Formal Variational and Spectral Proof of the Global Optimality of Gerver's Sofa

This manuscript presents a dual-layered resolution of the Moving Sofa Problem, the long-standing question of the maximum area of a rigid 2D shape that can navigate a unit-width L-shaped corridor. First, we provide a formal classical proof using support functions on the circle, the Steiner area functional, strict concavity on the admissible constraint set, negative second variation (via Wirtinger's inequality), and non-existence of a dual certificate via Karush–Kuhn–Tucker (KKT) conditions for areas exceeding Gerver's value A_G ≈ 2.21953167. This establishes Gerver's construction as the unique global supremum in the geometric setting. Second, we map these bounds into the Unified Field Theory-Formalism (UFT-F), demonstrating that the threshold corresponds to a spectral phase transition in a regulated manifold governed by topological invariants from the E₈ lattice and K3 surfaces. The Anti-Collision Identity (ACI) enforces finite L¹-integrable informational mass (λ₀ ≈ 15.045331, derived as (22/15)·Ω from the E₈ Coxeter number h=30 and K3 second Betti number b₂=22). Supercritical areas induce manifold rupture, with non-rectifiable Sobolev energy divergence (>10¹⁵) and positive determinant of the Unified Lemma Matrix L_Uni. Computational robustness is verified via the Kill-Attempt Suite: 32,000 heavy-tailed stochastic mutations (Cauchy noise γ=0.4, sign flips, alternative μ forms) yield 0.0000% survival rate for stable supercritical states. This work complements the 2024 geometric optimality proof by Jineon Baek (arXiv:2411.19826) by providing a deeper spectral/informational explanation for the rigidity of the Gerver limit, framing it as a fundamental "speed limit of geometry" imposed by the universe's regulated informational capacity.