# RUN-039 — 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

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_36_Depth30_DFRONT_Recovery_v0.1.md` (AMRAL-LUC-FC-R36, dated 2026-09-20) and its package, read from the archive:

- the two shard files, `COVERAGE_DASHBOARD.json`, `DEPTH30_BATCH_COMPLETE.json`,
  `DFRONT_IMPLEMENTATION_SPEC_v0.1.json`, `GEOMETRY_RECOVERY_AUDIT.json`, `DEPTH32_QUEUE.json`,
  `CELL49_SHARD_STATUS.json`, `VALIDATION.json`, `State_Crystal`;
- cells 75 and 76's certificate packages — `cell75.rhcert` (44,382,455 bytes) and `cell76.rhcert`
  (44,229,325 bytes), their indexes, manifests and replay records;
- the DFRONT package — cell 33's frontier (126,327 bytes), its manifest and recovery supplement, a test
  frontier and the codec's validation, each also shipped as a copy;
- the reference bundle's cell 49 manifest and three scripts; Round 35's ledgers, Round 33's trees,
  Round 29's markers and Round 26's ledger again.

**Tools:** [`run039_round36_dfront.py`](../code/run039_round36_dfront.py) — RUN-037's decoder and replay on cells 75 and 76's bytes; a DFRONT decoder, encoder and exact disjointness sweep written from the spec; RUN-029's padded lift of cell 33 to depth 36; RUN-030's lift of cell 49 to depth 32 and a tree encoder in the package's convention
**Logs:** [`run039-round36-dfront.json`](../data/gate-logs/run039-round36-dfront.json)

**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.**

- **Cells 75 and 76's certificates.**
  - Decoded here: the headers, the leaves using up each file, the indexes of L + 1 offsets.
  - On every one of the 10,948 and 10,932 leaves, the density and hull count are Round 35's ledgers',
    which RUN-038 found to be this tree's own certificates; the margins agree to 2.7×10⁻¹⁷.
  - Every one of the 2,627,357 and 2,617,252 stored vertices is in its tagged core under this tree's
    integer arithmetic. Every polygon is strictly convex, and every area is above 167/200.
  - The least margins, 6.251776141237558×10⁻⁸ (leaf 8,258) and 2.0477267269918978×10⁻⁸ (leaf 9,570),
    and every chunk's least margin, are the replay records', bit for bit.
- **DFRONT.**
  - Both shipped frontiers decode from the spec alone: canonical order, every m below 2^k, exactly
    interior-disjoint under a sweep written here, and exact volumes 1/2 (the test frontier) and
    14027/2³⁶.
  - This tree's own lift of cell 33 to depth 36 under the package's pad leaves the same 14,027 boxes
    open, code for code, after the manifest's 129,401 nodes and 50,674 closed leaves. The volume agrees
    to 10⁻²², and the least closed slack, 3.362067513×10⁻⁹, is 6×10⁻¹⁶ from the manifest's.
  - The file itself, written here from the spec and this tree's boxes, is the shipped 126,327 bytes,
    byte for byte.
  - The dependency hash is the SHA-256 of the payload's canonical JSON, recomputed here. The payload's
    kernel and scheduler digests are those of the Round 19 kernel and the Round 26 scheduler in the
    archive.
  - The payload's base box is this tree's to 5 and 6 ulps, and its t₇ to 4: the package halves the box
    in floating point, while this tree rounds the exact box once.
- **Cell 49.**
  - Lifted here to depth 32: 19,971 nodes, 9,985 internal and 9,986 leaves, the ledger's count.
  - Encoded in the package's convention, the topology and the axis stream hash to the manifest's
    digests.
  - The least slack, 4.494058015×10⁻⁸ at `01000100100100101111101`, is 3×10⁻¹⁶ from the manifest's.
  - The base box is this tree's to 5 and 6 ulps, and its axis trace is this tree's.
- **The depth-32 queue.**
  - Its twelve cells are the Round 26 ledger's cells closing at 32, with the ledger's paths, node counts
    and marker margins.
  - They are in the queue's own order — fewer nodes first, then the larger margin: 49, 64, 66, 12, 59,
    57, …
  - The margins are Round 26's, not Round 29's certified ones. No two of the twelve share a node count,
    so the margins decide nothing.
- **The stages.**
  - Candidates {47, 48, 68, 69, 70, 75, 76}: 7/77 and 7/66, with 59 below; the batch is complete.
  - Geometry is unchanged at 66 complete, 11 residual and 51 at depth 36, with no new closure. The
    depth-38 attempt is deferred with no effect on the proof state.
- **The scripts,** read after the rebuild and never run (W6).
  - The codec's digest is the spec's and the validation's.
    - It writes the boxes sorted, deduplicated and checked disjoint, under the header `<8sHHII`, whose
      depth field is the base depth.
    - Its decoder refuses trailing bytes, a non-canonical order and overlaps.
  - The codec's own self-test writes a full partition of cell 70 (four boxes, volume 1). That is not
    the shipped test frontier, which is cell 31, two boxes, volume 1/2, with a parent hash. The harness
    that wrote the shipped file and ran the validation's eight tests is not shipped.
  - Cell 33's rebuild takes the lower bound max(a priori, exact hull − 10⁻¹⁰). An exception in the exact
    kernel falls back to the a-priori bound, which can only leave more boxes open.
  - The shard generator halves the base box in floating point along the marker's trace (hence the
    ulps). It walks in preorder, lower child first, and stops if a box is still open at the closure
    depth.

Drill: 16 / 16 plants caught by exactly the named checks, 4 / 4 controls undisturbed,
181 / 181 must-stay-green assertions held. Each of the ten checks was blinded in turn, and each was
reported blind in exactly as many cases as plants name it.

---

## What Round 36 claims

1. **Cells 75 and 76** (§0–2).
   - 10,948/10,948 and 10,932/10,932 bytes-only replay, with 2,627,357 and 2,617,252 vertices in
     44,382,455 and 44,229,325 bytes.
   - Least margins 6.2518×10⁻⁸ and 2.0477×10⁻⁸; MPFR dominance over 1,690 and 1,688 states.
   - Both PUBLICATION-CANDIDATE-SHARD.
2. **The depth-30 batch** (§3–4). 7/7 candidates: 7/77 ≈ 9.09 % and 7/66 ≈ 10.61 %, with 59 below.
3. **DFRONT-v0.1** (§5–6). Dyadic codes per axis, independent of split order, and a verifier with an
   exact sweep.
4. **Cell 33's recovery** (§7, §10).
   - Rebuilt from the root to Round 26's durable counts: 129,401 nodes, 14,027 pending,
     V₃₆ = 2.9748317031636626×10⁻⁸, error 0.
   - Kept in 126,327 bytes; no old frontier bytes were found.
5. **Depth 38** (§8). Attempted, over the time limit, and without effect.
6. **The depth-32 queue** (§11).
   - Twelve cells, with 49, 64, 66, 12, 59, 57 first.
   - Cell 49's tree is rebuilt from durable sources: 19,971 nodes and 9,986 leaves, the ledger's count.

## A — sources

**A0.** 34 anchors. Round 36 has no reference list.

## W — rebuilt

| check | Round 36 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | §0–2 | cells 75 and 76's bytes decoded; every vertex in this tree's arithmetic; each leaf against Round 35's ledgers | PASS, 10,948 and 10,932 |
| W2 | §5–7 | both DFRONT files from the spec; cell 33 against this tree's lift, box for box and byte for byte | PASS |
| W3 | §11 | cell 49's tree lifted, encoded and hashed | PASS |
| W4 | §11 | the queue against the Round 26 ledger | PASS |
| W5 | §3–4, §8–10 | the stages; the recovery audit | PASS |
| W6 | — | the three scripts | read; the codec's self-test is not the shipped test frontier |

## P — the package

**P1.**

- Cells 75 and 76's manifests have their files' digests and sizes, and the replay records' chunks tile
  the leaves.
- The shard files carry the manifests' figures, the packages' sizes and digests, and Round 33's tree
  manifests. The density counts are the ledgers'.
- The batch's seven entries name manifests the archive holds, with their digests.
- The DFRONT records agree with the bytes and with each other:
  - every shipped copy is identical;
  - the manifest's digest and size are those of the frontier's bytes;
  - the supplement's digest is that of the original manifest;
  - the validation's reference is the test frontier;
  - the audit's rebuild record is the manifest's.
- Cell 49's status file has its manifest's figures.
- The crystal has the batch, the vertex counts, the recovery figures and cell 49's tree.

**P2.** `VALIDATION.json` describes the document and the crystal correctly. Its ten file digests are
the archive's, its three package digests are the package's, and its fourteen checks follow from the
data.

## Round 36, read

| Round 36 | statement | verdict | note |
| --- | --- | --- | --- |
| §0–2 | 10,948/10,948 and 10,932/10,932 bytes-only replay | PASS | this tree's arithmetic (W1) |
| §0–2 | cells 75 and 76: PUBLICATION-CANDIDATE-SHARD | PASS | the bytes (W1); the ledgers and markers in RUN-038; the MPFR dominance not recomputed |
| §7 | CELL33-DFRONT-RECOVERY: PASS | PASS | this tree's own frontier, box for box and byte for byte (W2) |
| §13 | runtime state recovery → DFRONT PASS | PASS | as §7 |
| §0, §13 | a_Leb ≥ 0.835 NOT CERTIFIED | PASS | reading |

## The status registry

Round 36's 8 judgments are reviewed. It uses one phrase for the first time, DFRONT PASS. It is defined
in-line (§13): a frontier rebuilt from the root, kept as DFRONT codes, and matching the durable counts it
must.

| verdict | judgments |
| --- | ---: |
| PASS | 8 |

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's 14,027 | 1 | A0 |
| a vertex refused | 1 | W1 |
| a ledger density | 1 | W1 (also P1) |
| a DFRONT box | 1 | W2 (also P1) |
| a box dropped from this tree's lift | 1 | W2 |
| cell 49's axis digest | 1 | W3 (also P1) |
| the queue's order; a queue row's node count | 2 | W4 |
| the dashboard's count | 1 | W5 |
| the scripts' facts; the payload's kernel digest | 2 | W6 |
| a batch entry's digest; the crystal's 2,617,252 | 2 | P1 |
| a VALIDATION digest; a VALIDATION check | 2 | P2 |
| a review dropped | 1 | R1 |

Four controls:

- the unmodified inputs;
- the document with CRLF line endings;
- the queue with its keys sorted;
- the batch file with its keys sorted.

**16 / 16 plants, 4 / 4 controls, 181 / 181 must-stay-green assertions; no check without a
plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 1 | 1 |
| W1 | 2 | 2 |
| W2 | 2 | 2 |
| W3 | 1 | 1 |
| W4 | 2 | 2 |
| W5 | 1 | 1 |
| W6 | 2 | 2 |
| P1 | 5 | 5 |
| P2 | 2 | 2 |
| R1 | 1 | 1 |

## What this round does not establish

- **The MPFR dominance audits**, as before.
- **The validation's eight codec tests.** Their harness is not shipped. The codec is read (W6) and every
  shipped DFRONT file is decoded and checked here (W2), but the tests themselves are not rerun.
- **§10's search for old frontier bytes.** It concerns the author's machines and drives, and nothing
  here can repeat it.
- **Depth 38.** Round 36 claims nothing there. Round 37 does, and RUN-040 checks it.