We construct a three-dimensional lattice by interlocking three mutually perpendicular families of regular octagons via shared axis-aligned edges. The resulting honeycomb has a primitive unit cell of exactly 6 nodes, uniform vertex degree 5, and regular octahedral voids whose 8 triangular faces naturally host a binary code. The Bloch Hamiltonian at the Γ point reduces to the complete graph K6, with eigenvalues +5 (×1) and−1 (×5). Under the full octahedral point group Oh, these decompose as A1g (scalar, massive) ⊕ T1u (3-component vector, massless) ⊕ Eg (symmetric tensor, massless). The characteristic polynomial along Γ– Xfactors exactly as (λ+1)3λ3 −3λ2 −9λ+3−8 cos kx = 0, yielding an analytical bare group velocity v= 2/3 for the T1u gauge branch and effective mass m∗ = 9/2 for the A1g matter branch. The mass gap ∆ ≥ 2 is maintained across the entire Brillouin zone. The octahedral voids sit on a simple cubic sublattice and are strictly disjoint, connected by single gauge- bridge edges. The two-void scattering geometry involves exactly 8 + 8 = 16 independent faces, preserving the microstate combinatorics of the companion 2D framework 1, 2. The face-adjacency graph of each octahedron is Q3, the 3-dimensional Boolean hypercube—the native topological home for an 8-bit information code. This extension eliminates holographic projection, upgrades the gauge boson from 2 to 3 spatial components, strengthens the mass gap from touching to permanent separation, and predicts sterile right-handed neutrinos as dynamically decoupled dark matter candidates. 2026-06-20 legacy canon revision: This is a canon-reconciled legacy version. Gauge-geometry claim needs comparison to current TCH/SMG status 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retains its historical derivation trail but carries a 2026-06-20 canon revision note identifying current status and superseded claims. 2026-06-21 canon refresh: This version incorporates the 2026-06-21 ANCHOR/DRIFT/PTMS canon refresh and rebuilt local PDF.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 Elliman (Sun,) studied this question.