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 ₀ (x, z) = v (x) \, z extended to a basin-curvature potential V (x, \z\) and a dimensional coupling invariant (d, ) = 1 + d/ (d +) with = 6/23, a feedback-gap residue (d d+1) = 138/ (23d+29) (23d+6) emerges as the substrate-architectural mass-gap. At d = 3 this yields 0. 01878, derivable equivalently as / (d+) (d+1+) ; 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 10^17 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).