Emergent Lorentz Invariance from a Discrete 4.8.8 Substrate: Chadha–Nielsen Renormalisation-Group Flow of the Velocity Anisotropy

The canonical Holographic Circlette substrate Z3 ⊗Q3 on the 4.8.8 Archimedean tiling carries a bifurcated bare group-velocity spectrum at the lattice scale: a rigorous 8×8 bi- partite Bloch construction gives leading-edge vfast ≈1.00 and vslow ≈0.78 (ratio 1.28), a ∼28% bare Lorentz anisotropy. The Velocity-Unification Conjecture (Holographic Circlette canonical reference 2, ANCHOR §7.4) asserts that this bifurcation flows to a common IR fixed point vfast = vslow ≡c under interaction-mediated drag from the substrate-emergent p·A vertex. We lift the Chadha–Nielsen 1983 leading-log argument 3 from continuum 4D QED to the discrete 4.8.8 substrate, obtaining β∆v =− c4.8.8 α π ∆v+ O(α2), c4.8.8 = 2 3 Rvertex Rphoton, where Rvertex, Rphoton are substrate-specific O(1) corrections from the bipartite Bloch vertex projection and the 4.8.8 photon propagator. (The baseline c4.8.8 = 2/3 recovers the stan- dard Chadha–Nielsen leading-log exponent 2α/(3π).) We prove a structural sign theorem (c4.8.8 > 0 is forced by substrate unitarity via the K¨all´en–Lehmann representation of the photon propagator), securing the qualitative IR-stable fixed point. We then reconcile the slow leading-log damping (∼3% over the chiral-to-visible-light range) with the framework’s (a0k)2 ∼10−17 Standard-Model-Extension cavity-resonator bound: macroscopic Lorentz invariance is secured by two structurally independent infrared suppressions — the 4.8.8 cos(4θ) cancellation theorem (within-branch isotropy, dominant at visible light) and the Chadha–Nielsen RG flow (cross-branch unification, qualitatively essential). Explicit numer- ical evaluation of Rvertex and Rphoton is flagged as the rigorous-closure target, with the Part 12 dressed-α Dyson–Schwinger machinery as the direct methodological precedent.