# RUN-028 — Round 25: all twelve B7 closures rebuild node for node; the recorded closure depths are checkpoints, and six cells close a level earlier

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-25
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_25_Deep_B7_Closure_Wave_v0.1.md` (AMRAL-LUC-FC-R25, dated 2026-09-20) and its package, read from the archive:

- `CLOSURE_LEDGER.json` (twelve cells, 58 checkpoints), `STOPPING_POLICY.json`,
  `VALIDATION.json`, `State_Crystal`;
- nine per-cell incremental records, and Round 24's certificate again;
- `r25_incremental_wave.py`, read only after this tree's reconstruction, and never run.

**Tools:** [`run028_round25_closure_wave.py`](../code/run028_round25_closure_wave.py) — RUN-027's exact-core B7 lift, recording the least slack of the boxes each depth closes; RUN-026's twelve cells
**Logs:** [`run028-round25-closure-wave.json`](../data/gate-logs/run028-round25-closure-wave.json)

**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.**

- **All twelve closures rebuild.**
  - The wave is the exact-core B7 lift again — the same trees as RUN-026 and RUN-027.
  - All twelve close, with the ledger's nodes and CERT leaves, 340,540 nodes in all, and
    N = 2L − 1 on every tree.
  - All 58 checkpoints of the nine incremental cells agree in cumulative nodes, CERT leaves,
    open boxes and volume.
  - The ledger's "worst new slack" at each checkpoint is this tree's plus 9.0×10⁻¹⁰, to
    10⁻¹⁴: the pad difference, 10⁻⁹ here against 10⁻¹⁰ there.
- **The closure depths are checkpoints.**
  - §1 defines D_j as the least depth at which the tree has no open box. The wave only looks
    at depths 30, 32, 34 and 36, and the ledger records the first of those at which a cell is
    complete.
  - Six cells close a level earlier than recorded: cells 1, 2 and 4 at 31, cells 6 and 10 at
    33, cell 7 at 35. The true depths are 30, 31, 31, 34, 31, 30, 33, 35, 32, 34, 33, 30.
  - So D_max = 35, not 36; the mean is 32.0, not 32.5; the median is 31.5, not 32.
  - The coverage curve at 30, 32, 34 and 36 (3, 7, 11, 12) is right, since it is read at the
    checkpoints. At 31, 33 and 35 it is 6, 9 and 12. The least common budget is 35.
- **Volume monotonicity holds, and cell 7 illustrates it.**
  - On every level of every tree, the open volume never grows. The open-box count grows on 259
    of them.
  - Cell 7's volumes at 22–36 are §11's. Its open count rises from 1,793 at depth 26 to 1,918
    at depth 30 while the volume falls ×14.96.
- **The singleton cover stands.** B7 alone closes all twelve cells, so the joint residual is
  empty.
- **What the package ships.** `r25_incremental_wave.py` is 64 lines. It computes the wave:
  - its bound is max(the a-priori disc-point area, the exact hull of the disc, the base cores and
    B7's core − 10⁻¹⁰);
  - it walks depth first with checkpoints at 22, 24, …, 38.
  But it imports Round 19's continuation probe and exact kernel from an absolute path outside
  the package, and writes its records there. The a-priori term never binds: every translation
  in the root lies within 0.29 of the origin, inside the disc of radius ½, where the a-priori
  area is π/4.

Drill: 14 / 14 plants caught by exactly the named checks, 4 / 4 controls undisturbed,
182 / 182 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 25 claims

1. **The wave** (§0–4). B7 run deeper on all twelve of Round 23's cells closes every one:
   closure depths 30 ≤ D_j ≤ 36; D_max 36, mean 32.5, median 32; coverage 3/12, 7/12, 11/12,
   12/12 at depths 30, 32, 34, 36.
2. **The singleton cover** (§5–6, §12). {B7} is a strict cover of the twelve; the joint residual
   is empty; nothing is claimed for the 77-cell atlas or globally.
3. **Unresolved volume** (§7–11). The pending count can grow while V_d falls. V_{d+1} ≤ V_d under
   partitioning splits. Cell 7 runs 3.33×10⁻⁵ → 0 from depth 22 to 36.
4. **Capacity** (§14–15). Γ_B7(30, 32, 34, 36) = 3, 7, 11, 12, and D⋆ = 36 is a common budget.
5. **Production** (§16–23). B7 first, portfolio on true residuals; 340,540 nodes in all, 45,401
   at most; Round 26 runs the 77-cell atlas and upgrades the arithmetic.

## A — sources

**A0.** 59 anchors. Round 25 has no reference list.

## W — rebuilt

| check | Round 25 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | §8–10, §14–15 | monotonicity on every tree; capacity; D⋆ | PASS |
| W2 | §2, §19–20 | the twelve closures | PASS, node for node |
| W3 | — | the ledger's 58 checkpoints | PASS |
| W4 | §1–4 | D_j as defined; the statistics | six cells a level earlier |
| W5 | §7, §11 | cell 7 | PASS |
| W6 | §5, §12 | the singleton cover | PASS |
| W7 | — | the runner | the exact lift; loads code from outside the package |

**W2 and W4 — the closures.**

| cell | recorded D_j | least complete depth | nodes | CERT leaves |
| ---: | ---: | ---: | ---: | ---: |
| 0 | 30 | 30 | 18,555 | 9,278 |
| 1 | 32 | 31 | 22,009 | 11,005 |
| 2 | 32 | 31 | 26,189 | 13,095 |
| 3 | 34 | 34 | 36,193 | 18,097 |
| 4 | 32 | 31 | 21,271 | 10,636 |
| 5 | 30 | 30 | 18,509 | 9,255 |
| 6 | 34 | 33 | 31,209 | 15,605 |
| 7 | 36 | 35 | 45,401 | 22,701 |
| 8 | 32 | 32 | 26,439 | 13,220 |
| 9 | 34 | 34 | 41,077 | 20,539 |
| 10 | 34 | 33 | 31,793 | 15,897 |
| 11 | 30 | 30 | 21,895 | 10,948 |

The ledger's markers are the cells' outer brackets, to 10⁻¹³, all below T.

**W3 — the checkpoints.** Every one of the 58 checkpoints for cells 3–11 is the exact tree cut at
that depth: cumulative nodes, cumulative CERT leaves, open boxes, and open volume to 10⁻¹².

**W5 — cell 7.**

| depth | 22 | 24 | 26 | 28 | 30 | 32 | 34 | 36 |
| --- | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| open volume | 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 |

**W7 — the runner.** Read after the rebuild, never run.

- It reconstructs each cell from its path with Round 19's official split and lazy cascade.
- It bounds a box by max(f_a-priori(d_min), the exact hull − 10⁻¹⁰), where d_min is the box's
  least translation. It splits by the least child δ, as RUN-023 does.
- It extends the open boxes depth first between checkpoints and writes the incremental
  records.
- Its two imported modules sit in Round 19's reference folder, loaded from an absolute path.

## P — the package

**P1.**

- §2's table is the ledger, row for row. The stopping policy's coverage is the ledger's.
- The crystal lists the ledger's depths and coverage.

**P2.**

- `VALIDATION.json` describes the document and the crystal correctly: SHA-256, bytes, `$$`
  176 and 10, both flags.
- Its digests of the ledger and the policy are the archive's.
- Its ten checks follow from the ledger. "max_depth_36" is the checkpoint's maximum.

## Round 25, read

| Round 25 | statement | verdict | note |
| --- | --- | --- | --- |
| §0, §5–6, §12 | 12/12 closed; singleton cover; empty residual | PASS | W2, W6 |
| §1–3 | D_j and its statistics | PARTIAL | the ledger records checkpoints; six cells close earlier (W4) |
| §4, §14–15 | coverage; capacity; D⋆ = 36 | PASS at the checkpoints | 35 is the least common budget (W4) |
| §7–11 | pending count misleads; volume monotone; cell 7 | PASS | W1, W5 |
| §13, §16–18 | budget-relative infeasibility; B7 first | PASS | RUN-026 found the harmonics cannot close these cells at all |
| §19–21 | ledger; validation; what is missing | PASS | W2, W3 |
| §22–23 | Round 26; handoff | n/a | |

## The status registry

Round 25's 5 judgments are reviewed, and its 2 first-used phrases defined: `EMPTY` and
`NOT YET COMPUTED`. Census v5 finds 2 of the 5: the set-off JOINT-REDUNDANT and the set-off
STILL COMPUTE-DEFERRED.

| verdict | judgments |
| --- | ---: |
| PASS | 4 |
| n/a | 1 |

The review of "12-CELL REFERENCE SINGLETON B_7 COVER: COMPLETE" records that the recorded
closure depths are checkpoints.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's ×15 | 1 | A0 |
| the policy's monotonicity | 1 | W1 |
| a cell's recorded nodes | 1 | W2 (also P1) |
| a checkpoint's open count; the slack offset said not to be the pad | 2 | W3 |
| the recorded depths claimed as least depths; D_max claimed 36 | 2 | W4 |
| cell 7's ratio | 1 | W5 |
| a residual cell added | 1 | W6 (also P2) |
| the runner said not to compute | 1 | W7 |
| a crystal coverage | 1 | P1 |
| the ledger'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 ledger with its keys sorted;
- the policy with its keys sorted.

**14 / 14 plants, 4 / 4 controls, 182 / 182 must-stay-green assertions; no check without a
plant.**

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

## What this round does not establish

- **Any 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.