Physics from Existence I: The Equation

No theory derives the 26 free parameters of the Standard Model simultaneously. From one binary degree of freedom and its occupation probability σ, we derive the Schrödinger equation, the basic equation of quantum theory, and a structure corresponding to the Standard Model. The paper posits only three axioms: existence is binary (n ∈ {0,1}, n² = n), existence is uncertain (σ ≠ 1), and non-existence is forbidden (σ ≠ 0). The Legendre transform of the logarithm of the partition function that gives Fermi–Dirac occupation statistics yields V = −H(σ) as an algebraic identity, where H is the binary Shannon entropy and −H is the canonical free energy of one binary degree of freedom. With the canonically normalised kinetic term (g = 1), a Schrödinger equation follows from the continuum limit of the transfer matrix. The depth ln 2 and the width of the well are fixed; it has exactly three bound states (N = 3), corresponding to the three fermion generations. The spatial dimension d = 3 follows from well preservation and gravity propagation. The Lie algebra on the finite geometry built from N = 3 corresponds to SU(3)×SU(2)×U(1). The 19 constants derived here agree with measurement to 0.0006%–2.6%; with the 7 derived in Paper II, the 26 physical constants are reproduced with no adjustable parameters. They include 1/α = 137.0368 (0.0006%), α_s(M_Z) = 0.1179 (0.07%), sin²θ_W = 3/13 = 0.23077 (0.23122 at M_Z with the next correction, <0.01%), the charged-lepton mass ratio R = ln(m_τ/m_e)/ln(m_μ/m_e) = 1.5293 (0.006%), and sinθ_C = 0.2245 (0.10%; 0.22435 after screening). Setting the absolute scale with the electron mass m_e, we predict the absolute neutrino masses without oscillation data (Σm_ν = 59.37 meV). Neutrinoless double beta decay is predicted to vanish exactly. All values are reproducible with the accompanying verification code.