Crystal Dynamics and the Ultraviolet-Complete Unified Theory

We present a UV-complete unified theory of all known physics derived from a single axiom: a binary oscillation field in four spacetime dimensions. The unique self-dual regular tessellation of R^4 — the 24-cell honeycomb {3,4,3,3} — emerges from this axiom and constitutes spacetime itself (not merely a model of it). Its Wilson-loop correlation matrix Q, with spectrum {+8^3, +4^12, 0^27, -2^24, -4^6}, generates the complete Standard Model Lagrangian plus general relativity through a single spectral action S[Q]. Building on Papers 1–3, Paper 4 closes all deferred derivations and establishes three new layers of structure. Dynamics: the Skyrme coupling β = π²/8 is proved from the 24-cell packing fraction; Newton's constant is fully derived with zero free parameters as G = ℏc/(m_SG² φ^191) where 191 = n_v·z − 1; the graviton is identified as the harmonic zero mode of the crystal graph Laplacian; and background independence is established by showing the D4 root lattice satisfies R_μν = 0. Spectrum: the group (Z/24Z)* ≅ (Z/2)³ classifies all Standard Model particles by three binary quantum numbers; the boson/fermion split is proved three independent ways (Weyl group, algebraic number theory, Legendre symbol); and the complete neutrino sector is derived geometrically from the three non-gauge distance graphs G2, G3, G4 of the 24-cell, with masses m_ν2 = (α/π)·m_e²/(m_W·cosθ_W) = 8.62 meV (−0.65%) and Δm²_31/Δm²_21 = N_c(z + N_c) = 33 both proved from first principles. Moonshine and quantum gravity: the six-step constructive chain 24-cell → D4 → Golay → Leech → Conway → Monster establishes the Monster group M as the symmetry group of the crystal quantum gravity, with Tr(Q³) = j(i) = 1728 (exact), 744 = n_v · M_F5, and the graviton-stimulated-emission gain G = z²/(z² − M3²) = 64/15 identified as a Monster-invariant. In total, 28 physical quantities are derived from the crystal numbers {N_c, n_v, z, L4, F5, M3} and the single physical bridge m_SG = 134.59 MeV with no free parameters. Nine falsifiable predictions are stated, including Δρ = 0 exactly (testable at FCC-ee), λ_H = (1/6)cos²θ_W (HL-LHC), Majorana neutrinos (0νββ experiments), and the neutrino mass ratio = 33 (a group-theoretic integer, unambiguous).