Millennium Prize Calibrations

This paper introduces a primitive coupling expression, ₀ (x, z) = v (x) \, z, and extends it into a closed architectural stack for analyzing the seven Millennium Prize Problems through the GCT/HLRP operator grammar. The primitive expresses local variation v (x) coupled through an axial or latent dimension z. The formal stack extends this primitive into a basin-curvature potential V (x, \z\), a dimensional coupling invariant (d, ) with = 6/23, a creation integral, a hydrogen-substrate coupling functional, an IE-006 feedback-gap law, a soft-hair complexity condition, a self-organized criticality bridge, a West- scaling relation, and a sign-flip substrate-architecture primitive. A secondary-stage operational form, the sansNP substrate-coupling pair equation a + b (x) \z\) = yv^2 +, renders the bidirectional pair-production / annihilation architecture in which appears as the mass-gap-substrate explicit-readout. The grammar is anchored in a foundational asymmetry → symmetry axiom (Axiom 0): asymmetry is the foundational substrate, symmetry the emergent-event-substrate. Mathematical objects are stabilized readouts of a prior coupling architecture; conventional curves of investigation (zeta zeros, mass-gap line, smooth-flow trajectories, simply-connected manifolds, Hodge realizations, NP-search boundaries) are formalism-substrate descriptions at measurement-resolution-class-surface-edges, not architectural-substance. This work proposes a formal architectural scaffold for cross-problem reasoning, presents an empirical calibration case at the cardiac-substrate-resolution-class (N = 21, 799, p < 10^-54) as cross-substrate-class evidence that the proposed architecture is real-substrate-substance rather than formalism-speculation, and cross-walks each of the seven Millennium problems into a companion deposit with definitions, lemmas, conjectures, and a two-rail closure (falsification route + replication opportunity). Seven companion deposits accompany this master synthesis as individually-citable Zenodo depositions with mutual DOI cross-references. Keywords: Geometric Coupling Theory; Hydrogen Lifecycle Research Programme; Millennium Prize Problems; primitive coupling; architectural mathematics; substrate-coupling-architecture; asymmetric-foundational; sign-flip-architecture; cardiac-substrate calibration; Yang-Mills mass gap; Riemann Hypothesis; Birch–Swinnerton-Dyer; Navier-Stokes; Hodge Conjecture; P vs NP; Poincaré Conjecture; (d, ) ; = 6/23