Internal proof review

Henry Zweiman. September 15, 2026. Developed as Q57.

This is the originating assistant's audit of its own argument. No independent mathematician or subagent reviewed the proof.

1. Physical normalization. The load is integrated over the filled domain. Constants on separate holes vary independently. The Riesz equation gives both the torsion integral and energy. The annular value is (A squared minus B squared)/(8 pi). All-Dirichlet torsion on the perforated set would be a different problem.
2. Lipschitz boundary. The admissible space is closed by gradient restriction. Harmonic comparison and finite weak charges give positive hole values. A containing-ball comparison bounds the solution. Exterior-cone barriers give boundary continuity with no estimate uniform in the Lipschitz character needed. Every positive regular level distinct from the hole values is compactly contained in the physical region.
3. Distribution. The gradient critical set has zero area, since weak derivatives vanish almost everywhere on their zero level sets and the Laplacian is -1. Reciprocal-gradient coarea exhausts the non-atomic distribution; only the finitely many hole atoms remain. Absolute continuity between atoms and Sard's theorem justify excluding nonregular mass ranks.
4. Flux and signs. Differentiating the admissible truncation identity gives boundary flux equal to the entire superlevel area, including enclosed hole plateaus. Both Cauchy-Schwarz and isoperimetric deficits are nonnegative.
5. Plateau bookkeeping. Telescoping the square of the area distribution includes a possible top jump. The first-moment penalty is nonnegative by the one-dimensional bathtub principle. The symmetric difference of the hole union and the top B ranks is exactly the symmetric difference of their rank intervals. Artificial cuts within plateaus require no perimeter estimate.
6. Essential calibration input. The global Hensel-Laux theorem, not its confined theorem, supplies a center. A smooth radial bump is pointwise dominated in relative energy by the original calibration. Both normal conventions are switched together. The first center derivative of the reference-disk energy vanishes; bounded derivatives of the smooth divergence control the bulk translation error, including remote tentacles.
7. Change of variable. The active-radius integral equals the time integral of the two deficits. Plateaus are excluded from that integral and charged by their separate first-moment penalty. Setting their velocity to zero inside the same active integral would incorrectly double-count them.
8. Absolute transport estimate. Compact smooth regular levels have normal graphs for either sign of the increment. Their discrepancy from the frozen-field flow is the integral of the absolute normal-velocity difference to first order. The disk evolves with unit normal speed. A bounded flow Jacobian and the reverse triangle inequality prove the absolute derivative bound; no derivative of a boundary intersection is assumed.
9. Velocity oscillation. Expansion of the weighted variance yields exactly (2 pi rho/P squared) times the Cauchy-Schwarz deficit. Weighted Cauchy-Schwarz then gives the square-root forcing term. The remaining comparison error is bounded by perimeter excess plus calibration energy.
10. Exceptional sets and bootstrap. Small velocities and large isoperimetric deficit occupy total length O(delta); plateau ranks above B occupy O(sqrt(delta)). Nestedness makes the discrepancy globally Lipschitz, so these exceptional sets can be traversed. Start at a good level with small deficit, fix its center, and apply Gronwall in both directions. A connected-component bootstrap removes the temporary smallness assumption. The zero-deficit and large-deficit cases are explicitly included.
11. Constants. Rescaling a mass window [a,b] with a at least B makes its constant depend only on b/a. Windows [2A/3,A] and [B,3B/2] have the same fixed ratio. They give the universal individual-asymmetry estimate with separately chosen centers. The single window [B,A] supplies concentricity.
12. Sharpness. The squared-radius second-harmonic perturbation preserves area exactly. The harmonic-energy identity has the correct sign. A C1 radial pullback bounds the energy multiplicatively, yielding a cubic remainder without a shape-differentiability assertion. Orthogonality to translation modes forces linear asymmetry, whereas the deficit is quadratic. Boundary collars establish the same obstruction for annular asymmetry.

No mathematical gap was identified in this audit. The constants are not effective; nonlinear, higher-dimensional and singular area-ratio limits remain open here. No finite computation substitutes for proof. The PDF and MathJax records concern only artifact integrity.