# RUN-053 — LESR Paper 03: the calculus holds along affine support paths — every formula, the step caps, the contact variation; but on the kernel, where the container is the hull of its witnesses, a witness move turns the container's edges and the first-order change is the edge average, not the affine maximum — so the finite LP finds no descent even where the area's gradient has length 0.39

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** *LESR Paper 03 — Support-Function Variational Calculus* (LESR-CALC-03-v0.1, 2026-09-20), with its
schema, sanity-check record, compiler scaffold, state record and VALIDATION. It is evidence, not a canonical
document.

**Tools:** [`run053_lesr_paper03.py`](../code/run053_lesr_paper03.py)

- sympy for the formulas;
- this tree's exact hull (RUN-009, with RUN-031's segments) for the bodies, and RUN-052's segments, first variation
  and mirror-symmetric polish Z of q†;
- scipy's HiGHS for sec. 39's LP.

**Logs:** [`run053-lesr-paper03.json`](../data/gate-logs/run053-lesr-paper03.json)

**Verdict: PASS for everything Paper 03 derives. Its first-order model of a witness move is the affine one, and on
the kernel that model is not the first-order behaviour:**

- **A witness move turns the container's edges.**
  - On the kernel the container is the hull of the placed witnesses, so each witness is active on whole sectors of
    normals, not at finitely many contacts (§33's hypothesis).
  - When a witness moves, the hull's edges turn. At first order an edge turning is a dipole in μ_h, which no
    direction h + tφ with Lφ a measure (§14) produces.
  - The true first-order change is the trapezoid rule: each edge of length ℓ whose ends move out at normal speeds
    r_a, r_b changes the area by ℓ(r_a + r_b)/2, and each arc by ∫ρ ċ·u. At a generic placement this matches the
    measured rates in four random directions, to 2×10⁻⁷ at step 10⁻⁸ (the difference shrinks with the step).
  - The affine model of §27–32 moves each edge parallel to itself, so it charges ℓ·max(r_a, r_b). It is larger by
    0.66 to 0.94 in those four directions, and in three of them it has the wrong sign.
  - At the polish Z it overcharges by a factor of 2.9 to 5.5: y3 costs 0.133 per unit, and the model says 0.732.
- **So the finite LP sees no descent where there plainly is one.** §39's LP puts the container free, re-places the
  witnesses (|ξ|∞ ≤ 1), and minimizes the affine first-order change.
  - At the generic placement it has optimum 0, at ξ = 0. Yet there the area is smooth, with no ties, and its gradient
    (central differences, which match the edge-average rule to 1.1×10⁻⁹) has length 0.390. Along the steepest descent
    the area falls at 0.390 per unit, while the affine model charges +0.107 for that move.
  - The reason is that each moving witness owns hull pieces on several sides (B5 has arcs on the right and a corner on
    the left). So every rigid move pushes some edge end outward, and the affine model makes the whole edge pay for
    it.
  - At q† it also has optimum 0, and the area falls toward Z at 6.3×10⁻⁶ per unit (Z is 9.03×10⁻¹¹ lower, RUN-052).
    §61 would issue a "finite-mode first-order closure certificate" at both points. Paper 04's projected flow (§14)
    would not move from either.
  - The gap is not the one §62 disclaims (modes above M, localized directions). No admissible container direction
    (continuous, with Lφ a measure), from any library, lowers the cost of those witness moves below zero.
  - None of §60's upgrades (mode order, witness family, second order, charts, continuum) closes it, because the
    missing descent is of first order.
- **Two statements are stronger than the text makes them.** §19's lower bound is the exact cap: on the negative set
  of ν_φ, μ_h + tν_φ ≥ 0 exactly when t·q ≤ 1, so ε⁺ = 1/‖q‖. §18's "may" is "does": if ν_φ⁻ has mass where μ_h
  has none, every positive step fails.

- **The formulas (W1).** In sympy, over trigonometric polynomials with free coefficients:
  - B is symmetric, B(f, g) = ½∫f(g + g″), and A[h + tφ] = A + 2tB(h, φ) + t²B(φ, φ) exactly;
  - δA = 2B(h, φ) = ∫φ(h + h″), δ²A = 2B(φ, φ) = ∫(φ² − φ′²);
  - the circle's Steiner formula; δA[h; 1] = the perimeter; translations give 0 at both orders, and the rotation
    tangent 0 at first order; ∫φ dμ_h = ∫h dν_φ;
  - the L² gradient is h + h″, and the H¹ gradient solves (1 − ℓ²∂²)g = h + h″;
  - §13's warning is right. For h = 1 + a cos 3θ the affine path h − th′ has area A − 36πa²t², while the rotation
    itself keeps A. The affine path along the rotation tangent shows a descent that is not there.
  - The sanity record's eight numbers are recomputed exactly: the circle of radius 2, the translation mode, and
    cos 2θ with its cap 1/3.
- **Bodies (W2).**
  - A(K + sB) = A + P s + π s² holds to 3×10⁻¹⁵ on the kernel's hull at q† and on the Reuleaux triangle and pentagon
    (s = 0.01, 0.1, 0.5).
  - μ_h(S¹), summed from the segments' densities and edges, is the perimeter to 1.3×10⁻¹⁵.
  - Along RT − t cos 3θ (t ≤ 1/8, admissible: RUN-052), the drawn boundary's area is A₀ + 2t − 4πt² to 2×10⁻⁹ (the
    polygon's own error). The width stays 1 to 4×10⁻¹⁶. The area gains, 0.0759 at t = 1/16 and 0.0537 at 1/8, are
    positive, as Blaschke–Lebesgue requires of a constant-width body other than the Reuleaux triangle.
- **Step caps (W3).**
  - The unit circle with cos 2θ: 1/3 (§20).
  - The Reuleaux triangle with −cos 3θ: Lφ = 8 cos 3θ is negative only on the arcs, where q = 8|cos 3θ|, so §19 gives
    1/8. That is also the largest admissible step RUN-052 found directly.
  - The unit square with ±cos 2θ: Lφ is negative on open arcs where μ_h has no mass, so the cap is 0 for both signs
    (§18).
- **The contact variation (W4).** A triangle is held in the unit disc at ψ = 0, t = (0.2, 0.1), with two corners on
  the circle. For six random (φ, ω, v), the least of C_ε near each contact, divided by ε, is §28's
  φ(θ_j) + ωh_W′(θ_j − ψ) − v·u_j to 9×10⁻⁷. The ω term reaches 0.16, so it is exercised.
- **The kernel (W5).**
  - Measured one-sided rates (A(v + tξ) − A(v))/t at steps 10⁻⁷ and 10⁻⁸, against the two models.
  - At a generic placement near q† (q† + (0.002, 0.001, 0.05, −0.001, 0.002)) there are no ties. The edge average
    matches every rate, and the affine model does not.
  - There the gradient is (−0.0791, 0.0330, 0.0425, 0.2089, 0.3149), by central differences and by the edge average
    alike. The LP optimum is 0, at ξ = 0.
  - At Z, both ways along y3, φ5 and y5: measured 0.1327, 0.0442 and 0.2733; affine 0.7320, 0.1274 and 0.7950.
  - At q† toward Z: measured −6.3×10⁻⁶ (−6.6×10⁻⁶ at step 10⁻⁸); affine +0.020. The LP optimum is 0, at ξ = 0.
- **The records (W6).**
  - The schema's variation formulas, exact path, contact formula and five cones are the text's.
  - Each of the state's 15 completed items has its section.
  - All five "not claimed" items are the text's own disclaimers (§1.1/54, §46, §59, §62, §40).
  - Both records name Paper 04 as the text does.
- **The scaffold (W7), read and never run.** It has 61 lines.
  - Its FourierMode gives δ²A = π(1 − m²)(a² + b²), and 2πa² for m = 0.
  - Its contact row has §29's signs (h′_W, −cos θ_j, −sin θ_j).
  - Its line search is §51's.
  - It solves no LP, and it takes δA as an input rather than computing it. It says so: "not a publication proof
    kernel".
- **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, 163 / 163 must-stay-green
assertions held. Each of the nine checks was blinded in turn, and each was reported blind in exactly as many cases
as plants name it.

---

## What the paper claims

1. Area is the mixed-area form B(h, h), exactly quadratic along affine support paths. Its first variation is
   ∫φ dμ_h (= ∫h dν_φ), and its second ∫(φ² − φ′²), independent of h. The constant direction gives the perimeter.
   Translations and rotations are gauge, and the affine path along the rotation tangent is not a rotation.
2. Convexity along a path is μ_h + tLφ ≥ 0, with a one-sided cap. It is 0 when ν_φ⁻ has mass where μ_h has none,
   and at least 1/‖q‖ when ν_φ⁻ = q μ_h.
3. A fixed-placement witness contact varies by φ(θ_j) + ωh_W′(θ_j − ψ) − v·u_j. With finitely many active contacts
   and strict margins this gives local feasibility (a "local sufficient principle").
4. First-order feasibility and descent over a finite mode library are finite LPs. An optimum ≥ 0 is a finite-mode
   first-order closure certificate, not continuum stationarity. There is an exact line search along one direction.
5. Shape velocity, integration with pathwise convexity and topology gates, and hybrid dynamics. Gradients need a
   metric (L², H¹), left to Paper 04. The continuum constraint is left to Paper 05.

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the calculus in sympy; the sanity record | all hold; the rotation-tangent path lowers A by 36πa²t² |
| W2 | Steiner and perimeter on three bodies; the triangle's admissible path drawn | 3×10⁻¹⁵; 2×10⁻⁹ |
| W3 | three step caps | 1/3; 1/8 by §19's density; 0 on the square |
| W4 | the contact variation against finite differences | 9×10⁻⁷ |
| W5 | witness moves on the kernel: measured, edge average, affine; the gradient; the LP at a generic placement and at q† | the average is right; the affine model overcharges; the LP finds no descent at either, though the gradient has length 0.39 |
| W6 | the schema and the state | as the text |
| W7 | the scaffold, read | row signs and δ²A right; no LP |

## P — the package

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

## The paper, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §4–9 | area, mixed form, exact quadratic, first and second variation | PASS | W1 |
| §10–13 | Steiner, perimeter, translation and rotation gauge | PASS | W1, W2; the rotation-tangent affine path lowers A at second order, as §13 warns |
| §14–15 | the measure lane; ∫φ dμ_h = ∫h dν_φ | PASS | W1, and RUN-052's two computations on bodies |
| §16–21 | tangent cone, step caps, polygon warning | PASS | W3; §19's bound is the exact cap, and §18's "may" is "does" |
| §26–29 | fixed-placement contact variation; compensation cone | PASS | W4 |
| §30–32 | finite LP feasibility; the feasible cone | PASS for finitely many isolated contacts | on the kernel every witness is active on sectors, and its moves turn edges (W5) |
| §33 | strict first-order feasibility | PASS as labelled | its hypotheses (finite active set, stable stratum) fail on the kernel |
| §37–39 | cone stationarity; the LP descent test | PASS as definitions | on the kernel the affine cone misses first-order descents, generic ones included (W5) |
| §40–41 | constrained second order; breaking modes | PASS | stated as incomplete |
| §42–47 | velocity, integration, pathwise gates, hybrid dynamics | PASS | definitions |
| §50–52 | exact area along a step; line search | PASS | W2 on a body; the scaffold implements §51 (W7) |
| §54–57 | gradients need a metric; L², H¹ | PASS | W1 |
| §60–62 | the gate on UESFCM's outputs; finite-mode closure certificate | NOT SOUND on the kernel | the certificate would be issued at a generic placement with gradient 0.39, and at q†; §62's disclaimers do not cover it |

## The status registry

Not touched: Paper 03 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the contact variation's sign in the text | 1 | A0 |
| the rotation-tangent path; a sanity record value | 2 | W1 |
| the Steiner residual; the drawn path | 2 | W2 |
| the triangle's cap; the square's cap | 2 | W3 |
| the contact against finite differences; the ω term idle | 2 | W4 |
| the edge average off; an LP descent at the generic placement | 2 | W5 |
| the schema's first variation; an item added to "not claimed" | 2 | W6 |
| an LP solver in the scaffold; its contact row's sign | 2 | W7 |
| a VALIDATION digest; the sanity record's bytes | 2 | P1 |

Three controls:

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

**17 / 17 plants, 3 / 3 controls, 163 / 163 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 |
| W7 | 2 | 2 |
| P1 | 2 | 2 |

## What this run does not establish

- **A corrected first-order test.** The edge-average rule suggests one for hull-type containers, where each edge is
  charged ℓ(r_a + r_b)/2 and ties are handled by their upper envelope. It is not built or verified here.
- **Anything for containers with isolated contacts**, where §33's hypotheses can hold and the affine model can be
  exact.
- **Whether the LP ever finds a descent on the kernel.** It found none at the three placements tried. A witness that
  owns hull pieces on one side only could move inward and be seen. Where each owns pieces on several sides, as here,
  the affine model makes every rigid move pay.
- **The continuum constraint and the metric-dependent flows**, which the paper leaves to Papers 04–05.