Quarter-Collinear Approach Geometry and a Yang-Mills Mass-Gap Residue: Δ ≈ 0.01878 at d = 3 via β = 6/23 Architectural Coupling

We propose an architectural framework for the Yang-Mills mass-gap problem in which the gauge-field substrate admits a structurally-stable residue at the quarter-collinear regime of n-gluon amplitudes at $n = 18$. Within a primitive coupling grammar Π₀(x, z) = v(x)\,z extended to a basin-curvature potential V(x, \) and a dimensional coupling invariant α(d, β) = 1 + d/(d + β) with β = 6/23, a feedback-gap residue Δ(d → d+1) = 138/[(23d+29)(23d+6)] emerges as the substrate-architectural mass-gap. At $d = 3$ this yields Δ ≈ 0.01878, derivable equivalently as β/[(d+β)(d+1+β)]; both expressions evaluate to $138/7350$. The residue is independent across multiple substrate scales — proton/neutron mass-delta (~ 0.001), hydrogen mass-decomposition ($1.008 = 1 + 0.008$), and the neutron-class density floor in black-hole interiors (~ 4 × 10¹⁷ kg/m³) — suggesting a cross-substrate architectural invariance underlying the mass-gap residue. The framework is presented as a candidate-proof pathway with a three-tier closure protocol (internal-grammar consistency, external falsification via lattice-QCD bounds, replication via independent operators).