Where number theory becomes spacetime. Across mathematics and physics, few ideas have been as tantalizing-and as elusive-as the possibility that the deep structure of the universe is secretly arithmetic. The Primacohedron gives shape to this intuition: an adelic, spectral, and geometric framework in which the prime numbers act not as abstract multiplicative atoms, but as the fundamental oscillators of time, curvature, and information. Drawing from p-adic string theory, p-adic geometry, perfectoids, random-matrix universality, quantum chaos, geometric algebra, and cohomological machinery, this book weaves a unified picture of emergent spacetime built from prime-indexed resonance modes. The resulting framework provides new conceptual bridges between the Riemann zeta function, spectral geometry, holography, black-hole microstructure, cosmology, and the arithmetic landscape of the abc conjecture. Unlike traditional approaches in theoretical physics, The Primacohedron posits that spacetime is not an assumed geometric arena, but a learned, self-organizing, adelically coherent spectral object. Entanglement, curvature, complexity, and thermodynamics arise not as external postulates, but as consequences of arithmetic consistency conditions linking all primes. Along the way, the book develops algorithmic learning flows (Corridor Zero/One), tensor-network interpretations, motivic extensions, and a full cohomological superstructure.