# RUN-055 — LESR Paper 05: the finite duality is Farkas and holds; the smooth No-Slack theorem holds and carries the curvature hypothesis Paper 00 lacked; but on the kernel the finite dual certificate exists wherever it is solved — at a generic placement whose area gradient has length 0.39 too

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** *LESR Paper 05 — Contact Pressure, Dual Multipliers, and the No-Slack Boundary Principle*
(LESR-DUAL-05-v0.1, 2026-09-20), with its dual schema, pressure-graph schema, sanity-check record, verifier scaffold,
state record and VALIDATION. It is evidence, not a canonical document.

**Tools:** [`run055_lesr_paper05.py`](../code/run055_lesr_paper05.py)

- scipy's HiGHS (primal LPs, and the dual simplex, which returns basic solutions);
- RUN-053's hull structure on the kernel, and RUN-052's placements.

**Logs:** [`run055-lesr-paper05.json`](../data/gate-logs/run055-lesr-paper05.json)

**Verdict: PASS for everything Paper 05 proves. §68 says the finite dual can be solved now on the Lebesgue line's own
witnesses; solved, it certifies a placement that is not stationary:**

- **The finite certificate exists on the kernel wherever it is solved.**
  - Following §5–8, the rows are contacts on each witness's own hull sectors, sampled at spacing π/(4M), plus every
    piece touching at each edge normal. The columns are Fourier modes up to M, and each witness (disc, B3, B5) turns
    and translates freely.
  - With M = 4 and 8, the primal has no descent and the dual has a certificate: λ ≥ 0, g = Aᵀλ, Bᵀλ = 0. This holds
    at the generic placement, at q† and at the polish Z. The residuals are at most 5×10⁻¹¹, with 14–23 nonzero
    multipliers, never more than the number of equations (§49).
  - At the generic placement the area's gradient has length 0.390, so §54's compiler would print DUAL_CERTIFIED at a
    placement that is not stationary.
  - This is Farkas applied to RUN-053's affine model, which finds no descent there. §11's theorem is right; its
    premise, the affine linearization, is what fails on a hull-type container.
- **The finite model needs care the paper does not state.**
  - Contacts must be sampled finer than the modes. At Z, a strict local minimum, M = 16 with six contacts per segment
    gives a "descent" of −0.107 that is not there.
  - The rotation gauge must be fixed. At the generic placement, M = 16 with every witness free to turn gives
    −6.2×10⁻⁴, shrinking to −2.2×10⁻⁴ at four times the contact density. It is gone once B3's turn is fixed (turning
    every witness together is a turn of the container, which a finite library can only approximate). Then the
    certificate exists again (38 multipliers, 40 equations).
- **§33 carries the hypothesis Paper 00 §29 lacked.**
  - With ρ_h ≥ κ₀ > 0 on the arc, an inward bump lowers the area. On h = 1 + 0.05 cos 3θ + 0.02 sin 2θ (κ₀ = 0.548),
    a C^∞ bump on (−0.4, 0.4) has convexity cap 0.0159, δA = −0.114 and δ²A = −0.971, and the area drops at the cap
    and at half of it.
  - At a corner, where ρ = 0 on the arc, RUN-050's admissible inward move has δA = 0. So the hypothesis is needed,
    and §34's measure form asks for ∫φ dμ_h < 0 outright.

- **The duality (W1).** On 300 random finite systems (8–40 rows, 3–12 modes, 3–9 placement columns), exactly one of
  a primal descent and a dual certificate exists every time: 213 certificates (residuals ≤ 5×10⁻¹⁴) and 87
  descents. The sanity record's three contacts at 120° balance to 3×10⁻¹⁶.
- **The No-Slack arc (W2).** As above.
- **The kernel (W3).** As above. The contact matrices' g₀ is the perimeter of the verified hull to 10⁻⁹.
- **The records (W4).**
  - The dual schema's three conditions are §10's. Its four graph-layer names are §50–53's, and the pressure graph's
    four contact states are §58's.
  - §54's seven compiler outputs are read from the text.
  - Each of the state's 14 completed items has its section, and all four "not claimed" items are the text's own
    disclaimers (§20, §22, §29, §59).
  - The state and the dual schema name Paper 06 "Lebesgue Extremal Shape Closure", as §73 does. Paper 06's own title
    is "Actual Lebesgue Shape State and First Closure Graph".
- **The scaffold (W5), read and never run.** It has 72 lines and verifies a supplied certificate: λ ≥ −tol,
  ‖Aᵀλ − g‖ ≤ tol, ‖Bᵀλ‖ ≤ tol.
  - It does not solve for one.
  - It does not check §48's complementarity (λ = 0 off the active constraints).
  - It accepts multipliers down to −tol.
  - Its balance helpers are §13 and §15. The generic placement's certificate above satisfies everything it checks.
- **The package (P1).** All six digests match the bytes and the archive's manifest. The text is UTF-8, with `$…$`
  and `$$…$$` only.

Drill: 13 / 13 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 99 / 99 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. Paper 03's linearized contacts stack into A c + B ξ ≥ 0. With no feasible descent, Farkas gives λ ≥ 0 with
   g = Aᵀλ and Bᵀλ = 0: each witness's contact pressure balances in translation and rotation. μ_h matches the
   pressure measure on the library's moments, not as measures.
2. The continuum version is conditional: a positive measure on the contact index space, its pushforward, and
   placement balances.
3. The No-Slack principle. A smooth arc with ρ ≥ κ₀ > 0 and nothing active on it admits an inward cut (proved). The
   measure form follows almost from the definition. At an optimum, supp μ_h lies in the closure of the dual supports.
4. Continuous curvature needs continuum pressure, while polygonal or finite-arc optima can be matched by finitely many
   contacts. Binding contacts (λ > 0) define the skeleton, and a dual gap Δ_dual joins the closure state.
5. The finite dual on the Lebesgue line's own witnesses is a low-compute task (§68).

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | Farkas' alternative on 300 random systems; the sanity record | exactly one alternative every time |
| W2 | §33 on a smooth body; the corner case | the cut lowers the area; at a corner δA = 0 |
| W3 | §5–11 on the kernel at three placements, M = 4, 8, 16 | certificates everywhere, the generic placement included; sampling and gauge artifacts at M = 16 |
| W4 | the schemas and the state | as the text; Paper 06 named by another title |
| W5 | the scaffold, read | verifies algebra only; no complementarity |

## P — the package

**P1.** The paper's, both schemas', the scaffold's, the sanity record's and the state's digests match the bytes and
the archive's manifest.

## The paper, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §1 | Minkowski's first variation | PASS | classical |
| §2–9 | the finite active-contact model | PASS | definitions; on the kernel contacts are sectors, sampled here |
| §10–11 | finite contact-pressure theorem | PASS | Farkas (W1) |
| §13–17 | translation and rotation balance; corners need subdifferentials | PASS | parts of Bᵀλ = 0 |
| §18–22 | moment matching, not measure equality; other active constraints | PASS | stated so |
| §27–31 | continuum pressure measure | PASS as conditional | stated so |
| §32–33 | local No-Slack arc theorem | PASS | W2; the hypothesis ρ ≥ κ₀ > 0 answers RUN-050's corner |
| §34–37 | measure version; support inclusion; strict-interior corollary | PASS | as conditional statements |
| §41–44 | continuous curvature needs continuum pressure; atoms need matching mass | PASS | by moments |
| §47–49 | certificate, verifier, Carathéodory sparsity | PASS | W3: nonzeros ≤ equations; the scaffold omits complementarity |
| §54 | the No-Slack compiler | PASS as a specification | on the kernel it would print DUAL_CERTIFIED at a non-stationary placement (W3) |
| §68 | the finite dual on the Lebesgue line is low-compute now | PASS | it is; its certificate does not mean stationarity there |

## The status registry

Not touched: Paper 05 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the theorem's classification in the text | 1 | A0 |
| both alternatives at once; the sanity record's norm | 2 | W1 |
| no decrease; the first variation's sign | 2 | W2 |
| no certificate at the generic placement; a flat gradient | 2 | W3 |
| a dual condition dropped from the schema; an item added to "not claimed" | 2 | W4 |
| a complementarity check in the scaffold; its sign test | 2 | W5 |
| a VALIDATION digest; the sanity record's bytes | 2 | P1 |

Three controls:

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

**13 / 13 plants, 3 / 3 controls, 99 / 99 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 |
| P1 | 2 | 2 |

## What this run does not establish

- **A dual that means stationarity on the kernel.** The trapezoid rule of RUN-053 would change A's rows at the edges.
  That dual is not built here.
- **The continuum pressure measure**, which the paper leaves conditional.
- **Anything about convexity multipliers.** The kernel's hull has corners, and its convexity constraints are active.
  Like §11's base model, the system here leaves them out, which only enlarges the primal.