Octahedral Parity-Symmetric 8-State Quantum Walk and 1 ⊕ 3 Discrete Effective Field Theory

We present a self-contained discrete effective field theory and induced quantum gravity framework based on a unitary quantum walk (Quantum Cellular Automaton, QCA) over the 6-direction generators ({±e_x, ±e_y, ±e_z}) and dual 8-state sign space ({±1}³) of a three-dimensional cubic-octahedral lattice. First, we prove that for an arbitrary number of binary axes D, the simultaneous algebraic conditions of parity-pair decomposition (2^(D-1) = 1 + D), rotational bivector closure (D(D-1)/2 = D), and Hurwitz normed division algebra closure admit the unique natural-number solution D = 3 (2D = 6 directions, 2^D = 8 states, quaternion algebra H), while pure causal graph growth without directional conservation collapses to a Myrheim-Meyer dimension d_MM → 1 via small-world percolation. Second, in the parity-decomposed 1 ⊕ 3 basis, the left-handed quaternionic kinetic operator J_{L,i} = G_i + A_i satisfies (k · J_L)² = |k|² I_4 to all orders, eliminating 99.94% of cube-corner superluminal propagation on a 64³ lattice and yielding both the four vacuum Maxwell equations (in the Riemann-Silberstein basis) and spin-1/2 Weyl spinor duality (V_spin† J_{L,i} V_spin = I_2 ⊗ σ_i). Third, we prove that the orthogonal-complement operators excluded from spacetime translation—the commuting right-handed mirror quaternion J_{R,i} = G_i - A_i ([J_{L,i}, J_{R,j}] = 0) and the symmetric shear block S_i—convert into the internal electroweak-like isospin algebra su(2)_iso ⊕ u(1)_Y and the eightfold Gell-Mann color algebra su(3)_c with quadratic Casimir eigenvalues C_2 = diag(0, 4/3, 4/3, 4/3). Fourth, across the (3 choose k) = 1 : 3 : 3 : 1 Brillouin zone corners, the Wilson second-order lattice Laplacian Δ_W(k) assigns ultraviolet mass gaps in the exact ratio 0 : 2r_W/a : 4r_W/a : 6r_W/a, decoupling the seven high-momentum mirror doublers and isolating a single low-energy fundamental fermion at k = 0. Fifth, we prove that two-particle exchange symmetry on H_8 ⊗ H_8 yields exact Bose-Einstein bunching (2.0× enhancement) and fermionic Pauli exclusion (P(x_1 = x_2) = 0.000%), while the Z_3 cyclic permutation operator yields the general Koide identity Q(r) = (1 + r²)/3 (r = 1 ⟺ Q = 2/3) and SU(3)_c lattice link variables produce Cornell linear quark confinement (V_qq(R) = σR - 4α_s/(3R) + V_0, R² = 0.9994). Sixth, we prove that the bilinear commutators Σ_{ab} = -(i/4)[Γ_a, Γ_b] close the exact spin(4) Lorentz Lie algebra, from which the Microscopic Weak Equivalence Principle (zero polarization birefringence, Δω = 2.66 × 10⁻¹⁵), spin-2 gravitational wave polarizations (h_+, h_×) and quadrupole radiation (h_+ ∝ Ï_+/r), the nonlinear Einstein-Hilbert action ∫ R√(-g) d⁴x (via O(a⁴) plaquette holonomy and Seeley-DeWitt heat kernel expansion), general-relativistic light deflection Δφ = 4GM/(c²b) under the isotropic optical metric n_opt = cos⁻²θ, and Planck-core singularity resolution (θ → π/2) emerge with machine precision (< 10⁻¹⁴). Seventh, across an acoustic-optical lattice event horizon (c_eff → 0), we demonstrate Parikh-Wilczek quantum tunneling with a Hawking-like thermal Boltzmann spectrum proportional to surface gravity (T_eff ≈ κ/π, R² = 1.0000), lossless 1:1 entropy transfer between spatial position and internal electromagnetic/parity degrees of freedom, and full unitary Page curve restoration (S_Page(t) → 0.0025) under dynamic evaporation backreaction.