# RUN-035 — Round 32: cells 31, 27 and 7 close here too, continued from this tree's own frontier at 40 — at depths 42, 43 and 45, inside the even checkpoints 42, 44 and 46 the round reports; cells 69 and 47 are the shipped trees node for node, every leaf passes, and cell 69's MPFR ledger is its own rule rebuilt, the escalated leaf included

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_32_Bulk_Migration_Wave_v0.1.md` (AMRAL-LUC-FC-R32, dated 2026-09-20) and its package, read from the archive:

- `COVERAGE_DASHBOARD.json`, `PUBLICATION_CANDIDATE_SHARD_69.json`, `ARITHMETIC_PROTOTYPE_SHARD_47.json`,
  `BULK_MIGRATION_POLICY.json`, `VALIDATION.json`, `State_Crystal`;
- cells 69 and 47's topologies, 2-bit axis streams and manifests, cell 69's mpmath and MPFR leaf ledgers and cell 47's
  mpmath ledger (7,856 + 7,856 + 9,228 records), and Round 26's to Round 29's files again;
- five scripts, read only after this tree's replays were computed, and never run.

**Tools:** [`run035_round32_bulk_wave.py`](../code/run035_round32_bulk_wave.py) — RUN-030's continuation, resumed from
its boxes open at 40 under the package's pad; RUN-030's lift and RUN-031's migration on the intended boxes; RUN-033's
winner rule; RUN-032's marker certificates and RUN-034's polygon under the package's normals; the corrected exact kernel
**Logs:** [`run035-round32-bulk-wave.json`](../data/gate-logs/run035-round32-bulk-wave.json)

**Verdict: PASS for all nine of Round 32's judgments — the three closures, both trees, cell 69's two lower ledgers and
its upper bound, its promotion, cell 47's prototype status, the dashboard, and the global claim not made.**

- **The three closures.**
  - Cells 31, 27 and 7 are continued here from the boxes this tree's own continuation (RUN-030) left open at 40,
    split and lifted under the package's pad (hull − 10⁻¹⁰), one depth at a time.
  - Cell 31 closes at 42, cell 27 at 43 and cell 7 at 45. §1 says 42, 44 and 46: the depths are checked in steps of
    two, and each cell is closed at the even checkpoint §1 names, as in Round 26's ledger. The trees reach 158,811,
    199,375 and 248,823 nodes in all.
  - Cell 7's open volume at 40, 42, 44 and 46 is 9.435×10⁻¹⁰, 1.342×10⁻¹⁰, 1.93×10⁻¹² and 0, §1's figures.
  - With the 63 cells this tree's atlas closes by 40 (RUN-034), the cells closed by each depth are 57, 63, 64, 65
    and 66 at 38, 40, 42, 44 and 46: every lower bound §0 gives. The continuation took three minutes.
- **The trees of cells 69 and 47.**
  - Lifted here to depth 30 from Round 26's paths, they are the shipped trees node for node: 15,711 nodes and 7,856
    leaves, and 18,455 and 9,228. The leaf paths are equal, the axis streams decode high bit first to this tree's
    axes, and both digests match the manifests.
  - The least slacks are 4.4068×10⁻⁹ on `011000100000100101100` and 1.5290×10⁻⁸ on `10001101101111110111`, 4 and
    6×10⁻¹⁶ from the manifests'.
- **The mpmath ledgers (§3, §6).**
  - Every leaf of both cells passes here on its intended box at the ledger's density, no candidate refused, with the
    ledger's hull count and its margin to 2.2 and 2.6×10⁻¹⁶. The least margins are the ledgers': 1.6525×10⁻⁹ at
    32,768 on cell 69's thinnest leaf, 6.7502×10⁻⁹ at 16,384 on cell 47's.
  - Each density is the larger of the start band chosen by the leaf's reference slack and the least that passes on
    this tree's ladder: on all 9,228 leaves of cell 47, and on 7,855 of cell 69's 7,856. The exception is the
    thinnest leaf, which passes only at 32,768, past this tree's ladder.
  - The kernel scan (RUN-031's three-way comparison) finds the old grid envelope's sub-step loss again: on one leaf of
    cell 69 (1.474×10⁻⁹ low) and two of cell 47 (the same path, 1.474×10⁻⁹, and another, 2.2×10⁻¹⁰). The corrected
    kernel and the verified envelope agree there, as on every other leaf.
- **Cell 69's MPFR ledger (§4, §8).**
  - It is its own rule, rebuilt here as in RUN-033: per direction, only the winning body's contact point. On all
    7,856 leaves the first density that passes is the ledger's, and so is the hull count; every margin is the
    ledger's plus 1.57–1.67×10⁻¹², and no candidate is refused.
  - The density counts are 512: 3,805; 1,024: 3,832; 2,048: 175; 4,096: 32; 8,192: 8; 16,384: 3; 65,536: 1.
  - The escalated leaf is `011000100000100101100`, cell 69's thinnest. Its margin climbs the ladder from −2.6×10⁻⁴ at
    512 to −3.07×10⁻⁹ at 32,768 and +2.535×10⁻⁹ at 65,536, as §4 says. The shard's escalation record is the
    ledger's.
  - That this leaf needs 65,536 is the rule's loss again: under every contact point it passes at 32,768 (above).
- **The markers (§5–6).**
  - The shard files' marker figures are Round 29's files, and marker 47's replay row is in Round 29's audit.
  - Marker 69's polygon with every constraint, under the package's normals and ceiling, has 2,611 active
    constraints, an area of 0.8349723974082344 (the shard's) and 42,639 / 42,639 bits; float-reduced, it is Round
    29's file. No separate A1 record for marker 69 is shipped: the MPFR script is itself the exact full hull.
  - This tree's certificates need 128 normals for marker 69 and 256 for marker 47; the exact margins are
    6.74×10⁻⁵ and 1.857×10⁻⁵.
- **The dashboard (§0, §7, §9).**
  - 66 complete: the 63 closed by 40 and the three above. The 11 residual cells are the rest; each record's volume,
    ratio and pending count at its last depth is this tree's continuation's, to 10⁻¹².
  - 51 at the common depth 36. Candidates {69, 70}, closed or better {47, 69, 70}, 63 = 66 − 3 pending, and every
    fraction is right.
- **What the package ships**, read after this tree's rebuild and never run.
  - `r32_generate_shard.py` (60 lines) lifts with the package's pad and swallows an exception in the exact kernel,
    which leaves that box on its a-priori bound, open. It reads Round 26's full-atlas results and its own cell
    folders, neither of them in the archive.
  - `r32_mpmath_emitter_chunk.py` (121 lines) works at 100 digits, starts from the slack's band, rationalizes at 17
    digits and loads decimals into intervals.
  - `r32_mpfr_lower_chunk.py` and `r32_mpfr_lower_chunk_hi.py` (149 lines each) differ only in their ladder, the
    second going on to 65,536 and 131,072.
  - `r32_mpfr_marker.py` (102 lines) writes an output that is not in the archive.
  - All five load from an absolute path.

Drill: 18 / 18 plants caught by exactly the named checks, 4 / 4 controls undisturbed, 220 / 220 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.

This run uses the exact kernel as corrected after RUN-031 (every envelope segment checked in closed form); the
correction is committed before it.

---

## What Round 32 claims

1. **Geometry** (§0–1). New strict closures: cell 31 at 42, cell 27 at 44, cell 7 at 46, cell 7's unresolved volume
   9.43×10⁻¹⁰ → 1.34×10⁻¹⁰ → 1.93×10⁻¹² → 0. So 66/77 are known complete and 11 residual. The common budget is still
   Γ(36) = 51/77; the higher budgets have only lower bounds, Γ(38) ≥ 57 … Γ(46) ≥ 66.
2. **Cell 69** (§2–5). A tree of 15,711 nodes and 7,856 leaves at depth 30, as the geometry ledger has it; 7,856 /
   7,856 PASS under mpmath, least margin 1.6525×10⁻⁹; under MPFR one leaf fails closed at 32,768 (−3.07×10⁻⁹) and
   passes at 65,536 (+2.5329×10⁻⁹), with no resplit; marker 69's upper bound 0.8349723974082344, margin
   2.76026×10⁻⁵, replayed both ways. So cell 69 is a PUBLICATION-CANDIDATE-SHARD.
3. **Cell 47** (§6). 18,455 nodes and 9,228 leaves; 9,228 / 9,228 PASS under mpmath, least margin 6.7502×10⁻⁹;
   no MPFR lower yet, so ARITHMETICALLY-CLOSED-PROTOTYPE.
4. **The dashboard** (§7–9). Geometry complete 66, closed or better 3, candidates 2, pending 63, residual 11;
   coverage 2/77 and 2/66. The fallback order is density, then precision, then resplit; geometry and arithmetic keep
   separate queues.

## A — sources

**A0.** 31 anchors. Round 32 has no reference list.

## W — rebuilt

| check | Round 32 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | §1 | cells 31, 27 and 7 continued from this tree's frontier at 40 | PASS; closed at 42, 43, 45, inside the checkpoints 42, 44, 46 |
| W2 | §2, §6 | cells 69 and 47 lifted to depth 30 | PASS; node for node, both digests |
| W3 | §3, §6 | every leaf on its intended box at the ledger's density | PASS; densities from the start bands |
| W4 | §4, §8 | cell 69's MPFR ledger against its own rule | PASS; leaf for leaf, the escalation included |
| W5 | §5–6 | the markers | PASS; marker 69's MPFR polygon is the one with every constraint |
| W6 | §0, §7–9 | the dashboard from W1–W5 and the atlas | PASS; every count, bound and fraction |
| W7 | — | the scripts | two inputs not in the archive; kernel errors leave boxes open |

## P — the package

**P1.**

- The shard files' three lower records are their ledgers': totals, passes, inconclusive, density counts, the least
  margin and its path, the ledger digest. The escalation record is the MPFR ledger's leaf.
- Both shards' geometry is their manifests', and each base path is Round 26's and its marker file's.
- The dashboard's pipeline is the shard files' statuses, and its eleven residual records are Round 27's, figure for
  figure; Round 27's other three are the cells closed here.
- The crystal carries 66/77, both stage sets, both leaf counts, the escalation and its margin.

**P2.**

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

## Round 32, read

| Round 32 | statement | verdict | note |
| --- | --- | --- | --- |
| §0–1 | the three closures; 66/77; the lower bounds | PASS | closed at 42, 43 and 45; the checkpoints 42, 44, 46 (W1, W6) |
| §2 | cell 69's tree | PASS | node for node (W2) |
| §3 | 7,856 / 7,856 under mpmath | PASS | W3 |
| §4 | 7,856 / 7,856 under MPFR after one escalation | PASS | its own rule, rebuilt leaf for leaf (W4) |
| §5 | marker 69's upper bound, replayed both ways | PASS | W5 |
| §6 | cell 47: 9,228 / 9,228; a prototype | PASS | W2, W3; its MPFR replays pending, as the shard says |
| §7–9 | the dashboard; the fallback order; two queues | PASS | W6; the order is a policy |
| §10–11 | Round 33; the handoff | n/a | |

## The status registry

Round 32's 9 judgments are reviewed, all PASS. It uses no phrase for the first time, and census v5 adds none.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's 2.5329 | 1 | A0 |
| the geometry claim | 1 | W1 |
| a continuation left open | 1 | W1, W6 |
| a topology bit; an axis-stream bit | 2 | W2 |
| an emitter margin; this tree's certificate failing a leaf | 2 | W3 |
| the escalated leaf's MPFR density | 1 | W4 (also P1) |
| marker 69's area | 1 | W5 |
| the dashboard's complete count | 1 | W6 (also P2) |
| a higher budget's lower bound | 1 | W6 |
| a residual record's pending count | 1 | W6 (also P1) |
| the scripts' facts | 1 | W7 |
| a shard's ledger digest; the crystal's leaf count | 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 dashboard with its keys sorted;
- cell 69's shard with its keys sorted.

**18 / 18 plants, 4 / 4 controls, 220 / 220 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 | 4 | 4 |
| W7 | 1 | 1 |
| P1 | 4 | 4 |
| P2 | 3 | 3 |
| R1 | 1 | 1 |

## What this round does not establish

- **The MPFR side.** MPFR is not on this machine. Cell 69's MPFR ledger is rebuilt here from its rule on this tree's
  arithmetic, not replayed in MPFR.
- **A common budget above 36.** The continuations here, like the round's, are per cell; 51/77 at 36 is still the
  only complete common budget.
- **Anything global.** a_Leb ≥ 0.835 is not certified.