Arithmetic Switch Dynamics III: Integrated Gravity, Border Microcount, Perforation Geometry, Spectral Renormalization, and Matter Resonances

Wepresent the third installment of the Unified Theory of Arithmetic Structures (TUSA),a two-chart dynamical paradigm in which evolution alternates between two mutually exclusive temporal regimes—Real Time (ϕ > 0, ω = 0) and Complex Time (ϕ = 0, ω > 0)—separated by a switch border Σ where ϕ = ω = 0. Using prime-gap observables as operationalproxies, we formulate a falsifiable program with four components. (i) A switch-entropy microcount based on integer partitions: Ωswitch(x) = p(N(x)) with N(x) = απ2(x)2 (twin-primecount) and α = 3/(32π2) fixed so that the leading term reproduces the Bekenstein–Hawkingcoefficient η = 1/4, yielding a logarithmic correction of the expected sign and pilot numerical convergence up to x ≤ 107. (ii) A sign-safe real-chart geometry defined by the couplingCk = gk/logpk and curvature proxy Kk = |∆logCk|, with perforations gk > (1 + ε)logpkacting as switch triggers. (iii) A loop-frequency observable f(T)loop (switches per accumulatedReal Time) and a preregistered curvature-controlled protocol whose pilot test on primes upto 2×106 yields a strong negative trend (permutation p ≈ 0.003). (iv) A finite toy spectralmodel for a switch Hamiltonian Hswitch evaluated via invariant spacing diagnostics; current Poisson-like statistics identify explicit upgrade targets. We also include a constrainedborder-resonance matter ansatz consistent with Koide accuracy for charged leptons.