# RUN-060 — LESR Paper 09B: every wall constant, the critical velocity and the transition speeds reproduce from the barrier's geometry, and on the primary chart the first area variation is exactly γ_w·w — the u and v terms cancel identically; the full Φ₁ stays non-negative along −w only by 0.035 per unit, and along the critical velocity the second-order slivers open at the antipodal walls, not where §18 looks

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-05
**Subject:** *LESR Paper 09B — Analytic Primary Blow-Up Equations* (LESR-LUC-09B-v0.1, 2026-09-21), with its result
record, certificate schema, verifier, state record and VALIDATION. It is evidence, not a canonical document.

**Tools:** [`run060_lesr_paper09b.py`](../code/run060_lesr_paper09b.py)

- the barrier's geometry at 70 digits (RUN-057's construction: B3, B5, the disc, θ*, x₅*, α*);
- the area of the forced 15-piece chain, differentiated at ρ = 0 at 90 digits;
- this tree's verified hull along the critical velocity, and RUN-059's one-sided Φ₁;
- the gate log of RUN-059.

**Logs:** [`run060-lesr-paper09b.json`](../data/gate-logs/run060-lesr-paper09b.json)

**Verdict: PASS for the wall constants, the collapse equations and their solution, w* = 0, the transition speeds, the
primary 15-piece structure and Φ₁^primary = γ_w·w. Two things are noted: how thin the full Φ₁'s margin is along −w, and
where the second-order slivers actually open.**

- **The wall constants (§4–5).**
  - A piece with centre p has support p·u(θ) + r. At a wall normal θ₀ its first-order line is a + bτ, with
    a = ṗ·u(θ₀) and b = p·u′(θ₀); the disc's line is 0. B3 moves at (0, su) on its circle, and B5's points move at
    (w, v) + J(p − C₅) as it turns.
  - At the top wall (π/2) the pieces are B3's arc about v2 and B5's corner v1; at the left wall (π − θ*), B3's arc
    about v0 and B5's corner v2.
  - The six constants rebuilt this way are the text's and the result record's to binary64: relative differences
    10⁻¹⁷ to 2×10⁻¹⁵. To 20 digits, β₃T = 0.30965396669380288225, β₅T = −0.1822307944644007122, γ_T =
    0.16245984811645316307, L_θ = 0.24402699852741554812, L_α = 0.09982945996422126522, γ_L = 0.09115783454691888673.
  - γ_T is B5's corner v1 lever arm, x(v1) − x₅*.
- **The critical velocity (§3, §6–9, §16).**
  - The three collapse equations are linear in (u, v, w), with determinant −1.28×10⁻⁴: full rank.
  - E_R − E_L is exactly 2L_θ cos θ*·w, so w* = 0.
  - The solution is u* = 5.15712624526841370759…, v* = −0.1892281899066409745…, and the closed forms give it to all
    70 digits.
  - The printed u* = 5.157126245268359 is 5×10⁻¹⁴ off. That is binary64 through a system whose determinant is about
    10⁻⁴; v* is right to 3×10⁻¹⁶.
  - RUN-059's numerical critical point (5.1572, −0.18923, 0) and its branch limit (5.157, −0.1892, 0) agree to their
    printed digits.
  - §20's "w* = 0: PROVED" is correct algebra, conditional on the branch keeping all three collapses at first order.
    That is CERT-09B.4, which is open. RUN-059's minimizer has w within 2×10⁻⁷ of 0.
- **The transition speeds (§10).**
  - τ_T = −0.14689252641883796, τ_L = τ_R = 0.081754472733397221 and τ_α = 2.5703350860764422 are the text's and the
    record's to binary64; τ_bottom = τ_T.
  - On the true hull along the critical velocity (ρ = 10⁻⁴ and 10⁻⁵), the hand-overs move at −0.1469 (top and
    bottom), 0.0817–0.0818 (π − θ*, and θ*) and 2.5701–2.5703 (α*).
  - The bottom equals the top exactly: the crossings near π/2 and 3π/2 solve one equation (B5's corner v1 against
    B3's arc about v2, and B3's corner v2 against B5's arc about v1). That is why RUN-059 found B5's arc exactly ρ
    wider than B3's.
- **The primary structure (§11, §18).**
  - Along the critical velocity the true hull has 18 pieces: RUN-059's 15 primary ones and three slivers of width
    O(ρ²):
    - the disc at π/2, 1.79ρ²;
    - B5's arcs about v3 and v2 at θ* and 2π − θ*, 2.72ρ² each.
  - Nothing opens at π ∓ θ*, where §18 (and Paper 09A's diagnostic) puts B5's corner slivers. Each body has constant
    width 1, so a wall and its antipode carry one equation, and the B5 sliver of the right wall shows at θ*.
  - At RUN-059's minima, where the velocity carries its O(ρ) correction, there are no slivers at all: 15 pieces.
  - So §11's primary 15-piece structure holds, and §18's list for the secondary analysis is incomplete. The second
    scale must look at all six walls, and the minimizing velocity closes them all.
- **The primary first variation (§12–14).**
  - The forced 15-piece chain (crossings where two bodies' pieces have equal support, each body's own breakpoints
    elsewhere) has at ρ = 0 the barrier's area, RUN-057's m(0).
  - Its first variation in (1, u, v, w) has coefficients (0, 0, 0, γ_w): below 5×10⁻⁴⁰ for the first three, and
    γ_w = 0.18389963923727719444… for w.
  - It is affine to 2×10⁻³⁹ at two more points. So the u and v terms do cancel identically, and so does the pure
    rotation.
  - The record's binary64 coefficients (5.6×10⁻¹⁷, 8.7×10⁻¹⁹, −4.4×10⁻¹⁶, 0.1838996392372776) are this to their
    precision.
- **The full Φ₁ (§15–16, §20).**
  - At the critical velocity the true hull's one-sided rate is zero to the resolution. Along all 14 directions tried it
    is positive, and so is what the slivers add over γ_w·d_w.
  - Per unit, the rates are 0.0012 along ±u, 0.2619 along ±v, 0.4032 along +w and 0.0354 along −w.
  - Along ±w the slivers add the same 0.2193. So Φ₁ = ±γ_w|w| + 0.2193|w|, and the full Φ₁ is non-negative along −w
    only because 0.2193 exceeds γ_w = 0.1839. That margin is what CERT-09B.4's "positive first-order penalty" has to
    certify.
  - The smallest rate, 0.0012 per unit u, is the first-order cost of a velocity off the collapse manifold in u.
- **The records.**
  - The result record says ANALYTIC_PRIMARY_BLOWUP_DERIVATION; INTERVAL_AUDIT_OPEN. The schema lists four interval
    obligations, including "full multi-chart Phi1 nonnegativity".
  - The state and schema name Paper 09C, "Secondary Sliver Asymptotics and Explicit Phi2", next. It is not in the
    archive.
  - VALIDATION's replay prints PASS with u*, v*, w* = 0 and γ_w.
- **The verifier**, read after the rebuild and never run, has 55 lines.
  - It carries the six constants and γ_w as numbers, derives none of them from the geometry, and solves the 3×3
    system in floating point.
  - It checks u and v against values it also carries, and takes Φ₁ = γ_w·w as "supplied by the analytic 15-piece
    summation". That summation is not shipped.
  - So it verifies the linear algebra, not the constants or γ_w.
- **The package (P1).** All five digests match the bytes and the archive's manifest. The text is UTF-8, with `$…$`
  and `$$…$$` only.

Drill: 17 / 17 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 183 / 183 must-stay-green
assertions held. Each of the ten checks was blinded in turn, and each was reported blind in exactly as many cases as
plants name it.

---

## What the paper claims

1. At a triple wall the three first-order support lines must meet at one transition speed, or a third-body sliver
   opens at O(ρ): a₃b₅ − a₅b₃ = 0 (§2–3).
2. The top, left and right walls give E_T, E_L, E_R with six constants. E_R − E_L forces w* = 0, and the closed
   forms give u* = 5.157126245268359, v* = −0.1892281899066407 (§4–9).
3. The transition speeds τ_T = τ_bottom, τ_L = τ_R and τ_α (§10), and a primary 15-piece structure with the two O(ρ)
   arcs (§11).
4. On the primary chart Φ₁^primary = γ_w·w, γ_w = 0.18389963923727759, the u and v terms cancelling identically; so
   Φ₁ = 0 at the topology-selected velocity, and C₁ = {(u*, v*, 0)} (§12–16).
5. Four interval obligations remain; full multi-chart Φ₁ ≥ 0, the second epi-derivative and the local gate are open;
   Paper 09C next (§17–21).

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the six constants from the geometry | the text's and the record's to binary64 |
| W2 | the system, its solution, w* = 0 | full rank; u* to 70 digits (the printed one 5×10⁻¹⁴ off); RUN-059 agrees |
| W3 | the speeds; the true hull's hand-overs | as printed; the hull moves at them |
| W4 | the signature along the critical velocity | 15 primary pieces + three O(ρ²) slivers at π/2, θ*, 2π − θ* |
| W5 | the forced chain's first variation | (0, 0, 0, γ_w) exactly |
| W6 | the true hull's one-sided rates | all positive; −w keeps 0.0354 |
| W7 | the record, schema and state | as the text; Paper 09C not archived |
| W8 | the verifier, read | linear algebra only |

## P — the package

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

## The paper, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §1–3 | blow-up coordinates; first-order lines; the collapse equation | PASS | W1, W3 |
| §4–6 | E_T, E_L, E_R and their constants | PASS | W1 |
| §7 | w* = 0 from E_R − E_L | PASS | exact (W2); conditional on CERT-09B.4 |
| §8–9 | u*, v* in closed form; agreement with 09A | PASS | the printed u* to 5×10⁻¹⁴ (W2) |
| §10 | the transition speeds | PASS | W3 |
| §11 | the primary 15-piece structure | PASS | W4 |
| §12–14 | Φ₁^primary = γ_w·w; u and v cancel | PASS | exactly (W5) |
| §15–16 | penalties off the collapse; C₁ one point | PASS | numerically; the −w margin is 0.0354 (W6) |
| §17–18 | the secondary slivers to classify | PARTIAL | along the critical velocity they open at π/2, θ*, 2π − θ*, not π ∓ θ* (W4) |
| §19–22 | obligations; status; Paper 09C | PASS / n/a | 09C is not in the archive (W7) |

## The status registry

Not touched: Paper 09B is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's u* | 1 | A0 |
| a constant here; a constant in the result record | 2 | W1 |
| the solution here; RUN-059's critical point | 2 | W2 |
| a hand-over speed on the hull; τ_α in the result record | 2 | W3 |
| a corner sliver along the critical velocity; RUN-059's piece count | 2 | W4 |
| a primary coefficient; γ_w in the result record | 2 | W5 |
| a negative one-sided rate | 1 | W6 |
| the state's next node; Paper 09C in the archive | 2 | W7 |
| the verifier's solve | 1 | W8 |
| a VALIDATION digest; the verifier's bytes | 2 | P1 |

Three controls:

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

**17 / 17 plants, 3 / 3 controls, 183 / 183 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 | 1 | 1 |
| W7 | 2 | 2 |
| W8 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **Any interval obligation.** The constants, the solution and Φ₁^primary are high-precision floating point, not
  interval enclosures.
- **The full Φ₁ ≥ 0.** It is positive along 14 directions at the critical velocity, with a −w margin of 0.0354 per
  unit; the neighbouring charts are not enumerated.
- **The secondary scale.** Paper 09C is not in the archive; the slivers here are measured, not derived.