Physics from Existence I: The Equation

No theory has simultaneously derived all 26 free parameters of the Standard Model. In this paper, we show that three axioms about existence — existence is bivalent (n ∈ {0,1}, i.e. n² = n), existence is uncertain (σ ≠ 1), and non-existence is forbidden (σ ≠ 0) — uniquely determine the single equation V = −H as an algebraic identity, and that the structure of the Standard Model and more than 20 physical quantities are derived from this equation with zero free parameters, all agreeing with experiment to 0.0006%–2.4% accuracy. V = −H is the canonical free energy of a single binary degree of freedom n ∈ {0,1}, determined as the Legendre transform of the Fermi-Dirac distribution. The Schrödinger equation is derived as a transfer matrix from this potential and the unit-mass kinetic term from the e-connection fixed by axiom A3 (§2.4). This equation has exactly three bound states (discrete solutions), corresponding to the three generations of fermions. The spatial dimension d = 3 follows from three conditions — bound-state preservation, spin-statistics, and propagating gravity. The mathematical structure unfolding from this coincidence reproduces the gauge group SU(3)×SU(2)×U(1). More than 20 physical constants are derived from this structure, including 1/α = 137.0368 (0.0006%), α_s = 0.1179 (0.08%), sin²θ_W = 3/13 (0.2%), the charged-lepton mass ratio R = 1.5293 (0.006%), and sinθ_C = 0.2245 (0.05%). The results further predict the absolute neutrino masses (Σm_ν = 59.3 meV), normal mass ordering, and strictly zero neutrinoless double-beta decay, yielding eight testable predictions. These quantities are not independent input parameters — they are all computed from the same mathematical structure fixed by V = −H.