# RUN-027 — Round 24: the first complete-edge certificate rebuilds bit for bit; the tail numbers mix two bounds, and cell 3 closes too

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-25
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_24_Residual_Joint_Admission_v0.1.md` (AMRAL-LUC-FC-R24, dated 2026-09-19) and its package, read from the archive:

- `RESIDUAL_CORE.json`, `JOINT_ADMISSION_SPEC.json`, `VALIDATION.json`, `State_Crystal`;
- the reference certificate `cell0_B7_complete_edge/` (manifest, replay summary,
  `topology.bin`), and five edge records;
- no code is shipped.

**Tools:** [`run027_round24_residual_joint.py`](../code/run027_round24_residual_joint.py) — RUN-023's exact-core B7 lift, keeping the tree, its least slack and its closest call; RUN-023's stream emitter; RUN-026's 96-direction pilot and cells
**Logs:** [`run027-round24-residual-joint.json`](../data/gate-logs/run027-round24-residual-joint.json)

**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.**

- **The certificate rebuilds bit for bit.**
  - The package's deep B7 is RUN-023's exact-core lift. Its manifest says so: "exact-inner-hull
    of the common base cores and the B7 eroded core".
  - On cell 0, this tree's lift to depth 30 has 18,555 nodes, 9,278 CERT leaves and none
    pending, so the forest identity holds.
  - Emitted with Round 20's convention (lower child first, most significant bit first), its
    topology is `topology.bin`, all 2,320 bytes.
  - Its least slack is 4.1174×10⁻⁸ with the pilot's 10⁻⁹ pad. That is the manifest's
    4.2074×10⁻⁸ at its 10⁻¹⁰ pad, to 10⁻¹⁷.
  - No box's exact area comes within 4.15×10⁻⁹ of T, so neither pad decides anything.
- **Cells 1–3 rebuild at every recorded depth.**
  - At depth 30 the exact tree leaves 5, 152 and 1,098 boxes open: 6.79×10⁻¹⁰, 2.06×10⁻⁸
    and 1.49×10⁻⁷ of volume.
  - Cells 1 and 2 close at depth 31, with 22,009 and 26,189 nodes, inside the recorded depth-32
    budget. Cell 3's depth-30 record is 31,797 nodes, 14,801 CERT leaves and 1,098 open.
  - The recorded B7 depth-24 edge on cell 0 (12,501 nodes, 799 open) is the same exact tree.
- **The tail numbers mix two bounds.**
  - §6's cell-0 tail runs 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. The first two are Round 23's 96-direction pilot (420 and 1,488 open). The
    other four are the exact tree's.
  - With exact cores throughout, depths 16 and 22 give 7.61×10⁻⁴ and 2.51×10⁻⁵. The collapse
    from 22 to 24 is ×3.6, not ×7.4.
  - §9's cell 3 is the same: 9.70×10⁻⁴ and 5.85×10⁻⁵ from the pilot, 1.49×10⁻⁷ from the exact
    tree. The quoted ×393 is ×217 with exact cores alone.
- **Cell 3 closes.** Run on, the exact tree closes it at lift depth 34 with 36,193 nodes. At
  depth 32 it has 445 boxes open. So the four hardest cells' residual is {3} at the tested
  budgets — as §22 says — and empty by depth 34. RUN-026 closed all twelve cells by depth 35.
- **The theorems are right.**
  - Joint dominance J_B ≥ max_K J_K is monotonicity of the hull. On 16 random bases and
    placements of B7 and B11, the joint hull exceeds the larger single hull by at least
    5.0×10⁻⁴.
  - The joint lower hull of §17 is sound: eroded cores lie inside their bodies, so the hull of
    the cores is never the larger.
  - Pointwise joint necessity (§11) speaks only of single-witness values, and §12 says so.
- **The H19 record.** Cell 0's H19 depth-24 edge is the 96-direction pilot's. This tree has
  38,037 nodes and 11,985 open, against the package's 38,015 and 11,987 — the harmonic gap
  RUN-026 could not close.

Drill: 16 / 16 plants caught by exactly the named checks, 4 / 4 controls undisturbed,
203 / 203 must-stay-green assertions held. Each of the eleven checks was blinded in turn, and
each was reported blind in exactly as many cases as plants name it.

---

## What Round 24 claims

1. **Residual collapse** (§0, §5–10).
   - Deepening B7 on Round 23's four hardest cells closes cell 0 at depth 30 and cells 1 and 2
     at depth 32.
   - Cell 3 is left at depth 30 with 1.49×10⁻⁷ open, contracting ×393 since depth 22.
   - Round 23's residual was budget-relative.
2. **The first complete edge** (§7). Cell 0, B7, depth 30: 18,555 nodes, 9,278 CERT leaves,
   none pending; 18555 = 2(9278) − 1; the replay passes. REFERENCE-COMPLETE-EDGE-PILOT, not
   publication-grade.
3. **Joint witnesses** (§1–4, §11–13, §16–18).
   - J_B(H(q)) ≥ max_K J_K(H(q)), so a strict single edge makes a joint proof redundant.
   - 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; 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.
4. **Admission** (§14–15, §19–24). REDUNDANT, DEFERRED-CONTRACTING, CANDIDATE,
   NECESSARY-RELATIVE-TO-PORTFOLIO. Cells 0–2 are REDUNDANT, cell 3 DEFERRED-CONTRACTING. The
   residual {0,1,2,3} → {3} is a 75% reduction; cells 4–11 are untested.
5. **Round 25** (§25): an adaptive deep single-witness wave over all twelve, then 77, cells.

## A — sources

**A0.** 39 anchors. Round 24 has no reference list.

## W — rebuilt

| check | Round 24 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | §1–4, §11–13, §17–18 | dominance; the joint lower hull; the spec | PASS |
| W2 | §6, §9 | each tail value against both trees | two bounds in one sequence |
| W3 | §7 | cell 0's tree to depth 30; its topology | PASS, bit for bit |
| W4 | §8–10 | cells 1–3 at depth 30 and to closure | PASS; cell 3 closes at 34 |
| W5 | §15, §22–23 | the states and the residual at depth 36 | {3} at the budgets; empty at 36 |
| W6 | — | the H19 depth-24 record | the harmonic gap of RUN-026 |
| W7 | — | what is recorded and shipped | the lower rule stated; no code |

**W1 — the theorems.** For 16 random base configurations, placements of B7 and B11:

- the exact joint hull exceeds the larger single hull by at least 5.04×10⁻⁴;
- the hull of the base and the two cores eroded to radius 0.99 is never larger than the joint
  hull.

The admission spec's monotonicity and its warning ("does not prove the joint portfolio
succeeds") are §3's and §12's.

**W2 — the tails.**

| cell | depth | package | 96-direction pilot | exact tree | the package's is |
| ---: | ---: | ---: | ---: | ---: | --- |
| 0 | 16 | 9.34×10⁻⁴ | 9.340×10⁻⁴ (420) | 7.605×10⁻⁴ (342) | the pilot's |
| 0 | 22 | 5.17×10⁻⁵ | 5.170×10⁻⁵ (1,488) | 2.512×10⁻⁵ (723) | the pilot's |
| 0 | 24 | 6.94×10⁻⁶ | | 6.941×10⁻⁶ (799) | the exact tree's |
| 0 | 26 | 1.77×10⁻⁶ | | 1.770×10⁻⁶ (815) | the exact tree's |
| 0 | 28 | 9.39×10⁻⁸ | | 9.393×10⁻⁸ (173) | the exact tree's |
| 0 | 30 | 0 | | 0 | the exact tree's |
| 3 | 16 | 9.70×10⁻⁴ | 9.696×10⁻⁴ (436) | 8.428×10⁻⁴ (379) | the pilot's |
| 3 | 22 | 5.85×10⁻⁵ | 5.855×10⁻⁵ (1,685) | 3.238×10⁻⁵ (932) | the pilot's |
| 3 | 30 | 1.49×10⁻⁷ | | 1.490×10⁻⁷ (1,098) | the exact tree's |

Open boxes in brackets. The residual core's depth-16 and depth-22 volumes for all four cells are
the pilot's to 10⁻¹².

**W3 — the certificate.**

- The manifest's base cell is Round 23's cell 0, to 10⁻¹⁵.
- The lift: RUN-023's root and split, exact cores, run to depth 30, keeping the tree. It has
  18,555 nodes, 9,278 CERT leaves and none pending.
- Re-emitted in preorder, lower child first and most significant bit first, its 18,555 topology
  bits are `topology.bin` byte for byte (SHA-256 `a9139473…`).
- Least CERT slack: 4.1174202×10⁻⁸ with a 10⁻⁹ pad, so 4.2074202×10⁻⁸ with the manifest's
  10⁻¹⁰, which is the manifest's value.
- The closest call over all boxes the tree bounded, |area − T|, is 4.15×10⁻⁹.

**W4 — cells 1–3.**

| cell | open at 30 | volume at 30 | closes at | nodes |
| ---: | ---: | ---: | ---: | ---: |
| 1 | 5 | 6.79×10⁻¹⁰ | 31 | 22,009 |
| 2 | 152 | 2.06×10⁻⁸ | 31 | 26,189 |
| 3 | 1,098 | 1.49×10⁻⁷ | 34 | 36,193 |

The four edge records (cells 0–3, at depths 30, 32, 32, 30) are the exact trees cut at those
depths: nodes, CERT leaves, open boxes and volumes.

**W5 — the residual.** At the tested budgets, only cell 3 has open boxes, so the states are the
package's: three REDUNDANT, one DEFERRED-CONTRACTING. At depth 36 none has.

**W6 — H19.** The record's B7 row is the exact tree at depth 24. Its H19 row is the 96-direction
pilot's. This tree has 38,037 and 11,985 there, against 38,015 and 11,987 — 22 more nodes, 2
fewer open. That is RUN-026's unresolved harmonic convention again.

**W7 — what is recorded.** Thirteen files, no code. Unlike Round 23, the certificate states its
lower rule — "APR max exact-inner-hull(common base cores, B7 eroded core) − 1e-10" — and its
split rule. The residual core does not say that its depth-16 and depth-22 volumes come from a
different bound.

## P — the package

**P1.**

- The text's tail values are the residual core's and the records', rounded.
- The three strict edges added are the three COMPLETE records. The admission spec's current
  states are §15's.
- The replay summary matches the manifest. The crystal carries the same numbers.

**P2.**

- `VALIDATION.json` describes the document and the crystal correctly: SHA-256, bytes, `$$`
  208 and 6, both flags.
- Its digests of the residual core, the spec, the manifest and the replay summary are the
  archive's.
- Its eight checks follow from the data.

## Round 24, read

| Round 24 | statement | verdict | note |
| --- | --- | --- | --- |
| §0 | summary; boxed verdicts | PASS | the certificate rebuilds (W3); cell 3 closes too (W4) |
| §1–5 | single and joint values; dominance; budget-relative residual | PASS | W1 |
| §6 | cell 0's tail collapse | PARTIAL | two bounds in one sequence (W2) |
| §7 | the certificate | PASS | W3, bit for bit |
| §8 | cells 1 and 2 | PASS | W4 |
| §9–10 | cell 3 contracting; residual interpretation | PARTIAL | ×393 mixes bounds; ×217 exact; it closes at 34 (W2, W4) |
| §11–13 | pointwise necessity; limitation; switching first | PASS | W1 |
| §14–15 | admission states | PASS at the tested budgets | cell 3 is REDUNDANT at 34 (W5) |
| §16–21 | joint cost; grammar; counterexample; gates | PASS | W1 |
| §22–24 | residual {3}; untested cells; queue | PASS as computed | empty at depth 34; RUN-026 closed cells 4–11 |
| §25–26 | Round 25; handoff | n/a | |

## The status registry

Round 24's 20 judgments are reviewed, and its 6 first-used phrases defined: the four admission
states, `CONTRACTING` and `REFERENCE-COMPLETE-EDGE-PILOT`. Census v5 finds 8 of the 20: the
set-off verdicts and states.

| verdict | judgments |
| --- | ---: |
| PASS | 11 |
| PARTIAL | 2 |
| n/a | 7 |

The PARTIALs are cell 3's "CONTRACTING" and "JOINT-DEFERRED-CONTRACTING". Each is true at
depth 30. Their ×393 evidence mixes the pilot with the exact tree. And the cell closes at 34.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's ×393 | 1 | A0 |
| the admission spec's warning | 1 | W1 |
| a tail value's source; a recorded depth-22 volume | 2 | W2 |
| a topology bit; the recorded slack | 2 | W3 |
| cell 1's recorded nodes; cell 3's closing depth | 2 | W4 |
| the residual at depth 36; cell 3's recorded state | 2 | W5 (the second also P2) |
| H19's recorded open count | 1 | W6 |
| code said to be shipped | 1 | W7 |
| a crystal count | 1 | P1 |
| the residual core's recorded digest; a VALIDATION check | 2 | P2 |
| a review dropped | 1 | R1 |

Four controls:

- the unmodified inputs;
- the document with CRLF line endings;
- the residual core with its keys sorted;
- the certificate manifest with its keys sorted.

**16 / 16 plants, 4 / 4 controls, 203 / 203 must-stay-green assertions; no check without a
plant.**

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

## What this round does not establish

- **Any 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.
- **The joint-witness theorems' use.** No joint tree is built, here or in the package.
- **H19's node counts.** The harmonic convention is still unresolved (RUN-026).