# RUN-032 — Round 29: all 77 necessity markers certify again, with 128–256 normals once the hull's bridging edges get their own; the package's polygons lose first order at those edges, so its thinnest marker is not the thinnest

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-26
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_29_Marker_Whole_Shard_ABI_v0.1.md` (AMRAL-LUC-FC-R29, dated 2026-09-20) and its package, read from the archive:

- `MARKER_MIGRATION_SUMMARY.json`, `MARKER_UPPER_SPEC.json`, `WHOLE_SHARD_ABI.json`,
  `WHOLE_SHARD_70.json`, `A1_MARKER_AUDIT.json`, `VALIDATION.json`, `State_Crystal`;
- `marker_00.json` … `marker_76.json`, and Round 26's ledger and Round 28's files again;
- two scripts, read only after this tree's certificates were computed, and never run.

**Tools:** [`run032_round29_marker_upper.py`](../code/run032_round29_marker_upper.py) — outer polygons from integer-interval support upper bounds (RUN-031's Machin π and Taylor enclosures), uniform integer normals plus the normals of the hull's bridging edges, intersected through their exact polar hull; RUN-031's directed cells; the corrected exact kernel
**Logs:** [`run032-round29-marker-upper.json`](../data/gate-logs/run032-round29-marker-upper.json)

**Verdict: PASS for the marker points, the outer-polygon theorem, all 77 upper certificates, the three-marker replay and the whole shard; PARTIAL for the rankings and densities of §11–13 — the package's outer polygons lose first order in the normal spacing at the hull's straight bridging edges, so its thinnest certificate (marker 6, 6.1×10⁻⁷) is not its thinnest marker (marker 47, exact margin 1.86×10⁻⁵, marker 6's being 2.11×10⁻⁵), marker 47's "tighter" bound is tighter only by more directions, and its densities are start points chosen from the Round 26 reference margins.**

- **The marker points.** Every point lies strictly inside its directed cell, rebuilt from RUN-031's
  directed root along the official axes; no path has an override. The ids are the paths' sorted
  order. Each point is the package's stored centre rationalized to 17 digits, within 7 ulps of this
  tree's centre (its t₃ and t₅ are a few ulps lower, RUN-031).
- **This tree's certificates.**
  - The support upper bounds are integer intervals at 2⁻⁴⁰⁰: the disc's |n|/2; each Reuleaux body's
    corners n·V_j; and n·V_k + |n| for each arc whose range of normals may hold n — excluded only
    when n·c_k < |n| cos(π/2m) at both ends.
  - The normals are a uniform grid, plus the normals of the hull's bridging edges: the verified
    envelope's hand-overs between bodies, 11 for every marker.
  - The half-planes are intersected through the exact polar hull of n_i/H_i, and the area is exact.
  - All 77 markers clear 167/200: 51 at 128 normals, 26 at 256. The thinnest certificate is marker
    47's, 1.50×10⁻⁶; so is the thinnest exact margin, 1.857×10⁻⁵.
- **The package's bounds.**
  - All 77 are upper bounds of the exact areas.
  - This tree's uniform normals at the package's densities give its active-constraint counts,
    77 of 77, and its areas to 1.3×10⁻¹⁰ (normals rounded at 10⁹ there, 2³⁰ here).
  - The losses run 1.1×10⁻⁵ to 1.1×10⁻⁴. At a straight bridging edge of length L the two support
    lines on either side of its normal meet outside it: the loss is up to L²·d/8, first order in the
    spacing d. Halving the density multiplies the loss by 1.3–5.5, as the edges' normals fall
    between grid directions, not by the 4 of a smooth body.
  - So the package's ranking is the loss's. Its thinnest marker, 6, has an exact margin of
    2.11×10⁻⁵. The thinnest is marker 47, whose certificate margin there is 7.57×10⁻⁶ at 32,768
    normals.
  - §13's marker 47: the Round 26 reference, a 20,000-direction polygon of support lines, loses
    1.83×10⁻⁵; the new one, at 32,768, loses 1.10×10⁻⁵. It is tighter because it has more
    directions; both losses are the bridging edges'.
  - The emitter picks its first density from the Round 26 reference margin in bands (above 5×10⁻⁴
    2,048; 10⁻⁴ 4,096; 2×10⁻⁵ 8,192; 3×10⁻⁶ 16,384; otherwise 32,768). 73 markers passed at their
    start and 4 went from 8,192 to 16,384. "No marker needs 65,536" says the ladder never went past
    its highest start.
- **The three-marker replay.**
  - Markers 6, 47 and 70 reproduce their files' densities, areas and margins.
  - The replay counts more active constraints (4,962, 10,255, 1,317 against 4,900, 10,204, 1,284).
    These are near-collinear ones the float hull dropped; the areas agree to the printed digits.
  - The replay script carries a copy of the emitter's support routine and normals, and skips only
    its candidate reduction: a second polygon path, as §14 says, not a second arithmetic.
- **The whole shard.**
  - Marker 70 is Round 28's cell: path, point and axes.
  - The shard's upper half is marker 70's file, by digest.
  - Its lower half is Round 28's certificate: the root, topology, axis-stream and ledger digests,
    the leaf counts and the least margin — the certificate RUN-031 replayed leaf by leaf.
  - Marker 70's exact margin is 1.98×10⁻⁴ (the package's certificate 9.25×10⁻⁵, this tree's
    1.37×10⁻⁴ at 128 normals).
- **What the package ships.**
  - `r29_marker_migrate.py` (190 lines) emits the markers; `r29_marker_a1_verify.py` (92) is the
    replay. Both load from an absolute path; everything they load is in the archive (Round 26's
    ledger, Round 28's root file, Round 25's runner).
  - The support upper bound finds the sector of a normal by an interval atan2. Normals are
    round(10⁹ cos θ, 10⁹ sin θ) reduced by their gcd.

Drill: 15 / 15 plants caught by exactly the named checks, 4 / 4 controls undisturbed,
173 / 173 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.

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 29 claims

1. **Rational marker points** (§1, §15). Each q_j is a rationalized reference centre inside
   [ℓ_k^hi, u_k^lo] of its directed cell; 77/77 lie safely inside. No path has a CPG override.
2. **Outer certificates** (§2–9).
   - Integer normals. The disc's support is ½‖n‖; the Reuleaux supports come by sectors of π/n,
     taking every possibly active piece.
   - The base hull's support is the maximum, rounded up to 10⁻²⁶. The half-planes contain the hull,
     so A(H(q)) ≤ A(P), and A(P) < 167/200 certifies the marker.
   - Dropping constraints is safe. The geometry is an exact polar hull.
3. **Results** (§10–13).
   - 77/77 PASS, none inconclusive.
   - Densities 4,096: 21, 8,192: 39, 16,384: 16, 32,768: 1; none needs 65,536.
   - The weakest margin is marker 6's, 6.1062×10⁻⁷; the median 3.5043×10⁻⁵.
   - Marker 47's new bound (7.5672×10⁻⁶) is "tighter" than its reference (2.89×10⁻⁷).
4. **A1 sample** (§14). Markers 6, 47 and 70 replayed without candidate reduction:
   A1-MARKER-SAMPLE-PASS, not 77/77.
5. **Whole shard** (§16–22).
   - A five-layer ABI.
   - Marker 70 is Round 28's cell: upper margin 9.24985×10⁻⁵, joined with the 9,278/9,278 lower
     leaves.
   - Status ARITHMETICALLY-CLOSED-WHOLE-SHARD-PROTOTYPE, publication_candidate = false.
6. **States and scope** (§23–27). ARITHMETICALLY-CLOSED-PROTOTYPE, PUBLICATION-CANDIDATE,
   THEOREM-READY. a_Leb ≥ 0.835 NOT CERTIFIED.

## A — sources

**A0.** 32 anchors. Round 29 has no reference list.

## W — rebuilt

| check | Round 29 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | §1, §15 | the 77 points in their directed cells; axes | PASS |
| W2 | §2–10 | this tree's 77 certificates | PASS, 128–256 normals with the bridging normals |
| W3 | §10–13 | the package's bounds against the exact areas | valid; first-order losses; the ranking is the loss's |
| W4 | §14 | the three-marker replay | PASS; more active constraints, same areas |
| W5 | §16–22 | the whole shard | PASS |
| W6 | — | the scripts | the replay shares the emitter's support code |

**W2 and W3 — the thinnest markers, three ways.**

| marker | exact margin | the package's certificate | this tree's (normals) |
| ---: | ---: | ---: | ---: |
| 47 | 1.857×10⁻⁵ | 7.57×10⁻⁶ (32,768) | 1.50×10⁻⁶ (256) |
| 6 | 2.112×10⁻⁵ | 6.11×10⁻⁷ (16,384) | 5.57×10⁻⁶ (256) |
| 70 | 1.982×10⁻⁴ | 9.25×10⁻⁵ (4,096) | 1.37×10⁻⁴ (128) |

This tree's certificates stop at the first density that passes, so their margins are not tight: at
256 normals marker 47's still gives away 1.7×10⁻⁵ of its exact margin, now second order in the
spacing.

## P — the package

**P1.**

- The summary is its 77 files': counts, density counts, the least margin (marker 6, its area and
  density), the median, the largest, and the override statistics.
- Each file's final record is its last attempt's, and every margin is 0.835 − area.
- The spec says ceil to 10⁻²⁶, fail-closed, and "dropping constraints enlarges the polygon".
- The crystal carries 77/77, 6.1062, 70 and 9,278/9,278.

**P2.**

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

## Round 29, read

| Round 29 | statement | verdict | note |
| --- | --- | --- | --- |
| §0, §10 | 77/77, none inconclusive | PASS | W2, W3 |
| §1, §15 | points inside directed cells; no override | PASS | W1 |
| §2–9 | the outer-polygon construction and theorem | PASS | reading; W2 |
| §11 | final densities; none needs 65,536 | PARTIAL | start points from reference-margin bands; 128–256 normals suffice with the bridging normals |
| §12 | the weakest marker is 6, 6.1062×10⁻⁷ | PARTIAL | the weakest certificate; the weakest marker is 47 (W3) |
| §13 | marker 47's bound is tighter than its reference | PARTIAL | by more directions; both losses are first order at the bridging edges |
| §14 | A1 sample PASS | PASS | a second polygon path over the emitter's support code (W4, W6) |
| §16–22 | the ABI; marker 70; the first whole shard | PASS | W5 |
| §23–27 | states; Round 30; handoff | PASS / n/a | |

## The status registry

Round 29's 19 judgments are reviewed, and its 7 first-used phrases defined: `UPPER-INCONCLUSIVE`,
`A1-MARKER-SAMPLE-PASS`, `ARITHMETICALLY-CLOSED-WHOLE-SHARD-PROTOTYPE`, `ARITHMETICALLY CLOSED`,
`ARITHMETICALLY-CLOSED-PROTOTYPE`, `PUBLICATION-CANDIDATE-SHARD` and `THEOREM-READY`. Census v5 adds
none.

| verdict | judgments |
| --- | ---: |
| PASS | 15 |
| n/a | 4 |

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's 6.1062 | 1 | A0 |
| a point moved out of its cell; two axes swapped | 2 | W1 |
| the certificate claim | 1 | W2 |
| an area below its exact area | 1 | W3 (also P1) |
| the package claim | 1 | W3 |
| an audited area | 1 | W4 |
| the shard's margin | 1 | W5 |
| the shard's leaf count | 1 | W5 (also P2) |
| the scripts' facts | 1 | W6 |
| the summary's density counts; the crystal's 77/77 | 2 | P1 |
| a recorded digest; a VALIDATION check | 2 | P2 |
| a review dropped | 1 | R1 |

Four controls:

- the unmodified inputs;
- the document with CRLF line endings;
- the summary with its keys sorted;
- the shard with its keys sorted.

**15 / 15 plants, 4 / 4 controls, 173 / 173 must-stay-green assertions; no check without a
plant.**

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

## What this round does not establish

- **Any marker beyond its point.** A marker certifies one configuration per cell, as §17 says;
  nothing here speaks for the rest of a cell.
- **An independent lower replay.** Round 30's; RUN-031 has replayed cell 0's 9,278 leaves with this
  tree's own arithmetic.