The Ulam Spiral Diagonal Selectivity Theorem: A First-Principles Derivation from the Prime Lattice Coherence Framework

This deposit contains the research paper “The Ulam Spiral Diagonal Selectivity Theorem: A First-Principles Derivation from the Prime Lattice Coherence Framework”, published in June 2026 as part of the CTF Framework Research Series (ctftheory.com). The paper provides an elementary, rigorous explanation for one of the most visually arresting unsolved patterns in number theory: the diagonal prime-clustering observed in the Ulam spiral.

Abstract:
We prove that the diagonal prime-clustering pattern of the Ulam spiral follows as a direct corollary of three existing theorems in the Prime Lattice Coherence Framework (PLCT): the Hard Wall Theorem, the Mod-9 Ratio Theorem, and the Zone Structure of the Z9 lattice. No new axioms are required. The proof proceeds in three parts:

1. Spine Exclusion Lemma: A diagonal is prime-rich if and only if its generating quadratic polynomial never produces a multiple of 3.

2. Zone Polarity Lemma: The leading coefficient of the quadratic modulo 3 determines whether its primes concentrate in the Hard Wall or Temporal zone in the exact 2:1 ratio predicted by the Mod-9 Ratio Theorem.

3. Visual Pattern Corollary: Every bright and dark line in the spiral image is the geometric shadow of the PLCT three-zone structure cast onto a quadratic sampler.

All lemmas and corollaries are verified computationally across millions of values with zero violations. The result transforms an empirical mystery into a structural consequence of the mod-9 prime lattice.

Key Findings:

• The Ulam spiral’s bright diagonals correspond exactly to quadratic polynomials that never produce a multiple of 3 (Spine‑free quadratics).

• Polynomials of the form n2±n+cn2±n+c (leading coefficient $a\equiv 1(\mathrm{mod}3)$) exhibit a Hard Wall : Temporal prime ratio of 1:2.

• Polynomials of the form 2n2±n+c2n2±n+c ($a\equiv 2(\mathrm{mod}3)$) exhibit the reverse ratio of 2:1.

• The visual pattern of alternating bright and dark diagonals is a 2D map of the three-zone mod‑9 structure of the prime lattice.

Methods:

• First-principles derivation using finite differences modulo 3 and complete finite enumeration (27 cases).

• Application of Dirichlet’s theorem on arithmetic progressions for equidistribution within coprime residue classes.

• Computational verification in Python, included in the appendix and available on ctftheory.com.

Files included in this deposit:

• Ulam_Spiral_Diagonal_Selectivity_Theorem.pdf – the full paper (LaTeX source available upon request)

• README.txt – summary of the deposit and linking to online resources

• (Optional) verification_code.py – Python script reproducing all numerical checks in the paper

References and Related Works:

• The full Prime Lattice Coherence Framework master document (Parts I–III) is available at https://ctftheory.com

• The proof relies on earlier PLCT theorems: Hard Wall Theorem and Mod‑9 Ratio Theorem, both rigorously established in the framework.