Reduction of the Born Rule to a Liouville–Holonomy Equivariance Problem in TS5D

What if the Born rule were not a postulate, but a theorem? Since 1926, the Born rule — which assigns probability ∣⟨φ∣ψ⟩∣2|||² ∣⟨φ∣ψ⟩∣2 to a quantum measurement outcome — has been imposed as an axiom. No derivation from non-quantum premises is universally accepted: neither Bohm, Nelson, Valentini, nor Deutsch-Wallace has closed the debate. This paper proposes a radically new approach. Within the framework of the **Five-Dimensional Solitonic Theory (TS5D) ** — an extended classical theory on the manifold M4×S1/Z6M⁴ S¹/Z₆ M4×S1/Z6 —, we prove that the derivation of Born *reduces* to a hierarchized system of **two precise mathematical locks**: Technical lock: regularization of the Liouville measure on field-theoretic phase space (three routes identified: modal truncation, cylindrical measures, Gaussian regularization). Dynamical lock: Liouville–holonomy equivariance under soliton-apparatus interaction dynamics. If both locks are removed, Gleason's theorem automatically locks in the ∣ψ∣2||² ∣ψ∣2 form. Four partial results structure the program: Emergence of Hermitian structure via Noether U (1) U (1) U (1) — *established*. Identification of the 5D holonomy phase as a geometric hidden variable — structurally established. Preservation of the Liouville measure by Hamiltonian flow — classical. Gleason's theorem applied to the effective Hilbert space — established under conditions. Decisive contribution: Appendix E presents a Bernoulli reference model in which the full chain leading to Born is explicitly proved — strong mixing ⟹ exponential thresholds ⟹ rates Γk∝∣ck∣2ₖ |cₖ|² Γk∝∣ck∣2 ⟹ pointer competition ⟹ P (k) =∣ck∣2P (k) = |cₖ|² P (k) =∣ck∣2. Numerical validation with Monte Carlo error ∼10−3 10^-3 ∼10−3 over 50, 000 samples. Candidate experimental signatures distinguishing TS5D from standard quantum mechanics and from Bohm-Valentini: Topological mass splitting in ratios 13: 15: 17 2π/62/6 2π/6 periodicity in Bell-type correlations Gaussian decay of CHSH visibility with spectral marking What this work claims to do: transform a vague question — "how does TS5D derive Born? " — into a precise, hierarchized mathematical problem, attackable by the standard tools of functional analysis and ergodic theory. Following Hilbert's tradition of structuring mathematics in 1900, this work proposes a theorematic reduction rather than a closure, and unambiguously identifies the remaining research directions. Compatibility with no-go theorems: Bell (5D geometric exit), Kochen-Specker (natural contextuality), PBR (ontic field in 5D) — all explicitly analyzed.