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

Why does the Born rule hold? Every deterministic approach to quantum mechanics eventually collides with the same wall: deriving the |ψ|² statistics of measurement outcomes instead of postulating them. This work confronts that wall deliberately — not by claiming to break through it, but by mapping it stone by stone. Within the TS5D program (a classical, deterministic five-dimensional solitonic field framework), this paper constructs the complete conditional chain that would yield the Born rule via Gleason's theorem: a non-injective 5D→4D projection hiding an internal phase profile, a Noether-derived candidate inner product, a conditional Liouville measure on the fiber of the preparation, a soliton–apparatus interaction dynamics, and an outcome functional whose σ-additivity, non-contextuality, anchoring and equivariance would force |ψ|² with no quantum postulate. It then does what most programmatic papers avoid: it audits its own chain, and localizes exactly the five locks — geometric, reductive, measure-theoretic, dynamical, probabilistic — that remain to be opened, each with its attack tools and known obstacles (KAM islands, the Gleason anchoring gap, the impossibility of attractors in volume-preserving flows, Bell nonlocality vs. no-signalling). The result is deliberately classified by the audit's own standard: a conditional mechanism architecture without dynamical realization or predictive closure — controlled underdetermination, stated as such. An appendix proves a conditional exponential-race outcome law in a fully explicit Bernoulli reference model, with a deposited deterministic script (fixed seed) that regenerates every Monte Carlo table verbatim. Readers interested in hidden-variable programs, the Dürr–Goldstein–Zanghì equivariance strategy, Gleason-type derivations, or simply in what intellectual honesty looks like when applied to one's own theory, will find here a complete worked example. New in version 2: full alignment with the TS5D foundational postulates (P1–P6) and the predictive-closure audit monograph (doi:10.5281/zenodo.21435537); corrections from several rounds of external critical review (fiber topology, phase/monodromy distinction, covariant Noether charge, measure disintegration over the joint preparation, state anchoring, corrected Bell analysis); withdrawal of all underived numerical signatures; exactly reproducible numerics (bernoulli_race.py, seed 20260720, reference output included). No claim was strengthened at any revision; several were weakened or withdrawn.