# RUN-051 — LESR Paper 01: the topology layer's two theorems hold on the Lebesgue kernel — at most two crossings per piece pair, and an area that is analytic inside a contact stratum and kinks where the stratum changes; the compiler scaffold implements the monotone envelope but not the stability test the text states

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** *LESR Paper 01 — Topology Envelope and Stratified Shape Space* (LESR-TOPO-01-v0.1, 2026-09-20), with
its schema, compiler scaffold, state record and VALIDATION. It is evidence, not a canonical document.

**Tools:** [`run051_lesr_paper01.py`](../code/run051_lesr_paper01.py)

- the kernel's own support pieces (the disc, and B3's and B5's corners and arcs);
- this tree's exact hull and its envelope's segments (RUN-009, RUN-031, RUN-046).

**Logs:** [`run051-lesr-paper01.json`](../data/gate-logs/run051-lesr-paper01.json)

**Verdict: PASS. Paper 01 is a framework: definitions, two small theorems, and lemmas on monotonicity and
stability. The theorems hold where the series uses them, on the Lebesgue kernel. One thing is noted:**

- **The scaffold does not do what §33 asks of it.** Its `stable_stratum` compares each margin with the
  perturbation bound directly. The text's lemma needs ‖δh‖ < m/C, with a sensitivity constant C; the scaffold has
  none.
- Its gate also returns only three of the schema's five statuses (never `INCONSISTENT_SIGNATURE` or `REOPEN`).
- It computes no crossing. It is labelled a specification, not a kernel, and it is one.

- **Finite signatures (W1).** §10 assumes each pair of pieces crosses finitely often.
  - For the kernel's pieces the difference is h_i − h_j = A cos(θ − β) + Δr, so a pair crosses at most twice a
    turn, unless the two pieces coincide. That bound does not depend on the placement, which is what the finiteness
    of signatures actually needs; §10's wording ("finitely many roots") is weaker.
  - At 300 random placements the most sign changes of any pair is 2, and no pair coincides.
  - The hull has at most 18 segments, against the 288 that the crossing bound allows.
- **Analyticity and the kink (W2).** §2 and §14, on the symmetric slice (x3*, 0, 0, x5, 0) through Cycle 12's
  contact switch x5*:
  - On each side, 21 points 10⁻⁴ to 5×10⁻⁴ away keep one piece sequence. A degree-6 polynomial fits the area
    there to 1.2×10⁻¹⁵ and 1.6×10⁻¹⁵: analytic, as §14 says.
  - The two sides' sequences differ, and across the switch the same fit leaves 3.7×10⁻⁶: the kink of §2, "shape
    coordinates may vary smoothly while contact topology jumps".
- **The records (W3).** The schema's six signature components are §3's. Its five statuses are §32's four and §44's
  `REOPEN`. The state lists fifteen completed items and the text's three disclaimers, and names the next paper as
  the text does.
- **The package (P1).** All four digests match the bytes and the archive's manifest. The text is UTF-8, with
  `$…$` and `$$…$$` only.

Drill: 10 / 10 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 68 / 68 must-stay-green
assertions held. Each of the six checks was blinded in turn, and each was reported blind in exactly as many cases
as plants name it.

---

## What the paper claims

1. Shape coordinates can vary smoothly while the contact combinatorics jump, so calculus belongs inside contact
   strata. A signature Γ = (W, P, O, T, I, G) cuts them out.
2. Theorems:
   - a finite support-piece system has finitely many signatures on a compact region;
   - the hull area is analytic on a regular stratum;
   - a regular transition is a codimension-one wall, and symmetry can lower a transition's codimension.
3. The topology envelope. The maximal carrier T_max (the admissible strata's closures) can only shrink. The
   mandatory core F_min (features every admissible signature shares) can only grow, and it generates the minimal
   carrier T_min.
4. A stability lemma (margins m, sensitivity C) and a trust radius. Topology before Hessians. A monotone
   research state, reopened only for an invalidated proof or a widened model.

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | pair crossings and hull segments at 300 placements | ≤ 2 per pair; ≤ 18 segments |
| W2 | the area through a contact switch | analytic inside (10⁻¹⁵), a kink across (3.7×10⁻⁶) |
| W3 | the schema and the state | as the text |
| W4 | the scaffold, read | no C, three statuses, no crossings |

## 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 | the shape space H_conv / E(2) | PASS | a definition; its E(2) = SO(2) ⋉ R² is SE(2), as Paper 02 §1 corrects (RUN-052) |
| §3–8 | signatures, contact graph, strata, frontier order and condition | PASS | definitions; the frontier condition is a requirement, not verified here |
| §10 | finitely many signatures for a finite piece system | PASS | needs a placement-uniform root bound; the kernel has 2 (W1) |
| §12–14 | regular crossings are analytic; so is the area on a regular stratum | PASS | W2 |
| §17–18 | codimension-one walls; symmetry lowers codimension | PASS | by the implicit-function theorem |
| §20–26 | T_max, F_min, T_min, the envelope; their monotonicity | PASS | set algebra |
| §33 | stability under perturbations below m/C | PASS | a Lipschitz argument; the scaffold drops C (W4) |
| §39 | observed signatures are not global strata | PASS | RUN-048 found Cycle 14's four skeletons resolution-dependent |
| §43–44 | monotone research state; reopening | PASS | rules |

## The status registry

Not touched: Paper 01 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the frontier order in the text | 1 | A0 |
| three crossings for a pair; segments past the bound | 2 | W1 |
| no kink across the switch; the same pieces on both sides | 2 | W2 |
| a status added to the schema; a completed item dropped from the state | 2 | W3 |
| a status returned in the scaffold | 1 | W4 |
| 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.

**10 / 10 plants, 3 / 3 controls, 68 / 68 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 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **The frontier condition** (§8) for any actual family of Lebesgue strata. The text asks for it to be verified
  in the finite chart model; neither the paper nor this run does so.
- **A contact atlas of the continuum problem.** The paper does not claim one either.
- **Whitney regularity**, disclaimed by the paper.