Arithmetic Quantum Waveguides: The Ouroboros Phase Transition in Safe Prime Polynomials

This repository contains the paper, figures, and analysis scripts for our study of arithmetic phase transitions in prime-generating polynomials. We demonstrate that the polynomial Q(n) = n^47 - (n-1)^47 acts as a "transparent waveguide" that preserves prime clustering coherence, exhibiting behavior analogous to Bose-Einstein Condensation. KEY FINDINGS: 1. Phase Transition Framework: We introduce "Arithmetic Temperature" T_A ∝ I(q) where I(q) = d(q-1) - 2 is the interference potential. Polynomials with I(q) < 3 undergo a symmetry-breaking transition from thermal gas to condensed crystal phase. 2. Experimental Confirmation: Direct deep-space probing at n ≈ 10^44 confirms a Structural Boost Factor of 113× relative to uncorrelated Poisson expectations—the first evidence for arithmetic crystallization at cosmological scales. 3. Riemann Zero Correlation: The 15 quadruplet positions correlate with Riemann zeta zero ordinates at r = 0.994 under n^(1/2.74) scaling, identifying them as standing wave nodes. 4. Magic Number Bound: Information-theoretic analysis reveals k_max = 28 as the maximum admissible constellation length—a structural isomorphism with nuclear shell magic numbers. REPOSITORY CONTENTS:- paper/: Full paper (PDF + LaTeX source)- figures/: All figures in PDF and PNG formats- scripts/: Python scripts for enhancement calculation and figure generation RAW DATA: The underlying dataset (18,356,706 verified Q47 primes) is available at https://doi.org/10.5281/zenodo.18305185 RELATED WORK:- Math Paper: https://doi.org/10.5281/zenodo.18305185- Information Entropy Paper: https://doi.org/10.5281/zenodo.18259473