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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.20242473 |
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- <div>We study three structures attached to the modulus 5 in analytic number theory: the Chebyshev bias E(x) = #{p ≤ x : p ≡ ±2 mod 5} − #{p ≤ x : p ≡ ±1 mod 5}; its spectral reconstruction from the non-trivial zeros of the Dirichlet L-function L(s, χ₅); and the golden phase lock of Paper 125 (DOI 10.5281/zenodo.19022277), in which the ratio of two character projections P(χ₂)/P(χ₃) at zeros of ζ(s) is phase-locked to the line at angle arctan(1/φ) in the complex plane.</div> <div> </div> <div>The phase lock, originally numerically observed at 50 zeta zeros to 40-digit precision, admits a textbook derivation from the L-function functional equation for the odd conjugate-pair characters χ₂, χ₃ mod 5. We give that derivation explicitly. The locked angle arctan(1/φ) is forced by the elementary identity sin(2π/5)/sin(π/5) = φ in the Gauss sum τ(χ₂), with the golden ratio entering through the regular pentagon's geometry.</div> <div> </div> <div>Three implications follow. (i) The lock is a property of every point on the critical line Re(s) = ½, not specifically of zeta zeros. (ii) The Hurwitz N = 120 mod-5 projections P(χ_k)(s) used in Paper 125 are equal, up to overall scaling, to Dirichlet L-functions: P(χ_k)(s) = 120^s · L(s, χ̄_k). (iii) Off the critical line the lock breaks; the off-line residual is symmetric in |Re(s) − ½|, an empirical signature of the functional equation made visible.</div> <div> </div> <div>We confirm (i) by direct numerical test at 50 random critical-line points (residuals statistically indistinguishable from zero-based residuals). We confirm (iii) at the first 10 zeros (off-line/on-line ratio ~10⁴⁶, residuals at Re(s) = 0.5 + δ identical to those at Re(s) = 0.5 − δ to 1.5 × 10⁻⁴⁹). Two zero-specific conjectures from earlier formulations — magnitude-distribution distinctness and a +/− sign imbalance at zeros — are dissolved under null comparison: K-S test on log-magnitudes returns p = 0.527 against random critical-line points; the 80/120 sign split at zeros is matched by 85/115 at random points, locating the imbalance as a critical-line property rather than a zero-specific one.</div> <div> </div> <div>The Chebyshev bias and density work (Layers 1, 2, density module) provide independent context: direct count to 10⁸, explicit-formula reconstruction with the partial-summation systematic offset diagnosed against theory, and a three-path density bracket placing δ(5; N, R) in [0.996, 0.999].</div> <div> </div> <div>Paper 125's Theorem 1 (DERIVED badge) is honoured by the Gauss-sum derivation in §5; Paper 125's §5 FRAMEWORK claim that the lock breaks off-line is upgraded to OBSERVED. The structural picture of the phase lock is now complete. What remains open is whether the lock — together with the constraint that character projections must cancel at a zero — forces zeros onto the critical line; that step is unchanged from Paper 125.</div>