Primacohedron: A p-Adic String & Random-Matrix Framework for Emergent Spacetime, Perfectoids, p-Adic Geometry, and a Proposal Towards Solving Riemann Hypothesis and abc Conjecture

We present the Primacohedron, a unified framework linking p-adic string resonances, zeta-function spectra, and emergent spacetime geometry. By extending non-Archimedea string amplitudes and constructing a spectral correspondence that maps Riemann–Dedekind zeros to a Hermitian operator, the model reproduces GUE-type fluctuations temporally and closed-string coherence spatially. A curvature–spectral duality yields emergent geometry, holographic behaviour, and dynamically Bekenstein-saturating learning. The framework further incorporates Diophantine geometry: radicals and height functions arise as spectral-energy sums of prime resonances, and the abc-inequality emerges as a curvature-stability condition on an adelic manifold. An adelic operator pair encodes analytic zeros and heights simultaneously, suggesting a geometric route toward Riemann Hypothesis (RH) and abc via curvature regularity. Finally, we extend the structure using perfectoid geometry and p-adic Hodge theory. Perfectoid tilting links mixed- and equal-characteristic layers through a curvature-preserving duality, while Hodge filtrations provide a cohomological interpretation of spectral dimensionality and arithmetic time. Together, these developments position the Primacohedron as a geometric, cohomological, and operator theoretic paradigm for understanding analytic and Diophantine phenomena within a single adelic spacetime.