The Kernel Deficit Dominates Twice the Hull Deficit: A Sharp Strengthening of Nakano's Inequality

For a simple polygonal region F in the plane, let K(F) be its visibility kernel, the set of points that see all of F, and let C(F) be its convex hull. We prove the sharp inequality: area(F) · (area(F) − area(K)) ≥ 2 · area(K) · (area(C) − area(F)) In terms of Sibley's guard-point ratio G and exterior-area ratio A, this becomes: G ≤ A / (2 − A) strictly improving Nakano's inequality G ≤ A for every nonconvex polygon. The coefficient two is optimal: an explicit one-parameter family of nonconvex equality examples is exhibited. Empty and zero-area kernels are included. The proof passes through a convex-body cap-union theorem. Cyclically sorting the directed boundary edges of a polygonal cap union U produces a convex companion H. A boundary-reversal inequality gives: area(H) + area(U) ≥ 2 · area(conv U) while a support-function identity and Minkowski's mixed-area inequality give: area(U)² ≥ area(K) · area(H) Inner polygonal approximation extends the result to every positive-area compact convex body, and a separate branch handles points, segments, and all other null-area cases. Both main theorems have machine-checked Lean 4/Mathlib proofs whose final statements were audited against the manuscript after kernel checking. The proof-complete formalization is available at:https://github.com/SilverAsh7/p5-kernel-deficit-lean, commit e62a622ed1ff1a70f4a10a860b4088ee35ee6b68 Both the production and research aggregates are admission-free, and the public theorem closures use only propext, Classical.choice, and Quot.sound. The reader-facing proof derives the required hull order directly from polygonal-Jordan and crosscut facts; the former unsupported interleaved-arc/Jordan–Schoenflies step has been removed. A final section carefully separates proved bounds, exhaustive exact finite certificates, corroborating computation, and conjectures concerning Nakano's still-open perimeter optimum α*; it makes no claim to determine that optimum. The manuscript includes a declaration of near-total generative-AI assistance.