Audit note. The 4 4 primitive-cell Bloch construction of §3. 2 and its dispersion-integral result (equation (10), I = 0. 172277, and the resulting dressed coupling ^-1 = 137. 0359995 at 7 significant figures, displayed in the table at §3. 4) are withdrawn as canonical framework predictions. The 4 4 Bloch Hamiltonian breaks the bipartite chiral symmetry via sublattice aliasing on the main diagonal; the chirality-respecting 8 8 construction reveals a second zero-energy Dirac cone at the M-point with logarithmically divergent integrand. The subtraction term 8/| k|² in equation (11) regularises only the -point divergence, leaving M-point unregulated; the 4 4 model's artificial gapping at M (D₄ ₄ (, ) = 2) acted as an unintended UV cutoff, making the integral finite. The 7-significant-figure agreement is therefore coincidental, not structural. The strengths of the four fundamental forces span nearly forty orders of mag- nitude, yet the Standard Model cannot derive these coupling constants from first principles. The Combinatorial Hierarchy (CH) of Parker-Rhodes, Noyes and Bastin (1960s–70s) generated the correct numerical scales via the recursive sequence 3 → 7 → 127 → 2127 −1, but was dismissed as numerology for lack of a geometric substrate. We show that the CH is the strictly mandated computational capacity of an 8-bit error-correcting code on the 4. 8. 8 Archimedean lattice established in Parts I–IV of this work. We prove that the trivalent vertex geometry of the 4. 8. 8 tiling uniquely seeds the hierarchy at Level 1, and that the 4. 8. 8 is the only Archimedean tiling satisfying all necessary constraints. From the lattice we derive: (i) the gravitational coupling αG = 1/2127 ≈5. 877× 10−39, in 99. 5% agreement with experiment, with zero free parameters; (ii) the bare electromagnetic coupling 1/α = 137 from recursive topological additivity; (iii) the dressed fine-structure constant via a Brillouin-zone dispersion integral normalised by the bridge-corrected fermion cell area, yielding 1 = 137. 035 999 5 α against the experimental 137. 035 999 084—agreement to seven significant figures with no free parameters; and (iv) the weak coupling at the lattice scale αW = 1/28 = 1/256 from anti-phase error-correction transmission through the square bridge plaquette. The coupling inverses form the sequence 20, 21, 28, 137, 2127—the Hierarchy Problem reduces to counting bits in an 8-bit code. The strong coupling is evaluated via non-perturbative heat-kernel step-scaling, confirming asymptotic freedom in the ultraviolet and topological confinement in the infrared, with the peak dispersion coupling αpeak = 0. 1168 matching the experimental αs (MZ) = 0. 1179 to 0. 9% 2026-06-20 legacy canon revision: This is a canon-reconciled legacy version. Force-coupling constants and dressed alpha status changed The paper retains its historical derivation trail but carries a 2026-06-20 canon revision note identifying current status and superseded claims. 2026-06-21 canon refresh: This version incorporates the 2026-06-21 ANCHOR/DRIFT/PTMS canon refresh and rebuilt local PDF.
David Graham Elliman (Sun,) studied this question.