Informational Foundations of Physics: A Complete Axiom System

We propose an axiomatic framework for fundamental physics in which information is the sole primitive substance. Space, time, matter, and forces emerge as derived structures from an arithmetic informational substrate. The informational space is structured by eight prime numbers {2,3,5,7,11,13,17,19} as fundamental modes, with phases modulated by the golden ratio φ = (1+√5)/2. All observable physics derives from a single generating function: 𝒵 = Σ_p Σ_k p^(-k) exp(i·2πk·ln(p)·φ), where factorizability F = |𝒵|/ΣW yields correlation density Φ = 1-F. The irreducible structural incoherence in 𝒵 is identified with Planck's constant ℏ = Φ_struct; quantum mechanics emerges from this arithmetic impossibility. The theory rests on 13 axioms organized in three groups (structure, manifestation constraints, dynamics) and proves 26 structural theorems, including: uniqueness of φ as phase modulator, derivation of d=3 spatial dimensions by elimination, Born rule as projective quadratic density, spectral cutoff k_max=2 with configuration (8,2), Standard Model gauge structure U(1)×SU(2)_L×SU(3)_c, Weinberg angle sin²θ_W = 3/13 (0.19% accuracy), tree-level fine-structure constant α_0 = 1/(2π²N_eff), and uniqueness of the action functional. No spacetime, Hilbert space, Lagrangian, or physical constant is presupposed. No free parameter is adjusted to match experiment. This paper serves as the axiomatic reference for all companion papers deriving specific physical observables.