# RUN-023 — 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

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-25
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_20_Mixed_Witness_Escalation_v0.1.md` (AMRAL-LUC-FC-R20, dated 2026-09-19) and its package, read from the archive:

- `BENCHMARK_SUMMARY.json`, `MIXED_SHARD_SCHEMA.json`, `ESCALATION_POLICY.json`, `VALIDATION.json`, `State_Crystal`;
- the reference mixed shard: `manifest.json`, `verification_summary.json`, and for each of four cells
  the topology and leaf-tag streams of its B7, B11 and B19 lift trees.

The package contains no code.

**Tools:** [`run023_round20_mixed_escalation.py`](../code/run023_round20_mixed_escalation.py) — RUN-019's B7 probe for any odd n, with the pilot's sampled cores or exact ones; a preorder emitter for the lift streams; this tree's exact hull and RUN-022's cancellation-free 20k bracket for four bodies; a multi-start placement search; Round 19's segmentation from RUN-022
**Logs:** [`run023-round20-mixed-escalation.json`](../data/gate-logs/run023-round20-mixed-escalation.json)

**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.**

- **The reference shard rebuilds, but nothing in the package can replay it.**
  - The four marked cells are Round 19's four lowest depth-40 centres (RUN-022). Boxes,
    marker points, exact areas and brackets agree to 1.5×10⁻¹⁴, and every outer bracket
    is below 0.835.
  - All twelve lift trees rebuild with RUN-019's probe (B7, B11, B19 to depth 12 on each
    cell): nodes, CERT and PENDING leaves exactly, worst CERT slacks to 4×10⁻¹⁶. Every
    PENDING leaf is at depth 12.
  - The streams re-emit byte for byte under exactly one of eight conventions: lower
    child first, most significant bit first, a CERT leaf 0 and a PENDING leaf 1. The
    text, the schema and the manifest state none of the three.
  - The schema types nothing below `base_cells`.
  - The package ships no code. The "independent mixed-shard replay" of §16 — a
    hand-written monotone-chain hull, every CERT leaf, stream exhaustion — exists here
    only as its summary file. This tree's rebuild is the only replay there is.
- **The negative filter is right, and B23 is disqualified on cell-00.**
  - §4 is one line: if one legal placement of K has an outer bound below T, then J_K is
    below T, so K cannot close the cell alone.
  - B23 at (0.22598188, −5.68×10⁻⁵, −3.14×10⁻⁴): exact 0.8349254026, outer
    0.8349443918, below T by 5.6×10⁻⁵ (pilot floating arithmetic).
- **The recorded placements rebuild; the minima they are called are not minima.**
  - All nine of §9's placements give their recorded exact, inner and outer areas, to
    3×10⁻¹⁴.
  - This tree's own search — 28 random starts and a polish of the package's point, per
    witness — finds lower placements for every witness. By little for B7, B9, B15 and
    B23. By 3.7×10⁻⁵ for B11, 8.0×10⁻⁶ for B13, 1.24×10⁻⁴ for B17, 3.17×10⁻⁴ for B19 and
    5.9×10⁻⁵ for B21.
  - B19 is at 0.8350921, not 0.8354089: 9.2×10⁻⁵ above T, not 4.1×10⁻⁴. B21 is at
    0.8350840. So the "geometric forcing" axis of §10's Pareto frontier, and the policy's
    "thinner margin: B19–B21", are measured by a search that stopped early.
  - No placement below 0.835 turns up for any witness but B23: no new disqualification.
- **The efficiency scans, the deep comparison and the tails rebuild exactly.**
  - §7's eight cells are the eight lowest by outer bracket. By exact centre instead, the
    means would be 160.375 and 154.375, not 160.625 and 154.0. The four-cell scan uses
    the four lowest exact. Neither choice is stated.
  - The benchmark's 16 deeper probes (depth 14 and 16 on two cells) rebuild, and so does
    every depth-50 tail count, 14 of them.
- **The sampled bound shows no path to closure; exact cores close.**
  - On the depth-50 cell, B7's open leaves grow with depth under the pilot's bound: 653,
    1,015, 1,046, 1,397, 2,143 at depths 18–26. §12's fractions fall only because 2^d
    grows faster.
  - With exact cores, B7 closes that cell at lift depth 21 (2,509 nodes).
  - B7 with exact cores also closes the track's depth-46 and depth-43 cells (depths 23,
    25) and all four reference cells (depths 30, 31, 31, 34; 18,555 to 36,193 nodes).
    The worst slack is 6.6×10⁻⁹, on cell-03.
  - This is floating arithmetic and this tree's own exact hull, so it is not a
    certificate. But it moves the obstacle. What Round 20 escalates away from — B7
    "not yet" closing, the horizon ranking, the need for other witnesses — is the
    pilot's 40-direction lift bound, not B7.
- **The package.**
  - `VALIDATION.json` is right: digests, bytes, `$$` counts, and its four key checks.
  - The escalation policy's states are not §2's. It adds `BASE-UNRESOLVED` and
    `MULTI-WITNESS-REQUIRED` and drops `BASE-CERTIFIED`.
  - The manifest uses three statuses in neither list: `NECESSITY-MARKED-PILOT`,
    `PARTIAL-UNRESOLVED` and `WITNESS-DISQUALIFIED-BY-COUNTEREXAMPLE-PILOT`.
  - The crystal's list of search minima omits B9, B15 and B17 — three of the four
    lowest after B23.
  - §11's depth-18 list omits B13, which has the largest fraction there (0.314%).

Drill: 20 / 20 plants caught by exactly the named checks, 4 / 4 controls undisturbed,
266 / 266 must-stay-green assertions held. Each of the twelve checks was blinded in
turn, and each was reported blind in exactly as many cases as plants name it.

---

## What Round 20 claims

1. **The mixed shard** (§1–2, §15–16).
   - A base leaf carries its id, path, box, necessity marker, lift attachments and
     negative evidence.
   - Seven states, `BASE-CERTIFIED` to `PENDING`. A shard is `COMPLETE` only when every
     base obligation is base-certified or covered by a complete lift.
   - The reference: four depth-40 marked cells, each with B7, B11 and B19 depth-12
     partial lifts. Markers re-verified by outer bracket, pending leaves kept,
     `theorem_ready = false`.
   - An independent verifier replays it all: `REFERENCE-MIXED-SHARD-VERIFIED`.
2. **The negative semantics** (§3–6).
   - Open leaves only make a certificate incomplete.
   - The counterexample theorem: one legal placement with an outer bound below T
     disqualifies a sole closer.
   - B23 on cell-00: exact 0.83492540, outer 0.83494439.
   - A search that finds nothing proves nothing (`NO-COUNTEREXAMPLE-FOUND`, never
     `VIABLE-PROVED`).
3. **Efficiency and forcing** (§7–10).
   - Depth-12 mean pending on eight cells: B7 160.6, B9 154.0, B11 119.1. On four
     cells: B13 89, B15 68, B17 69, B19 64, B21 64, B23 64.
   - Near-disc witnesses certify easily without forcing more.
   - 3-D search minima on cell-00, B7 0.83707775 to B23 0.83492540.
   - Witness choice is a Pareto problem.
4. **Horizon** (§11–13). On the tracked branch's depth-50 cell (τ ≈ 3.83×10⁻⁴):
   - at lift depth 18, B7 0.249%, B11 0.297%, B19 0.281%, B21 0.276%;
   - at 20, B7 0.0968% against 0.14–0.19% for the rest;
   - B7's tail u₁₈ … u₂₆ from 2.491×10⁻³ to 3.193×10⁻⁵, still not closed.

   So B7 is costly shallow but contracts faster, and a scheduler should estimate
   ρ_K(d₁, d₂) = u_{K,d₂} / u_{K,d₁}.
5. **Escalation** (§14, §17–20): the state machine N0–N5; a counterexample cache; odd
   n is no ladder; Fourier witnesses last.
6. **The ledger and Round 21** (§21–23).

## A — sources

**A0.** 48 anchors. Round 20 has no reference list.

## W — rebuilt

| check | Round 20 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | §1–2, §15 | the four cells and markers; the manifest against the schema | PASS |
| W2 | §15–16 | twelve lift trees; their streams | PASS; the one matching convention is stated nowhere |
| W3 | §3–6 | the theorem; B23 | PASS |
| W4 | §9 | nine placements; this tree's search | PASS as recorded; every minimum is lower |
| W5 | §7–8 | eight-cell and four-cell scans | PASS; the eight are chosen by outer bracket |
| W6 | benchmark | 16 deeper probes | PASS |
| W7 | §11–13 | the depth-50 tails; B7 with exact cores | PASS as computed; exact cores close all seven cells |
| W8 | §16 | what is shipped | no code, no encoding, an untyped schema |

**W1 — the markers.**

- The four cells are the four lowest exact centres of Round 19's depth-31 → 40
  segmentation, rebuilt here with RUN-022's code: paths, and boxes to 10⁻¹⁷.
- Each marker's point is its cell's centre. Its exact area is this tree's to
  7.8×10⁻¹⁶, its inner and outer brackets to 1.5×10⁻¹⁴.

  | cell | exact | outer |
  | --- | ---: | ---: |
  | cell-00 | 0.8348018420 | 0.8348206176 |
  | cell-01 | 0.8348116722 | 0.8348279638 |
  | cell-02 | 0.8348238282 | 0.8348435895 |
  | cell-03 | 0.8348266675 | 0.8348497057 |

- The manifest's header meets the schema's required fields and its magic constant.
  The schema's `base_cells` is an untyped array.

**W2 — the lift trees.**

- The probe is RUN-019's B7 probe with 7 replaced by n:
  - the root is φ ∈ [0, 2π/n), |x|, |y| ≤ tₙ(0.8365);
  - each box's bound is the hull of the disc and the three erosion cores, with the
    pilot's 40-direction sampling;
  - a box splits by the least child δₙ, ties to φ then x.
- On all twelve trees the nodes, CERT and PENDING counts are the manifest's and the
  summary's, and the worst CERT slacks agree to 4.4×10⁻¹⁶:

  | cell | B7 | B11 | B19 |
  | --- | --- | --- | --- |
  | cell-00 | 867 / 273 / 161 | 715 / 239 / 119 | 511 / 192 / 64 |
  | cell-01 | 867 / 275 / 159 | 715 / 241 / 117 | 509 / 191 / 64 |
  | cell-02 | 873 / 277 / 160 | 715 / 238 / 120 | 515 / 194 / 64 |
  | cell-03 | 867 / 273 / 161 | 715 / 239 / 119 | 509 / 191 / 64 |

  (nodes / CERT / PENDING.) Every PENDING leaf is at depth 12.
- Each tree was written out in preorder under all eight combinations of child order,
  bit order and tag meaning. Exactly one gives all 24 files byte for byte: the lower
  child first, the most significant bit first, CERT 0 and PENDING 1. The manifest's
  digests and bit counts agree. The format is otherwise Round 18's, but Round 18's tags
  were 2 bits, and this 1-bit tag is defined nowhere.

**W3 — the negative filter.**

- §4: `J_K(H(q)) = min_g Area(conv(H(q) ∪ gK)) ≤ Area(conv(H(q) ∪ g₀K)) ≤ U(…) < T`. A
  certificate for the cell would need a lower bound ≥ T at every placement, including
  g₀. PASS.
- §3 and §6 are the same logic read the other way: open leaves bound nothing from
  above, and a failed search bounds nothing from below.
- B23 on cell-00, at the manifest's placement (its phase inside [0, 2π/23)):

  | | exact | inner | outer |
  | --- | ---: | ---: | ---: |
  | manifest | 0.8349254025739498 | 0.8349253775010099 | 0.8349443917869217 |
  | rebuilt | 0.8349254025739500 | 0.8349253775010098 | 0.8349443917869175 |

  The outer bound is below T by 5.56×10⁻⁵. The 4.2×10⁻¹⁵ between the outer values is
  the two bracket formulas (RUN-022).

**W4 — the search.**

- The nine placements recorded in the benchmark rebuild: exact to 2×10⁻¹⁶, inner and
  outer to 3×10⁻¹⁴, and §9's eight digits.
- This tree's search: for each witness, Nelder–Mead on the exact hull area over (φ, x,
  y), from 28 random starts (φ over one period, |x|, |y| ≤ 0.02) and from the package's
  own point:

  | witness | package (§9) | this search | lower by |
  | --- | ---: | ---: | ---: |
  | B7 | 0.8370777454 | 0.8370777154 | 3.0×10⁻⁸ |
  | B9 | 0.8356295743 | 0.8356295728 | 1.5×10⁻⁹ |
  | B11 | 0.8361731203 | 0.8361364467 | 3.7×10⁻⁵ |
  | B13 | 0.8359229101 | 0.8359149272 | 8.0×10⁻⁶ |
  | B15 | 0.8352910727 | 0.8352910602 | 1.2×10⁻⁸ |
  | B17 | 0.8352710872 | 0.8351473006 | 1.24×10⁻⁴ |
  | B19 | 0.8354088638 | 0.8350921012 | 3.17×10⁻⁴ |
  | B21 | 0.8351431025 | 0.8350840426 | 5.9×10⁻⁵ |
  | B23 | 0.8349254026 | 0.8349253208 | 8.2×10⁻⁸ |

- Polishing the package's own points gains almost nothing, so the package's minima are
  local minima. For B11, B13, B17, B19 and B21 there are deeper basins elsewhere.
  B19's, at φ ≈ 0.233, is the largest correction.
- B23 goes below T from 18 of the 29 starts. No other witness goes below T from any
  start, and no best outer bound falls below T: no new disqualification.
- The ordering moves: B19 and B21 now come just after B23, and B17 is close behind.
  The policy file's "certificate-efficient, thinner margin: B19–B21" is truer than its
  numbers said. "Balanced shallow: B11–B13" is unchanged.

**W5 — efficiency.**

- The eight cells of §7 are the eight lowest by outer bracket among Round 19's 77
  marked cells: the lowest exact centres 1–7 and 9.

  | | B7 mean / max | B9 mean / max | B11 mean / max |
  | --- | ---: | ---: | ---: |
  | benchmark | 160.625 / 162 | 154 / 158 | 119.125 / 120 |
  | eight lowest by outer | 160.625 / 162 | 154 / 158 | 119.125 / 120 |
  | eight lowest by exact | 160.375 | 154.375 | — |

- The four-cell scan (§7's second line) uses the four reference cells, the four lowest
  by exact. Its per-cell nodes and pending counts rebuild for B7 to B19, and B21 and B23
  give mean pending 64.

**W6 — the deeper comparison.** The benchmark's `deep_certificate_comparison` is
reported nowhere in the text. B7, B9, B11 and B13 to lift depths 14 and 16 on cell-00 and
cell-01 give all 16 recorded node, CERT and PENDING counts.

**W7 — the tails.**

- The depth-50 cell is the tracked branch's (RUN-022), τ = 3.825×10⁻⁴. Each witness's
  tree is run once to its deepest depth and cut. A max-depth-d probe is that tree
  stopped at d.
- Every count of the benchmark's `fine_depth50_tail` rebuilds:

  | depth | B7 | B11 | B13 | B19 | B21 |
  | ---: | ---: | ---: | ---: | ---: | ---: |
  | 18 | 653 (0.249%) | 778 (0.297%) | 823 (0.314%) | 736 (0.281%) | 724 (0.276%) |
  | 20 | 1,015 (0.0968%) | 1,986 (0.189%) | 1,959 (0.187%) | 1,543 (0.147%) | 1,473 (0.140%) |
  | 22 | 1,046 | 4,453 | | | |
  | 24 | 1,397 | | | | |
  | 26 | 2,143 (3.19×10⁻⁵) | | | | |

  (open leaves, and the fraction of the 2^d grid.) §11's depth-18 list leaves out B13,
  the largest there.
- B7's open leaves grow — 653, 1,015, 1,046, 1,397, 2,143 — while the fraction falls,
  because the grid grows by 4 every two levels and the leaves by less. A tail that keeps
  more open leaves at every depth is not contracting toward closure.
- The same trees with exact cores:

  | cell | closes at lift depth | nodes | worst slack |
  | --- | ---: | ---: | ---: |
  | track depth 50 | 21 | 2,509 | 1.1×10⁻⁶ |
  | track depth 46 | 23 | 3,671 | 7.8×10⁻⁷ |
  | track depth 43 | 25 | 5,369 | 1.7×10⁻⁷ |
  | cell-00 (depth 40) | 30 | 18,555 | 4.1×10⁻⁸ |
  | cell-01 | 31 | 22,009 | 5.0×10⁻⁸ |
  | cell-02 | 31 | 26,189 | 4.3×10⁻⁸ |
  | cell-03 | 34 | 36,193 | 6.6×10⁻⁹ |

  On the depth-50 cell the exact tree has 83 open leaves at depth 12, 100 at 18, 35 at
  20 and none at 21.
- What this does and does not show:
  - The bound is sound: each box's hull of the disc and the three cores lies inside
    every placement's hull.
  - The root is complete for T ≤ 0.8365: translations beyond t₇(0.8365) put a corner of
    B₇ at least d⋆ from the centre, which costs at least 0.8365 (Round 07's tₙ,
    RUN-010; the capacity reading is RUN-018's).
  - The areas are floating-point values of this tree's exact envelope, less the pilot's
    10⁻⁹ pad. At cell-03 the slack is 6.6 times that pad.
  - So these are not certificates. They do say that on every marked cell tried, B7
    alone would close, given an exact or interval lift bound. The escalation Round 20
    builds answers the pilot's sampling.

**W8 — what is shipped.** The package's 32 files are the document and its crystal, three
JSON files (benchmark, policy, validation), a schema, the manifest, the summary and 24
streams. No code: the emitter, the verifier, the search and every probe behind the
benchmark are absent. The manifest records each
stream's bits and digest, but not its bit order, child order or tag meaning.

## P — the package

**P1.**

- The benchmark's `mixed_shard_verification` is the summary file, and the summary's
  cells, lifts and B23 values are the manifest's.
- §9's digits are the benchmark's, and §12's u_d are its fractions.
- The crystal's search list is the benchmark's, rounded, but lists only B7, B11, B13,
  B19, B21 and B23. It omits B9, B15 and B17, three of the four lowest after B23.
- The escalation policy's states are `BASE-UNRESOLVED, NECESSITY-MARKED, LIFT-PROBE,
  LIFT-ACTIVE, LIFT-CERTIFIED, WITNESS-DISQUALIFIED, MULTI-WITNESS-REQUIRED, PENDING`.
  §2's are `BASE-CERTIFIED, NECESSITY-MARKED, LIFT-PROBE, LIFT-ACTIVE, LIFT-CERTIFIED,
  WITNESS-DISQUALIFIED, PENDING`.
- The manifest's statuses `NECESSITY-MARKED-PILOT`, `PARTIAL-UNRESOLVED` and
  `WITNESS-DISQUALIFIED-BY-COUNTEREXAMPLE-PILOT` are in neither list.

**P2.**

- `VALIDATION.json` describes the document and the crystal correctly: SHA-256, bytes,
  `$$` 98 and 4, both flags.
- Its digests of the manifest, the summary, the benchmark, the schema and the policy
  are the archive's.
- Its four key checks follow from the data.

## Round 20, read

| Round 20 | statement | verdict | note |
| --- | --- | --- | --- |
| §0 | summary; boxed verdicts | PASS / PARTIAL | the grammar as specified is PARTIAL (W2, W8) |
| §1–2 | the shard record; seven states | PASS | the policy file's states differ (P1) |
| §3 | incomplete, not failed | PASS | W3; W7 shows it in both directions |
| §4 | the counterexample theorem | PASS | W3 |
| §5 | B23 on cell-00 | PASS | W3 |
| §6 | no positive value in absence | PASS | W3 |
| §7–8 | shallow efficiency; disc-like witnesses | PASS | W5; the eight cells' choice unstated |
| §9 | search minima | PASS as recorded | W4: every one is lower; B19 by 3.2×10⁻⁴ |
| §10 | Pareto choice | PASS as a principle | its forcing axis is mismeasured (W4) |
| §11–12 | horizon ranking; B7's tail | PASS as computed / PARTIAL as a finding | W7: the sampled tails do not contract toward closure; exact cores close |
| §13–14 | ρ_K; the state machine | PASS | definitions |
| §15–16 | the reference shard; its replay | PASS | W1–W3; the replay code is not shipped (W8) |
| §17–20 | cache; odd-n lesson; Fourier; production rule | n/a | policy |
| §21–23 | ledger; Round 21 | n/a | the ledger is reviewed in the registry |

## The status registry

Round 20's 18 judgments are reviewed, and its 8 first-used phrases defined:

- `CURRENT CERTIFICATE INCOMPLETE`, `IMPLEMENTED + REPLAYED`, `NOT YET ACHIEVED`,
  `REJECTED AS INFERENCE`;
- `NO-COUNTEREXAMPLE-FOUND`, `VIABLE-PROVED`, `WITNESS-DISQUALIFIED`,
  `REFERENCE-MIXED-SHARD-VERIFIED`.

Census v5 finds 12 of the 18; Round 20's status ledger is written as label lines.

The four PARTIALs are:

- "Mixed shard grammar: CLOSED" — the convention is unstated, the schema untyped, no
  code shipped;
- "Mixed shard grammar: IMPLEMENTED + REPLAYED" — replayed here, not by anything in the
  package;
- "Horizon-dependent ranking: NUMERICAL-EVIDENCE" — it ranks the sampled lift bound;
- "B7 full closure of marked cell: NOT YET ACHIEVED" — true of the pilot's bound only.

| verdict | judgments |
| --- | ---: |
| PASS | 9 |
| PARTIAL | 4 |
| n/a | 5 |

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's B23 outer bound | 1 | A0 |
| a marker's outer area | 1 | W1 (also P1) |
| a lift's pending count; a stream bit; another convention said to match | 3 | W2 (the first also P1) |
| the text's B23 outer | 1 | W3 |
| a recorded placement; a new disqualification claimed; this search's B19 minimum | 3 | W4 |
| the text's eight-cell mean; the eight said to be the lowest exact | 2 | W5 |
| a deeper probe's count | 1 | W6 |
| a tail count; an exact-core closure depth; "open leaves grow" denied | 3 | W7 |
| code said to be shipped | 1 | W8 |
| the policy made to match §2 | 1 | P1 |
| the schema's recorded digest; a VALIDATION key check | 2 | P2 |
| a review dropped | 1 | R1 |

Four controls:

- the unmodified inputs;
- the document with CRLF line endings;
- the manifest's cells in reverse order;
- the benchmark with its keys sorted.

**20 / 20 plants, 4 / 4 controls, 266 / 266 must-stay-green assertions; no check
without a plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 1 | 1 |
| W1 | 1 | 1 |
| W2 | 3 | 3 |
| W3 | 1 | 1 |
| W4 | 3 | 3 |
| W5 | 2 | 2 |
| W6 | 1 | 1 |
| W7 | 3 | 3 |
| W8 | 1 | 1 |
| P1 | 3 | 3 |
| P2 | 2 | 2 |
| R1 | 1 | 1 |

## What this round does not establish

- **Any 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.
- **That the search minima are the minima.** This tree's search finds lower placements.
  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.** B7 with exact cores closed all seven cells it was run
  on: three on the track and the four reference cells. It was not run on the rest of
  Round 19's 77.