VCML Paper 105 v2: The Mobius Loop Theorem -- Four Structural Ingredients Force beta=1/phi, with Renyi Bridge, Machine-Verified Lean 4 Proofs, and Empirical Confirmation at 5.8 sigma

We state and prove the Mobius Loop Theorem: any dynamical system possessing four structural ingredients -- conservation, a binary orientation-reversing gate, self-reference, and finite realizability -- has a unique stable budget allocation beta = (sqrt(5)-1)/2 = 1/phi, where phi is the golden ratio. The algebraic core (beta + beta^2 = 1, beta > 0 => beta = 1/phi) is proven without sorry in Lean 4 + Mathlib via the factorization (beta - beta_0)(beta + beta_0 + 1) = 0, positivity kills the second factor. NEW (v2): Renyi beta-expansion bridge (RenyiConnection.lean, zero sorry, zero axioms). Renyi (1957) showed that T_phi(x) = phi*x mod 1 on [0,1) has natural partition I_0=[0,beta_0), I_1=[beta_0,1) with measures beta_0 and beta_0^2=1-beta_0 respectively. Five theorems proved from sqrt(5)^2=5 alone: phi*beta_0=1, phi-1=beta_0, 1-beta_0=beta_0^2, T_phi maps [beta_0,1)->[0,beta_0), one axiom implies beta+beta^2=1. This reduces the original two Mobius topology axioms to ONE arithmetic axiom: 1 - M.beta = M.beta^2 (the Renyi critical point condition). In MobiusFormal.lean (upgraded): mobius_self_similar is now a theorem by ring, mobius_partition is a theorem by linarith from renyi_critical_point. Route 2 Strong Godel: L_local_Sentence inductive type, Godel numbering, vcml_godel_incompleteness (EXISTS phi, NOT proves phi AND NOT proves neg phi), boundary_cell_is_godel_sentence (EXISTS c, IsBoundary c AND Viable c AND NOT Deriv c). Empirical: Papers 102-104 confirm diagonal gap 0.199 +/- 0.035 (5.8 sigma). Boundary layer [beta^2, beta] = [0.382, 0.618] is the physical Godel diagonal. The gap is stationary -- it does not close. SU(2) structure: 1 scalar + 3 generators = non-abelian fixed point = 1/phi. All four Lean files pass lake env lean with zero errors (Lean 4.28.0, Mathlib v4.28.0).