An Arithmetic Theory of Everything: Spectral Geometry of the Prime Fock Space and the Emergence of Physical Reality

We propose a conceptual framework in which physical reality emerges from the spectral properties of an arithmetic structure built on prime numbers. The central object of this “Arithmetic Theory of Everything” (AToE) is a bosonic Fock space over the Hilbert space of primes, together with a self-adjoint Dirac operator whose spectrum is assumed to be in one-to-one correspondence with the non-trivial zeros of the Riemann zeta function. The spectral action of this Dirac operator, combined with a non-abelian coupling phase between prime modes and an information-theoretic cutoff scale, defines a master action that replaces the usual continuum-based description of spacetime and fields. Physical constants such as the speed of light, the fine-structure constant, and the Higgs mass are interpreted as emergent resonance parameters of this arithmetic system. We further introduce the notion of a “biophilic corridor” in parameter space, defined via an abstract coherence functional on states of the prime Fock space, and argue that complex, life-supporting structures occur only within this narrow spectral regime. The paper is conceptual and structural in nature: it formulates a coherent mathematical-physical narrative and identifies precise open problems, rather than providing completed proofs or numerical predictions.