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. n 2=n), existence is uncertain (σ6=1), and non-existence is forbidden (σ6=0)—uniquely determine the single equation V = −H as an algebraic identity. The mathematical structure generated by this equation corresponds to that of the Standard Model, reproducing more than 20 physical quantities with no adjustable parameters, all agreeing with experiment to 0.0006%–2.6% 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 kinetic term of the canonical normalisation (g = 1, §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—preservation of the well, 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%), αs = 0.1179 (0.01%), sin2 θW = 3/13 (0.2%), the charged-lepton mass ratio R = 1.5293 (0.006%), and sin θC = 0.2245 (0.10%). The results further predict the absolute neutrino masses (Σmν = 59.2 meV) and strictly zero neutrinoless double-beta decay, yielding seven testable predictions. These quantities are not independent input parameters—they are all computed from the same mathematical structure fixed by V = −H.