Chain Complex Structure on the 4D Hypercubic Lattice: Categorical Isomorphism between Kihara Cube-Packing and Wilson Lattice Gauge Theory (Paper 8)

This paper clarifies the mathematical positioning of the self-consistent identity α⁻¹ = N(1) + V_4(1)·α = 137 + (π²/2)α (predicting α to 8.7 ppb precision) observed in Paper 7. We establish the categorical isomorphism between Wilson lattice gauge theory and Kihara cube-packing on the chain complex of the 4D Euclidean integer lattice (Theorem 6.2). The isomorphism is given by Schläfli duality (tesseract ↔ 16-cell) and is equivariant under the hypercubic symmetry group B_4. Both formalisms exhibit the same bulk-boundary decomposition via the discrete Stokes theorem. The main implications: (1) Paper 7s inside/outside decomposition 1 = 137α + V_4(1)α² is naturally interpreted as a bulk-boundary normalisation on the chain complex; (2) Wilson theorys analytical methods (strong-coupling expansion, RG, Monte Carlo) become applicable to the Kihara programme; (3) Kihara-side number-theoretic identities (Lagrange-Jacobi four-square theorem) may provide novel computations for Wilson theory; (4) The α identity is positioned as a natural manifestation of the topological structure of the 4D lattice, not mere numerology. Open problems explicitly stated: first-principles derivation of αs numerical value, the privilege of R=3, geometric identification of the residual c_3 ≈ 1.6×10⁻³, generalisation to other gauge groups, projection to physical (3+1)D spacetime, connection with the Higgs mechanism, and numerical verification. Companion paper to Paper 7 (Concept DOI: 10.5281/zenodo.19869266).