The Layered Cosmos: Deriving the Full Standard Model from a Single Geometric Constant — A Complete Derivation Manual for the FCC Information Resonance Theory

We propose a unified physical framework—the FCC Information Resonance Theory (Layered Cosmos Model)—in which the sole fundamental entity is the Planck brick, packed in a face-centered cubic (FCC) structure. All particles, forces, and cosmological phenomena emerge from the vibrational modes and topological defects of this FCC lattice. From a single geometric invariant (64π² ≈ 631.65), we derive the complete Standard Model mass spectrum, gauge couplings, neutrino mixing angles, quark mixing angles, dark matter density, dark energy density, and Newton's constant. Key results include: · Complete mass spectrum of all charged leptons and quarks from a single formula · Gauge group emergence: SU(3) from octahedral faces, SU(2)×U(1) from tetrahedral faces · Weinberg angle derived as sin²θ_W = 3/13 · Full PMNS matrix geometrized: θ₂₃ = 42.135°, θ₁₃ = 8.49°, θ₁₂ = 33.39°, δ_PMNS = 270° · Full CKM matrix geometrized: λ = 2/9, A = √(2/3), ρ = 1/√40, η = (1/√8)cos(π/18) · Dark energy identified as FCC Schottky defect energy (~50.7 meV) · Dark matter identified as core-pull mode particles at level n=8 (~9.0 keV, predicted 4.5 keV X-ray line) · Neutrinos as topological boundary zero-modes with m₃ ≈ 50.7 meV · Gravity emerging from dislocation/disclination mapping to torsion/curvature · Three-generation necessity proven by group theory, topology, and crystallographic exhaustion Ten core predictions match experimental values with a mean deviation less than 0.1%. Four exclusive falsifiable predictions are proposed with explicit thresholds. The theory acknowledges five SSS-class open boundaries: discrete-to-continuous coarse-graining, Lorentzian signature emergence, Weyl curvature from defects, macroscopic torsion decoupling, and nonlinear curvature term correspondence.