# RUN-037 — 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

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-26
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_34_RHCERT_Publication_Candidates_v0.1.md` (AMRAL-LUC-FC-R34, dated 2026-09-20) and its package, read from the archive:

- `RHCERT_FORMAT_SPEC.json`, `PUBLICATION_CANDIDATE_SHARD_47.json` and `_48`,
  `ARITHMETIC_PROTOTYPE_SHARD_68.json`, `COVERAGE_DASHBOARD.json`, `CERTIFICATE_SIZE_ECONOMICS.json`,
  `MARKER68_A1.json`, `CELL68_LOWER_MIGRATION.json`, `VALIDATION.json`, `State_Crystal`;
- the two certificate packages — `cell47.rhcert` (36,506,484 bytes) and `cell48.rhcert`
  (35,657,092), their indices, manifests and replay summaries — and Round 33's trees and ledgers;
- three scripts, read only after this tree's replay was computed, and never run.

**Tools:** [`run037_round34_rhcert.py`](../code/run037_round34_rhcert.py) — an RHCERT decoder written from §2–5 and the spec file; every stored vertex checked in its tagged core with RUN-031's integer-interval enclosures on the intended boxes; RUN-031's certificates under the emitter's start bands for cell 68; RUN-032's and RUN-034's outer polygons for marker 68
**Logs:** [`run037-round34-rhcert.json`](../data/gate-logs/run037-round34-rhcert.json)

**Verdict: PASS for the format, both certificates, the sizes (but §11–12's packaged "about 8.1 MiB" and "about 7.6 MiB" are 8.02 and 7.54), cell 68's lower bound, marker 68's upper bound, the stage counts and the promotion of cells 47 and 48 — whose certificate bytes pass here under an arithmetic that shares nothing with mpmath or MPFR; UNMEASURED for marker 68's MPFR replay and the MPFR conditions of the promotion, not recomputed here.**

- **The format.**
  - Decoded from §2–5 and the spec file alone. The header is `<8sHBBI`: `L34RHC01`, version 1,
    exponent 18, one tag byte, the leaf count. The leaves use up the file exactly.
  - The index is `L34RHIDX`, version 1, with L + 1 offsets: each leaf's start, then the end. The
    spec does not give the index header's layout; its size gives it — `<8sHI` with the offset count,
    14 + 8(L + 1) bytes.
  - The leaf order is the topology's preorder and the ledger's order. Every leaf's density and hull
    count is the Round 32/33 ledger's, and all four tags occur.
- **The replay.**
  - Each leaf's box comes from this tree's root and the shipped topology, and its cores from this
    tree's integer intervals: no mpmath, no MPFR.
  - Of 2,163,069 + 2,109,700 stored vertices, none is refused its tagged core. Every polygon is
    strictly convex, and every area is above 167/200.
  - The least margins are 6.75024723062513×10⁻⁹ (leaf 4820) and 1.0316417842561089×10⁻⁷ (leaf
    6986): the replay summaries', bit for bit, since the area is exact from the bytes.
  - The margins are within 2.7×10⁻¹⁷ of the ledgers', since the 10⁻¹⁸ grid replaces 17-digit
    decimals.
  - This is the concrete certificate verified by a second, independent arithmetic — directly on the
    bytes, the job Round 33's dominance audit was to do through MPFR.
- **The sizes.**
  - 16.877 and 16.902 raw bytes a vertex.
  - The packages are 8,411,289 and 7,909,374 bytes, their digests the shard files', at 3.889 and
    3.749 bytes a vertex: the economics file to the last digit.
  - §11–12 round the packages to "about 8.1 MiB" and "about 7.6 MiB"; they are 8.02 and 7.54 MiB.
    The raw sizes, 34.8 and 34.0 MiB, are right.
- **Cell 68.**
  - No leaf ledger is shipped, only a summary. On this tree's lift (RUN-036), each leaf certified
    from its slack's start band upward gives the summary's density counts exactly: 820, 3,429, 3,632,
    1,140, 181, 46 and 7.
  - The least margin falls on the same leaf, `00111011001001010000010` at 16,384, within 2.6×10⁻¹⁷.
- **Marker 68.**
  - The replay row is Round 29's file.
  - The full polygon under the package's normals and ceiling has its 1,317 active constraints and
    its area, and the float-reduced one is Round 29's (1,284).
  - This tree's certificate needs 128 normals and has a margin of 9.96×10⁻⁵; the exact margin is
    1.61×10⁻⁴.
- **The stages.** {47, 48, 69, 70} are candidates, {47, 48, 68, 69, 70} are closed or better, and
  61 = 66 − 5 are pending. Every fraction is right.
- **What the package ships.**
  - The emitter reruns Round 32's with binary64 values rounded to the 10⁻¹⁸ grid, starting from the
    ledger's density. Its reference slacks and base boxes come from two files not in the archive,
    which the verifier does not need.
  - The verifier reads only the bytes. It rebuilds each box from Round 28's root and the topology,
    quantizes its mpmath cores outward on a 10⁻⁴⁰ grid (sound), and requires strict convexity.
  - Its mpmath enclosures still pass through decimal strings; this tree's replay does not.

Drill: 16 / 16 plants caught by exactly the named checks, 4 / 4 controls undisturbed,
201 / 201 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 34 claims

1. **RHCERT-v0.1** (§1–10).
   - A binary certificate: a header, per leaf a density and a vertex count, and per vertex
     zigzag-varint deltas on a 10⁻¹⁸ grid and a body tag, in the topology's leaf order, with an index
     of L + 1 offsets.
   - The verifier trusts only the bytes. It quantizes the directed cores inward to a 10⁻⁴⁰ grid and
     checks D²_max < R²_lower, strict convexity, and 200|S| > 167(2·10³⁶).
2. **The replays** (§11–12).
   - Cell 47: 9,228/9,228, 2,163,069 vertices, least margin 6.75024723×10⁻⁹, 36,506,484 raw bytes,
     about 8.1 MiB packaged.
   - Cell 48: 9,285/9,285, 2,109,700 vertices, least margin 1.03164178×10⁻⁷, 35,657,092 bytes,
     about 7.6 MiB.
3. **Promotion** (§13–14). The eight conditions — tree, topology, independent and MPFR marker
   replays, the bytes, their replay, the MPFR dominance, strict inequalities — make cells 47 and 48
   PUBLICATION-CANDIDATE-SHARD.
4. **Size economics** (§15). About 16.9 raw bytes a vertex, 3.5–3.9 packaged: full persistence is
   practical.
5. **Cell 68** (§16). 9,255/9,255 mpmath lower, least margin 2.23292×10⁻⁸; its marker's independent
   and MPFR replays pass: ARITHMETICALLY-CLOSED-PROTOTYPE.
6. **Stages** (§17–18). Candidates {47, 48, 69, 70}, 4/77; closed or better {47, 48, 68, 69, 70},
   5/77. Geometry is unchanged.

## A — sources

**A0.** 35 anchors. Round 34 has no reference list.

## W — rebuilt

| check | Round 34 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | §2–5 | both certificates decoded from the spec; the index; order, densities, hull counts | PASS; the index header's layout not in the spec |
| W2 | §6–12 | every vertex in this tree's enclosures; convexity; exact areas | PASS, 0 of 4,272,769 refused |
| W3 | §11–12, §15 | raw and packaged sizes | PASS; 8.02 and 7.54 MiB, not 8.1 and 7.6 |
| W4 | §16 | cell 68's summary from this tree's certificates under the start bands | PASS, exactly |
| W5 | §16 | marker 68's replay row, polygons and this tree's certificate | PASS |
| W6 | §13–14, §17–18 | the stage counts; which promotion conditions hold here | PASS; the MPFR ones not recomputed |
| W7 | — | the three scripts | the verifier sound; two emitter inputs not in the archive |

## P — the package

**P1.**

- Each certificate's manifest has its file's digests and sizes. Its replay summary has the
  manifest's counts, digests, density counts and least margin, and its shard file has the same and
  the package's size.
- The density counts are the ledger's.
- Cell 68's prototype file carries its migration record, its manifest and marker 68's margin.
- The dashboard's cells are the shard files', and the crystal carries both byte counts, both replays,
  the four candidates and Q = 10¹⁸.

**P2.** Its ten digests are the archive's, and its eleven checks follow from the data.

## Round 34, read

| Round 34 | statement | verdict | note |
| --- | --- | --- | --- |
| §1–10 | RHCERT-v0.1 and its verifier | PASS | decoded from the spec (W1); the verifier's quantization sound (W7) |
| §11–12 | 9,228/9,228 and 9,285/9,285; vertices; least margins; raw bytes | PASS | replayed here in integers (W2) |
| §11–12 | packaged about 8.1 and 7.6 MiB | PARTIAL | 8.02 and 7.54 MiB (W3) |
| §13–14 | promotion of 47 and 48 | PASS | the bytes pass under two arithmetics; the MPFR conditions not recomputed (W6) |
| §15 | size economics | PASS | W3 |
| §16 | cell 68 and marker 68 | PASS / UNMEASURED | the summary rebuilt, the polygon rebuilt; the MPFR replay not here |
| §17–18 | stages; geometry unchanged | PASS | W6 |

## The status registry

Round 34's 12 judgments are reviewed. It uses no phrase for the first time, and census v5 adds none.

| verdict | judgments |
| --- | ---: |
| PASS | 11 |
| UNMEASURED | 1 |

The UNMEASURED judgment is marker 68's MPFR replay.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's 2,109,700 | 1 | A0 |
| a header's exponent byte; an index offset | 2 | W1 |
| a vertex refused | 1 | W2 (also W6) |
| a replayed margin | 1 | W2 |
| a package's size | 1 | W3 (also P1) |
| the economics' rate | 1 | W3 |
| cell 68's density counts | 1 | W4 (also P1) |
| marker 68's area | 1 | W5 |
| the dashboard's count | 1 | W6 |
| the scripts' facts | 1 | W7 |
| a summary's digest; the crystal's 9,285/9,285 | 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 economics file with its keys sorted;
- the dashboard with its keys sorted.

**16 / 16 plants, 4 / 4 controls, 201 / 201 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 | 2 | 2 |
| W4 | 1 | 1 |
| W5 | 1 | 1 |
| W6 | 2 | 2 |
| W7 | 1 | 1 |
| P1 | 4 | 4 |
| P2 | 2 | 2 |
| R1 | 1 | 1 |

## What this round does not establish

- **MPFR anything.** Marker 68's MPFR replay and the dominance audits are not recomputed. For the
  lower bounds of cells 47 and 48, this tree's replay of the bytes is itself a second arithmetic.
- **Cell 68's certificate bytes.** They come in Round 35. Here only its summary is rebuilt.