Geometry from connectivity: the remainder of a fixed lattice, curvature from defects, and the scale axis as a puncture

Companion to Spectral Quantum Gravity (main text v455.0.1, doi:10.5281/zenodo.23966562).Induced gravity on a lattice requires that the lattice carry the geometry rather than serve as a fixed background. This note tests that requirement on explicit networks:On a fixed spatial lattice with continuous time the lapse sector is exactly clean, but a traceless shear remainder of +0.54 to +0.63 ε₀ survives where covariance requires −ε₀. It is not of Hořava type and cannot be read as Mukohyama's dark matter.With equal bonds and no metric, a honeycomb closed by twelve pentagons reproduces the conical heat-trace constant (0.366 against 0.3667) and excludes the smooth value 1/3.The 120-cell has the exact 3-sphere multiplicities (l+1)² through l = 5, and its effective radius equals its geometric circumradius.The window operator −∂² + c₀/ω² of the scale axis is the angular channel of a puncture. On the honeycomb a face with n sides threaded by a flux φ gives ν = 6|m − φ|/n to within 2 %. A puncture is never a complete mirror (ν_min ≤ 3/5), so the main text's condition c₀ ≥ ¾ remains an input.Every statement is graded (proved / computed / argued); the predictions were registered before the computations.