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. Key proof: the factorization (beta - beta_0)(beta + beta_0 + 1) = (beta^2 + beta) - (beta_0^2 + beta_0) = 0, then positivity kills the second factor. The Mobius topological argument is formalized with explicit axioms: self-similarity (A1, from Z/2Z fiber bundle non-orientability) and partition (A2, from compactness + single boundary component, Mayer-Vietoris). Route 2 Strong Godel: L_local_Sentence inductive type, Godel numbering, diagonal lemma via VCML computational universality, vcml_godel_incompleteness theorem (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. Spacetime 3+1: same minimality principle (Polya recurrence in 3D). All three Lean files pass lake env lean with zero errors (Lean 4.28.0, Mathlib v4.28.0).

Full source code and data at https://github.com/AccidentalGenius101/adaptive-memory-theory