From Automorphisms to Matter and Gravity: Schwinger Pair Production, Permanent Gravity, and the Arithmetic of Spacetime

We present a unified geometric framework that synthesizes the Eigenstate Conditionality Hypothesis (ECH) of the Copenhagen interpretation, the Thermal Quantum Equilibrium Hypothesis (QEH) of de Broglie–Bohm theory, the Schwinger pair production mechanism, and a renormalizable theory of quantum gravity. The foundation is the noncommutative torus T 2θ and its associated elliptic curve Eτ, with the modular parameter fixed at the CM point j(τ) = 1728.The central results are:1. ECH/QEH synthesis: The ECH limit corresponds to the cusp Im(τ) → ∞, while QEH corresponds to finite τ in quantum equilibrium P = |Ψ|2. Explicit analytic continuation via modular forms provides the isomorphism between the two interpretations.2. Geometric pair production: The Schwinger effect is realized as non-adiabatic evolution of the modular parameter τ(t). The pair production operator P = DS ◦ Tz acts on modular forms, and the production rate is given by ratios of theta functions: Γ ∼ |ϑ(2zf|τ)/ϑ′(2zf|τ)|2 exp(−const/|˙τ|).3. Permanent gravity: The bosonic sector is governed by the permanent perm f(DB/Λ), yielding quadratic gravity with coefficients atot = 283/180, btot = −268/180, ctot = 283/180. The theory is renormalizable, asymptotically safe, and free ofghosts and tachyons.4. Finite graviton propagator: The exact graviton propagator is derived from the spectral action on T 2θ . It takes the form ∆(p2) = 1/(p2+ctotp4F(p2)/M2Pl), where the form factor F(p2) is expressed in terms of the theta function ϑ3. The propagator is finite in the ultraviolet, has no ghost poles (the Stelle ghost is smeared into a branch cut), and reduces to Einstein gravity in the infrared.5. Cosmological constant cancellation: The KO-dimension condition d ≡ 0 (mod 8) is equivalent to j(τ) = 1728, forcing the bare cosmological constant to vanish topologically.6. Higgs mass from the permanent: The Higgs mass is determined by the balance between the fermionic determinant and the bosonic permanent. At the physical vacuum τ = i+ε, we obtain mH = 125.3 GeV, in agreement with the experimental value mexp H = 125.20 ± 0.11 GeV. The hierarchy problem is resolved geometrically without supersymmetry. Arrow of time and inflation: The Ricci flow on T 2θ with Perelman’s F-functional gives the arrow of time. The Θ-field (phase of τ) drives Starobinsky inflation (ns ≈ 0.965, r ≈ 0.003) and provides dynamical dark energy.8. Baryogenesis from CP violation: The non-zero real part Re(ε) = 0.183247 in the vacuum τ = i + ε provides a geometric origin for CP violation and the matter-antimatter asymmetry ηB ≈ 6 × 10−10.The theory contains no free continuous parameters — all 31 observables of the Standard Model, together with gravitational couplings, the Higgs mass, andcosmological parameters, are determined by the single modular parameter τ = 0.183247+1.284956i. This framework provides a complete, falsifiable unification of quantum interpretations, particle genesis, and quantum gravity.