Proof review

Henry Zweiman. September 16, 2026.

This review was performed by the originating assistant. It is not independent human verification or a formal proof certificate.

The complete consolidated manuscript was reread, including its weak-curvature hypotheses and the passages to singular limits. No fatal gap was found in this pass. The following checks identify the actual proof obligations rather than treating numerical agreement as proof.

• Section 2: differentiation gives the harmonic radial equation. Angular-potential minimization orders the positive Steklov branches. The barrier for the logarithmic derivative remains strict at the hemispherical endpoint by an integrating factor; a non-strict endpoint limit would not suffice.
• Section 3: the potential is regular near the closed proper convex domain because no pair of its points is antipodal. An inward directional derivative excludes a boundary minimum. The balanced multiplier is positive before the positive-polar conclusion is used. The weighted divergence identity retains its normal-product term. A strictly decreasing kernel and the strictly negative support value of an interior point replace a strict-curvature assumption. At a farthest boundary point from any point of the closed domain, the support value is strictly negative, which makes the final polar inequality strict. The Hessian argument proves uniqueness only among interior centers.
• Section 4: the cotangent radial function is the support function of the polar of the gnomonic image. Its mean controls its maximum and first derivative without a second-derivative bound. Exact harmonic centering makes the degree-one component quadratically small in the deviation. The remaining harmonics give a uniform positive quadratic perimeter term. The area density has vanishing third derivatives at zero; its Taylor error is bounded by the squared mean times the deviation energy. The equal-perimeter radius increases by at least the deviation energy divided by the mean. The trial quotient changes only quadratically, whereas the ball eigenvalue has strictly positive endpoint slope. These estimates establish the stated uniform deficit for all sufficiently small means. Zero spectral deficit, rather than an unspecified equality in an intermediate bound, is the rigidity hypothesis.
• Section 5: the two rotated restrictions have matching traces at the seam, hence define an H1 function with the sum of their energies. Their boundary trace and norm agree with the hemisphere eigenfunction. The removed angular region has a strictly positive explicit energy lower bound for every n at least three. A smooth extension of the one-variable kernel makes the fixed trial globally Lipschitz. No harmonic equation across the seam is assumed.
• Section 6: Crofton is calibrated against an equator. The cap-intersection probability is the tail of a beta distribution with parameters 1 and (n-1)/2. It bounds local boundary area and prevents boundary measure from concentrating at a point. Summing centered coordinate Rayleigh bounds controls collapsed-volume sequences. In the full-dimensional case, three independent positive dual normals produce a great circle that robustly misses the limit, contradicting maximal perimeter. One- and two-dimensional pointed dual cones give hemispheres and proper lunes.
• The boundary-integral passage in Section 6 uses convex graphs on compact face patches and the cap estimate in a shrinking neighborhood of the edge. Uniformly small convex graph heights imply uniformly small gradients on smaller patches by secant bounds. The edge neighborhood has uniformly vanishing boundary area. Interior convergence and the zero-volume boundary of a full-dimensional convex limit justify volume dominated convergence. Only a fixed-trial limsup is used; full spectral continuity is neither assumed nor claimed.
• Section 7 exhausts collapsed, proper-lune and hemisphere limits. Positive-polar containment forces harmonic centers to converge to the center of a hemispherical limit. Inner convex convergence then makes the mean cotangent tend to zero. The endpoint deficit is strict for every nonball, including hypothetical nonball equality cases.

The dimension restriction is substantive: the endpoint slope and the displayed lune energy lower bound used here do not give the same argument in dimension two. The perimeter interval is dimensional but nonexplicit. The full spherical fixed-perimeter conjecture is not solved.

Artifact review

The publication copy has complete Markdown, portable LaTeX, and a 17-page PDF. All exact final pages were rendered and visually inspected after the bibliography and equation numbering were corrected. There is no clipping, overlapping text, or missing rendered mathematical glyph in the inspected copy. The main theorem statement is kept together. Text extraction, equation-reference checks and the 462-expression Markdown check supplement this visual inspection. These are artifact checks, not mathematical certification.

The three diagnostic scripts check algebra and finite examples. The manuscript's universal conclusions rest on the hand proofs. The paper does not compute the actual first eigenvalue of a lune numerically.