A NEW ARCHITECTURE FOR PRIME NUMBER THEORY: FROM EXACT CALCULATION TO PRIME COUNTING VIA SPECTRAL CALIBRATION OF THE ZETA FUNCTION - Fonseca , Murillo (Journal of Interdisciplinary Debates, 2026)
This research presents a paradigm shift from the stochastic tradition within Analytic Number Theory by establishing a deterministic framework for the derivation of prime numbers and their associated prime-counting functions. The methodology is centered on the Spectral Connection Function, an operator that employs the non-trivial zeros of the Riemann Zeta Function as a harmonic basis for the interpolation of modular cardinality sequences over a continuous domain. Through the Sine Equation of Cardinalities, we demonstrate the mapping of this spectral interpolation onto discrete, exact prime values. Local instability is resolved via Spectral Measures, involving dynamic sliding-window calibration, and the minimization of Spectral Energy through Golden Ratio heuristics. Numerical validation yields a vanishing error in both prime determination and cardinality counting across controlled intervals.