Primacohedron: A p-Adic String and Random-Matrix Framework for Emergent Spacetime

Abstract

Sandi Setiawan*

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-Archimedean amplitude formalism for open/closed p-adic strings, develop a spectral correspondence mapping Dedekind/Riemann zero 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: The Primacohedron thus establishes a spectral route from numbertheoretic operators to spacetime dynamics, blending p-adic strings, zeta-function operators, random matrices, and holographic complexity into a single coherent synthesis.

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