# RUN-050 — LESR Paper 00: every formula the bridge uses holds, symbolically and on bodies; its one programmatic principle needs a curvature hypothesis at corners, where the first variation of an inward move is zero and the area drops only at second order; the predecessors it builds on are not in the archive

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** *Lebesgue Extremal Shape Reconstruction, Paper 00 — From Skew Fields to Variational Shape Closure*
(LESR-SFVC-00-v0.1, 2026-09-20), one of the 42 canonical documents (RUN-001), with its state record and VALIDATION.

**Tools:** [`run050_lesr_paper00.py`](../code/run050_lesr_paper00.py)

- sympy: every formula as an identity over trigonometric polynomials with free coefficients;
- this tree's exact hull (RUN-009, with RUN-031's segments) for the bodies;
- the status registry (R1).

**Logs:** [`run050-lesr-paper00.json`](../data/gate-logs/run050-lesr-paper00.json)

**Verdict: PASS. The paper is a bridge, not a result: it states definitions and classical facts, and marks its
principles as programmatic. Every formula it uses holds. Two things are noted:**

- **The No-Slack Boundary Principle (§29) needs a hypothesis.** Its conclusion δA[h; φ] < 0 holds when the sector
  carries curvature, μ_h(I) > 0. On a corner's sector of normals, an inward convexity-preserving move has δA = 0
  exactly, and the area drops only at second order. The principle survives through δ²A, but not as written.
- **§1–6 build on earlier EveMissLab documents** (SKEW-CALC v0.2, the Moser skew-field initialization, Subtractive
  Topology v3.0, GSC, RVSC, TGSA, Geometric Skew). None of them is in the archive, so the paper's account of what
  they already contain is not checked here.

- **The identities (W1).** Each holds in sympy over trigonometric polynomials with free coefficients:
  - A[h] = ½∫(h² − h′²) = ½∫h(h + h″) (§16);
  - the first variation δA[h; φ] = ∫φ(h + h″) (§17);
  - the second variation ∫(φ² − φ′²) (§19), which for cos kθ is π(1 − k²): 0, −3π, −8π, −15π for k = 1…4;
  - the translation kernel (a cos θ + b sin θ)″ + (a cos θ + b sin θ) = 0, and the area unchanged by it (§18);
  - the rotation generator ∂_φ h(θ − φ) = −h′ at 0, and the area unchanged by rotation (§12);
  - the multiplier (∂² + 1) cos mθ = (1 − m²) cos mθ (§13).
  - The placement formula h_{gγ}(θ) = h_γ(θ − φ) + t·u_θ (§2) holds on random point sets to 10⁻¹⁵.
  - On bodies: the Reuleaux triangle's area by its curvature measure (density 1 on its arcs, none at its corners)
    is (π − √3)/2 to 10⁻¹¹; the seed hull's area 0.834780945912 by the kernel agrees with a 200,000-vertex polygon
    of its contact points to 2.6×10⁻¹⁰, the polygon's inscribed loss.
- **The principle at a corner (W2).**
  - Take the unit square and the sector [0, π/2] of normals its corner (½, ½) carries, where μ_h has no mass.
  - Move it inward by εφ with φ = −sin 2θ on the sector and 0 elsewhere. Inside, φ + φ″ = 3 sin 2θ ≥ 0.
  - At the sector's ends φ′ jumps by 2, which the edges' atoms (length 1) absorb while ε ≤ ½. So the move is
    admissible, and φ < 0 on the sector.
  - Its first variation is 0: φ vanishes at the edge normals, and μ_h has no mass inside.
  - The area is exactly quadratic in ε: A(ε) = 1 − (3π/8)ε².
  - So at corners the extremality contradiction comes from δ²A = ∫(φ² − φ′²) < 0 (sectors narrower than π), not
    from δA < 0. §29 is labelled programmatic, and it is right to be.
- **The sources (W3).** No leaf of the archive is named for any of the seven predecessors.
- **The state record (W4).** Its eleven "new bridge" components are the paper's sections, one each: §7, 8, 10, 12,
  11, 13, 17, 19, 20, 22 and 25. Its next gate is §35's. It lists seven recovered components from the predecessors.
- **The package (P1).** Both digests match the bytes and the archive's manifest. The document is UTF-8, and it
  uses no \( \) or \[ \] delimiters, as VALIDATION's `math_delimiters: PASS` says.
- **The registry (R1).** The census found one judgment: `PROGRAMMATIC PRINCIPLE` (§29), first used here. It is now
  defined and reviewed: PASS by reading, with the corner hypothesis noted.

Drill: 9 / 9 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 75 / 75 must-stay-green
assertions held. Each of the seven checks was blinded in turn, and each was reported blind in exactly as many cases
as plants name it.

---

## What the paper claims

1. Value closure (the area gap) does not close the shape. It adds shape closure: support function h, curvature
   measure μ_h = h + h″ ≥ 0, skew-admissible perturbations, the E(2) quotient, group orbits and Fourier modes, the
   area functional and its first and second variations, contact strata, a support corridor, and a multi-gap state
   (Δ_A, Δ_h, Δ_Γ, Δ_G).
2. The skew-field line already had the support skew K, the best-placement skew E_C, a conceptual shape update and
   contact topology. What was missing is the legal variation.
3. Two programmatic items: the No-Slack Boundary Principle, and infinite-dimensional KKT measures. Both are stated
   as not yet proved.
4. The next gate is a Support-Function Admissible Variation Compiler.

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the formulas in sympy; placement on point sets; areas of two bodies | all hold |
| W2 | §29 at a square's corner | δA = 0; A(ε) = 1 − (3π/8)ε² |
| W3 | the predecessors in the archive | none |
| W4 | the state record's components | one section each |

## P — the package

**P1.** The paper's and the state's digests match the bytes and the archive's manifest; the text is UTF-8 with
only `$…$` and `$$…$$` for mathematics.

## The paper, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §2 | h_{gγ}(θ) = h_γ(θ − φ) + t·u_θ | PASS | W1 |
| §3 | E_C(γ) = 0 exactly when some placement fits | PASS by cited theorem | h_γ ≤ h_C is containment (classical) |
| §8 | h + h″ ≥ 0 as a measure characterizes support functions | PASS by cited theorem | classical |
| §10 | skew-admissible perturbation | PASS | a definition |
| §12–13 | the E(2) quotient; the Fourier multiplier 1 − m² | PASS | W1; its E(2) = SO(2) ⋉ R² is SE(2), as Paper 02 §1 corrects (RUN-052) |
| §16–19 | the area, its first and second variations; translation modes vanish | PASS | W1 |
| §20–25 | strata, envelope, corridor, multi-gap state | PASS | definitions |
| §29 | No-Slack Boundary Principle (`PROGRAMMATIC PRINCIPLE`) | PASS as labelled | needs μ_h(I) > 0 for δA < 0 (W2) |
| §30 | infinite-dimensional KKT | PASS | stated as future work |
| §1–6 | what the earlier skew-field documents contain | not checked | not in the archive (W3) |

## The status registry

R1 PASS: one judgment reviewed, and one phrase defined (`PROGRAMMATIC PRINCIPLE`). Both are recorded for LESR by
this run.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the first-variation formula in the text | 1 | A0 |
| an identity reported false; a second-variation value | 2 | W1 |
| the corner's first variation | 1 | W2 |
| a predecessor planted in the archive's names | 1 | W3 |
| a state component dropped | 1 | W4 |
| the state's bytes; a \( \) delimiter in the document's bytes | 2 | P1 |
| the review dropped | 1 | R1 |

Three controls:

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

**9 / 9 plants, 3 / 3 controls, 75 / 75 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 | 1 | 1 |
| W4 | 1 | 1 |
| P1 | 2 | 2 |
| R1 | 1 | 1 |

## What this run does not establish

- **The predecessors' content.** §1–6 describe earlier documents that are not in the archive.
- **Anything about the Lebesgue extremal shape.** Paper 00 does not claim anything about it either; the papers that
  follow do.
- **The classical theorems it cites** (support functions and convexity, containment by support functions), used
  here as known.