# RUN-036 — Round 33: the dominance theorem is right and the four new trees rebuild node for node; its audits rest on MPFR, not recomputed here — and the certificates the ledgers do not keep, this tree regenerates from density and rule

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-26
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_33_Cross_Backend_Dominance_v0.1.md` (AMRAL-LUC-FC-R33, dated 2026-09-20) and its package, read from the archive:

- `CROSS_BACKEND_DOMINANCE_SPEC.json`, `CELL47_STATUS.json`, `CELL48_STATUS.json`,
  `DEPTH30_BATCH_STATUS.json`, `COVERAGE_DASHBOARD.json`, `SCOPE_LEDGER.json`, `VALIDATION.json`,
  `State_Crystal`;
- the two dominance audits, marker 48's replay, and the trees, manifests and ledgers of cells 47,
  48, 68, 75 and 76; Round 29's marker 48 and Round 32's cell 47 again;
- one script, read only after this tree's lifts and certificates were computed, and never run.

**Tools:** [`run036_round33_dominance.py`](../code/run036_round33_dominance.py) — RUN-030's lift and RUN-031's certificates on four more cells; the witness-core states counted on this tree's lifts; RUN-032's and RUN-034's outer polygons for marker 48; the corrected exact kernel
**Logs:** [`run036-round33-dominance.json`](../data/gate-logs/run036-round33-dominance.json)

**Verdict: PASS for the dominance theorem and its polygon corollary (read), the audits' state counts, the trees of cells 48, 68, 75 and 76, all 9,285 leaves of cell 48, marker 48's upper bound, the ledgers' vertex totals and the stage counts; UNMEASURED for the audits' own results, which compare mpmath enclosures with MPFR ones and MPFR is not on this machine (PARTIAL for their root containment: t₃, t₅, t₇ yes, π and √3 not recomputable); PASS for keeping cells 47 and 48 below publication-candidate, with one correction to its reason — the missing certificates are reproducible without their bytes.**

- **The theorem.** A supremum over a subset is no larger, and ρ_A ≤ ρ_B, so a point that A
  certifies inside every defining disk is certified by B. The polygon corollary follows vertex by
  vertex, and the exact area does not depend on the backend. Both are correct as stated.
- **The audits.**
  - Each audit checks every (rotation, half-widths) state once and every leaf's translation
    separately. That is sound, since a core's corner box is its unit box plus the translation.
  - The states are 1,564 and 1,574 on this tree's own lifts of cells 47 and 48, as the audits say:
    5.9 leaves per state.
  - Their results compare the Round 28 root's mpmath enclosures with the Round 31 root's MPFR ones,
    and are not recomputed here: MPFR is not on this machine, and bringing it in would mean a
    download. Nor is the least radius gain, 2.15×10⁻⁶⁴.
  - The part of the root containment that needs no MPFR holds: the MPFR t₃, t₅, t₇ lie inside Round
    28's (RUN-034 W1).
  - Every leaf of both cells is certified here directly, with no mpmath and no MPFR (below, and
    RUN-035).
- **The trees.**
  - Cells 48, 68, 75 and 76, lifted here to depth 30, are the shipped trees node for node and axis for
    axis: 18,569/9,285, 18,509/9,255, 21,895/10,948 and 21,863/10,932 nodes and leaves.
  - Their least slacks are the manifests' to 2×10⁻¹⁶, and cell 47's three files are Round 32's
    bytes.
- **Cell 48's ledger.**
  - Every one of the 9,285 leaves passes on its intended box at the ledger's density, with the
    ledger's hull count and its margin to 2.3×10⁻¹⁶.
  - The densities are the first that pass from the start band on every leaf. The least margin is
    1.0316×10⁻⁷, on `110101101001101011111010` at 16,384.
- **Marker 48.**
  - The replay row is Round 29's marker file.
  - Its full polygon under the package's normals and ceiling, rebuilt here from this tree's support
    bounds, has the replay's 2,601 active constraints and its area. The float-reduced polygon is
    Round 29's (2,560 constraints; 41,557 bits over 41,557).
  - This tree's certificate needs 128 normals and has a margin of 4.03×10⁻⁵; the exact margin is
    1.02×10⁻⁴.
  - The MPFR replay's record is not shipped.
- **The certificate entity.**
  - The ledgers keep index, path, status, density, hull count and margin, and the hulls total
    2,163,069 and 2,109,700 vertices — 34.6 and 33.8 MB at 16 bytes a vertex. So §8–10 are right
    that no polygon is stored.
  - But this tree regenerates every leaf's polygon from its density and the emitter's rule, with its
    hull count on 9,228 and 9,285 of the leaves and its margin to 2.6×10⁻¹⁶. The regenerated vertices
    are the emitter's up to its 17-digit rounding.
  - What the ledgers lack is a byte-level transcript, not the certificate: an independent verifier
    can recover it without trusting the PASS bit, as §9 asks, only not bit for bit.
- **The stage counts.** Two candidates, four arithmetic-closed-or-better, and 62 = 66 − 4 at a lower
  stage. The batch's leaf counts and slacks for 68, 75 and 76 are this tree's lifts'.
- **What the package ships.** `r33_enclosure_dominance.py` (108 lines) loads backend A through
  120- to 150-digit decimal strings and backend B through Round 31's wrapper, which rounds its inputs
  to nearest (RUN-034). It reads Round 32's working folder, which is in the archive under other names.

Drill: 16 / 16 plants caught by exactly the named checks, 4 / 4 controls undisturbed,
202 / 202 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 33 claims

1. **Cross-backend enclosure dominance** (§1–3). R^B_j ⊆ R^A_j and ρ_B ≥ ρ_A give G_A^cert ⊆
   G_B^cert. A polygon certified vertex by vertex under A transfers to B, and the exact area does not
   depend on the backend. The audit is by unique core states, about 6× fewer than the leaves.
2. **The audits** (§4, §7).
   - Cell 47: root containment (t₃, t₅, t₇, π, √3), base box, base cores and witness boxes all PASS.
     1,564 unique states, 0 failures, least radius gain 2.15×10⁻⁶⁴.
   - Cell 48: the same, with 1,574 states and 0 failures.
3. **Cell 48** (§5–6).
   - 18,569 nodes, 9,285 leaves, depth 30; 9,285/9,285 mpmath PASS.
   - Densities 256: 824, 512: 3,438, 1,024: 3,725, 2,048: 1,065, 4,096: 183, 8,192: 43, 16,384: 7.
     Least margin 1.03164×10⁻⁷.
   - Marker 48: an independent exact polar replay and an MPFR replay both PASS. Outer
     0.8349325735415322, margin 6.74265×10⁻⁵.
4. **The certificate entity** (§8–10, §13).
   - The Round 32 ledgers keep no hull vertices, source tags or exact areas, so dominance has nothing
     to transfer, and cells 47 and 48 stay below publication-candidate.
   - Future emitters keep either the full rational hull or a deterministic transcript.
   - The hulls total about 2.16×10⁶ and 2.11×10⁶ vertices.
5. **The batch and the stages** (§11–12).
   - Cells 68, 75 and 76 have trees: 18,509/9,255, 21,895/10,948, 21,863/10,932.
   - Arithmetic-closed-or-better: {47, 48, 69, 70}, 4/77. Publication-candidate: {69, 70}, 2/77.
   - Geometry is unchanged at 66/77. a_Leb ≥ 0.835 is NOT CERTIFIED.

## A — sources

**A0.** 36 anchors. Round 33 has no reference list.

## W — rebuilt

| check | Round 33 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | §1–4, §7 | the theorem; the unique witness-core states on this tree's lifts | PASS; the MPFR results not recomputed |
| W2 | §5, §11 | cells 48, 68, 75, 76 lifted here; cell 47's files | PASS, node for node |
| W3 | §5 | cell 48's 9,285 leaves on their intended boxes | PASS; the start bands on every leaf |
| W4 | §6 | marker 48's replay row, full and reduced polygons, this tree's certificate | PASS |
| W5 | §8–10 | the ledgers' fields, the vertex totals, a regeneration | the totals as stated; the certificates regenerate |
| W6 | §11–13 | the stage counts and the batch | PASS |
| W7 | — | the dominance script | sound audit; A by decimal strings, B by Round 31's wrapper |

## P — the package

**P1.**

- The spec's two audits are the two dominance files, and so are the status files' copies.
- Each status file matches its ledger (counts, least margin and path, digest, density counts) and
  its manifest. Both are prototypes, not publication candidates.
- The dashboard and the batch carry the status files' states.
- The scope ledger closes the certificate-entity finding as an ABI requirement, and the geometry
  scope unchanged.
- The crystal carries the two state counts, {47, 48, 69, 70} and 9,285/9,285.

**P2.**

- `VALIDATION.json` describes the document and the crystal correctly: SHA-256, bytes, `$$` counts,
  both flags.
- Its eight digests are the archive's, and its twelve checks follow from the data.

## Round 33, read

| Round 33 | statement | verdict | note |
| --- | --- | --- | --- |
| §1–2 | the dominance theorem and its polygon corollary | PASS | reading (W1) |
| §3 | about 6× fewer states than leaves | PASS | 5.9, on this tree's lifts (W1) |
| §4, §7 | the audits: containment, cores, witness boxes; 0 failures; gain 2.15×10⁻⁶⁴ | UNMEASURED | MPFR not here; root t₃, t₅, t₇ contained (RUN-034); every leaf certified here directly |
| §5 | cell 48's tree and 9,285/9,285 | PASS | W2, W3 |
| §6 | marker 48's replays | PASS / UNMEASURED | the full polygon rebuilt; the MPFR run's record not shipped |
| §8–10 | no certificate entity; the vertex totals | PASS | as stated (W5); but the certificates regenerate from density and rule |
| §11 | cells 68, 75, 76's trees | PASS | W2 |
| §12–13 | the stage counts; no promotion | PASS | W6 |

## The status registry

Round 33's 18 judgments are reviewed. It uses no phrase for the first time, and census v5 adds none.

| verdict | judgments |
| --- | ---: |
| PASS | 7 |
| PARTIAL | 2 |
| UNMEASURED | 9 |

The nine UNMEASURED judgments are the audits' own PASSes and the MPFR marker replay. The two PARTIAL
ones are the root-containment judgments: t₃, t₅ and t₇ are shown here, π and √3 are not.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's 1,574 | 1 | A0 |
| an audit's state count | 1 | W1 (also P1) |
| a tree's bit; a manifest's slack | 2 | W2 |
| a ledger margin | 1 | W3 (also W5) |
| a certificate here | 1 | W3 |
| marker 48's area | 1 | W4 |
| a ledger hull count | 1 | W5 |
| the dashboard's count; a batch slack | 2 | W6 |
| the script's facts | 1 | W7 |
| a status file's margin; the crystal's 1,574 | 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 spec with its keys sorted;
- the batch with its keys sorted.

**16 / 16 plants, 4 / 4 controls, 202 / 202 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 | 1 | 1 |
| W5 | 2 | 2 |
| W6 | 2 | 2 |
| W7 | 1 | 1 |
| P1 | 3 | 3 |
| P2 | 2 | 2 |
| R1 | 1 | 1 |

## What this round does not establish

- **The MPFR half of any audit here.** The enclosures B computes are not recomputed.
- **Cells 68, 75 and 76's leaves.** Only their trees are checked. The package has no ledger for them
  yet.