# RUN-030 — Round 27: at a common depth 40 the residual's lower bounds are exact, 57 by 38 and 63 by 40; the five thinnest leaves certify again under this tree's own integer intervals, and seven more

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-26
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_27_Residual_Interval_Replay_v0.1.md` (AMRAL-LUC-FC-R27, dated 2026-09-20) and its package, read from the archive:

- `RESIDUAL_PROGRESS.json`, `INTERVAL_TIGHT_LEAF_AUDIT.json`, `INTERVAL_POLYGON_REPLAY.json`,
  `INTERVAL_BACKEND_SPEC.json`, `VALIDATION.json`, `State_Crystal`;
- `cell0_tight12_reference.json`, and Round 26's ledger again;
- three scripts, read only after this tree's reconstruction, and never run.

**Tools:** [`run030_round27_interval_replay.py`](../code/run030_round27_interval_replay.py) — an inner-polygon certificate in integer intervals; RUN-027's exact lift keeping each leaf's path; RUN-029's lifts under the package's pad, continued from depth 36
**Logs:** [`run030-round27-interval-replay.json`](../data/gate-logs/run030-round27-interval-replay.json)

**Verdict: PASS throughout.**

- **The residual.** Run to a common depth 40, it gives Γ_B7(38) = 57 and Γ_B7(40) = 63 — the
  package's lower bounds are exact. Its 14 cells without a known edge are the 14 still open at 40,
  each record exact at its last tested depth.
- **The certificates.** Cell 0's twelve thinnest leaves certify under this tree's own
  integer-interval polygons, on its boxes and on the shipped ones. On the package's five the counts
  are the same and the margins agree to 4.8×10⁻¹⁷.
- **The stored state.** §13's boxes are 4 ulps from this tree's. The exact t₇ lies below both
  roots, but the φ root falls 3.5×10⁻¹⁷ short of 2π/7, so the adjacent-float enclosure carries the
  top edge.

- **The certificates, rebuilt independently.**
  - This tree's certificate shares nothing with the package's arithmetic.
    - Every real in a core's definition is enclosed by integer intervals at 2⁻⁴⁰⁰, rounded
      outward at every step: π (cut after its 60th decimal), R_n, each corner V_j, δ and
      ρ = 1 − δ.
    - cos and sin come from their Taylor sums, each term enclosed, plus the Lagrange remainder.
    - Candidate points are kept as exact binary floats. Their hull and shoelace are computed in
      exact integers.
  - The package uses mpmath.iv at 80 digits and rounds its candidates to 17-digit decimals.
  - Cell 0's twelve thinnest leaves all clear 167/200, on this tree's boxes and on the shipped
    ones, with no membership failure.
  - On the package's five, the candidate, verified and hull-vertex counts are its own (3170,
    3149, 3736, 3363, 3536), and the margins agree to 4.8×10⁻¹⁷. The candidates differ only in
    rounding: the package nudges by (1 − ε)p + εc over `linspace` directions and rounds to
    decimals.
- **The verifier refuses what it must.**
  - The thinnest leaf's candidates pushed outward by the same 10⁻¹²: 0 of 3170 verified.
  - At 4,096 directions every point verifies, but the polygon falls 1.80×10⁻⁷ short: not a pass.
  - Each certified polygon lies 0.99–1.41×10⁻⁸ below the exact hull of its cores, as §7's
    theorem requires.
- **The stored state (§13).**
  - The shipped boxes are not this tree's to the bit. The paths and φ agree exactly; x and y differ
    by up to 4 units in the last place.
  - The cause is the root radius t₇: 0x1.9c9e1c2550f72p-3 shipped, 0x1.9c9e1c2550f76p-3 here. At
    60 digits the exact t₇ is 0.2014734457862874141…, below both (by 3.39 and 7.39 units), so
    neither root misses a translation.
  - The φ root is another matter. 2·fl(π/7) falls 3.5×10⁻¹⁷ short of 2π/7, less than one unit of
    π/7. The ±1-unit enclosure of §13 is what covers that sliver on the top-edge leaves, so it is
    load-bearing there, not a hedge. It stays so until Round 28's exact root.
- **The residual at a common depth 40.**
  - The 26 residual trees are continued from the 258,764 boxes RUN-029 left open at 36, under the
    package's pad. That is 6,111,066 nodes in all.
  - Γ_B7 at 37, 38, 39 and 40 is 56, 57, 63 and 63. Round 27 reports 57 and 63 only as lower bounds,
    since several broad cells were not advanced; at a common budget they are exact.
  - The 14 cells without a known edge are exactly the 14 open at 40. Each of the package's records
    — open boxes, volume, and ratio to the previous wave at its last tested depth (36, 38 or 40) —
    is this tree's, the volumes and ratios to 10⁻¹².
  - The other twelve close at 37 (cells 4, 14, 35, 36, 51), at 38 (cell 37) and at 39 (cells 0, 16,
    18, 28, 29, 44). Round 26's ledger records cells 4 and 14 at 38 and cell 0 at 40: the next even
    checkpoint again, as in RUN-028.
  - At 40 the 14 still contract: V₄₀/V₃₈ is at most 0.608. That is cell 23, the slowest — its ratio
    runs 0.576, 0.634, 0.608 over 34→36, 36→38, 38→40. Their open boxes grew to 825,168 while the
    volume fell (§11).
- **What the package ships.** Three scripts; each loads its inputs from an absolute path.
  - `interval_polygon_replay.py` (171 lines) and `tight_leaf_interval_batch.py` (72) build the
    certificates as §15–16 describe. Each reads its leaf boxes from a file the archive does not
    hold (`r27_cell0_worst_leaf.json`, `r27_cell0_tight12.json`), besides Round 24's manifest and
    Round 25's runner.
  - `fast_negative_prefilter.py` (46) takes a box as closed only from the a-priori value or the
    exact hull − 10⁻¹⁰, so it cannot create a false positive, as §5 says. Its fast estimate comes
    from `r26_fastgeom`, which the archive does not hold under that name (nor `r26_wave_scheduler`,
    presumably Round 26's `wave_scheduler.py`). Whether that estimate lies inside or outside the
    hull — how many false negatives it makes — cannot be told.

Drill: 16 / 16 plants caught by exactly the named checks, 4 / 4 controls undisturbed,
161 / 161 must-stay-green assertions held. Each of the nine checks was blinded in turn, and each
was reported blind in exactly as many cases as plants name it.

---

## What Round 27 claims

1. **Residual continuation** (§1–5). Only the depth-36 residual is extended. The common curve stays
   Γ_B7(36) = 51. Higher budgets give lower bounds only: Γ_B7(38) ≥ 57 and Γ_B7(40) ≥ 63, "not a
   common-depth curve". 14 cells have no known closure: 5 last tested at 36, 8 at 38, 1 at 40. A
   fast negative prefilter "can create false negatives, but cannot create false positives".
2. **Rational inner polygons** (§6–12). Points verified inside certified cores span a hull inside
   the cores' hull (Theorem 7.1). The disc is checked exactly, a core by interval discs with
   d²_hi < (ρ²)_lo. mpmath.iv is a prototype backend.
3. **The stored state** (§13). The boxes are the stored binary64 values, each enclosed by its
   neighbouring floats; the exact root is not yet rebuilt.
4. **The certificates** (§14–18). Cell 0's worst leaf (path 1101011010010011111001110, depth 25,
   reference slack 4.20742×10⁻⁸): 16,384 directions per body, inward 10⁻¹², 3170/3170 verified,
   exact area 0.8350000319489227, margin 3.1949×10⁻⁸. The five thinnest: 5/5 PASS, margins 3.19,
   6.42, 11.6, 15.7, 20.3 ×10⁻⁸.
5. **Scope** (§19–25). No exact root, no full replay, no second backend: PUBLICATION THEOREM: NOT
   YET; a_Leb ≥ 0.835 NOT CERTIFIED.

## A — sources

**A0.** 35 anchors. Round 27 has no reference list.

## W — rebuilt

| check | Round 27 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | §0, §3 | the residual at a common depth 40 | PASS; the bounds are exact |
| W2 | §13–14, §17 | cell 0's leaves; the stored boxes; the root | PASS; boxes to 4 ulps |
| W3 | §10–11, §15–17 | the twelve thinnest, certified here | PASS, 12/12 on both box sets |
| W4 | §7 | the theorem; the enclosures; refusals | PASS |
| W5 | — | the scripts | two inputs and the prefilter's modules not in the archive |

**W1 — the 14 cells open at 40.** The continuation joins RUN-029's trees: the same counts at 36,
and every box open there split at 37.

| cell | package: last tested | open at 40 | V₄₀ | V₄₀ / V₃₈ |
| ---: | ---: | ---: | ---: | ---: |
| 1 | 38 | 96,156 | 1.27×10⁻⁸ | 0.572 |
| 2 | 38 | 44,739 | 5.93×10⁻⁹ | 0.465 |
| 6 | 38 | 165,812 | 2.20×10⁻⁸ | 0.557 |
| 7 | 38 | 7,118 | 9.43×10⁻¹⁰ | 0.164 |
| 13 | 38 | 26,147 | 3.47×10⁻⁹ | 0.299 |
| 23 | 36 | 186,202 | 2.47×10⁻⁸ | 0.608 |
| 24 | 36 | 110,456 | 1.46×10⁻⁸ | 0.526 |
| 25 | 38 | 18,201 | 2.41×10⁻⁹ | 0.242 |
| 26 | 38 | 38,259 | 5.07×10⁻⁹ | 0.383 |
| 27 | 38 | 4,026 | 5.34×10⁻¹⁰ | 0.102 |
| 31 | 40 | 2,244 | 2.97×10⁻¹⁰ | 0.079 |
| 32 | 36 | 39,750 | 5.27×10⁻⁹ | 0.270 |
| 33 | 36 | 40,934 | 5.43×10⁻⁹ | 0.378 |
| 34 | 36 | 45,124 | 5.98×10⁻⁹ | 0.294 |

Cell 31 is the one record at 40, and it is this tree's to the box: 2,244 open, 2.97×10⁻¹⁰.

**W2 — the twelve thinnest leaves.** Cell 0 has 9,278 CERT leaves. The twelve thinnest are the
shipped reference's twelve, path for path, and their bounds agree to 7.8×10⁻¹⁶.

| leaf | depth | slack (reference) | margin here | margin, shipped box |
| ---: | ---: | ---: | ---: | ---: |
| 1 | 25 | 4.21×10⁻⁸ | 3.1949×10⁻⁸ | 3.1949×10⁻⁸ |
| 2 | 27 | 7.39×10⁻⁸ | 6.4154×10⁻⁸ | 6.4154×10⁻⁸ |
| 3 | 24 | 1.29×10⁻⁷ | 1.1567×10⁻⁷ | 1.1567×10⁻⁷ |
| 4 | 29 | 1.71×10⁻⁷ | 1.5724×10⁻⁷ | 1.5724×10⁻⁷ |
| 5 | 27 | 2.16×10⁻⁷ | 2.0269×10⁻⁷ | 2.0269×10⁻⁷ |
| 6 | 23 | 2.34×10⁻⁷ | 2.2388×10⁻⁷ | 2.2388×10⁻⁷ |
| 7 | 23 | 3.15×10⁻⁷ | 3.0345×10⁻⁷ | 3.0345×10⁻⁷ |
| 8 | 25 | 3.36×10⁻⁷ | 3.2344×10⁻⁷ | 3.2344×10⁻⁷ |
| 9 | 26 | 3.85×10⁻⁷ | 3.7109×10⁻⁷ | 3.7109×10⁻⁷ |
| 10 | 28 | 3.89×10⁻⁷ | 3.7546×10⁻⁷ | 3.7546×10⁻⁷ |
| 11 | 19 | 3.99×10⁻⁷ | 3.8722×10⁻⁷ | 3.8722×10⁻⁷ |
| 12 | 22 | 4.14×10⁻⁷ | 4.0147×10⁻⁷ | 4.0147×10⁻⁷ |

The polygon costs each leaf about 10⁻⁸ against its exact hull, so the reference slack is not the
certificate's margin. It is the margin's ceiling.

**W3 — against the package's five.** The counts are the package's, and the margins agree to
4.8×10⁻¹⁷ on either box set. Certified at the stored values themselves, without the enclosure, the
five give the same polygons and areas.

**W4 — the enclosures.**

- π lies inside its enclosure (checked against 600-bit mpmath, as a guard against a typo in the
  digits, not as the source of the bound).
- R₃, R₅ and R₇ lie inside theirs, to widths near 10⁻⁶².
- The thinnest leaf's seven B7 corners lie inside their boxes, each at most 2.2×10⁻⁵⁸ wide.

**W5 — the scripts.** Read after the rebuild, never run.

- The certificate scripts:
  - They generate candidates from Round 25's runner (loaded from an absolute path).
  - They nudge by (1 − ε)p + εc and rationalize each coordinate with `format(v, '.17g')`.
  - They enclose every stored float by its neighbours in mpmath.iv at 80 digits, and accept a point
    only when d²_hi < (ρ²)_lo — the rule of §11.
  - They take the exact hull with Fractions.
- The replay reads the worst leaf from `r27_cell0_worst_leaf.json`, the batch its five from
  `r27_cell0_tight12.json`. Neither file is in the archive; the shipped
  `cell0_tight12_reference.json` holds the same paths and slacks.
- The prefilter extends a cell's pending boxes depth first from a pickled state. Below T − 2×10⁻⁵
  on a 256-direction estimate it splits without the exact hull.

## P — the package

**P1.**

- The progress file's totals are its own cells': 14 records, depths 5 / 8 / 1, the volume minimum,
  median and maximum. Each volume is pending / 2^d times the root volume.
- The five records last tested at 36 are Round 26's ledger rows.
- The audit's five paths and reference slacks are the shipped reference's first five.
- The replay's record of the worst leaf is the audit's first row:
  - its exact fraction prints as the audit's area and margin;
  - its lower bound and path are the reference's;
  - 16,384 directions, inward 10⁻¹².
- §17's margins round the audit's. The backend spec says 80 digits and fail-closed.
- The crystal carries the bounds, the worst leaf and 5/5.

**P2.**

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

## Round 27, read

| Round 27 | statement | verdict | note |
| --- | --- | --- | --- |
| §0, §2–3 | Γ(36) = 51; Γ(38) ≥ 57, Γ(40) ≥ 63 | PASS | exact at a common depth (W1) |
| §0, §3 | the 14 cells without a known edge | PASS | each record exact; all 14 open at 40 (W1) |
| §4 | residual scheduling by volume | PASS | reading |
| §5 | the prefilter makes no false positive | PASS | by the code (W5) |
| §7 | the inner-polygon theorem | PASS | W4 |
| §10–12 | exact disc test; interval core test; prototype backend | PASS | W3, W5 |
| §13 | stored binary64 state, enclosed | PASS | the shipped boxes are 4 ulps from this tree's; the φ root needs the enclosure (W2) |
| §14–17 | the worst leaf; 3170/3170; exact area; 5/5 | PASS | W2, W3 |
| §18–19 | what it proves and does not | PASS | |
| §20–22 | migration flow; adaptive density; Round 14 | PASS | reading |
| §23–25 | joint queue deferred; Round 28; handoff | n/a | |

## The status registry

Round 27's 6 judgments are reviewed, and its one first-used phrase defined: `NOT YET`. Census v5
adds none.

| verdict | judgments |
| --- | ---: |
| PASS | 6 |

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's 3.19489 | 1 | A0 |
| a cell's open count; the depth-40 bound raised to 77 | 2 | W1 (also P1) |
| the claimed open cells | 1 | W1 |
| a shipped box's φ one unit up; the certificate cell's path | 2 | W2 |
| an audited area | 1 | W3 (also P1) |
| an audited candidate count | 1 | W3 |
| the claimed hull containment | 1 | W4 |
| the scripts' facts | 1 | W5 |
| the replay's exact fraction; the backend's precision; the crystal's area | 3 | 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 progress file with its keys sorted;
- the tight-leaf audit with its keys sorted.

**16 / 16 plants, 4 / 4 controls, 161 / 161 must-stay-green assertions; no check without a
plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 1 | 1 |
| W1 | 3 | 3 |
| W2 | 2 | 2 |
| W3 | 2 | 2 |
| W4 | 1 | 1 |
| W5 | 1 | 1 |
| P1 | 6 | 6 |
| P2 | 2 | 2 |
| R1 | 1 | 1 |

## What this round does not establish

- **Any bound on a whole cell.** Twelve leaves of cell 0 are certified; its other 9,266 and every
  other cell's are Round 28's and later.
- **The exact root.** Both roots lie outside the exact t₇, and the φ sliver is covered only by the
  enclosure; an exact root is Round 28's.