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: Weextendthenon-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 Beken stein bound dynamically. Conclusions: The Primacohedron 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.

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.

The extraction of rotational energy from a Kerr black hole admits several celebrated descriptions, including the Penrose process, wave superradiance, and the Blandford-Znajek mechanism. Traditionally treated as distinct, these processes are shown here to be unified through a single criterion: the existence of negative Killing energy flux across the horizon, locally manifested as negative phase velocity (NPV) in the ergosphere. I demonstrate that (i) the Penrose process corresponds to particle trajectories with Eχ < 0, (ii) superradiance corresponds to wave modes with ω < mΩH, and (iii) the Blandford–Znajek mechanism corresponds to electromagnetic field lines with ΩF < ΩH. In all cases, the horizon absorbs negative energy while positive energy escapes to infinity. By expressing these mechanisms through the NPV condition S · k < 0, I present a unified theoretical framework for black hole energy extraction.

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.

The Primacohedron provides a unifying adelic framework in which number-theoretic, spectral, and geometric structures arise from prime-indexed resonance modes. Building on its interpretation of the non-trivial zeros of the Riemann zeta function as the spectrum of a Hilbert–Pólya–type operator, this work extends the construction to Diophantine geometry and the abc conjecture. We show that radicals and height functions naturally correspond to spectral-energy sums of prime resonances, while the abc inequality emerges as a curvature-stability condition on an underlying adelic manifold. Within this spectral–Diophantine duality, violations of RH or abc manifest as curvature singularities of a unified spectral–height geometry. We further introduce an adelic operator pair (H_spec, H_ht) encoding L-function zeros and arithmetic heights simultaneously, propose a curvature-anomaly correspondence linking analytic and Diophantine pathologies, and outline a programme suggesting how a completed Primacohedron—extended to motivic L-functions and Vojta theory—could imply both RH and abc through a single geometric regularity principle. This synthesis positions the Primacohedron as a candidate framework for an arithmetic spacetime whose curvature governs both the analytic behaviour of zeta and L-functions and the Diophantine behaviour of rational points, offering a geometric route toward long-standing conjectures in number theory.