# RUN-058 — LESR Paper 08: the two-gate reduction and its arithmetic hold, the 10 / 141 / 44 split reproduces, and the rotation is first-order critical on the true hull; but not for the reason given — in the finite active-contact model the paper names, turning B5 costs +0.029 at first order — and the "honest" 195-root shape ledger leaves out the 14.6 % of root 6217's box that Cycle 11 closed for A₃⁺ only

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** *LESR Paper 08 — Positive-Rotation Certificate Reduction* (LESR-LUC-08-v0.1, 2026-09-20), with its
195-root ledger, diagnostic, certificate schema, verifier, state record and VALIDATION. It is evidence, not a
canonical document.

**Tools:** [`run058_lesr_paper08.py`](../code/run058_lesr_paper08.py)

- RUN-057's 65-digit m(0);
- Cycle 12's rotation profile and Cycle 11's 200 subroots of root 6217, as RUN-048 reads them;
- RUN-048's profile minimizer for two samples;
- RUN-053's affine model, and this tree's verified hull;
- the gate logs of RUN-048 and RUN-057.

**Logs:** [`run058-lesr-paper08.json`](../data/gate-logs/run058-lesr-paper08.json)

**Verdict: PASS for the reduction, its arithmetic and its partition. Two things are not as the paper states them:**

- **First-order criticality holds, but not by the argument given.** §3 says that in the finite active-contact
  variational model, inf{δA[d] : d ∈ K_feas, ρ̇ = 1} = 0, and that this is derived from Paper 07's pressure/KKT
  structure.
  - In that model (RUN-053's affine LP at the barrier, B5 turning at unit rate, B3 free to enter the branch or slide
    along it, B5 free to translate), the least first-order change is +0.0293. It is reached with B5 translating by
    (−0.022, −0.1625) per unit of turn. Turning B5 with no compensation costs 0.0474.
  - Paper 07's pressure split is not a subgradient (RUN-057).
  - On the true hull, the one-sided rate along the compensations of the re-minimized profile is 5.3×10⁻⁵ (from the
    sample at ρ = 0.001) and 1.1×10⁻⁴ (at 0.002). That is proportional to ρ, so it goes to zero: the rotation is
    first-order critical, numerically. Neither the model nor Paper 07's split shows it.
- **The "honest" shape ledger is not the whole domain.** §12–13 rebuild the barrier-shape ledger from Cycle 11's 200
  subroots of root 6217, prune five by A_ref, and keep 195.
  - Those 200 fill 55.9 % of root 6217's box. Of the rest, 29.5 % is NOT-F3 (d₃ < s: never in the far branch,
    rightly irrelevant).
  - But 14.6 % lies in the branch and was closed only by Cycle 11's search, for A₃⁺. Its trees are not shipped
    (RUN-049).
  - The paper's own rule reopens anything closed for A₃⁺ only, unless A_ref prunes it or it is impossible. By that
    rule this 14.6 % belongs in the shape ledger as well: 3.3 % of the box with ρ < 0.003 and 11.3 % with
    ρ ≥ 0.003. "200 → 195" leaves it out, and so does the verifier.

- **The arithmetic (W1).**
  - From m(0) to 65 digits: A_ref − m(0) = 3.8944077590×10⁻⁷, the crossing radius is 0.00197342538724, and the
    margin at ρ = 0.003 is 5.1055922410×10⁻⁷, as §7–8 print them.
  - The diagnostic and the replay agree to double precision.
  - The composition (§11) and κ₀ ≥ 0.2 ⇒ ½κ₀ ≥ 0.1 (§20) are arithmetic.
- **The profile (W2).**
  - The first eight samples of Cycle 12's profile give ratios (m(ρ) − m(0))/ρ² from 0.22406 to 0.24571, as §5 says.
  - The diagnostic lists them against the profile's own ρ = 0 search value, which is 7.3×10⁻¹¹ above the exact m(0)
    it quotes.
  - Re-minimized as in RUN-048, samples 1 and 2 give 0.2254 and 0.2253. Sample 2's stored value is 8.3×10⁻⁸ high: its
    0.2457 is a search artefact. The local ratio is about 0.225 (m″(0⁺) ≈ 0.45), so 0.1 has a factor of 2.25 of
    slack.
  - RUN-048's re-minimized ratios over the whole profile run from 0.1738 to 0.2254. So every sample satisfies both
    gates, numerically.
- **Criticality (W3).** As above.
- **The ledger (W4).**
  - From the subroots' ρ ranges: 10 local-only, 141 tail-only and 44 crossing roots, exactly §15's.
  - The ledger file lists all 200 with this tree's boxes, the five marked SHAPE_PRUNED_BY_AREF.
  - The box shares come from 400,000 uniform points, as above.
- **The records (W5).**
  - The state calls the first-order rotation "CRITICAL_GIVEN_P07_PRESSURE_KKT", and the schema "critical by Paper07
    contact-pressure/KKT architecture; interval dual audit still required". The gates are OPEN, and the diagnostic
    says NUMERICAL_DIAGNOSTIC_NOT_PROOF.
  - The state names the next node "Critical-Cone Second Variation at rho=0+". Paper 09A's own title is "Multi-Scale
    Topology-Wall Correction and the One-Sided Blow-Up Critical Cone".
- **The verifier (W6), read and never run.** It has 86 lines.
  - It rebuilds the ledger from the 200 subroots alone.
  - It takes the five prunes as a constant, without checking their certificates.
  - It checks the arithmetic, with a 10⁻¹⁵ tolerance at the cut. It cannot see the part of the box outside the 200.
- **The package (P1).** All six digests match the bytes and the archive's manifest. The text is UTF-8, with `$…$`
  and `$$…$$` only.

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. ρ = 0 is a topology wall. The mirror makes the profile even, but evenness alone allows a cusp maximum.
2. The rotation is first-order critical, derived from Paper 07's pressure/KKT structure. So the local question is
   second order, and a reduced critical-cone value κ₀ > 0 would give quadratic growth.
3. Two gates: LOCAL-QUADRATIC, m(ρ) ≥ m(0) + 0.1ρ² on [0, 0.003], and AREF-TAIL, m(ρ) ≥ 0.8349865 on
   [0.003, π/120]. They overlap by 5.1×10⁻⁷ at ρ = 0.003, and compose to m(ρ) ≥ m(0).
4. The barrier-shape ledger restarts from Cycle 11's 200 subroots: 5 pruned by A_ref, 195 active, split
   10 / 141 / 44 at ρ = 0.003.

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the arithmetic from m(0) to 65 digits | as printed; the doubles to their precision |
| W2 | the profile's ratios; two samples re-minimized | as stated; one stored value is a search artefact; true ratio ≈ 0.225 |
| W3 | ρ̇ = 1 in the affine model and on the true hull | +0.0293 in the model; ∝ ρ → 0 on the hull |
| W4 | the split; the ledger file; root 6217's box | 10 / 141 / 44 exact; 14.6 % of the box closed for A₃⁺ only is missing |
| W5 | the ledger, diagnostic, schema and state | as the text; Paper 09A named by another title |
| W6 | the verifier, read | 200 subroots only; prunes taken as given |

## P — the package

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

## The paper, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §1–2 | ρ = 0 is a wall; the mirror makes m even; evenness does not give a minimum | PASS | as RUN-048 saw |
| §3 | first-order criticality derived from Paper 07's pressure/KKT | holds, not derived | the named model gives +0.0293; Paper 07's split is not a subgradient; the true hull gives ∝ ρ (W3) |
| §4–6 | the local question is second order; the ratio 0.224–0.246; the target 0.1 | PASS | W2; the 0.246 is a search artefact, the true ratio ≈ 0.225 |
| §7–11 | arithmetic; the two gates; composition | PASS | W1 |
| §12–14 | the shape ledger from 200 roots; 200 → 195 | incomplete | 14.6 % of root 6217's box closed for A₃⁺ only is not in it (W4) |
| §15–17 | 10 / 141 / 44 | PASS | W4, exact |
| §18–24 | chart family; next nodes; counter-search | PASS | stated as open |

## The status registry

Not touched: Paper 08 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the ledger count in the text | 1 | A0 |
| the gap; the diagnostic's crossing radius | 2 | W1 |
| a profile sample; a re-minimized sample | 2 | W2 |
| the affine value made 0; Paper 07's subgradient made to hold | 2 | W3 |
| a subroot's ρ range; the box shares | 2 | W4 |
| the state's first-order status; the schema's local domain | 2 | W5 |
| the verifier's prunes; a box share in the verifier | 2 | W6 |
| a VALIDATION digest; the ledger's bytes | 2 | P1 |

Three controls:

- the unmodified inputs;
- the document and verifier 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

- **Either gate.** They are OPEN in the paper, and they hold only at the 26 profile samples here.
- **Which part of the 14.6 % Cycle 11 closed by bounds that would also clear A_ref.** Its trees are not shipped. Some
  of it may already lie above 0.8349865, and a replay would tell.
- **An exact first-order statement.** The criticality is a floating-point observation along two compensations.