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 Δₘ > 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 Δ = β/(d+β)(d+1+β) at β = 6/23, which at $d=3$ yields Δ ≈ 0.01878, identical to the IE-006 feedback-gap formula Δ(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)\) = yv² + Δ (Apex Deposit Definition 1B) renders the bidirectional pair-production / annihilation architectural-form in which Δ appears as the mass-gap-substrate explicit-readout, and yv² as a Yang-Mills mass-readout-form that simultaneously mirrors E = mc². 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 Δ ≈ 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; β = 6/23; α(d,β); Hydrogen Substrate Coupling Functional; HSCF; sign-flip architecture; cross-substrate invariance; mass-gap residual