An emergent hypercharge photon on a diamond stack of honeycomb layers: kinematics, origin of the link variables, and a no-go for scale hierarchies

This companion to the Spectral Quantum Gravity (SQG) framework asks how a compact U(1) gauge field with a single light cone can arise on a substrate of honeycomb layers, and what such a field must look like to carry Standard-Model hypercharge.Kinematics: stacking the layers directly requires four relations between the lattice couplings, and all four follow from one discrete metric. Stacking them with tetrahedral bonds gives the diamond lattice. There the curl operator is exactly a scalar times the transverse projector, C†C = (16 − |γ|²) Π_T, so the lattice photon has no birefringence at any wave vector [proved].Coulomb phase: Monte Carlo simulations of compact U(1) on the explicit diamond cell complex (4³ × 4) show a transition between β = 1.0 and 1.2, with a monopole density below 10⁻² for β ≳ 1.5, consistent with a Coulomb phase.Field range: the lattice Coulomb law gives α = √(U/K′)/4π. With the elementary charge ⅙ of one family, the link field fluctuates by about ±12 quanta per bond. The link variable must therefore be a continuous phase, not a few-level degree of freedom such as a spin-ice bond boson.Origin of the link variables: the phase of the substrate field is a pure gradient, and a rigidly rotating site is a spin connection. A single-site interaction of one family, built from operators invariant under SU(3)×SU(2)×U(1)Y×Z₄^X, has no accidental one-body symmetry beyond U(1){B−L}. In a parton reading with this group as Gauss-law redundancy, the hopping phases of the partons are candidate link variables. The gauge group is an input of this reading, not a result.Induced photon: integrating out the partons at one loop induces a Maxwell term with photon speed ≈ 0.4 t·a. For a parton gap not large compared with the hopping, this term places the compact link field on the deconfined side of the measured threshold [computed at one loop]. The induced hypercharge coupling is larger than the measured value by a factor of order ten, so the coupling at the cutoff remains input.Hierarchy no-go: for the constructions studied, a scale hierarchy of coarse-grained honeycombs carries three-dimensional static fields and a causal light cone, but no compact photon. The monopole threshold grows by about a factor three per level.Appendix: a recomputation of the spin-ice route (ring exchange 12J±³/Jzz², transverse-field coefficient 252Γ⁶/Jzz⁵, checked by exact diagonalisation) shows why a bond variable with few states cannot carry hypercharge.Every statement is graded as proved, computed (on the stated lattices) or argued. Open points are listed explicitly:the ratio of parton gap to hopping;deconfinement beyond one loop;the value of the coupling;the equality of photon, fermion and Goldstone speeds;stability of the Coulomb phase with the full chiral matter at strong light–mirror coupling.