Complete particle physics from an exactly solvable 21-state vacuum: 27 Standard Model predictions with zero free parameters and quantum hardware validation

<title>Abstract</title> <p> Recent work on emergent gauge symmetries in discrete lattice systems [1–6] demonstrates that continuous gauge groups can arise from finite combinatorial structures. Several recent frameworks—including G ₂ -holonomy compactifications, primorial lattices, and hypergraph rewriting—have attempted to derive Standard Model parameters from discrete postulates; however, all rely on continuous manifolds or infinite lattices with intractable path integrals. We present an exactly solvable alternative. Starting from a Hamiltonian on the 6-dimensional hypercube {0,1} ⁶ , we apply an exact Feshbach–Schur projection to integrate out 43 high-energy states, obtaining a unique 21 × 21 effective infrared operator <bold>A</bold> . The physical transition graph of the resulting 21-state vacuum manifold is isomorphic to the dodecahedron (Aut = A ₅ × ℤ ₂ ). The CP involution partitions the vacuum into 13 gauge channels with block dimensions 8 ⊕ 3 ⊕ 2, matching the adjoint dimensions of SU(3) × SU(2) × U(1). The spectral invariants of <bold>A</bold> then determine, with zero adjustable parameters: α ⁻¹  = 137.033 (22 ppm from PDG), sin ² θ _W  = 0.2312 (exact), v _H = 246.17 GeV (0.02%), the complete meson spectrum (m _η′ at 0.02%), quark and lepton masses, CKM mixing elements, the baryon asymmetry, and Newton’s gravitational constant—27 observables across 13 orders of magnitude with a median error of 0.33%. The discrete path integral is a convergent finite sum verified to machine precision (10 ^− 14 ). Crucially, the same Hamiltonian has been independently implemented on IBM quantum hardware (ibm_fez, 48–156 qubits, 230 + jobs) and validated across four unrelated application domains. Two falsifiable BSM predictions are reported: a topologically stable 180.5 GeV dark matter candidate and a 5.17 TeV boundary state. </p>