# RUN-061 — the LESR midpoint handoff: every number it records is this tree's to the printed digit, and its three ledgers and target-specific closure are right; of what it lists as done in Paper 07, B3's subgradient is not; run here, the counter-certificate line finds no lower branch, finds the new topology, and finds the finite dual uninformative

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-05
**Subject:** *Lebesgue / LESR / UESFCM 中繼交接文件 — Certificate 主線 × Counter-Certificate 本地端分工*
(LESR-LUC-MID-HANDOFF-v0.1, 2026-09-20), with its VALIDATION record. It is a work contract, not a theorem, and
evidence, not a canonical document.

**Tools:** [`run061_lesr_midpoint_handoff.py`](../code/run061_lesr_midpoint_handoff.py)

- RUN-057's 70-digit construction of the barrier;
- this tree's verified hull for the mirror check;
- the gate logs of RUN-002, RUN-048, RUN-049, RUN-055, RUN-056, RUN-057, RUN-059 and RUN-060.

**Logs:** [`run061-lesr-midpoint-handoff.json`](../data/gate-logs/run061-lesr-midpoint-handoff.json)

**Verdict: PASS for the barrier state, b* and g₃, the three ledgers and target-specific closure, cell 131, the
artifact index and the package. Four things are not as the document states them, and the counter-certificate line it
assigns has, in this tree, a first answer.**

- **The barrier state (§5).** x₃* is x₃⁰ − s exactly. θ*, x₅* and α* agree with the 70-digit construction to all
  34 printed decimals, and the barrier's area to all 37.
- **Paper 07, as the document lists it (§6).**
  - b* and g₃ agree to their printed digits, and 0 < b* < L_θ.
  - The 8 non-zero bridges (four L_θ, two L_α, two L_vert), the 6 triple-support lines, and B5's force and rotation
    balances are RUN-057's.
  - **§6 I is not done as stated.** "B3 constrained subgradient g₃ < 0, balanced by η₃ = −g₃" fails: (g₃, 0) is not
    a subgradient, short by 0.0663 over 72 directions. The first component that does hold lies in
    [−0.0477, −0.0474] (RUN-057).
  - §6 J's conclusion still holds numerically: the barrier is the constrained minimum, with a least feasible rate of
    0.0354.
- **The ledgers (§3–4).**
  - The public bracket 0.8344 ≤ a_Leb ≤ 0.8440935944 is Mishra's and Gibbs' (RUN-002), cited and not verified here.
  - A₃⁺ = 0.834781191 is Paper 06's ceiling (RUN-056).
  - Cell 131 is closed for A₃⁺ by its prefix-tree certificate, and its candidate is the only one below 0.8349865
    (RUN-049). So §4's REOPEN_FOR_SHAPE is right, and so is the rule that closure does not carry across targets.
- **Paper 08's attack order (§9).**
  - **Evenness gives no derivative.** m is even: every one of five profile samples has its mirror image's area to
    10⁻¹⁵. But evenness gives m′(0⁺) = −m′(0⁻), not m′(0) = 0, and a kink stays even.
    - Paper 08 (§1–2) and Paper 09A (§0–2) later say this themselves.
    - m′(0⁺) = 0 does hold numerically: the ratio (m − m₀)/ρ² stays near 0.2256 (RUN-059).
  - **The Cycle 14 "transition" is one sample.** ρ ≈ 0.01409689011226 is Cycle 12's sample 14, the only degenerate
    one. The sampled contact sequence changes twice:
    - from generic A to generic B between samples 8 and 9 (ρ 0.00806 to 0.00906);
    - at that degenerate sample, inside B.
    Neither change is located, and both lie in the tail gate's range [0.003, π/120], where Paper 08 differentiates
    no chart.
- **The counter-certificate line (§10–12), as this tree has run it.**
  - **CC-08A, a lower positive-rotation branch: none.**
    - RUN-048's re-minimized profile has ratios 0.1738 to 0.2254, positive at all 26 samples.
    - The re-minimized branch at four ρ ≤ 0.002 has 0.2252 to 0.2255 (RUN-059).
    - No COUNTER_CANDIDATE.
  - **CC-08B, a new contact topology: found.**
    - The positive branch's signature is not the barrier's: 13 at ρ = 0, 18 at the stored sample, 15 at the minima
      (RUN-059).
    - Along the critical velocity, O(ρ²) slivers open at θ*, π/2 and 2π − θ* (RUN-060).
  - **CC-08C, dual infeasibility: the finite dual does not discriminate.**
    - It is feasible even at a generic placement whose area gradient has length 0.39 (RUN-055).
    - Paper 07's pressure split is not a subgradient (RUN-057).
  - **HC-03**, the barrier-shape tree rebuild: Paper 08's 195-root ledger leaves out the 14.6 % of root 6217's box
    closed for A₃⁺ only (RUN-058).
  - HC-01 and HC-02, the interval audit and the pressure rationalization, are not done here: everything in this tree
    on the barrier is high-precision or floating point.
- **The artifact index (§25).** All 12 files are in the archive.
- **The package (P1).**
  - VALIDATION's digest is the document's and the archive's. The text is UTF-8, with `$…$` and `$$…$$` only.
  - Its next node is "Paper 08 — Positive-Rotation Barrier Closure". Paper 08's own title is "Positive-Rotation
    Certificate Reduction", as RUN-058 found for Paper 07's state record.

The handoff's counter-certificate line is the side this arm takes; §10–14 are read here as its first task list.

Drill: 13 / 13 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 114 / 114 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 document claims

1. From now on there are two lines with one owner each: certificate and analytic closure in the cloud, and
   counter-certificates and heavy computation locally. A replayed counter-candidate stops the proof effort on the old
   claim (§0–2).
2. Three ledgers stay apart: the global value (bracket 0.8344 to 0.8440935944), the root-6217 barrier shape, and the
   global extremal shape (unbound). Proof closure does not imply shape irrelevance, as cell 131 shows (§3–4).
3. The canonical barrier and what Paper 07 completed: the signature, bridges, triple lines, b*, B5's balances, B3's
   subgradient, and the zero-rotation minimizer (§5–6). What remains: CERT-07A and CERT-07B (§7).
4. Paper 08's claim m₆₂₁₇(ρ) ≥ m₆₂₁₇(0) on [0, π/120], and its attack order: evenness ⇒ m′(0) = 0, the second
   variation, chart continuation, a profile derivative, a finite partition, the transition at ρ ≈ 0.0141 (§8–9).
5. The local line's priorities CC-08A to CC-GLOBAL-A and tasks HC-01 to HC-05; a return format; grades for
   counter-certificates and certificates; mistakes to avoid; a roadmap; an artifact index; two checklists (§10–29).

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the barrier state | every printed digit |
| W2 | Paper 07's list | b*, g₃, bridges, lines, balances as listed; B3's subgradient fails |
| W3 | the ledgers and cell 131 | as stated; REOPEN_FOR_SHAPE right |
| W4 | CC-08A, B, C in this tree | no lower branch; new topology; the dual uninformative |
| W5 | evenness; Cycle 14 | even, with no derivative from it; two sequence changes, one sample named |
| W6 | the artifact index | all 12 archived |

## P — the package

**P1.** The VALIDATION digest matches the document and the archive. Its next node is not Paper 08's title.

## The document, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §0–2 | the two lines; counter-certificate first | n/a | a work rule |
| §3–4 | three ledgers; PROOF-CLOSED ⇏ SHAPE-IRRELEVANT; cell 131 | PASS | W3 |
| §5 | the barrier state | PASS | W1 |
| §6 A–H | Paper 07's signature, bridges, lines, b*, balances | PASS | W2 |
| §6 I | B3's subgradient g₃, multiplier −g₃ | FAIL as stated | not a subgradient; the valid component is −0.0477 to −0.0474 (W2) |
| §6 J | the zero-rotation translation minimizer | PASS | numerically (W2) |
| §7 | CERT-07A, CERT-07B open | PASS | open here too |
| §8 | m(ρ) ≥ m(0) on [0, π/120] | open | holds at every sample (W4) |
| §9 A | evenness ⇒ m′(0) = 0 | FAIL as an inference | even, but a kink stays even (W5) |
| §9 F | the transition at ρ ≈ 0.01409689011226 | PARTIAL | one sample of two sequence changes (W5) |
| §10–12 | CC-08A, B, C | answered here | W4 |
| §13–29 | tasks, formats, grades, roadmap, index, checklists | PASS / n/a | the index complete (W6) |

## The status registry

Not touched: the handoff is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's cell-131 candidate | 1 | A0 |
| θ* here | 1 | W1 |
| b* here | 1 | W2 |
| Paper 07's subgradient made to hold | 1 | W2, W4 |
| the cell-131 ranking; A₃⁺ | 2 | W3 |
| a negative profile ratio; a corner sliver along the critical velocity | 2 | W4 |
| a mirror area; Cycle 14's classes | 2 | W5 |
| an artifact removed | 1 | W6 |
| the VALIDATION digest; Paper 08's title | 2 | P1 |

Three controls:

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

**13 / 13 plants, 3 / 3 controls, 114 / 114 must-stay-green assertions; no check without a plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 1 | 1 |
| W1 | 1 | 1 |
| W2 | 2 | 2 |
| W3 | 2 | 2 |
| W4 | 3 | 3 |
| W5 | 2 | 2 |
| W6 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **Any certificate.** CERT-07A and CERT-07B and the interval tasks are not done here.
- **That no lower branch exists.** The search here is the profile's 26 samples and four small ρ, re-minimized; it is
  not a covering of [0, π/120].
- **The bracket.** Mishra's and Gibbs' bounds are cited, not verified.