; 37 |
| §6 A–H: Paper 07 completed the signature, bridges, triple lines, b*, and B5's force and rotation balances. | reading | PASS | numerical only | 8; four L_θ; two L_α; two L_vert; 6 |
| §6 I: Paper 07 completed 'B3 constrained subgradient g₃ < 0, balanced by η₃ = −g₃'. | theorem | FAIL | refuted | 0.0663; 72; [−0.0477, −0.0474] |
| §6 J: the zero-rotation translation minimizer. | computation | PASS | numerical only | 0.0354 |
| §7: CERT-07A and CERT-07B remain open. | reading | PASS | open |  |
| §8: Paper 08's claim m₆₂₁₇(ρ) ≥ m₆₂₁₇(0) on [0, π/120]. | theorem | PARTIAL | open | [0, π/120] |
| §9 A: evenness of m implies m′(0) = 0. | reduction | FAIL | refuted | five; 10⁻¹⁵; 0.2256 |
| m′(0⁺) = 0 on the positive-rotation branch. | computation | PASS | numerical only | 0.2256 |
| §9 F: the Cycle 14 transition at ρ ≈ 0.01409689011226. | computation | PARTIAL | corrected | 0.01409689011226; sample 14; samples 8 and 9; ρ 0.00806 to 0.00906; [0.003, π/120] |
| CC-08A: search for a lower positive-rotation branch (counter-candidate). | computation | PASS | numerical only | 0.1738; 0.2254; 26; ρ ≤ 0.002; 0.2252; 0.2255 |
| CC-08B: a new contact topology on the positive branch. | computation | PASS | numerical only | 13; 18; 15; θ*; π/2; 2π − θ* |
| CC-08C: dual infeasibility as a counter-certificate. | computation | PASS | numerical only | 0.39 |
| HC-03: the barrier-shape tree rebuild. | reading | PASS | numerical only | 195; 14.6 % |
| HC-01 and HC-02: the interval audit and the pressure rationalization. | specification | UNMEASURED | not measured |  |
| §13–29: tasks, formats, grades, roadmap, artifact index, checklists. | specification | PASS | label only | 12 |
| Package: VALIDATION digest and next-node title. | reading | PARTIAL | label only |  |

Open: Any certificate: "CERT-07A and CERT-07B and the interval tasks are not done here."; That no lower branch exists: "it is not a covering of [0, π/120]."; The bracket: "Mishra's and Gibbs' bounds are cited, not verified."; §8 m(ρ) ≥ m(0) on [0, π/120]: "open \| holds at every sample (W4)"; HC-01 and HC-02: "are not done here"

Unclear: The report's verdict for §0–2 is 'n/a' and for §10–12 'answered here'; these have no exact enum value and are recorded as UNMEASURED and PASS (per CC answer) respectively.; The report's §8 verdict is 'open' with the note 'holds at every sample'; recorded as PARTIAL / open.; The report says the CC-08A search covers 'the profile's 26 samples and four small ρ, re-minimized'; it is not a covering, so 'none found' is a numerical search result only.

### RUN-062 — 雙向圓核重疊分解法：從切分、分離到資訊重組 — Bidirectional Circular Overlap Decomposition and Recomposition (BCODR-v0.1, 2026-09-21), one of the 42 canonical documents, with its VALIDATION record

Report: `reports/RUN-062-BCODR.md`

> Verdict: PASS. BCODR is a methodology: it computes nothing, and it says so (§78). Its one status phrase in a judgment position is a label. Its mathematical statements hold where they can be instantiated, and its links to UESFCM and to the certificate grades are consistent. One thing is noted: the root-6217 map it proposes as the next step starts from an incomplete ledger.

**For the paper.** BCODR is a methodology that computes nothing; where its mathematics can be instantiated it holds: D_μ and D_J are metrics on discs (20,000 random triples), and the arrangement's cell count is O(N²), tight at N(N − 1) + 2 when every pair crosses. Its proposed root-6217 map would start from the 195-root ledger that omits 14.6 % of root 6217's box.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §0–9: a closure architecture of outward circles from inner seeds and inward circles from outer constraints, overlapped, with overlap kept as information. | definition | UNMEASURED | label only |  |
| §10–11: minimum distinguishability by D_μ(A, B) = μ(A △ B) or the normalized D_J(A, B) = μ(A △ B)/μ(A ∪ B). | theorem | PASS | numerical only | 20,000; 10⁻¹²; [0, 1]; 3; 9 |
| §11: normalizing is needed because D_μ scales with size while D_J is scale-free. | computation | PASS | numerical only | 3; 9 |
| §12–17: multiplicity m, balance b = m₊ − m₋, novelty, certificate strength and a default information field. | definition | UNMEASURED | label only |  |
| §18–22: the arrangement cuts the domain into atomic cells of constant membership. | theorem | PASS | numerical only | F = V + 1 + C; 36; 3 to 5; 200,000; 22; 8 |
| §59: the number of arrangement cells is O(N²). | theorem | PASS | analytic (proof checked) | N(N − 1) + 2; 134; N = 12 |
| §79 B: does the arrangement capture all membership transitions? | theorem | PASS | analytic (proof checked) |  |
| §23–38: separation, recomposition under provenance-keeping rules, graph, surviving spine; conflict cells first. | definition | UNMEASURED | label only |  |
| §38: CONFLICT CELL, the document's one status phrase in a judgment position. | definition | PASS | label only |  |
| §16: six certificate weights (0.1, 0.3, 0.5, 0.7, 0.9, 1.0). | specification | PASS | label only | 0.1, 0.3, 0.5, 0.7, 0.9, 1.0; five |
| §39–46: the kernel-deformation gate; relations to topology, groups, the skew field, UESFCM and LESR (§45 maps UESFCM's Expand, LinkBack, FiniteClose, Recenter). | reading | PASS | label only |  |
| §47–50, §84: a root-6217 parameter-space map as the next step, starting from '195 shape-active parent roots' and a 'positive-rotation candidate branch'. | specification | PARTIAL | corrected | 195; 14.6 % |
| §51–58 and §61–77: colours, stop conditions, schedules, thresholds, data model, algorithm, certificate outputs, principles. | specification | UNMEASURED | label only |  |
| §78–83: the honest boundary (BCODR is not a convergence theorem); open questions; extensions. | reading | PASS | open |  |
| Package: VALIDATION digest and fields. | reading | PASS | label only |  |

Open: That BCODR works: "It is an architecture; its case studies are RUN-063 onward."; §79's open questions apart from B: "§79's other questions (overlap connectivity, packing, information monotonicity, spine convergence) stay open."; A convergence theorem: "BCODR is not a convergence theorem, and nothing here tests one."

Unclear: Many sections are given verdict 'n/a' (definitions, rules, labels); these are recorded as UNMEASURED / label only since the enum has no n/a value.; The report notes its own first grid-count attempt over-counted (10 cells for 3 circles, where 8 is the maximum); this concerns the verification method, not a claim of BCODR, and is not listed as an item.

### RUN-063 — BCODR-LUC-01 — Root-6217 參數空間的雙向圓核閉包 / Bidirectional Kernel Closure of the Root-6217 Shape Space (BCODR-LUC-01-v0.1, 2026-09-21), with its arrival information ledger (JSON and CSV), kernel-metric schema, frontier graph (DOT and PNG), verifier, state record and VALIDATION

Report: `reports/RUN-063-BCODR-LUC01.md`

> Verdict: PASS for the domain, every arrival time, balance, entropy and score, the nexus, the finite-radius cells, the most balanced eight, and the records. Not as stated: the active/prune boundary contacts are 64 pairs, not 35. Noted: the frontier is set mostly by where the ray is cut.

**For the paper.** LUC 01's arrival-time field over Paper 08's 195 cells rebuilds cell for cell (T₋ to 3×10⁻¹⁶, T₊ to 2×10⁻⁹), with cell 131 the unique zero-radius nexus holding the whole critical ray and the most balanced eight as stated; on the exact dyadic grid the active/prune boxes touch in 64 pairs, not 35. The frontier is largely set by the ray's cut at ρ = 0.003 (the nearest ray point for 178 of 195 cells), and no area is computed.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1–3: the 200 subroots fix a normalized 5-D space; the first kernel is the Euclidean ball there. | definition | PASS | numerical only |  |
| §4–13: outward field from Paper 09B's critical ray (0 ≤ ρ ≤ 0.003), inward field from the five A_ref-pruned boxes; arrival times T₊ and T₋, meeting radius, balance and balance entropy. | computation | PASS | numerical only | 3×10⁻¹⁶; 195; 2×10⁻⁹; Nine; 10⁻¹² |
| Nearest prune roots recorded in the ledger. | computation | PASS | numerical only | Nine; two |
| §31: the scheduling score uses median(T∩). | computation | PASS | corrected | 140; 0.766444; 0.696945 |
| §14: no positive-measure active/prune overlap. | computation | PASS | exact-arithmetic certificate | 2²⁰ |
| §15: there are 35 active/prune boundary-contact pairs. | computation | FAIL | refuted | 35; 64; 40; 14; 8; 2; 55; 54; 30; 10⁻¹⁸ |
| §16: contact is not overlap. | definition | PASS | label only |  |
| §17–19: root 131 is a zero-radius contact nexus, not a prune. | computation | PASS | numerical only | 131; 0 ≤ ρ ≤ 0.003 |
| §20–24: the most balanced finite-radius cells are 166, 167, 164, 165, 127, 101, 104 and 103. | computation | PASS | numerical only | 140; 166, 167, 164, 165, 127, 101, 104 and 103 |
| Dependence of the finite-radius frontier on the ray's cut at ρ = 0.003. | computation | PASS | numerical only | 178; 195; ρ = 0.003; 0.001; 0.002; 0.004; 160; 168; 103; 104; 0.006; 100 |
| §25: the 200 boxes are measure-atomic cells. | computation | PASS | exact-arithmetic certificate | 200; 2²⁰ |
| §26–34: cuts on kernel events, merges under rules, the scheduling score, colours, Φ₂ only on the surviving responsibility. | specification | PASS | label only |  |
| §35: 195 = 10 + 44 + 141. | computation | PASS | numerical only | 195; 10; 44; 141; 55.9 %; 14.6 % |
| §36–43: the first graph; what it does not claim; LUC 02 next. | reading | PASS | label only |  |
| The state, CSV, schema and graph records. | reading | PASS | label only | 195 |
| The verifier (read after the rebuild, never run). | reading | PARTIAL | label only | 32 lines; 35 |
| Package integrity: paper, ledger, CSV, schema, PNG, verifier and state digests. | reading | PASS | label only |  |

Open: Anything about areas: "The field is geometric bookkeeping in a parameter space; no cell's area is computed here."; That the normalized Euclidean metric means anything for shapes: "The document does not claim it does (§3, §38)."; The 14.6 % of root 6217's box closed for A₃⁺ only: "has no cell in this field (RUN-058)."

Unclear: The overlap/contact checks are done 'on the exact grid of root 6217's box (2²⁰ units a side, every box edge an integer)'; recorded as exact-arithmetic certificate for §14 and §25, though the report does not use the word 'certificate'.; §31's item: the report says the score 'follows its formula' with the finite cells' median; whether this is a correction of the text or of the reading of median(T∩) is recorded as 'corrected' but the report's table verdict for §26–34 is 'PASS / n/a'.

### RUN-064 — BCODR-LUC-02 — Atomic Cut and Structural Recomposition (BCODR-LUC-02-v0.1, 2026-09-21), with its atomic cut ledger, children CSV, structural recomposition, overlap persistence, spine graph (DOT and PNG), verifier, state record and VALIDATION

Report: `reports/RUN-064-BCODR-LUC02.md`

> Verdict: PASS for the cut, all 288 signatures, the 17 classes and the seven BOTH classes, the second cut's counts, and the records. Not as stated: the provenance flip at 127, and 127 as a thick, provenance-switching corridor. Noted: at its parent's radius a BOTH child is one that holds a point where the later field first touches the parent; the 17 classes need corner contacts and the parent's ρ class.

**For the paper.** LUC 02's 288 children, their signatures (187 / 88 / 4 / 9) and the 17 recomposed classes rebuild exactly, the 17 requiring corner adjacency and the parent's ρ class. 127's provenance 'flip' is a tie (1/8 from both 144 and 155), and a BOTH child is simply one holding the point or face where the later-arriving field first touches its parent, so 127's 'thick corridor' is a face against a point; no area is computed.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1–3: the nine frontier parents of LUC 01 are cut along all five dimensions into 288 children (2⁵ each), with t_B from LUC 01. | computation | PASS | exact-arithmetic certificate | nine; 288; 2²² |
| §4–6: each child is signed by the fields that reach it at its parent's meeting radius; the children split 187 / 88 / 4 / 9, so 9/288 = 3.125 % is the first two-way interaction region. | computation | PASS | numerical only | 187; 88; 4; 9; 9/288; 3.125 %; 10⁻⁹; 4×10⁻¹⁶; 10⁻⁵ |
| Characterization of a BOTH child at its parent's radius (report's analysis of §4–6, §21). | computation | PASS | analytic (proof checked) | seven; 12; 1/8; 19 |
| §7–9, §31: root 131's children are 2 / 8 / 22 / 0; 131 is a coarse bicontact container, not a resolved overlap. | computation | PASS | numerical only | 2; 8; 22; 0 |
| §10–16: a strict merge (same signature, ρ class and prune provenance; adjacent closures) recomposes the 288 into 17 classes, 7 of them BOTH; 166 ↔ 167 is the one merge across parents. | computation | PASS | numerical only | 17; 7; 6; 2; 2; 19; 22; 16; 16 |
| §17: the cut changes 127's provenance from coarse nearest 144 to local nearest 155. | computation | FAIL | refuted | 1/8; 0.12499999999999978; 8; 24 |
| §18: the cut changes 101's provenance from coarse 28 to local 144. | computation | PASS | numerical only | 28; 144 |
| §19: a cut can change provenance. | theorem | PASS | numerical only |  |
| §20–21: cut again, most BOTH children keep one BOTH grandchild. | computation | PASS | numerical only | one; seven |
| §22–23, §29: 127's two BOTH children keep nine (2 → 9); persistence 4.5; 9/64 against 1/32. | computation | PASS | numerical only | 6 + 3; 4.5; 9/64; 1/32; 8; 4 |
| §23, §32: 127 is a thick, branched, provenance-switching corridor and the next target. | reading | NOT AS STATED | corrected | 4.5 |
| §24–27: the spine has three families; grouping is not merging. | definition | PASS | label only | three |
| §28–30: H_prov, P_refine, the information vector. | definition | PASS | label only |  |
| §33–35: the routes. | specification | UNMEASURED | label only |  |
| §36–37: not a global 195 → 7; 288 → 17. | reading | PASS | label only | 195; 7; 288; 17 |
| §38–40: LUC 03 next; Results A, B and C. | reading | PARTIAL | label only |  |
| The CSV, recomposition, persistence record, state and graph. | reading | PASS | label only | 288 |
| The verifier (read after the rebuild, never run). | reading | PARTIAL | label only | 40 lines; 15 |
| Package integrity: VALIDATION digests. | reading | PASS | label only |  |

Open: Anything about areas: "The cut is bookkeeping in a parameter space; no cell's area is computed here."; That 127 is not worth a deeper cut: "The switching surface between 144 and 155 inside 127 is real, and LUC 03 may find something there."

Unclear: T₋ is computed as an exact fraction and T₊ to 50 digits on a 2²² grid; recorded as exact-arithmetic certificate for the halving and the tie analysis, numerical only where the report compares to the ledger's floats.; Several table verdicts are 'n/a' or 'PASS / n/a' or 'PASS / not as stated'; mapped to UNMEASURED, PASS and PARTIAL respectively.

### RUN-065 — BCODR-LUC-03 — True Common-Point Overlap and Provenance Resolution (BCODR-LUC-03-v0.1, 2026-09-21), with its ledger, gap CSV, plot, verifier, state record and VALIDATION

Report: `reports/RUN-065-BCODR-LUC03.md`

> Verdict: PASS for the theorem, every true overlap time and gap, the modes, 131's midpoint, and 127's value, provenance and reduction, and for the records. Not as stated: the provenance of 101, 103 and 104, where four boxes tie exactly. Noted: the seven outward-limited gaps are exactly zero, and the ledger's negative ones come from LUC 01's ray grid; the printed r* and λ* are right only to about 10⁻¹¹; at 127 the provenance 155 is right, but no flip led to it.

**For the paper.** LUC 03's true overlap times for all nine parents (to 8×10⁻¹³), the balanced/outward-limited modes, 131's midpoint value 0.0473333498919450 and 127's value 0.12695366779092488 with its one-dimensional reduction rebuild to 40 digits; the seven outward-limited gaps are exactly zero, the printed r* and λ* are right only to about 10⁻¹¹, and 101, 103, 104 are exactly provenance-degenerate (four-way tie), not 'close to'. 127's provenance 155 is right but no flip led to it; no area is computed.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §1: the true overlap time of a box is the least radius at which the two fields cover one same point of it. | definition | PASS | label only |  |
| §2: T_true ≥ T_box. | theorem | PASS | analytic (proof checked) |  |
| §5 (table): the true overlap times of the nine parents. | computation | PASS | numerical only | 8×10⁻¹³; 40 digits |
| §3: gaps within 2×10⁻⁹ are numerical (from independent numerical optimizers). | computation | PASS | corrected | 2×10⁻⁹; −5×10⁻¹¹; −1.3×10⁻⁹; 5×10⁻¹³ |
| §4, §6–7, §16: modes — 131 and 127 balanced, the other seven outward-limited. | computation | PASS | numerical only | μ = 1/2; μ = 0.2394…; μ = 1; 0.026 |
| §5 (table) provenance: '28' is the earliest prune box at 101, 103 and 104. | computation | NOT AS STATED | corrected | 28, 68, 144 and 155; 1.4×10⁻¹³; 2×10⁻¹³; 4×10⁻¹⁶; 0.18 to 0.24 |
| Provenance 155 at 131, 166, 167, 164, 165 and 127. | computation | PASS | numerical only | 2×10⁻¹² |
| Second-place provenance at 164. | computation | NOT AS STATED | corrected | √(5/32); 28; 68; 144 |
| §8: root 131's true time 0.0473333498927312, at a midpoint. | computation | PASS | numerical only | 0.0473333498919450; 0.0473333498927312; 7.9×10⁻¹³; μ = 1/2; ρ = 0.002540688875; 2.8×10⁻⁸ |
| Root 131's second-earliest box (144) and its time. | computation | PASS | numerical only | √2/8; 1/8; 0.17677669529537773; 1.3×10⁻¹² |
| §9: root 127's true time 0.126953667790942, balanced. | computation | PASS | numerical only | 0.12695366779092488; 0.00195366779092488; 1/8; 0.126953667790942; 1.7×10⁻¹⁴ |
| §10: at 127, 155 is earliest, ahead of 144 by 0.0420149036170295, so LUC 02's flip holds. | computation | PARTIAL | corrected | 0.04201490361714857; 0.0420149036170295 |
| §11–13: 127's active face and the one-dimensional reduction of its 5-D minimax. | reduction | PASS | numerical only | −1/8; −1/4; −7/16; 3/8; 40 digits |
| §14: r* = 0.0025452863775066, λ* = 0.760555242947675, T = 0.126953667790925; the 5-D value is 1.707e-14 high. | computation | PASS | numerical only | 0.126953667790925; 1.7×10⁻¹⁴; r* = 0.0025452863775066; λ* = 0.760555242947675; 3.7×10⁻¹¹; 7.1×10⁻¹²; r* = 0.0025452863403046; λ* = 0.76055524294058; 2×10⁻¹⁸ |
| §15: a low-dimensional germ. | reading | PASS | numerical only |  |
| §17: 101, 103 and 104 are close to PROVENANCE_DEGENERATE; 127 resolved. | reading | NOT AS STATED | corrected | four |
| §18–22: information vector, scheduler, new terms, LUC 04. | specification | UNMEASURED | label only |  |
| §23: the conclusions. | reading | PASS | label only |  |
| The CSV, state and ledger records. | reading | PASS | label only | nine |
| The verifier (read after the rebuild, never run). | reading | PARTIAL | label only | 46 lines |
| Package integrity: VALIDATION digests. | reading | PASS | label only |  |

Open: Anything about areas: "The true overlap is still bookkeeping in a parameter space."; That the true overlap time is the better scheduler: "This run computes the time but does not test the argument."

Unclear: Values are computed 'to 40 digits' via the dual of a convex problem on RUN-064's exact grid; the report does not call these interval enclosures, so strength is recorded as numerical only.; §5's table verdict is 'PASS / not as stated'; split here into a PASS item for the times and a NOT AS STATED item for the '28' provenance.

### RUN-066 — BCODR-LUC-04 — True Overlap Germ Tangent Decomposition (BCODR-LUC-04-v0.1, 2026-09-21), with its result record, plot, verifier, state record and VALIDATION

Report: `reports/RUN-066-BCODR-LUC04.md`

> Verdict: PASS for the meeting point, the KKT relation, the rank and the critical tangent space, D₊, H± and Q±, the stiffness as §14 defines it, and the isolation of the birth germ. Not as stated: that stiffness as the overlap's second-order law, and the growth law built on it (§16–19, §22). Noted: the printed z*, g±, α* and slope carry the printed r*'s error; the printed T* is LUC 03's 5-D value.

**For the paper.** At 50-digit numerics, LUC 04's meeting point, KKT relation, rank 2, three-dimensional critical tangent space, Q± and straight-line stiffness 0.45429481026314 rebuild, and the birth germ is isolated. The overlap's actual second-order law is the Lagrangian αQ₊ + (1 − α)Q₋ with softest stiffness 0.2366, so the lens grows to 1.036 √ε (not 0.7476 √ε) and the κ > 0.4 certificate target would fail; no interval certificate is established.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §0–3: 127's earliest common point z*, its radius T*, and the active x₃ face. | computation | PASS | numerical only | r* = 0.0025452863403045615; T* = 0.12695366779092487911; 3×10⁻¹⁰; 1.7×10⁻¹⁴ |
| §4–5: g₊ and g₋ satisfy a KKT relation with the active x₃ face, with multipliers α* and η*. | computation | PASS | numerical only | 10⁻⁹; 3×10⁻¹⁰; α* = 0.23944475705941908; 7.1×10⁻¹²; η* = 0.12035442698726; 3×10⁻¹³; 10⁻⁴⁵ |
| §6: the active normals have rank 2, for a structural reason. | computation | PASS | analytic (proof checked) | 1.0133; 0.07374; 0 |
| §7–8: a three-dimensional critical tangent space and its basis. | computation | PASS | numerical only | 3; −0.4041979698; −0.404197989002394; 1.9×10⁻⁸ |
| §9: first-order tangency is not enough. | reading | PASS | label only |  |
| §10–12: reduced Hessians H₊ (Schur complement over the ray parameter) and H₋ = diag(1, 0, 1, 0, 1), restricted to Q₊ and Q₋. | computation | PASS | numerical only | D₊ = 1706.98000390142; 4×10⁻¹¹; 6×10⁻¹¹; diag(0, 0, 1) |
| §13: Q₊ alone is indefinite; Q₋ is flat. | computation | PASS | numerical only | −0.0011924 |
| §14–15: the combined stiffness κ_germ = min max(vᵀH₊v, vᵀH₋v) = 0.45429481027797 and its direction. | computation | PASS | numerical only | 0.45429481026314; t = 0.4997; 1.5×10⁻¹¹; 1.7×10⁻⁷ |
| §16: the expansion d±² = T*² + h² vᵀH±v + o(h²) gives every critical direction a second-order rise in overlap radius, so the birth germ is isolated (with stiffness κ_germ). | theorem | PARTIAL | corrected | 0.236577; 0.239445; 0.763138; 0.2366; 0.4543 |
| Isolation of the birth germ. | computation | PASS | numerical only | 0.236577; 0.239445; 0.763138 |
| Direct slice measurement of the least-radius growth (no expansion). | computation | PASS | numerical only | 0.239445 h²; 0.236854 h²; 3×10⁻⁷; h = 10⁻⁴; 1; 0.9918; 0.4543 |
| §17: from dimension 3 to 0. | reading | PASS | numerical only | 3; 0 |
| §18–19: after birth the overlap grows as a lens with \|h\| ≲ 0.7476 √ε and ε ≳ 1.789 h². | computation | FAIL | refuted | 0.7476 √ε; 1.789 h²; 1.036 √ε; 1.030 √ε; 0.932 h² |
| §20–21: no hidden thick corridor; no Φ₂ needed for flatness. | reading | PASS | numerical only |  |
| §22: interval certificates CERT-04A–C with the target κ > 0.4 called 'generous'. | certificate | NOT AS STATED | corrected | 0.4; 0.2366 |
| §23: a lens growing as O(√(T − T*)). | computation | PASS | numerical only |  |
| §24–27: information, spine, 131 next, conclusion. | reading | PASS | label only |  |
| The result record and the state. | reading | PASS | label only |  |
| The verifier (read after the rebuild, never run). | reading | PARTIAL | label only | 44 lines; 0.4 |
| Package integrity: VALIDATION digests. | reading | PASS | label only |  |

Open: Anything about areas: "The germ lives in a parameter space."; Any interval certificate: "Everything here is 50-digit numerics, like the document's, and CERT-04A–C stay open."; Rigorous bounds on growth: "The slices measure the growth; they bound nothing rigorously."

Unclear: §6's rank-2 item is recorded as analytic because the report gives a structural reason valid 'for every r'; the singular values themselves are numerical.; §16's table verdict is 'PASS / not as stated'; split into a PARTIAL item for the second-order law and a PASS item for isolation.

### RUN-067 — BCODR-LUC-05 — Balanced Overlap Germ Taxonomy (BCODR-LUC-05-v0.1, 2026-09-21), with its result record, plot, verifier, state record and VALIDATION

Report: `reports/RUN-067-BCODR-LUC05.md`

> Verdict: PASS for 131's nearest pair, T*, the midpoint and its place inside 131, the rank-1 first order and the four-dimensional critical space, H±, κ₁₃₁, its C_R and C_H, and the isolation of both germs. Not as stated: the comparison of the two germs (§13–14), and §22's rule as a test of isolation. Noted: the midpoint theorem needs the nearest pair of all, and its proof the meeting point's own nearest points; §3 "reproduces" LUC 03 to 8×10⁻¹³; LUC 06 is not in the archive.

**For the paper.** At 50-digit numerics, root 131's interior-midpoint germ rebuilds (T* = 0.04733334989194496, rank 1, four-dimensional critical space, κ₁₃₁ = 0.45464980777815, C_R and C_H as printed), and at a midpoint the straight-line stiffness is the overlap's own; both germs are isolated. The 'same second-order mechanism' ratio 1.0008 is a ratio of straight-line constants: the overlaps' own constants differ by a factor 1.92, and the taxonomy's rule cannot reach its unstable class and does not certify isolation; the midpoint theorem needs the globally nearest pair.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §0: 127 and 131 are of different first-order class but the same second-order outcome. | reading | PASS | numerical only |  |
| §1–2: 127 is a face-constrained balanced germ (FCBG) and 131 an interior-midpoint one (IMBG). | definition | PASS | label only |  |
| §3: the nearest ray–155 pair at ρ* and T* = 0.0473333498919449, reproducing LUC 03. | computation | PASS | numerical only | ρ* = 0.0025406888749283; 0.00254068891729192; 4.2×10⁻¹¹; 0.04733334989194496; 0.0473333498919449; 0.0473333498927312; 7.9×10⁻¹³; 0.0095 |
| §4–5: the midpoint theorem — 131's meeting point is the midpoint of the nearest ray–155 pair — and its proof via \|p* − z\| + \|z − q*\| ≤ 2T. | theorem | NOT AS STATED | corrected | 0.0947; 0.2464 |
| §6–8: at 131 no face is active, g₋ = −g₊, the rank is 1 and the critical space four-dimensional. | computation | PASS | numerical only | 1; 4; D₊ = 1709.2110247947; diag(1, 0, 1, 0, 1) |
| §9: κ_germ decides isolation. | method | NOT AS STATED | corrected |  |
| §10–12: H₋, H₊, κ₁₃₁ = 0.454649807778291, birth dimension 0. | computation | PASS | numerical only | 0.45464980777815; t = 0.50003; 1.4×10⁻¹³; 0.0001098, 1, 1 and 1 |
| At 131 the straight-line constant equals the overlap's own second-order constant. | computation | PASS | numerical only | 1/2; 0.4546497978; 10⁻⁸; 0.5000 h²; 0.4918 h²; 3×10⁻⁷ |
| §13: κ₁₃₁ / κ₁₂₇ = 1.00078, less than a tenth of a percent apart. | computation | NOT AS STATED | corrected | 1.00078142539226; 3×10⁻¹¹; 0.2366; 0.4546; 1.92; 1.0008 |
| §14: the same second-order mechanism at both germs. | reading | PARTIAL | corrected |  |
| §15–19: C_R = 1.789 and 4.803, C_H = 0.748 and 0.456 (127 and 131). | computation | PARTIAL | numerical only | C_R = 4.802637134452; C_H = 0.4563101327193; 0.932; 1.036; 5.15; 2.68; 0.44; 0.61 |
| §20: the germ taxonomy (IMBG, FCBG, edge, corner, provenance-degenerate, flat, unstable). | definition | PASS | label only |  |
| §21–22: the rule κ_germ > 0 isolated, = 0 flat, < 0 unstable; future types. | method | NOT AS STATED | refuted | 0.9987; 0.4543; 0.24 |
| §23–24: isolated germs; no Φ₂ needed for flatness. | reading | PASS | numerical only |  |
| §25–26: LUC 06 next. | reading | UNMEASURED | not measured |  |
| §27: the conclusion, including 'κ ≈ 0.454 for both'. | reading | PARTIAL | corrected | 0.454 |
| The result record and the state. | reading | PASS | label only | 7.9×10⁻¹³ |
| The verifier (read after the rebuild, never run). | reading | PARTIAL | label only | 45 lines; 0.4 |
| Package integrity: VALIDATION digests. | reading | PASS | label only |  |

Open: Areas: "Anything about areas."; Any interval certificate: "the numerics are at 50 digits, and the slices measure the growth but bound nothing."; The taxonomy beyond these two germs: "Whether the other classes occur is not examined."; LUC 06: "The next node, LUC 06, is not in the archive"

Unclear: The verdict line gives '§3 "reproduces" LUC 03 to 8×10⁻¹³' while the body gives 7.9×10⁻¹³; both numbers are recorded.; §15–19's table verdict is 'PASS for 131'; recorded as PARTIAL because 127's values are corrected by RUN-066.

### RUN-068 — BCODR-LUC-07 — Root 131 Candidate/Germ Alignment and Mirror-Covariance Reduction (BCODR-LUC-07-v0.1, 2026-09-21), a single document without a package

Report: `reports/RUN-068-BCODR-LUC07.md`

> Verdict: PASS for the midpoint separation theorem, T* and κ₁₃₁, the mirror algebra and the support-function covariance, and the up/down symmetry of Paper 07's barrier (CERT-07.4, replayed body by body and at 50 digits). Not as stated: that Cycle 17's candidate is recorded by value only (§6, §20), and the positions of the B5 slivers (§22). Noted: the printed p*, q* and z* are LUC 03's optimizer values, 5×10⁻⁷ from the exact pair; LUC 06, and the schema and verifier that §23 describes, are not in the archive.

**For the paper.** LUC 07's midpoint separation theorem, mirror algebra and support-function covariance hold, and Paper 07's zero-rotation barrier is exactly up/down symmetric body by body, hence achiral (CERT-07.4 replayed at 50 digits, no interval certificate). Cycle 17's candidate vector for 131 is in the archive (0.109 from the IMBG point, 7.7×10⁻⁴ from the ray), and A_cand is not 131's lowest value: the barrier at 131's corner is 1.379×10⁻⁷ lower.

| claim | kind | finding | strength | numbers |
| --- | --- | --- | --- | --- |
| §0: LUC 06 narrowed the selected spine to 131; 131's six properties. | reading | PARTIAL | label only | six |
| §1: the nearest pair p*, q*, meeting point z*, T* = 0.0473333498919449, ρ*, κ₁₃₁. | computation | PASS | numerical only | 0.0473333498919449; 0.0025406761; 0.00254068891729192; 0.0025406888749283; 4.2×10⁻¹¹; 5×10⁻⁷; 2×10⁻⁷; 6×10⁻⁸; −0.24999999999999983; −0.25 |
| §2: three notions of alignment (point, seed, structural). | definition | PASS | label only |  |
| §3–4: midpoint separation theorem — a midpoint at T* > 0 lies T* from both seeds — and the printed distances. | theorem | PASS | analytic (proof checked) | 0.09466669978536; 1.5×10⁻¹² |
| §5: the provenance of 131's germ is the ray and box 155 (link to Paper 09B). | reading | PASS | numerical only |  |
| §6, §20: Cycle 17's candidate in 131, A_cand = 0.8349862484449208, keeps 131 open; its parameter vector is not recorded and needs export. | reading | NOT AS STATED | corrected | 0.8349862484449208; 68 |
| Position of Cycle 17's stored minimizer for 131 relative to the germ, the ray and 155. | computation | PASS | numerical only | 6.6×10⁻⁶; 0.109; 7.7×10⁻⁴; ρ = 1.7×10⁻⁴; 0.124 |
| Whether A_cand is 131's lowest point. | computation | PASS | numerical only | m₀ = 0.83498611055922; 1.379×10⁻⁷ |
| Root 131 cannot be closed by A > A_ref. | reading | PASS | numerical only |  |
| §7–8: S_x = R_π S_y; exact single-axis (up/down) symmetry makes a planar body achiral. | theorem | PASS | analytic (proof checked) |  |
| §9 (CERT-07.4): Paper 07's zero-rotation barrier is up/down symmetric, hence achiral. | theorem | PASS | analytic (proof checked) | 0, 2π/3 and 4π/3; 2πk/5; 1,440; 360; 10⁻⁴⁵; 50 digits |
| §10: marked and branch chirality. | definition | UNMEASURED | not measured |  |
| §11: support-function covariance h_{S_xK}(θ) = h_K(π − θ), h_{S_yK}(θ) = h_K(−θ). | theorem | PASS | analytic (proof checked) |  |
| §12–15: signature reversal; global, chart and dual covariance. | theorem | PASS | label only |  |
| §16: κ is invariant under an isometric mirror. | theorem | PASS | analytic (proof checked) |  |
| §17–18: the decision tree and certificates CERT-07.1–07.8. | specification | UNMEASURED | open | 07.1; 07.2; 07.4 |
| §19: closed items A–E. | reading | PASS | label only |  |
| §21–22: return to Paper 09C and its three slivers at π/2 and π ∓ θ*. | reading | PARTIAL | corrected | π/2; θ*; 2π − θ*; π ∓ θ* |
| §23: a companion JSON schema and verifier. | reading | NOT AS STATED | not measured |  |
| §24–25: the verdict; the conclusion. | reading | PASS | label only |  |
| Document integrity. | reading | PASS | label only |  |

Open: Structural alignment (CERT-07.3): "The candidate's support topology is not compared with the Paper 09B branch. Its nearness to the ray's start is a distance, not a mechanism."; Any interval certificate: "The symmetry is exact by construction and replayed at 50 digits; the distances are numerical."; Marked or nonzero-rotation chirality, and the root atlas's mirror map: "These stay open, as §20 lists them."; CERT-07.4 interval form: "This replays CERT-07.4 numerically. No interval certificate is attempted."

Unclear: The barrier's up/down symmetry is called exact 'body by body' (by construction) and also 'replays CERT-07.4 numerically' with 'No interval certificate'; strength recorded as analytic (proof checked) for the exact by-construction argument.; §0's table verdict is 'n/a / PASS' (no LUC 06; the six properties hold per RUN-063, 065, 067); recorded as PARTIAL.; §23's verdict is 'not as stated' because the files are absent; recorded as NOT AS STATED with strength not measured.