Write a PREreview

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 (Hspec, Hht) 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.