The Admissibility-Boundary Dimensional Bridge: Structural Identication, Uniqueness of ℓ, and Geometry-Constrained Bounds on the Bridge Functional

Within Universal Grid Mechanics (UGM), a dimensional bridge is constructed from the admissibility axioms alone, without importing gravitational data. The bridge connects the dimensionless gravitational coupling G = √3/4π², derived purely from hexagonal geometry, to the SI system through a single structural length scale ℓ. Route B, originally formulated through a cubic spectral slot, is shown to be degenerate: hexagonal Brillouin-zone inversion symmetry eliminates all odd-harmonic contributions, rendering the cubic invariant identically zero. The present work establishes spectral closure through the lowest non-degenerate invariant K₂BZ, which is second-order. K₂BZ is proved equal to 1/2 by two independent routes: numerical quadrature over the exact hexagonal Brillouin zone (0. 500000 ± 0. 000002, stable under grid refinement) and analytically via the identity cos²θ = ½ (1 + cos 2θ) combined with BZ inversion symmetry. Dimensional purity is established: K₂BZ is dimensionless and depends on dimensional quantities only through ℓ. The Route B amplitude factorises as Aₘax = K₂BZ · π · κₕex = π/ (6√3), where κₕex = 1/ (3√3) is the exact hexagonal shell-normalisation constant derived from the Brillouin-zone geometry. The prefactor π/ (3√3) is identified as the exact hexagonal geometric-spectral prefactor of the primitive six-direction class, appearing independently in the gravity sector as ΓU = π√3/3 and in Route B, indicating a common underlying hexagonal normalisation structure. The spectral closure chain K₂BZ ⟹ Aₘax ⟹ ℓ★ ⟹ (G, c, h) SI is fully established. Existence and uniqueness of ℓ★ > 0 follow from a monotonicity lemma (dL/dℓ > 0, dH/dℓ ≤ 0 under admissible redistribution) and a one-crossing argument. The bridge factor λ is shown to be a dimensionless admissible-response functional Λφ; S, not a free parameter. A primitive-shell non-collapse condition — grounded in Axiom 0's requirements for bounded change and local information consistency under repeated updates — yields the geometry-derived lower bound λ ≥ 1/ (3√3). A strict upper bound λ < 1 follows conditionally from the morphology-induced non-full-saturation of the multi-component response. Combined: 1/ (3√3) ≤ λ < 1. The bridge factor enters only the observational bridge a★ = λℓ/τ₀² and does not affect the spectral closure chain or the SI constants. The response function φ (x) is constrained to a geometry-aware, multi-component admissible class (core, band, filament, lobe) ; purely radial responses are formally excluded for anisotropic configurations. The structural constraints are illustrated using a high-resolution MeerKAT + Spitzer + WISE composite image of the Galactic Centre, with UGM structural regions identified against the observed morphology. The remaining programme is constrained inversion of φ (x) within the identified admissible class — not free fitting. This work completes the spectral closure of Route B. The dimensional bridge is now reduced to a single admissible function φ (x), whose determination is delegated to constrained observational inversion.