← Lebesgue Universal Covering Problem · LAYER 2

II · Computation and Reproduction

The Lebesgue case's second layer: all 42 documents independently recomputed, RUN-001 to RUN-068

The first layer, "I · Precursor", holds the 42 documents by Neo.K with Aletheia / ChatGPT, GPT-5.6 Sol. This second layer, "II · Computation and Reproduction", is their independent recompute by 懸衡 (Xuán Héng) (AMRAL's counter-proof arm, Claude), in the documents' own order: every finite claim recomputed, every construction reproduced, every theorem-level step read against its source. The author's scripts are never run, and are read only after the rebuild.

There are 68 runs (RUN-001 to RUN-068), each with one gate, one report and one mutation drill — planted faults, controls and must-stay-green assertions, with every check blinded in turn to confirm that each fault is caught by exactly the checks it should trip. Before each commit the run was repeated from a cold cache and its gate log compared byte for byte. Round 00's canonical-source rules and claim ledger were measured on all 42 documents, and every status phrase in them reviewed.

This layer does not rewrite the documents and does not claim the Lebesgue problem solved: the global lower bound a_Leb ≥ 0.835 stays NOT CERTIFIED, and nothing recomputed here conflicts with the published interval 0.8344 ≤ a_Leb ≤ 0.8440935944. It records what reproduces and to what precision, what is not as the documents state it, and the new readings. The third layer, "III · Crystallization", is the formal paper written from these two (a preprint), covering these 42 documents and 68 runs.

Three layers: I · Precursor (the 42 documents) → II · Computation and Reproduction (this page, 68 runs) → III · Crystallization (the formal paper).

New readings · the main counter-results

The most important of the 68 runs' findings; after each, the runs that decide it, whose reports open on their pages.

Gate 6217-A is not established. The ρ₅ = 0 reduction drops the F3 constraint, so Cycle 13's 2-D certificate leaves the arc side of the slice uncovered; it holds numerically (by 2.05×10⁻⁴) but is not proved. RUN-048, RUN-056
44 % of root 6217's box was closed only by Cycle 11's search, and Paper 08's "honest" 195-root shape ledger leaves out the 14.6 % closed for A₃⁺ only. RUN-049, RUN-058, RUN-062
LESR's first-order model fails on the kernel: when the container is the hull of its witnesses, a witness move changes the area by an edge average, not the affine maximum, so Papers 03–05's LP, flow and dual certify placements that are not stationary. RUN-053, RUN-054, RUN-055
Paper 07's translation KKT is not established as argued (the multiplier is 0.047, not 0.123), and Paper 08's reason for criticality is not the right one; both facts hold numerically. RUN-057, RUN-058, RUN-061
In the middle rounds several "B7 not yet closed" and "residual" readings belong to the pilot's sampled bounds: with exact cores B7 closes every marked cell, including cell 3, the one residual left. RUN-023, RUN-024, RUN-026, RUN-027
On Paper 09B's primary chart the first variation is exactly γ_w·w; the full Φ₁ stays non-negative along −w only by 0.035 per unit, and the second-order slivers open at the antipodal walls (θ*, 2π − θ*, π/2), not at π ∓ θ*. RUN-060, RUN-068
BCODR's numbers reproduce and its readings do not: 64 contact pairs, not 35; 127's "flip" is a tie; a germ's stiffness needs the Lagrangian weight (127's is 0.2366, and the two germs differ by 1.92); the candidate vector LUC 07 calls unrecorded is in Cycle 17's diagnostic, and 131's corner holds the lower barrier itself. RUN-063, RUN-064, RUN-066, RUN-067, RUN-068
Round 00's claim ledger: of 758 judgment statements only 89 use its ten statuses — a gap in bookkeeping, not in mathematics. RUN-003

All main-line rounds · 38

Each card stands for one layer-1 document; its tags sum up the verdicts of that page's runs.

The four methodologies · 4

Each methodology's page gathers its whole track's runs: the method itself and its cycles, papers or case studies.

All files

These files are byte for byte the research tree amral-research-trees's lebesgue-verification-xuanheng/ (branch agent/lebesgue-verification-xuanheng, commit 86e5452): the README, 72 programs, 84 data files (gate logs, the status registry, the source manifest) and the 68 reports. SHA256SUMS lists every file's SHA-256. Re-running needs the original packages, kept in AMRAL's private research archive.

lebesgue-verification-xuanheng.zip — The whole second layer (zip, SHA-256 be2da1185c04dd50…)
SHA256SUMS — Every file's SHA-256
README.md — The research tree's README: one line per run and the commands to re-run
RUN-001-CORPUS-IDENTITY.md — corpus identity: the forty-two curated documents are the archived originals
RUN-002-ROUND00-BASELINE-AND-INTERFACE.md — Round 00: the baseline it measures itself against, and the interface it fixes
RUN-003-ROUND00-GOVERNANCE.md — Round 00's governance, measured on the corpus it governs
RUN-004-ROUND01-REDUCTION-CHAIN.md — Round 01: the reduction chain holds, and the reflection branch is made explicit
RUN-005-ROUND02-FINITE-COMPILER.md — Round 02: the finite shape compiler holds, and its error bound is safe by a wide margin
RUN-006-ROUND03-PLACEMENT-CERTIFICATE.md — Round 03: the placement certificate holds, and its sanity example could not have failed
RUN-007-ROUND04-COVER-COMPILER.md — Round 04: the cover compiler holds; its sanity polygon is not quite the one it defines
RUN-008-ROUND05-SATURATION-GATE.md — Round 05: the saturation theory holds; its sanity test could not have caught a wrong bound
RUN-009-ROUND06-WITNESS-EXCHANGE.md — Round 06: the exchange compiler holds; its seed ladder is exact at the second rung and strict at the third
RUN-010-ROUND07-MINIMIZER-ATLAS.md — Round 07: the lifting theorems hold, every external number checks, and one recorded lift is not a minimum
RUN-011-ROUND08-CONDITIONAL-LIFT.md — Round 08: the proof grammar holds, two hard leaves close at 10⁻⁴, and the verifier has no root for a mirror image
RUN-012-ROUND09-ADAPTIVE-ATLAS.md — Round 09: the cost layer is sound, the common core closes cells 2.2× wider than the representative prune, and dominance pruning has nothing to prune
RUN-013-ROUND10-REFERENCE-DRY-RUN.md — Round 10: the reference certificate replays and re-emits byte for byte, the exact core clears every leaf by 3.8×10⁻⁴, and the split's tie-break lives only in the code
RUN-014-ROUND11-GLOBAL-SHARDS.md — Round 11: the frontier theorems hold and the reference shards merge, but the digests and boxes cannot be rebuilt from the text, and the depth forecast is not a bound
RUN-015-ROUND12-CHECKPOINT-CRYSTAL.md — Round 12: the checkpoint machinery works, but three stored aggregates contradict their ledgers and a rejected certificate comes back unchanged
RUN-016-ROUND13-AUTONOMOUS-SCHEDULER.md — Round 13: the merge rules are sound and the race replays, but the reference merger never writes the aggregate and cannot hold two valid certificates
RUN-017-ROUND14-SEMANTIC-REBASE.md — Round 14: the migration rules hold and compaction keeps readiness, but the reported capacity jumps above a failed target and the hot core drops the evidence that matters
RUN-018-ROUND15-CAPACITY-LADDER.md — Round 15: the capacity algebra is exact, but the local proof's capacity is 0.835, not 0.8350003545 — its B7 lift root was sized for 0.835
RUN-019-ROUND16-GLOBAL-PILOT.md — Round 16: every pilot number rebuilds, but its frontier profile measures its 40-direction sampling; APR can never fire on this root, and 19 HARD cells hold configurations below 0.835
RUN-020-ROUND17-D3-CPG.md — Round 17: the D3 quotient is exact and every run rebuilds; the quotient's gain is understated, and the tail window's edge comes from a changed override rule, not from starting at depth 10
RUN-021-ROUND18-SHARD-EMITTER.md — Round 18: both checkpoints decode, re-prove and re-emit byte for byte; the tail forest is not Round 17's tail window, three conventions of the format are unstated, and the shipped verifier never reads the pending frontier
RUN-022-ROUND19-NECESSITY-LIFT.md — Round 19: every tree, bracket and probe rebuilds; lazy evaluation changes nothing but cost; the depth-20 recheck is complete; the B7 floor is mostly the pilot's sampling
RUN-023-ROUND20-MIXED-ESCALATION.md — Round 20: every tree, marker and placement rebuilds; the package ships no code; its search minima are not minima; with exact cores B7 closes all seven marked cells it was run on
RUN-024-ROUND21-LIFT-TAIL-SYNTHESIS.md — Round 21: the tail atlas rebuilds box by box, and every box closes with exact cores; the synthesis search values sit below the true minima
RUN-025-ROUND22-FOURIER-PARETO.md — Round 22: the legality rules and the lift pilot rebuild exactly; with dense cores the Pareto map's volume axis inverts, and the "cheap" harmonics cannot close their cell
RUN-026-ROUND23-PORTFOLIO-CLOSURE.md — Round 23: the strict-incidence grammar is sound and the B7 edges rebuild exactly; with exact cores B7 closes all twelve cells, and no harmonic can close any
RUN-027-ROUND24-RESIDUAL-JOINT.md — Round 24: the first complete-edge certificate rebuilds bit for bit; the tail numbers mix two bounds, and cell 3 closes too
RUN-028-ROUND25-CLOSURE-WAVE.md — Round 25: all twelve B7 closures rebuild node for node; the recorded closure depths are checkpoints, and six cells close a level earlier
RUN-029-ROUND26-FULL-ATLAS.md — Round 26: all 77 necessity cells rebuild row for row under the package's pad; the thinnest leaf in the atlas clears T by 8.5×10⁻¹¹, not 2.9×10⁻⁹
RUN-030-ROUND27-INTERVAL-REPLAY.md — Round 27: at a common depth 40 the residual's lower bounds are exact, 57 by 38 and 63 by 40; the five thinnest leaves certify again under this tree's own integer intervals, and seven more
RUN-031-ROUND28-EXACT-ROOT.md — Round 28: the directed root and all 9,278 leaves of cell 0 check under this tree's own arithmetic; no leaf needs the 16,384-direction density; and this tree's exact kernel loses a piece on one leaf
RUN-032-ROUND29-MARKER-UPPER.md — Round 29: all 77 necessity markers certify again, with 128–256 normals once the hull's bridging edges get their own; the package's polygons lose first order at those edges, so its thinnest marker is not the thinnest
RUN-033-ROUND30-A1-REPLAY.md — Round 30: the whole shard passes again here, and the second implementation's lower ledger comes back leaf for leaf from its own rule; that rule keeps one contact point per direction, so its thinnest leaf measures the rule's loss, not the geometry
RUN-034-ROUND31-MPFR-REPLAY.md — Round 31: the second backend's root, marker and 9,278 leaves pass again here, and its lower ledger is Round 30's leaf for leaf — the same polygons, their membership decided again in MPFR; the queue is this tree's atlas; the wrapper rounds its exact inputs to nearest, so §3's directed rule holds for its outputs only
RUN-035-ROUND32-BULK-WAVE.md — Round 32: cells 31, 27 and 7 close here too, continued from this tree's own frontier at 40 — at depths 42, 43 and 45, inside the even checkpoints 42, 44 and 46 the round reports; cells 69 and 47 are the shipped trees node for node, every leaf passes, and cell 69's MPFR ledger is its own rule rebuilt, the escalated leaf included
RUN-036-ROUND33-DOMINANCE.md — Round 33: the dominance theorem is right and the four new trees rebuild node for node; its audits rest on MPFR, not recomputed here — and the certificates the ledgers do not keep, this tree regenerates from density and rule
RUN-037-ROUND34-RHCERT.md — Round 34: the certificates are real — both RHCERT files decode from their specification alone and all 4,272,769 stored vertices check out in this tree's own integer arithmetic; cell 68's summary comes back from this tree's certificates
RUN-038-ROUND35-RHCERT-PRODUCTION.md — Round 35: cell 68's certificate is this tree's own, leaf for leaf — every density and hull count agrees, and every one of its 2,126,336 vertices checks out here; cells 75 and 76 certify here too; the lost frontier is noted, not guessed
RUN-039-ROUND36-DFRONT.md — Round 36: cells 75 and 76's certificates check out here, vertex for vertex; cell 33's DFRONT file is this tree's own depth-36 frontier, written here byte for byte; cell 49's tree hashes to its manifest
RUN-040-ROUND37-DEPTH32-DFRONT.md — Round 37: the three depth-32 certificates are this tree's own, leaf for leaf; cell 33's continuation to depth 42 is this tree's, every checkpoint and all 166 segment files byte for byte; the merge never checks the premise its theorem rests on
RUN-041-DLMVC.md — DLMVC: the lagged line's mathematics holds and in two places goes further than this tree's first reading; its one "independent reproduction" re-ran the author's script and reproduced the script's error — the failure DLMVC exists to prevent
RUN-042-DLMVC-DEEP-PASSES.md — DLMVC's four deep passes on Round 04: every constant recomputes and the error bound holds on random bodies; two of Pass 04's quantities are grid samples on the unsafe side, and its figure for Pass 03's cell radius is a different quantity
RUN-043-UESFCM.md — UESFCM: seventeen Lebesgue cycles chain cleanly and never mark evidence as proof, but the target they anchor to is written down nowhere in the archive, the LinkBack types are used in one cycle, and the texts disagree on which states end the loop
RUN-044-UESFCM-CYCLES-01-03.md — UESFCM Cycles 01–03: the B7 separator LP reproduces to the last digit and the domain constants to the ulp; the "sharp root domain" is Mishra's Remark 20, which he notes saves no boxes; s7, the replacement family and the sublevel's thinness come out different from Cycle 03's figures
RUN-045-UESFCM-CYCLES-04-06.md — UESFCM Cycles 04–06: Gate A's dual certificate holds under this tree's own integer intervals, but its verifier skips two premises; the D3 quotient, the symmetric anchor and the box bounds are right; the far-branch minima reported are search values that sit above better points
RUN-046-UESFCM-CYCLES-07-09.md — UESFCM Cycles 07–09: the saved F3 frontier, its 24 skeletons and 1,332 signatures, the 12 reflection classes and the Taylor diagnostic all reproduce from this tree's own hull; the compilers they motivate ship as interfaces only
RUN-047-UESFCM-CYCLES-10-11.md — UESFCM Cycles 10–11: both erosion certificates decode exactly and all 56,149 of their area leaves hold in exact arithmetic with this tree's exact cores, by at least 2.28×10⁻⁵; the least slacks and η_model reproduce, the float split leaves 10⁻¹⁸ gaps that continuity closes, and Cycle 11's verifier cannot run from its own package
RUN-048-UESFCM-CYCLES-12-14.md — UESFCM Cycles 12–14: root 6217's three certificates replay and hold in exact arithmetic, and the symmetric triple contact reproduces to 80 digits; but the ρ₅ = 0 reduction drops the F3 constraint, so Cycle 13's 2-D certificate leaves the arc side of the slice uncovered — there the least area is again the contact point, so Gate 6217-A holds numerically and is not yet proved
RUN-049-UESFCM-CYCLES-15-17.md — UESFCM Cycles 15–17: the boundary-locking lemmas hold, both upper references and all three certificates reproduce, and all 115,511 area leaves hold in exact arithmetic; the references are loose to first order, Cycle 15's counterexample falls for only 1.6×10⁻⁷ along its ray, and "all other regions eliminated by finite certificates" includes 44 % of root 6217 that only a search closed
RUN-050-LESR-PAPER00.md — LESR Paper 00: every formula the bridge uses holds, symbolically and on bodies; its one programmatic principle needs a curvature hypothesis at corners, where the first variation of an inward move is zero and the area drops only at second order; the predecessors it builds on are not in the archive
RUN-051-LESR-PAPER01.md — LESR Paper 01: the topology layer's two theorems hold on the Lebesgue kernel — at most two crossings per piece pair, and an area that is analytic inside a contact stratum and kinks where the stratum changes; the compiler scaffold implements the monotone envelope but not the stability test the text states
RUN-052-LESR-PAPER02.md — LESR Paper 02: the group layer holds — its projectors, its gcd law, and its first-variation theorem, on symmetric bodies and on the kernel's own hull; but on the kernel's seed no pure Fourier mode survives the convexity test, and breaking the symmetry costs area at first order — the recorded seed's asymmetry is search noise
RUN-053-LESR-PAPER03.md — LESR Paper 03: the calculus holds along affine support paths — every formula, the step caps, the contact variation; but on the kernel, where the container is the hull of its witnesses, a witness move turns the container's edges and the first-order change is the edge average, not the affine maximum — so the finite LP finds no descent even where the area's gradient has length 0.39
RUN-054-LESR-PAPER04.md — LESR Paper 04: the metric layer and the projection theorem hold (for a cone the descent inequality is an equality); but on the kernel the projected velocity is zero at every placement tried, so the flow never starts — at a generic placement the area falls at 0.390 per unit along its steepest descent
RUN-055-LESR-PAPER05.md — LESR Paper 05: the finite duality is Farkas and holds; the smooth No-Slack theorem holds and carries the curvature hypothesis Paper 00 lacked; but on the kernel the finite dual certificate exists wherever it is solved — at a generic placement whose area gradient has length 0.39 too
RUN-056-LESR-PAPER06.md — LESR Paper 06: the barrier and its four-direction orbit reproduce, and target-specific closure is the right accounting; but Task B has a one-line answer — no measure on the four directions can match the barrier's curvature once cos 2θ is in the library — and the ledger still marks the ρ₅ = 0 slice closed, which RUN-048 found is not established
RUN-057-LESR-PAPER07.md — LESR Paper 07: every constant of the root-6217 barrier reproduces from the contact equations to the last printed digit, and so does the 13-interval signature; but the translation KKT is not established as argued — the claimed subgradient (g₃, 0) fails in a direction where B3 and B5 move left together; the barrier is still the constrained minimum, with a multiplier of 0.047, not 0.123
RUN-058-LESR-PAPER08.md — LESR Paper 08: the two-gate reduction and its arithmetic hold, the 10 / 141 / 44 split reproduces, and the rotation is first-order critical on the true hull; but not for the reason given — in the finite active-contact model the paper names, turning B5 costs +0.029 at first order — and the "honest" 195-root shape ledger leaves out the 14.6 % of root 6217's box that Cycle 11 closed for A₃⁺ only
RUN-059-LESR-PAPER09A.md — LESR Paper 09A v0.2: the 18-piece signature and both primary O(ρ) arcs reproduce, and the branch's velocity tends to (5.157, −0.1892, 0); but at properly minimized placements the three "secondary slivers" have zero width — the diagnostic's come from a stored sample 1.5×10⁻⁹ above the minimum — and the first epi-derivative has a single zero, with a slope in every direction
RUN-060-LESR-PAPER09B.md — LESR Paper 09B: every wall constant, the critical velocity and the transition speeds reproduce from the barrier's geometry, and on the primary chart the first area variation is exactly γ_w·w — the u and v terms cancel identically; the full Φ₁ stays non-negative along −w only by 0.035 per unit, and along the critical velocity the second-order slivers open at the antipodal walls, not where §18 looks
RUN-061-LESR-MIDPOINT-HANDOFF.md — the LESR midpoint handoff: every number it records is this tree's to the printed digit, and its three ledgers and target-specific closure are right; of what it lists as done in Paper 07, B3's subgradient is not; run here, the counter-certificate line finds no lower branch, finds the new topology, and finds the finite dual uninformative
RUN-062-BCODR.md — BCODR: a closure architecture, as it says of itself; its two differences are metrics on discs, its arrangement's cells are counted right (O(N²), exactly N(N−1) + 2 when every pair crosses), its certificate weights and its UESFCM link are consistent; its root-6217 map would start from the 195-root ledger that leaves out 14.6 % of the box
RUN-063-BCODR-LUC01.md — BCODR LUC 01: the arrival-time field over root 6217 rebuilds from the 200 subroots, cell for cell, and the most balanced eight are the text's; but the boxes touch the prune seeds in 64 pairs, not 35, and for 178 of the 195 cells the "distance to the critical ray" is the distance to its end ρ = 0.003
RUN-064-BCODR-LUC02.md — BCODR LUC 02: the 288 children, their signatures and the 17 classes rebuild exactly; but 127's provenance "flip" is a tie — it is 1/8 from 144 and from 155 alike — and a BOTH child is one that holds the point where the later field first touches its parent, which is all 127's "thick corridor" is
RUN-065-BCODR-LUC03.md — BCODR LUC 03: the true overlap times, the modes and 127's one-dimensional reduction rebuild to 40 digits; but 101, 103 and 104 have no single provenance — four prune boxes tie exactly there — and the "reverse numerical error" is LUC 01's ray grid, not the optimizers
RUN-066-BCODR-LUC04.md — BCODR LUC 04: 127's germ, its KKT relation, its critical tangent space and both reduced Hessians rebuild at 50 digits, and the germ is isolated; but the overlap's second-order law is the Lagrangian α H₊ + (1 − α) H₋, softest 0.2366, not the straight-line 0.4543 — the lens grows to 1.036 √ε, not 0.748 √ε
RUN-067-BCODR-LUC05.md — BCODR LUC 05: root 131's interior-midpoint germ rebuilds at 50 digits, and there the straight-line stiffness is the right one; but the "same second-order mechanism" — κ₁₃₁ / κ₁₂₇ = 1.0008 — is a ratio of 1.92 for the overlaps' own constants, and the taxonomy's rule can neither reach the unstable class it defines nor certify isolation
RUN-068-BCODR-LUC07.md — BCODR LUC 07: the midpoint-separation and mirror theorems hold, and Paper 07's barrier is exactly up/down symmetric, hence achiral; but the candidate vector that §6 says is not recorded is in Cycle 17's diagnostic — 7.7×10⁻⁴ from the ray, 0.109 from the IMBG point — and it is not 131's lowest point: 131's corner holds the barrier, 1.38×10⁻⁷ lower