# RUN-029 — Round 26: all 77 necessity cells rebuild row for row under the package's pad; the thinnest leaf in the atlas clears T by 8.5×10⁻¹¹, not 2.9×10⁻⁹

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-26
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_26_Full_Atlas_Arithmetic_v0.1.md` (AMRAL-LUC-FC-R26, dated 2026-09-20) and its package, read from the archive:

- `FULL_ATLAS_LEDGER.json` (77 cells), `WAVE_SCHEDULER_SPEC.json`, `ARITHMETIC_AUDIT.json` and its
  supplement, `ARITHMETIC_CONTRACT.json`, `ARITHMETIC_MIGRATION_SPEC.json`, `VALIDATION.json`,
  `State_Crystal`;
- `residual_depth36.json`;
- five scripts, read only after this tree's reconstruction, and never run.

**Tools:** [`run029_round26_full_atlas.py`](../code/run029_round26_full_atlas.py) — RUN-027's exact-core B7 lift on all 77 cells to depth 36, then again, with the pad as a parameter, on the residual and on every cell whose ledger row the first pass changes; RUN-026's segmentation for the cells
**Logs:** [`run029-round26-full-atlas.json`](../data/gate-logs/run029-round26-full-atlas.json)

**Verdict: PASS for the 77 cells and their markers, the closure curve, every ledger row, the residual's volumes, classes and ratios, the tight-leaf polygon audit and the radial audit; PASS as stated for §14's least slack, which is the least over the ten audited trees only — over all 51 complete trees the least is 8.47×10⁻¹¹ (cell 8), and 11 trees hold a leaf thinner than §14's 2.8982×10⁻⁹. The rejection of a larger global pad stands, more strongly than §14 argues it.**

- **Two pads.**
  - The package closes a box once its exact hull less 10⁻¹⁰ reaches T. This tree's lift has so far
    used the pilot's 10⁻⁹ (RUN-027, RUN-028), which did not matter on Round 23's twelve cells.
  - On the full atlas it does. Under 10⁻⁹, 15 of the 77 trees differ from the ledger: 2 or 4 more
    nodes, or 1 more open box at depth 36 — each a box whose hull clears T by more than 10⁻¹⁰ but
    not by 10⁻⁹, split here and closed there. No closure depth moves, so the curve is the same.
  - The residual cells and those 15 were lifted again under 10⁻¹⁰ (31 cells). Every one of the 77
    rows then agrees with the ledger: nodes, open boxes, status and volume. That is 4,481,119
    nodes, as the ledger says; 26 fewer than under 10⁻⁹.
- **The curve and the residual.**
  - Γ_B7 at 22, 24, …, 36 is 0, 0, 0, 0, 7, 19, 38, 51, under either pad. At the depths between
    (29, 31, 33, 35) it is 0, 15, 30, 45.
  - The 26 residual cells: 5 dust, 7 thin, 14 broad. Their volume quantiles and V₃₆/V₃₄ ratios
    agree with the ledger to 10⁻¹² — median 0.30094, largest 0.57615 (cell 23), every ratio below 1.
  - Under this tree's pad the same statistics move in the fifth digit (the largest ratio becomes
    0.57617): the residual's numbers are pad-dependent at that scale.
- **The least slack.**
  - The ten trees the audit sampled: least 2.8982×10⁻⁹ (cell 21), and 10, 9, 8, 6 of them above
    2, 3, 5 and 10 ×10⁻⁹, as the audit says.
  - All 51 complete trees: least 8.47×10⁻¹¹ (cell 8, a leaf at depth 35). Five trees are below
    10⁻⁹ — cells 8, 39, 62, 43 and 50, the pad-sensitive cells that close by 36. 29 trees hold a
    leaf below 10⁻⁸.
  - So §14's conclusion holds with room to spare: a 10⁻⁸ pad would reopen a leaf in 29 of the 51
    complete trees, and even 10⁻⁹ reopens leaves in 5 of them (each still closes by 36, with 2 or
    4 more nodes) and changes 10 residual trees.
- **The tight leaf and the radial core.**
  - The certificate cell's 9,278 CERT leaves; the thinnest has nominal lower 0.8350000421742021.
    Uniform contact polygons of its bodies give 0.8349998195 at 4,096 directions (below T) and
    0.8350000060 at 8,192 (above), as §22 says.
  - The radial fallback as the audit builds it — every body radial at N rays — keeps 60, 33, 228
    and 148 of the 9,278 leaves at N = 24, 32, 48, 64: the audit's fractions exactly. With only
    B7's core radial and the rest exact it keeps 507, 723, 1,278 and 1,790. Either way §20's
    rejection stands.
- **What the package ships.** Five scripts, 35–95 lines: the initial runner, the wave scheduler,
  an ancestor-prefix probe and two audits. All load Round 19's modules (the audits also Round
  24's manifest) from an absolute path outside the package. The runner's bound is max(the a-priori
  value, the exact hull − 10⁻¹⁰); the a-priori term never binds on these roots.

Drill: 17 / 17 plants caught by exactly the named checks, 4 / 4 controls undisturbed,
214 / 214 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 26 claims

1. **The atlas** (§1–4). Round 19's 77 necessity-marked depth-40 cells, each with U_base < 0.835,
   run in common-depth waves 22 → 36, resumable, and monotone in the budget.
2. **The curve and the residual** (§5–11). Γ_B7 = 0, 0, 0, 0, 7, 19, 38, 51; 51/77 = 66.23% close by
   36, 26 remain. Their volumes run 6.36×10⁻¹² to 6.99×10⁻⁸ (5 dust, 7 thin, 14 broad); every
   ratio to the previous wave is below 1, the largest 0.57615. No joint-witness necessity.
3. **Arithmetic** (§13–25). Every reference leaf is ARITHMETIC-PENDING. A larger global pad is
   rejected (least audited slack 2.8982×10⁻⁹). A per-leaf interval ledger, a selective
   revalidation theorem, a backend contract (δ up, ρ = 1 − δ down), eroded-Reuleaux cores kept
   (the radial fallback keeps a few percent), the tight-leaf polygon audit, marker margins
   (least 2.885×10⁻⁷, median 6.806×10⁻⁵), and theorem_ready = false.

## A — sources

**A0.** 58 anchors. Round 26 has no reference list.

## W — rebuilt

| check | Round 26 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | §1, §23 | the 77 cells; the marker margins | PASS |
| W2 | §5–6 | the curve at the waves and between | PASS, under either pad |
| W3 | §7–9 | the residual's volumes, classes, ratios | PASS, to 10⁻¹² under the package's pad |
| W4 | — | the ledger's 77 rows | PASS under the package's pad; 15 differ under 10⁻⁹ |
| W5 | §14 | the least slack | PASS as stated; the atlas holds thinner leaves |
| W6 | §20, §22 | the tight leaf; the radial core | PASS |
| W7 | — | the scripts | load code from outside the package |

**W1 — the cells.** The 77 are exactly Round 19's depth-40 cells whose outer marker is below T,
and the ledger's markers are this tree's outer brackets to 1.2×10⁻¹³. The margins T − U: least
2.885×10⁻⁷, median 6.806×10⁻⁵; 8, 13 and 30 below 10⁻⁵, 2×10⁻⁵ and 5×10⁻⁵ — the audit's.

**W4 — the rows this tree's pad changes.**

| cell | ledger: nodes / open at 36 | under 10⁻⁹ |
| ---: | ---: | ---: |
| 1 | 115,409 / 17,811 | 115,409 / 17,812 |
| 7 | 128,957 / 10,722 | 128,959 / 10,723 |
| 8 | 39,007 / 0 | 39,009 / 0 |
| 13 | 154,279 / 16,234 | 154,279 / 16,235 |
| 18 | 82,267 / 2,329 | 82,269 / 2,329 |
| 23 | 170,083 / 30,218 | 170,083 / 30,219 |
| 24 | 154,409 / 22,735 | 154,411 / 22,735 |
| 26 | 120,053 / 13,913 | 120,055 / 13,913 |
| 28 | 82,759 / 2,234 | 82,761 / 2,234 |
| 36 | 53,407 / 3 | 53,409 / 3 |
| 39 | 59,775 / 0 | 59,779 / 0 |
| 43 | 43,037 / 0 | 43,039 / 0 |
| 50 | 54,735 / 0 | 54,737 / 0 |
| 51 | 62,571 / 192 | 62,573 / 192 |
| 62 | 28,201 / 0 | 28,203 / 0 |

Lifted again under 10⁻¹⁰, all fifteen give the ledger's row. A tree under the smaller pad is a
subtree of the one under the larger, so the other 62 — whose rows already agree under 10⁻⁹ — hold
no box in between.

**W5 — the thinnest leaves of the complete trees** (the package's terms: hull − 10⁻¹⁰ − T).

| cell | least slack | at depth |
| ---: | ---: | ---: |
| 8 | 8.47×10⁻¹¹ | 35 |
| 39 | 1.74×10⁻¹⁰ | 30 |
| 62 | 1.77×10⁻¹⁰ | 32 |
| 43 | 7.48×10⁻¹⁰ | 29 |
| 50 | 8.76×10⁻¹⁰ | 31 |
| 10 | 1.05×10⁻⁹ | 30 |
| 5 | 1.11×10⁻⁹ | 30 |
| 22 | 2.22×10⁻⁹ | 31 |
| 53 | 2.43×10⁻⁹ | 33 |
| 9 | 2.62×10⁻⁹ | 30 |
| 15 | 2.68×10⁻⁹ | 31 |
| 21 | 2.8982×10⁻⁹ | 32 |

Cell 21 is the audit's minimum; the eleven above it were not among the ten it sampled.

**W6 — the tight leaf.** The thinnest CERT leaf of the certificate cell lies at depth 25, nominal
lower 0.8350000421742021 (the supplement's 0.835000042174202). Polygons: 0.8349998195 (4,096)
and 0.8350000060 (8,192), each within 5×10⁻¹² of §22.

**W7 — the scripts.** Read after the rebuild, never run.

- `initial_full77_runner.py` (95 lines) and `wave_scheduler.py` (93) take the 77 cells from Round
  19's segmentation file (those with the outer marker below T), rebuild each from its path, and
  run the waves with the bound above and the lift's split by the least child δ. The scheduler
  keeps each cell's state in a pickle outside the package.
- `ancestor_prefix_probe.py` (59) lifts a base prefix to checkpoints 16–36: §12's stopped
  ancestor experiment.
- `radial_core_audit.py` (73) and `sampled_inner_gap_audit.py` (35) are §20's and §22's audits, on
  Round 24's certificate cell.
- Each loads Round 19's reference modules — the two audits also Round 24's manifest — from an
  absolute path.

## P — the package

**P1.**

- The scheduler spec's curve, residual, classes and largest ratio are the ledger's.
- §7's and §9's printed values round the ledger's quantiles.
- The contract and the migration spec are marked formalized, not implemented. The contract rounds
  δ up and ρ = 1 − δ down, and keeps a marker only when its interval hull is below T.
- The crystal carries the curve, the classes and the least slack.

**P2.**

- `VALIDATION.json` describes the document and the crystal correctly: SHA-256, bytes, `$$`
  counts, both flags.
- Its six digests are the archive's, and the supplement's parent digest is the audit's.
- Its twelve checks follow from the ledger, the audit and the contract.

## Round 26, read

| Round 26 | statement | verdict | note |
| --- | --- | --- | --- |
| §0, §5–6 | 51/77 at d = 36; the curve; 26 residual | PASS | W2, W4 |
| §1 | 77 necessity-marked cells | PASS | W1 |
| §2–4 | waves; resumable state; budget monotonicity | PASS | under a fixed pad a deeper budget never reopens a tree |
| §7–9 | volumes; classes; all contracting | PASS | W3, to 10⁻¹² |
| §10–11 | joint queue deferred; pending count is not difficulty | PASS | W3 |
| §12 | the ancestor experiment stopped | n/a | no claim (W7) |
| §13, §15–21 | arithmetic pending; the ledger, theorem and contract | PASS | formal (P1) |
| §14 | least audited slack 2.8982×10⁻⁹; a larger pad rejected | PASS as stated | the atlas's least is 8.47×10⁻¹¹ (W5) |
| §20, §22 | radial fallback; tight-leaf polygons | PASS | W6 |
| §23 | marker margins | PASS | W1 |
| §24–25 | migration; theorem_ready = false | PASS | |
| §26–27 | Round 27; handoff | n/a | |

## The status registry

Round 26's 11 judgments are reviewed, and its 5 first-used phrases defined: `ARITHMETIC-PENDING`,
`INTERVAL-PASS`, `INTERVAL-INCONCLUSIVE`, `NOT CERTIFIED` and `ALL CONTRACTING`. Census v5 adds
none.

| verdict | judgments |
| --- | ---: |
| PASS | 6 |
| n/a | 5 |

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's Q₅₀ | 1 | A0 |
| a marker; the audit's median margin | 2 | W1 |
| a claimed curve value | 1 | W2 |
| a residual ratio; the class counts | 2 | W3 |
| a ledger row; the pad-sensitive cells | 2 | W4 |
| the audit's least slack; a slack claim | 2 | W5 |
| the 4,096-direction polygon | 1 | W6 |
| the scripts' facts | 1 | W7 |
| a contract rule; a crystal curve value | 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 ledger with its keys sorted;
- the audit with its keys sorted.

**17 / 17 plants, 4 / 4 controls, 214 / 214 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 | 2 | 2 |
| W5 | 2 | 2 |
| W6 | 1 | 1 |
| W7 | 1 | 1 |
| P1 | 2 | 2 |
| P2 | 2 | 2 |
| R1 | 1 | 1 |

## What this round does not establish

- **Any bound.** The trees are floating-point runs of the exact-core kernel, as the package's
  own scope says; nothing here is interval arithmetic.
- **The residual beyond 36.** Round 27's.