# RUN-056 — LESR Paper 06: the barrier and its four-direction orbit reproduce, and target-specific closure is the right accounting; but Task B has a one-line answer — no measure on the four directions can match the barrier's curvature once cos 2θ is in the library — and the ledger still marks the ρ₅ = 0 slice closed, which RUN-048 found is not established

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** *LESR Paper 06 — Actual Lebesgue Shape State, Target-Specific Ledgers, and the First Closure Graph*
(LESR-LUC-06-v0.1, 2026-09-20), with its ledgers, four-contact dual template, closure graph (JSON, DOT, PNG, SVG),
compute-deferred tasks, state record and VALIDATION. It is evidence, not a canonical document.

**Tools:** [`run056_lesr_paper06.py`](../code/run056_lesr_paper06.py)

- this tree's exact hull (RUN-009, with RUN-031's segments) and RUN-048's contact solution x₅*;
- RUN-055's finite contact system;
- the gate logs of RUN-048 and RUN-049, and RUN-052's polish of q†.

**Logs:** [`run056-lesr-paper06.json`](../data/gate-logs/run056-lesr-paper06.json)

**Verdict: PASS for what Paper 06 proves and computes. Its accounting (value proof, barrier shape and global shape
kept apart; proof closure not inherited by a shape target) is right. Three things are noted:**

- **Task B has a short answer.**
  - The four directions ±θ*, π ± θ* all have the same cos 2θ, namely cos 2θ*. So any measure carried by them,
    whatever its weights and signs, has cos 2θ-moment cos 2θ* times its mass.
  - For g = Aᵀλ, the mass must be the perimeter P = g₀, so the cos 2θ-moment would be P cos 2θ* = 2.05. The
    barrier's curvature measure has 0.124.
  - So no λ on the four contacts satisfies g = Aᵀλ for any library that holds 1 and cos 2θ. The least nonnegative
    residuals are 1.6, 2.0 and 3.0 for modes up to 2, 4 and 8.
  - "SHAPE_DUAL_STATIONARITY = OPEN" is too mild for the template. It is impossible there; a certificate would need
    contacts on the arcs (Paper 05 §41).
  - With every contact (RUN-055's system, M = 4) a certificate exists at the barrier. But releasing d₃ ≥ s, the area
    falls toward the anchor at 0.0331 per unit, so that certificate does not mean stationarity either (RUN-053).
- **The ledger marks the ρ₅ = 0 slice closed for A₃⁺.** It records `rho0_value_proof_status: CLOSED_FOR_A3PLUS`, and
  the graph's node RHO0 `INDEPENDENTLY_REPLAYED_THRESHOLD`. RUN-048 found that Cycle 13's certificate covers the
  rectangle side of the slice only: of the arc side, 1.3 % lies in Cycle 12's certified subroots. So the value
  closure of that slice is not established, though it holds numerically (by 2.05×10⁻⁴).
  - Likewise "68 → 61" counts the formal ledger's open cells. RUN-049 found that 44 % of root 6217's box was closed
    by Cycle 11's search only.
- **The graph's vocabulary is its own.**
  - Every one of its 24 edges carries a target id (§45). It is acyclic, and every node is reached from QSTAR.
  - Two of its roles are not among §44's twelve: SHAPE_REFERENCE and COMPUTE_DATA_GATE.
  - Of its 20 edge types, only CERT_PRUNE, REOPEN_FOR_SHAPE and UPPER_FEASIBLE are defined anywhere in the series
    (Paper 04's seven, §9's three reopenings, §51's seven roles).
  - VALIDATION hashes the JSON and the DOT but not the PNG or the SVG.

- **The barrier (W1).**
  - x₃* = x₃⁰ − s exactly. θ* = arccos(1/(2(R₃ + x₃*))) = 0.4540452162931835 agrees with the text, the ledger and
    the template, and x₅* with RUN-048's 80-digit solution.
  - At the four directions the disc, B3 and B5 each support at ½, and so does the hull, all to 10⁻¹⁵.
  - The exact hull's area is 0.834986110559 > A₃⁺. Read off the sampled hull, its stabilizer is exactly D₁
    (§27 claims "at least").
- **The template (W2).**
  - Equal weights on the four directions balance translation to 10⁻¹⁵.
  - For B3 and B5 at the barrier, Σ h′_W over the orbit is 0 at θ*, and at 200 random angles (§31's lemma), to
    10⁻¹⁴.
  - Task B is as above.
- **The graph (W3).**
  - 23 nodes and 24 edges, and the DOT and the SVG draw the same nodes (and the DOT the same edges).
  - The ledger's numbers are the text's: the bracket, A₃⁺, the 61 cells, A_ref, the outer polygon's
    0.8349864346145534, and the cell lists 28/36/68/144/155, 129/131 and 125/152/150.
- **Prior runs (W4).** As above. Also, the ledger's F₃ ceiling A₃⁺ = 0.834781191 stands, but RUN-052's polish of q†
  is 2.45×10⁻⁷ below it (a floating-point value of the verified hull, not an enclosure).
- **The records (W5).**
  - The state's six discoveries have their sections (§5, §6, §8, §17, §24, §31).
  - The four tasks are §48's A–D, and the template's statuses are §32's.
  - The state names Paper 07 "Root-6217 Barrier Shape Reconstruction as Controlled LESR Test". Paper 07's own title
    is "Root-6217 Barrier Shape Reconstruction".
- **The package (P1).** All seven digests match the bytes and the archive's manifest, and so does VALIDATION's
  recorded translation norm. 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. The value bracket is not a shape bracket. Finite-witness forcing is existential, so no pointwise support corridor
   exists yet (UNBOUND).
2. There are three target-specific ledgers (value proof, root-6217 barrier shape, global extremal shape). Proof
   closure is target-relative, and REOPEN_FOR_SHAPE does not undo a proof.
3. The root-6217 symmetric state is a barrier, not an F₃ minimizer nor the Lebesgue shape. It has explicit contact
   equations, a four-direction triple-support orbit, a cusp, a sample profile, a tight reference, and cell 131 as the
   canonical proof-closed and shape-reopen example.
4. Equal multipliers on the orbit give a placement-equilibrium template (translation and rotation, by reflection and
   constant width). g = Aᵀλ is open.
5. A first actual closure graph, and four compute-deferred tasks.

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the barrier's numbers, supports, area, stabilizer | as stated; the stabilizer is exactly D₁ |
| W2 | the template's balances; §31's lemma; Task B | balances hold; Task B is impossible on four contacts |
| W3 | the graph and the ledger | consistent; roles and edge types beyond the series' lists; images unhashed |
| W4 | the ledger against RUN-048, 049, 052 | ρ₅ = 0 marked closed, not established; 44 % search-only |
| W5 | the state and the tasks | as the text; Paper 07 named by a longer title |

## P — the package

**P1.** The paper's, the ledger's, the template's, the graph JSON's and DOT's, the tasks' and the state's digests
match the bytes and the archive's manifest.

## The paper, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §2–6 | value bracket ≠ shape bracket; existential forcing; corridor unbound | PASS | §4 is the standard argument |
| §7–9 | three ledgers; closure is target-relative; REOPEN split | PASS | the right accounting |
| §10 | A₃⁺ = 0.834781191 bounds Λ(F₃), not a_Leb | PASS | RUN-052's polish lies 2.45×10⁻⁷ below it |
| §11–17 | the far decomposition; the barrier; θ*, x₅*, the orbit; area > A₃⁺ | PASS | W1 |
| §18–20 | the cusp; the rotation profile; sample monotonicity is not a theorem | PASS | RUN-048 |
| §21 | the tight reference and its outer polygon | PASS | RUN-049 |
| §22–24 | 68 → 61; Type I and II closure; cell 131 | PASS | RUN-049; 61 is the formal ledger, 44 % of 6217 is search-only |
| §25 | Cycle 13 proved A > A₃⁺ on the ρ₅ = 0 slice | NOT ESTABLISHED | RUN-048: the arc side is left out (W4) |
| §27–31 | stabilizer ⊇ ⟨σ_x⟩; the orbit; the equilibrium lemma | PASS | W1, W2; the stabilizer is exactly D₁ |
| §32 | g = Aᵀλ open | stronger: impossible on the four contacts | W2 |
| §43–47 | the actual graph; target ids; spines | PASS | W3 |
| §48 | four tasks | PASS | Task B carried out here |

## The status registry

Not touched: Paper 06 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| x₃* in the text | 1 | A0 |
| the barrier below A₃⁺; the ledger's θ* | 2 | W1 |
| the lemma off; the cos 2θ moment matched | 2 | W2 |
| an edge without a target id; an edge dropped from the DOT | 2 | W3 |
| RUN-048's coverage changed; the ledger's ρ₅ = 0 status changed | 2 | W4 |
| a discovery added to the state; a task dropped | 2 | W5 |
| a VALIDATION digest; the template's norm | 2 | P1 |

Three controls:

- the unmodified inputs;
- the document and the DOT 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

- **Tasks A, C and D.** The shape-tree rebuild with A_ref, the import of an outer universal cover, and the
  near-anchor atlas are not attempted here. RUN-052's polish of q† is a first near-anchor point.
- **A dual certificate that means stationarity at the barrier.** It would need the far-branch constraint d₃ ≥ s with
  its own multiplier, and edge-turning moves (RUN-053).
- **The PNG and the SVG's pictures.** Only the SVG's node titles are read.