# RUN-057 — LESR Paper 07: every constant of the root-6217 barrier reproduces from the contact equations to the last printed digit, and so does the 13-interval signature; but the translation KKT is not established as argued — the claimed subgradient (g₃, 0) fails in a direction where B3 and B5 move left together; the barrier is still the constrained minimum, with a multiplier of 0.047, not 0.123

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** *LESR Paper 07 — Root-6217 Barrier Shape Reconstruction* (LESR-LUC-07-v0.1, 2026-09-20), with its
certificate, verifier, compute-deferred list, signature graph, plots, state record and VALIDATION. It is evidence,
not a canonical document.

**Tools:** [`run057_lesr_paper07.py`](../code/run057_lesr_paper07.py)

- mpmath at 70 digits, from the contact equations alone;
- this tree's verified hull (RUN-009, with RUN-031's segments) for the signature and for F(x₃, x₅);
- RUN-053's affine first-order model, for §15's premise;
- RUN-048's gate log.

**Logs:** [`run057-lesr-paper07.json`](../data/gate-logs/run057-lesr-paper07.json)

**Verdict: PASS for the reconstruction; NOT ESTABLISHED for the translation KKT as argued (§15–16); the conclusion
it draws holds numerically.**

- **The reconstruction reproduces.**
  - From x₃* = x₃⁰ − s and the two contact equations: θ*, x₅*, α* (the only other root in (θ*, 3π/10)), the three
    bridge lengths, b*, the B5 arc force, g₃, and the area to 65 digits. Each agrees with every digit the text
    prints, with the certificate to 60 places, and with VALIDATION's replay.
  - The 13 area contributions and every piece's centre and radius agree with the certificate to 60 places.
- **The signature is this tree's verified hull.**
  - At the barrier the hull's owners, cyclically, are the table's (B3's corner v0 is one segment across normal 0).
  - Its hand-overs are the symbolic ends to 10⁻⁷.
  - Its atoms are four L_θ, two L_α, two L_vert and four zeros at the intrinsic arc/corner switches. Three bodies
    touch at exactly the six normals of §5.
  - VALIDATION's "minimum midpoint winner margin" 0.00526 is the margin over the other bodies; over every other piece
    it is 0.00060, where B3's arc beats its own corner v1.
- **The KKT step fails as argued.**
  - §15 takes the area's directional derivative to be the integral, against μ_h, of the largest active rate at each
    normal. RUN-053 showed that this is the affine model and overstates the rate. At the barrier it overstates
    F′(v; d) by 0.265 to 0.451 in all 72 directions; along +x₅ it gives 0.689 where the area rises at 0.403.
  - So a pressure split dominated by it need not be a subgradient, and this one is not. Along d = (cos 202°,
    sin 202°), B3 and B5 moving left together, F′(v; d) = 0.044 while g·d = 0.114.
  - At finite steps, F(v + td) − F(v) − t g·d is −7×10⁻⁸, −6.9×10⁻⁶ and −2.5×10⁻⁴ for t = 10⁻⁶, 10⁻⁴ and 10⁻². The
    subgradient inequality fails outright, not as a derivative estimate.
- **The conclusion survives.**
  - On every feasible direction (d₃ ≤ 0) the area rises, at least at 0.0354 per unit (along −x₅). F is convex
    (Fáry–Rédei), so (x₃*, x₅*) does minimize F on x₃ ≤ x₃*.
  - Over 720 directions, (g₃′, 0) passes for g₃′ between −0.0477 and −0.0474. So the multiplier of the far-branch
    constraint is η₃ ≈ 0.047, about 39 % of the paper's 0.1228.
  - Moving B3 back toward the anchor (+x₃), the area falls at 0.0331 per unit, which is the constraint's work.
- **§17 rests on the symmetrization RUN-048 questioned.** Extending the slice minimum to all translations at ρ₅ = 0
  uses Cycle 12's y → 0, which does not keep d₃ ≥ s. RUN-048 found the arc side left out of Cycle 13's
  certificate (1.3 % certified), and its least area there numerically at the barrier itself (0.8349861106, at
  α = π). So §17's conclusion is numerically right, and not established by §17.

- **The verifier (W6), read and never run.** It has 152 lines.
  - It checks the winners at the 13 midpoints only.
  - Its KKT tests are the sign of g₃, b* ∈ (0, L_θ), and B5's residual. That residual is zero by the very line that
    defines b* (b* = F5x/(2 cos θ*)).
  - It evaluates the area once and takes no directional derivative, so it cannot see §15's step.
- **The records (W5).**
  - The certificate calls the KKT "DERIVED_FROM_PRESSURE_DECOMPOSITION GIVEN_SIGNATURE", and the state
    "CLOSED_GIVEN_SIGNATURE".
  - The state's nine facts have their sections, and there are three tasks.
  - The DOT has the 13 intervals. The state names Paper 08 "Positive-Rotation Barrier Closure"; Paper 08's own title
    is "Positive-Rotation Certificate Reduction".
- **The package (P1).** All seven digests match the bytes and the archive's manifest, and the text has no `\(`, `\)`,
  `\[` or `\]`. VALIDATION hashes the signature graph's PNG but not its DOT.

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 contact equations give θ*, x₅* and a second disk–B5 crossing α*.
2. The signature has 13 open normal intervals, eight nonzero bridges (lengths L_θ, L_α, L_vert), six triple-support
   lines with latent third contacts, and an exact area from the analytic primitive.
3. The surface-area measure splits into nonnegative witness pressures: B3 gets its arcs, B5 its arcs plus b* at
   π ± θ*, and the disk the rest. B5 then balances in translation and rotation.
4. B3's horizontal subgradient is g₃ = 2(sin θ* − ½) < 0, balanced by the far-branch normal η₃ = −g₃. By convexity
   (§16) the barrier minimizes the ρ₅ = 0 translation problem, and (§17) all translations at ρ₅ = 0 in the far
   branch.
5. What remains is the positive-rotation gate m(ρ) ≥ m(0).

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the constants from the contact equations, 70 digits | every printed digit; the certificate to 60 places; the replay |
| W2 | the signature on the verified hull | the table's, cyclically; atoms and triple lines as stated |
| W3 | F's directional derivatives against §15's premise and (g₃, 0) | premise overstates by 0.27–0.45; (g₃, 0) fails; F′ > 0 on feasible directions |
| W4 | §17 against RUN-048 | rests on the symmetrization that misses the arc side |
| W5 | the certificate's statuses, the state, the tasks | as the text; Paper 08 named by another title |
| W6 | the verifier, read | midpoints only; no derivative; B5's residual by definition |

## P — the package

**P1.** The paper's, the certificate's, the verifier's, both PNGs', the state's and the tasks' digests match the
bytes and the archive's manifest.

## The paper, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §1–2 | θ*, x₅*, α* | PASS | W1, to every printed digit |
| §3–7 | signature; bridges; six triple lines; latent contacts | PASS | W2; the interval audit is still open, as the paper says |
| §8–9 | area; surface-area measure | PASS | W1, 65 digits |
| §10–12 | pressure split; B5 balances | PASS | exact; B5's x-balance holds by b*'s definition |
| §13–14 | g₃ < 0, exactly | PASS | the sign |
| §15 | pressure splits give a convex subgradient | FAIL | its premise (F′ = ∫ max active rate dμ_h) is the affine model; (g₃, 0) fails the subgradient inequality (W3) |
| §16 | global KKT on the ρ₅ = 0 translation slice | conclusion holds, proof does not | F′ ≥ 0.0354 on feasible directions; η₃ ≈ 0.047 |
| §17 | all translations at ρ₅ = 0 in the far branch | NOT ESTABLISHED by its argument | RUN-048's symmetrization gap; numerically true |
| §18–19 | the positive-rotation gate open | PASS | stated so |

## The status registry

Not touched: Paper 07 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the KKT's normal cone in the text | 1 | A0 |
| a digit of α* in the certificate; the replay's area | 2 | W1 |
| an owner changed; a bridge's mass | 2 | W2 |
| the subgradient made to hold; the premise made exact | 2 | W3 |
| RUN-048's arc minimum; §17's appeal to Cycle 12 removed | 2 | W4 |
| the state's KKT status; a task dropped | 2 | W5 |
| a derivative in the verifier; its sample point | 2 | W6 |
| a VALIDATION digest; the DOT's bytes | 2 | P1 |

Three controls:

- the unmodified inputs;
- the document, verifier and DOT 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

- **An interval proof of the signature** (Task P07_INTERVAL_SIGNATURE_AUDIT). The verified hull's closed-form
  segment checks are floating-point.
- **A certified multiplier.** η₃ ≈ 0.047 comes from 720 measured directions. An exact subgradient would need the
  edge-average rule with the latent contacts' tents, which is not built here.
- **The positive-rotation gate.** That is Paper 08.