# RUN-052 — LESR Paper 02: the group layer holds — its projectors, its gcd law, and its first-variation theorem, on symmetric bodies and on the kernel's own hull; but on the kernel's seed no pure Fourier mode survives the convexity test, and breaking the symmetry costs area at first order — the recorded seed's asymmetry is search noise

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** *LESR Paper 02 — Group-Orbit Shape Expansion and Symmetry Breaking* (LESR-GROUP-02-v0.1, 2026-09-20),
with its schema, compiler scaffold, state record and VALIDATION. It is evidence, not a canonical document.

**Tools:** [`run052_lesr_paper02.py`](../code/run052_lesr_paper02.py)

- sympy for the formulas, and the O(2) action on Fourier coefficients up to mode 24;
- this tree's exact hull (RUN-009, with RUN-031's segments) for the Reuleaux polygons and the kernel;
- the kernel's family area (RUN-046) around Round 07's best placement q† (RUN-010).

**Logs:** [`run052-lesr-paper02.json`](../data/gate-logs/run052-lesr-paper02.json)

**Verdict: PASS. Paper 02 is a framework built from classical facts, and every fact it states holds. Three things
are noted. The first two concern the kernel itself:**

- **On the kernel's seed the pure-mode library is empty.** §54 proposes taking the current seed's stabilizer and
  generating preserving and breaking modes from it.
  - A pure mode cos m(θ − c) can be added to a convex body only if it keeps one sign wherever the curvature measure
    has no density (§14).
  - The kernel's hull has a stretch of 8π/15 of such normals: a B5 corner's cone and a B3 corner's cone, meeting at
    the top edge. That is wider than π/2.
  - So no pure mode with m ≥ 2 can be added at any phase, sign or amplitude. This holds on the mirror slice and at
    the recorded seed alike.
  - §14 says the smooth bound is only sufficient. On this seed it gives nothing. The scaffold computes the smooth
    cap, discards it, and would pass every mode on.
- **The recorded seed's asymmetry is noise, and breaking costs area at first order.**
  - Round 07's best placement q† has φ5 2.41×10⁻⁵ off 2π/5 and y5 = −3.9×10⁻⁶, so its hull has no symmetry.
  - Polished on the mirror slice (y3 = y5 = 0, φ5 = 2π/5), the area drops by 9.03×10⁻¹¹ to 0.8347809458213. That
    point is a strict local minimum.
  - Along every breaking coordinate the area rises linearly, at the same rate both ways: 0.133 per unit of y3, 0.044
    of φ5, 0.273 of y5.
  - The reason: on the mirror slice the disc, B3 and B5, each of width 1, all touch the hull's supporting lines at the
    normals ±π/2. A breaking move lifts one of them over a line.
  - So §39's test, run on the kernel, is decided at first order by the contacts, not by δ²A. §34's δA = 0 is about
    linear support perturbations, and W2 confirms it on the kernel's hull on the mirror slice. A placement move at a
    shared contact is not linear.
- **§15 defines the stabilizer on the wrong space.** On X = H_conv/E(2) every g ∈ O(2) fixes every class, so the
  formula as written gives O(2) for every shape. The stabilizer meant is the one on the translation gauge. §15's next
  sentence gives it, and every later section uses it.

- **The erratum (W1).** Paper 00 §12 and Paper 01 §1 do write E(2) = SO(2) ⋉ R². Paper 02's SE(2) and E(2) are the
  standard notation.
- **The group on support functions (W1).**
  - Placements by turns and mirrors, h_{gK}(u) = h_K(Rᵀu) + t·u and (σ_α h)(θ) = h(2α − θ), hold on random point
    sets to 10⁻¹⁵.
  - The translation gauge a₁ = b₁ = 0 puts the Steiner point at the origin: (1/π)∫h u dθ = Σ α_i v_i / 2π on random
    polygons to 10⁻¹⁵.
  - §7's matrices, L's eigenvalue 1 − m² on V_m, and L's equivariance under turns and mirrors hold in sympy.
- **The projectors (W1).** On the modes up to 24, the averages over C_n and D_n (n = 1…8) are orthogonal
  projections.
  - Their ranks are 1 + 2⌊24/n⌋ and 1 + ⌊24/n⌋: 49, 25, 17, 13, 9, 9, 7, 7 and 25, 13, 9, 7, 5, 5, 4, 4.
  - Their images are §21–22's modes, and with the mirror axis at α, §23's cos kn(θ − α).
- **The variations (W2).**
  - ∫(φψ − φ′ψ′) on 1, cos mθ, sin mθ (m ≤ 5) is diagonal: 2π, then π(1 − m²) twice for each m (§36–37).
  - §34 on bodies: on the Reuleaux triangle and pentagon, and on the kernel's hull on the mirror slice, every breaking
    mode up to m = 12 has |δA| below 4×10⁻¹⁵. It is computed as ∫φ dμ_h and as ∫(hφ − h′φ′), and the two agree.
  - Preserving modes need not have δA ≠ 0 (cos 6θ on the triangle has 0), and the theorem does not say they do.
  - At q†, whose hull is not symmetric, the sine modes' δA reach 1.5×10⁻⁴.
- **The gcd law (W3).**
  - D_n seeds (n = 2…8) plus ε cos mθ (m = 2…24): the stabilizer read off each sampled body on the translation gauge
    is D_gcd(n,m) in all 161 cases. §27's four examples are among them.
  - For m = 1 the law's D₁ is the stabilizer of the function, but the body is only translated and keeps D_n. So the
    law needs m ≥ 2, which §8's "m = 1 is gauge" supplies.
- **The caps (W4).**
  - A smooth seed, with ρ between 0.585 and 1.415: for m = 2…60, eight phases and both signs, the exact cap times
    (m² − 1)/κ₀ lies in [1.000, 2.004], and at m = 60 in [1.000, 1.009]. So §12's bound holds, and §13's m⁻² rate is
    the true rate.
  - The Reuleaux triangle: up to m = 24, the only admissible pure mode is m = 3, at one phase and one sign, with cap
    1/8.
  - The pentagon: only m = 5, with cap 1/24. On both, that is the lowest D_n-preserving mode, and every breaking mode
    is blocked.
  - The kernel's hull: no mode.
- **The seed (W5).**
  - q†'s mirror image has q†'s area to 4×10⁻¹⁶: the family is mirror-symmetric.
  - Random directions around the polish, 1,000 at each radius 10⁻⁷, 10⁻⁶ and 10⁻⁵, all raise the area (least
    0.0033 per unit).
  - The area falls monotonically along the segment from q† to the polish.
  - The stabilizer, read off the sampled hull, is D₁ at the polish and trivial at q†.
  - The six supports at ±π/2 (the disc, B3v1 and B5v0 above; the disc, B3v2 and B5v3 below) are all ½ to one ulp.
- **The records (W6).**
  - The schema's six outputs are §44's, and its mode classes are §8's. Its operators and gcd prediction are those of
    §9, §17 and §26.
  - Each of the state's 16 completed items has its section.
  - Three of its four "not claimed" items are the text's own disclaimers (§26, §38, §52). The first, that every
    extremal candidate has a finite stabilizer, is the state's own.
  - Both records name Paper 03 as the text does.
- **The scaffold (W7), read and never run.** It has 62 lines.
  - It can return three of §44's six outputs: PRESERVE_H, BREAK_TO_K and GAUGE_MODE.
  - It computes the smooth cap and never uses it, so it never returns CONVEXITY_BLOCKED.
  - It knows only m = 1 as gauge. For the seed 1 + a cos nθ, sin nθ is the turn's tangent (§46), and the scaffold
    would call it BREAK_TO_K.
  - It calls m = 0 SCALE_MODE. But h + a₀ adds a disc of radius a₀ (the parallel body), which is a dilation only for
    a disc.
- **The package (P1).** All four digests match the bytes and the archive's manifest. The text is UTF-8, with `$…$`
  and `$$…$$` only.

Drill: 17 / 17 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 163 / 163 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 the paper claims

1. A notation erratum: SE(2) = SO(2) ⋉ R², E(2) = O(2) ⋉ R². The rigid motions and reflections act on support
   functions, and the translation gauge is a₁ = b₁ = 0.
2. The Fourier modes V_m are the real O(2) modes. L = 1 + ∂² acts on them by 1 − m² and commutes with O(2).
   m = 0 is mean support, m = 1 is gauge, and m ≥ 2 are the intrinsic modes.
3. On a smooth seed with ρ ≥ κ₀ > 0 a pure mode is admissible up to κ₀ / ((m² − 1)‖φ_m‖∞), so high modes cost
   O(m⁻²). With atoms, only the measure inequality counts.
4. A finite stabilizer's average P_H splits a perturbation into its preserving and breaking parts. C_n keeps the
   modes m ≡ 0 (mod n), and D_n the cosines cos kn(θ − α). An aligned cos mθ on a D_n seed leaves at least
   D_gcd(n,m), and exactly that without accidental symmetry.
5. On an H-invariant seed a pure breaking mode has δA = 0, and δ²A = π(1 − m²)(a_m² + b_m²). So breaking is decided
   by the second variation and the constraints, and must be tested.
6. A mode compiler with six outputs, a step rule, an extended frontier relation, a finite symmetry-breaking graph,
   and a group-closure index.

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the erratum; placements; the Steiner point; §7's matrices; L; the projectors of C_n and D_n | all hold |
| W2 | the Fourier Gram matrix; δA of breaking modes on three symmetric bodies and on q† | diagonal; < 4×10⁻¹⁵ on the symmetric bodies, 1.5×10⁻⁴ at q† |
| W3 | the stabilizers of 161 perturbed D_n bodies | D_gcd(n,m) in every case; m = 1 keeps D_n |
| W4 | exact caps on a smooth seed; admissible pure modes on bodies with corners | ratio in [1, 2.004]; one mode each on the Reuleaux polygons; none on the kernel's hull |
| W5 | q† against its mirror-symmetric polish | the polish is 9.03×10⁻¹¹ lower, a strict local minimum with first-order rises |
| W6 | the schema and the state | as the text; one "not claimed" item is the state's own |
| W7 | the scaffold, read | 3 of 6 outputs; cap unused; no rotation gauge |

## P — the package

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

## The paper, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §1 | SE(2) and E(2) | PASS | the erratum is accurate: Paper 00 §12 and Paper 01 §1 wrote E(2) = SO(2) ⋉ R² |
| §2–6 | the group on support functions; the translation gauge | PASS | W1 |
| §7–11 | V_m as O(2) modes; L diagonal and equivariant | PASS | W1 |
| §8 | m = 0 "scale / mean support" | PASS | mean support, yes; h + a₀ is the parallel body, not a dilation |
| §12–14 | the smooth cap; O(m⁻²); the measure version | PASS | W4; on the kernel's hull every pure mode is blocked |
| §15 | the intrinsic stabilizer | PASS as its second sentence | on X = H_conv/E(2) the formula gives O(2) for every shape |
| §17–23 | projectors; the C_n and D_n rules | PASS | W1 |
| §24–27 | preserving modes; the gcd law | PASS for m ≥ 2 | W3; m = 1 is a translation and keeps D_n |
| §28–32 | isotropy lattice; orbits; orbit splitting | PASS | definitions; W5 shows a breaking move lifting one of a mirror pair of contacts |
| §34 | δA = 0 on pure breaking modes | PASS | W2 |
| §36–37 | δ²A on the Fourier modes | PASS | W2 |
| §39 | symmetry breaking must be tested | PASS | on the kernel's seed it is decided at first order: the symmetric point is a kinked minimum (W5) |
| §41–46 | skew basis, compiler, outputs, step rule, gauge | PASS | specifications; the scaffold implements part of them (W7) |
| §54 | extract the current seed's stabilizer, generate modes | PASS as a plan | the recorded seed is asymmetric only by noise; its polish has D₁; no pure mode is admissible on it (W4, W5) |

## The status registry

Not touched: Paper 02 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the gcd law in the text | 1 | A0 |
| the erratum's source; a projector's rank | 2 | W1 |
| a breaking mode's first variation; the Gram matrix off its diagonal | 2 | W2 |
| one case off the gcd law; the translation's stabilizer | 2 | W3 |
| an extra admissible mode on the triangle; the smooth ratio below 1 | 2 | W4 |
| q† below the polish; a flat breaking direction | 2 | W5 |
| an output added to the schema; a completed item added to the state | 2 | W6 |
| CONVEXITY_BLOCKED returned in the scaffold; its m = 0 class renamed | 2 | W7 |
| a VALIDATION digest; the state's bytes | 2 | P1 |

Three controls:

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

**17 / 17 plants, 3 / 3 controls, 163 / 163 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 |
| W7 | 2 | 2 |
| P1 | 2 | 2 |

## What this run does not establish

- **That the polish is the kernel's global minimum.** It is a local polish of q† on the mirror slice, with 3,000
  random directions around it. Its value is a floating-point evaluation of the verified hull, not an enclosure.
  Every bound that uses q† (Round 07's outer area 0.834781190917, the headroom to 0.835) stays valid, since the
  polish is lower.
- **The constrained second-order theory** (§40), which the paper leaves to Paper 05.
- **§29's frontier relation** in general, which the paper states for regular equivariant models only.
- **Anything about modes that are not pure.** Contact-localized modes (§55) can be admissible where pure modes are
  not. The paper does not build them here, and this run does not either.