The Unification of Matter, Forces, and Spacetime from the Bost–Connes System

Starting from the Bost–Connes system, we perform the p-adic reduction at a prime p and construct the state space H_p = L²(Z/p^K Z), K = p(p−1)−1, and argue that its structural properties naturally describe the physical world. The five axioms of quantum mechanics are derived, one by one, from the algebraic structure of the state space; the geometry of spacetime is determined by the observable space at the ground state, the involution pairing giving dimension 4 and signature (1,3); matter consists of the stable excitations that keep their form under propagation, the eigenmodes of the convolution operator K; the forces are the couplings between these eigenmodes, arising from frequency resonance and from the non-uniformity of the metric. When the characteristic quantities of the p channels are compared with observation—three generations of fermions, five elementary particles, three gauge forces, and the Standard Model gauge group—the unique compatible value is p = 7. The physical world we inhabit is described by the state space over Z/7Z. Fixing p = 7, the computation yields the electroweak scale 246.19 GeV (−0.011%), the Planck mass 1.221×10^19 GeV (+0.04%), the proton mass 937 MeV (−0.17%), the nine charged-fermion masses all within 1.5% (the three charged leptons at experimental precision), the electromagnetic coupling 1/α_EM = 137.036 (+0.0003%), the color coupling 1/α_s = 8.49 (+0.08%), the weak coupling 1/α_W = 29.60 (−0.004%), the nuclear coupling α_nuclear = 14.10 (+0.7%), and the gravitational constant G = 6.70×10^−39 GeV^−2 (−0.09%). Neutrinos are of Majorana type, with absolute masses 5.07, 9.96, 50.6 meV for the three generations and mass-squared-difference ratio 34.5 (experiment 33.8±0.9), a decisive prediction. Version 3 (this version): the appendix on the derivation and uniqueness of the evaluation rules is restructured along the four links of operation, record, readout and composition, and its opening now states that every structure of the evaluation background is unique; the former two external inputs are derived in place as corollaries of the modular structure and the phase-transfer criterion in a new section on evaluation points and evaluation levels; the completeness and uniqueness section is merged into a single three-layer statement; the non-degeneracy conditions and the per-cycle emission of the closed self-loop are given their structural provenance. 146 pages.