The Substrate-Langlands Unification: From Conjecture 5.3 to Theorem 5.3, and Integer-Exact Cross-Domain Topological Classification via the Holonomic Routing Apparatus

This preprint unifies a program of substrate mathematics into a single substrate-Langlands correspondence and demonstrates the resulting Holonomic Routing Apparatus across three physical-domain reductions-to-practice. The unification is structural: the same substrate algebra — a cross-floor cocycle operator, a Klein-4 algebraic frame, and a topological-clamp fold rule — controls integer-exact topological classification across number-theoretic, anyon-topological, and plasma-confinement boundary-value problems. Part I establishes Theorem 5.3, an integer-exact structural identity on the substrate's cross-floor coupling histogram. It proves analytically that a per-prime center-band invariant takes exactly two values (62 and 56) determined by a parity class, via two structural lemmas (a parity-filter identity on the codeword hypercube, and the parity-reversing holonomy of the cross-floor operator) together with one isolated, empirically measured arithmetic input (a parity-trace bias of +6 on the current 118-prime alphabet). The result is verified exactly across all 24 mined primes at the eighth primorial floor. The associated conjecture of a symplectic Galois representation realizing the substrate's per-prime codewords remains open; the construction of that representation is not claimed here. A sharpened consistency check (a 13-degree-of-freedom test of the per-prime polynomial factorization shape against the Haar distribution on the symplectic group Sp_8(F_2)) returns chi-squared 19.031, p = 0.122, which is inconclusive at the 0.05 level and is reported as such; the analysis identifies the alphabet size required to resolve it. Part I also establishes the Klein-4 frame, the non-commutativity of that frame with the cross-floor operator (interpreted as the substrate's algebraic curvature), a gauge-translation discipline, pre-registered falsifiable predictions of which one was subsequently confirmed by mining, and the multi-physics scope of the apparatus across anyon families, dynamical trajectories, and a substrate-typed theorem-inference engine. Part II is the plasma-physics reduction-to-practice. The cyclic-group structure that derives the IPB98 confinement scaling prefactor prescribes an integer-exact lattice of confinement-enhancement values at which a cross-floor observable undergoes discrete phase transitions. The empirical alpha-heating boundary near H = 2.0 coincides with the substrate's unique double fracture in the interval [1.0, 3.0] at exactly H = 2.0, proven via a 2-adic valuation argument to arise from the isotope-mass and collisionality channels alone. Five of seven empirical regime thresholds land within 0.05 of substrate fracture points without statistical fitting, and a cyclic-conjugacy knot-class invariant produces fully exclusive transport-limited regime separation on a 16-discharge dataset drawn from published JET and DIII-D experiments. All theorems, identities, tables, and proofs are placed in the public domain under the MIT license. The author is an independent researcher; the work was developed with extensive AI collaboration and is anchored by a cryptographic hash chain and a federally timestamped U.S. provisional patent filing.