Yang-Mills Existence and Mass Gap as Fiber-Field Closure Under the Hydrogen Substrate Coupling Functional

The Yang-Mills existence-and-mass-gap problem asks for a mathematically rigorous demonstration that quantum Yang-Mills theory exists on R⁴ and admits a positive mass gap ₘ > 0 above the vacuum. This companion deposit proposes that the Yang-Mills mass-gap problem is the formalism-substrate-resolution-class readout of an underlying substrate-coupling-architecture present at the quarter-collinear amplitude-regime (HLRP #1 Approach Geometry of Amplitude Zeros, Zenodo DOI 10. 5281/zenodo. 18727570) — the operator-corpus's primary architectural anchor. The mass-gap-substrate is the IE-006 residual feedback-gap of the substrate-coupling-architecture, given by = (d+) (d+1+) at = 623, which at d=3 yields 0. 01878, identical to the IE-006 feedback-gap formula (d d+1) = 138/ (23d+29) (23d+6) at d=3: 138/ (98 75) = 138/7350 0. 01878. Arithmetic identity verified; two expressions of the same surface. The sansNP substrate-coupling pair equation a + b (x) \z\) = yv^2 + (Apex Deposit Definition 1B) renders the bidirectional pair-production / annihilation architectural-form in which appears as the mass-gap-substrate explicit-readout, and yv^2 as a Yang-Mills mass-readout-form that simultaneously mirrors E = mc^2. The Hydrogen Substrate Coupling Functional (HSCF, Apex Deposit Definition 6) with operator-corpus BH-substrate-unification extension provides the cross-substrate-resolution-class invariant (Lemma YM-3): the YM mass-gap at quark-substrate-resolution-class equals the mass-delta at proton/neutron-substrate-resolution-class equals the mass-substrate at the neutron-class-decoupled-floor at the BH-center-substrate-resolution-class. Same architectural-substance across substrate-resolution-classes — this is the YM mass-gap-substrate-substance underneath the formalism-readout. The corpus's quarter-collinear anchor (HLRP #1) precedes the math-physics-establishment parallel-arrival at the half-collinear regime (Guevara, Lupsasca, Skinner, Strominger, Weil 2026, arXiv: 2602. 12176) in operator-corpus internal-derivation order — geometry-forward methodology, not formalism-derivation-substrate. The parallel-arrival is calibration-evidence at the math-physics-establishment-substrate-class, not the origin of the corpus claim. This work proposes a formal architectural scaffold for Yang-Mills mass-gap as substrate-stabilized residual; the specific value 0. 01878 at d=3 provides a clean external falsification surface. Keywords: Yang-Mills theory; mass gap; gauge field theory; substrate-coupling architecture; Hydrogen Lifecycle Research Programme; HLRP; approach geometry of amplitude zeros; quarter-collinear regime; half-collinear regime; double-copy; sansNP equation; IE-006 feedback gap; = 6/23; (d, ) ; Hydrogen Substrate Coupling Functional; HSCF; sign-flip architecture; cross-substrate invariance; mass-gap residual