Which network carries three-dimensional space? Dynamical connectivity in the SQG architecture: an ordered honeycomb, its ordering transition, and one geometry for the gauge fields

The first geometry companion of Spectral Quantum Gravity (SQG) showed which fixed networks carry curvature. This companion lets the connectivity fluctuate and asks which dynamical network carries a three-dimensional space together with the matter and gauge content of SQG. All computations were registered with their predictions before they were run; failed predictions are reported.Three-dimensional dynamical triangulations reproduce the known collapsed and branched-polymer regimes and give no extended three-dimensional phase; the bosonic SQG fields only shift the bare coupling [computed].The SQG architecture, a two-dimensional base times the kinetically activated scale axis, gives a spectral dimension of 2.7–2.8 without tuning [computed].A fluid base does not survive the SQG matter: the fermion zero modes alone give a central charge c ≥ 24, beyond the c = 1 barrier, and the base becomes a branched polymer [computed/argued].An ordered honeycomb with a defect core energy carries the matter: spectral dimension 3.07 of the product, Hausdorff dimension 2.0 of the base. The core energy comes from the band energy of the honeycomb fermions, which penalises every non-hexagonal ring, reinforced by a three-sublattice condensate [computed].Equilibrium: local Monte Carlo forms a defect glass from a disordered start and cannot decide. The exact count of sphere triangulations (Tutte) and the asymptotic count of fullerene isomers fix the entropy difference between the fluid and the ordered sector absolutely; thermodynamic integration along the ergodic fluid branch places a first-order ordering transition at λ_m ≈ 0.37–0.42 for 362 to 1442 vertices (0.30 with the condensate). At full coupling the perfect honeycomb is the equilibrium state; the glass is kinetic, and slow formation reduces it as a power of the formation time [computed].Gauge fields add no second metric: a flux on the honeycomb dome is a twisted fibre over the same geometry; a compact U(1) on honeycomb × scale axis × time has a Coulomb-like regime above β ≈ 1.15; with the Hodge weights of the cell complex its photon has a single light cone equal to the geometric one, to < 1 % deep in the Coulomb regime and within about 10 % near the threshold [computed].Every statement is graded as proved, computed or argued. Open points are listed explicitly: matter fields in the equilibrium analysis, the embedding of the core energy in the substrate, couplings that depend on the scale axis (local three-dimensional gravitational dynamics), the fermion zero-mode count on the base, and the birefringence of the compact photon in the base plane.