The TGIU-Prime Equivalence, A Relational Model Linking Informational Collapse and the Distribution of Primes

We propose the TGIU--Prime Equivalence (TPE): the occurrence of prime numbers along the integers is the arithmetic shadow of the same informational--entropic balance that, in the TGIU framework, governs $2\mathrm{D}\!\to\!3\mathrm{D}$ emergence and physical structure selection. Concretely, we map the UIRE collapse functional to a discrete arithmetic potential $\Phi_{\mathrm{arith}}$ whose stationary points, after a stability test, align with primes. This yields: (i) a one-parameter family of prime density laws interpolating between the Riemann--von~Mangoldt behavior and a TGIU collapse-angle modulation; (ii) a spectral test predicting correlations between the argument of the zeta function and a TGIU ``entropy curvature'' proxy; and (iii) falsifiable predictions on local prime irregularities (Maier-type regions) as collapse-resonance (CR) windows. We state the principle, derive consequences, show illustrative fits and correlations, and give a minimal computational protocol. The construction is consistent with classical number theory at leading order and be rejected if the predicted oscillatory components are not observed beyond noise.