Geometric Reconstitution of Arithmetic

This paper introduces a geometric framework in which the central objects of analytic number theory—prime numbers, the nontrivial zeros of the Riemann zeta function, and the explicit formula—arise as structural consequences of a recursive attractor field.

The framework is built on four primitive field variables (recursion depth τ, quantum ratio Q, coherence potential Φ, and spectral coordinate σ) together with a sign-based survivability (admissibility) functional. Within this setting, primes are characterised as minimal defect emissions of recursive closure, zeta zeros correspond—conditional on explicit hypotheses—to spectral equilibria on a distinguished stability spine σ = 1/2, and the classical explicit formula is reinterpreted as a conservation law balancing discrete emission against spectral response.

The Riemann Hypothesis is recast as a geometric stability condition: all admissible spectral fixed points are confined to the spine σ = 1/2, conditional on contractivity and spectral identification hypotheses that are stated explicitly. Apparent randomness in prime gaps is explained via curvature stress and witness-extractability regimes, reframing probabilistic models as resolution-limited approximations. In the infinite-recursion limit, the framework selects the E8 lattice as the terminal attractor under admissibility constraints.

All classical results of analytic number theory are preserved and reinterpreted, not replaced. The paper is foundational and field-defining in scope: it establishes a coherent geometric coordinate system for arithmetic and delineates a clear technical programme for verifying the underlying hypotheses and constructing explicit operator models in subsequent work.