# RUN-059 — LESR Paper 09A v0.2: the 18-piece signature and both primary O(ρ) arcs reproduce, and the branch's velocity tends to (5.157, −0.1892, 0); but at properly minimized placements the three "secondary slivers" have zero width — the diagnostic's come from a stored sample 1.5×10⁻⁹ above the minimum — and the first epi-derivative has a single zero, with a slope in every direction

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-05
**Subject:** *LESR Paper 09A v0.2 — Multi-Scale Topology-Wall Correction and the One-Sided Blow-Up Critical Cone*
(LESR-LUC-09A-v0.2, 2026-09-20), with its diagnostic, certificate schema, state record and VALIDATION. It supersedes a
v0.1 that is not in the archive. It is evidence, not a canonical document.

**Tools:** [`run059_lesr_paper09a.py`](../code/run059_lesr_paper09a.py)

- this tree's verified hull (RUN-009, with RUN-031's segments) for every winner interval;
- the branch re-minimized on the exact hull at four ρ, by Nelder–Mead restarted from shrinking simplices;
- Φ₁ by Richardson extrapolation on the exact hull;
- Cycle 12's rotation profile as RUN-048 reads it, and the gate log of RUN-058.

**Logs:** [`run059-lesr-paper09a.json`](../data/gate-logs/run059-lesr-paper09a.json)

**Verdict: PASS for the withdrawals, the blow-up chart, the 18-piece count, both primary arcs, the numerical centre and
ratio, and the box. One thing is not as the paper suggests: at the minimum, the three secondary slivers have zero
width.**

- **The withdrawals (§2).** Both are right to withdraw.
  - Paper 07's pressure split is not a subgradient (RUN-057).
  - In Paper 08's own finite model, turning B5 costs +0.029 at first order (RUN-058).
  - The positive branch does change its contact structure at first order (below).
- **The chart (§4, §12, §16).**
  - Cycle 12's 26 positive samples lie within 2.4×10⁻⁸ of the circle d₃ = s.
  - Re-minimized here at ρ = 0.002, at the anchor ρ₀ = 0.0010069…, at 0.0005 and at 0.00025, the branch's (u, v, w)
    is (5.2312, −0.18896, −0.00035), (5.1944, −0.18909, −0.00018), (5.1756, −0.18916, −0.00009) and (5.1664,
    −0.18919, −0.00004).
  - These are linear in ρ, with limit (5.157, −0.1892, 0). That is §12's u ≈ 5.16, v ≈ −0.1892, w ≈ 0, and also
    Paper 09B's closed form (5.157126…, −0.189228…, 0).
  - The anchor sample itself sits at (5.1967, −0.1891, −0.0002). Every point lies in §16's box.
  - A single Nelder–Mead simplex stalls in this kinked minimum: at ρ = 0.0005 and 0.00025 it stopped at ratios 0.232
    and 0.246, against 0.2255 when restarted.
- **The signature (§5).**
  - At the barrier the verified hull has 12 open winner intervals around the circle. That is 13 when B3's corner v0
    is cut at normal 0, which is how Paper 07 counts.
  - At the anchor sample, Cycle 12's first positive sample (the diagnostic records it bit for bit), it has 18 on the
    same convention. Each is the diagnostic's piece, with the ends within 2×10⁻⁸.
- **The newborn intervals (§6–7).**
  - At the anchor sample the five are the diagnostic's, widths to 6×10⁻⁹:
    - the disc near π/2, 8.5×10⁻⁹;
    - B3's arc v2, 0.14804ρ;
    - B5's corners v2 and v3 near π ∓ θ*, 3.7×10⁻⁷ and 7.0×10⁻⁷;
    - B5's arc v1, 1.14804ρ.
  - On the re-minimized branch the two primary arcs are O(ρ). B3's arc is 0.14903ρ, 0.14797ρ, 0.14741ρ and 0.14714ρ,
    tending to about 0.1469ρ (Paper 09B's |τ_T| is 0.146893).
  - B5's arc is exactly ρ wider at every ρ. Both bodies have constant width 1, so B5 wins at θ + π exactly where B3
    wins at θ; the extra ρ is B5's own turn.
  - **The three secondary slivers are below 10⁻¹¹ at all four minimizers: ties, not intervals.**
    - The minimizer keeps the three triple contacts the slivers would break.
    - They exist at the anchor sample because its area is 1.5×10⁻⁹ above the branch's minimum there.
    - So the "O(ρ²)" of §7 is not what the minimum shows. The class the diagnostic calls "secondary, smaller than ρ"
      is, at the minimum, absent. §7 was right not to claim a scaling.
  - At the minimum the signature has 15 pieces on the same convention, not 18: Paper 09B's "primary 15-piece
    structure".
- **The first epi-derivative (§10–11, §14, §16).** Φ₁(u, v, w) is measured as 2f(ρ) − f(2ρ), with
  f(ρ) = (A − A₀)/ρ and ρ = 10⁻⁶.
  - On a 280-point grid around the box it is non-negative. Its least value, 8.6×10⁻⁶, is at the grid point (5.16,
    −0.1892, 0). Outside §16's box its least is 5.4×10⁻⁴.
  - Its minimizer is a single point, (5.1572, −0.18923, 0). Φ₁ rises linearly in every direction there, per unit:
    1.3×10⁻³ / 1.2×10⁻³ along ±u, 0.26 along ±v, 0.40 and 0.036 along ±w. So C₁ is one point, numerically.
  - Extrapolated from ρ ≥ 10⁻⁶, the value there is −1.2×10⁻⁷. But along the ray through it A − A₀ is positive at
    every ρ from 3×10⁻⁸ to 10⁻³, and below about 3×10⁻⁷ the straight-line model fails. This is a kink that moves
    with ρ, not a first-order descent. The resolution here is about 10⁻⁷.
- **Growth (§13, §18–19).**
  - The ratio (m(ρ) − m(0))/ρ² on the branch is 0.225236, 0.225413, 0.225501 and 0.225544, tending to about 0.2256.
    That is §13's "≈ 0.22", with a factor 2.26 over the target 0.1.
  - The tail gate and the "200 → 195" ledger are Paper 08's. The state record retains the ledger, which RUN-058
    found leaves out 14.6 % of root 6217's box.
- **The records.**
  - The diagnostic says NUMERICAL_STRUCTURAL_DIAGNOSTIC_NOT_PROOF and marks both corrections WITHDRAWN.
  - Its critical branch is (5.16, −0.1892, 0, 0.22, 0.1).
  - The schema lists four certificates and warns against freezing the 18-piece signature.
  - The state names the next node "Paper 09B — Analytic Primary Blow-Up Equations", as 09B's own title does.
- **The package (P1).** All four digests match the bytes and the archive's manifest. The text is UTF-8, with `$…$`
  and `$$…$$` only.

Drill: 15 / 15 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 129 / 129 must-stay-green
assertions held. Each of the eight checks was blinded in turn, and each was reported blind in exactly as many cases
as plants name it.

---

## What the paper claims

1. The fixed-chart rotation derivative from Paper 07's pressure balance, and one smooth chart for all small ρ > 0,
   are withdrawn: contact boundaries move at first order, and thinner winner intervals are born on a smaller scale.
2. A one-sided blow-up chart: B3 on d₃ = s at α₃ = π − ρu, y₅ = ρv, x₅ = x₅* + ρw.
3. At Cycle 12's first profile sample, 18 merged open winner intervals against the barrier's 13. Two primary
   intervals of width about 0.1480ρ and 1.1480ρ; three thinner slivers whose O(ρ²) scale is a working hypothesis.
4. Φ₁, C₁ = argmin Φ₁, Σ₂ and c* = inf inf Φ₂. The numerical centre u ≈ 5.16, v ≈ −0.1892, w ≈ 0; the ratio ≈ 0.22.
5. Four certificates (primary chart, sliver classification, Φ₁ ≥ 0, Φ₂ ≥ 0.1), the box u ∈ [4.5, 6.0],
   v ∈ [−0.23, −0.15], w ∈ [−0.01, 0.01], and the tail gate and ledger unchanged.

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the samples on the circle; the branch at four ρ and its limit | (5.157, −0.1892, 0); every point in the box |
| W2 | the signature at the barrier and the anchor | 13 and 18, the diagnostic's pieces |
| W3 | the newborn intervals at the anchor and on the branch | primary O(ρ); secondary absent at the minimum |
| W4 | Φ₁ on a grid, at its minimizer, along the ray | ≥ 0 to 10⁻⁷; one zero, slopes in every direction |
| W5 | the ratio; the ledger | ≈ 0.2256; the 195 still omits 14.6 % |
| W6 | the diagnostic, schema and state | as the text; the anchor is Cycle 12's sample |

## P — the package

**P1.** The paper's, the diagnostic's, the schema's and the state's digests match the bytes and the archive's manifest.

## The paper, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §0–3 | ρ = 0 is a wall; the two withdrawals | PASS | RUN-057, RUN-058; the structure changes at first order (W2, W3) |
| §4 | the blow-up chart on d₃ = s | PASS | W1 |
| §5 | 18 merged open intervals against 13 | PASS | W2, on the cut-at-0 convention |
| §6 | primary widths 0.1480ρ and 1.1480ρ | PASS | W3; the coefficient drifts to about 0.1469 as ρ → 0 |
| §7 | three secondary slivers, O(ρ²) a hypothesis | PARTIAL | they exist at the stored sample; at the minimum they are ties (W3) |
| §8–11 | the nested blow-up; Φ₁, C₁, Σ₂, c* | PASS | definitions; Φ₁ measured (W4) |
| §12–13 | u ≈ 5.16, v ≈ −0.1892, w ≈ 0; ratio ≈ 0.22 | PASS | W1, W5 |
| §14–18 | four certificates; the box; the target | PASS | stated as open; Φ₁ > 0 outside the box on the grid (W4) |
| §19 | the tail gate and the 200 → 195 ledger unchanged | as stated | the ledger omits 14.6 % (RUN-058) |
| §20–22 | lessons; Paper 09B next | n/a | |

## The status registry

Not touched: Paper 09A is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's 0.1480ρ | 1 | A0 |
| a stored sample off the circle; a branch velocity | 2 | W1 |
| a piece in the diagnostic's signature; an interval dropped here | 2 | W2 |
| a width in the diagnostic; a sliver at a minimizer | 2 | W3 |
| a negative grid value; a negative point on the ray | 2 | W4 |
| a branch ratio; RUN-058's missing share | 2 | W5 |
| the diagnostic's status; the state's next node | 2 | W6 |
| a VALIDATION digest; the schema's bytes | 2 | P1 |

Three controls:

- the unmodified inputs;
- the document with CRLF line endings;
- the VALIDATION record with the keys sorted.

**15 / 15 plants, 3 / 3 controls, 129 / 129 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 | 2 | 2 |
| W5 | 2 | 2 |
| W6 | 2 | 2 |
| P1 | 2 | 2 |

## What this run does not establish

- **Any of the four certificates.** Everything here is in floating point: the exact hull in double precision,
  Nelder–Mead, Φ₁ from ρ ≥ 10⁻⁶.
- **Φ₁ ≥ 0 off the grid**, or below the resolution of about 10⁻⁷.
- **That the secondary slivers are absent for every small ρ.** They are ties at the four ρ ≥ 2.5×10⁻⁴ examined here.