Primacohedron: A p-Adic String & Random-Matrix Framework for Emergent Spacetime, and a Proposal towards solving Riemann Hypothesis

Background. Unifying number theory, string amplitudes, and spacetime emergence remains a central challenge in fundamental physics. Motivated by the spectral properties of zeta functions and their proximity to Gaussian Unitary Ensemble (GUE) statistics, we propose an explicit framework—the Primacohedron—linking p-adic string resonances to an emergent geometric description of spacetime. Methods. We extend the non-Archimedeanamplitude formalism for open/closed p-adic strings, develop a spectral correspondence mapping Dedekind/Riemann zeros to eigenvalues of a Hermitian operator H, and introduce a learning framework (Corridor Zero/One) for reconstructing spacetime spectra. Additional sections explore the arithmetic-holographic connection, spectral geometry, and cosmological implications. Results. The expanded model unifies arithmetic quantum chaos, random matrix theory, and holography. Temporal fluctuations arise from open p-adic resonances following GUE statistics, while spatial coherence emerges through closed zeta sectors. A curvature–spectral duality defines emergent geometry, black-hole microstructure yields porous horizons, and algorithmic learning saturates the Bekenstein bound dynamically. Conclusions. ThePrimacohedron thus establishes a spectral route from number theoretic operators to spacetime dynamics, blending p-adic strings, zeta-function operators, random matrices, and holographic complexity into a single coherent synthesis. In addition, Primacohedron also suggests a concrete pathway toward a Hilbert–P´olya-type operator and offers a physically motivated set of sufficient conditions under which Riemann Hypothesis would follow.