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) = n⁴7 - (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" TA ∝ 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 ≈ 10⁴4 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 kₘax = 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