Dual Ulam Spiral Overlay

How to read this: The Ulam spiral places integers in a square outward coil, centre = 1. Primes cluster along diagonal lines — Euler's quadratic polynomials n² + n + 41 etc. Overlaying two spirals at angle 0° aligns them exactly: white spots are positions that are prime in both labellings. Rotating Spiral B by a few degrees shifts which diagonals align and produces moiré-like interference bands — the contact-geometric fingerprint of two lattice rotations. At 0° the coincidence set is dense along the main diagonals; at irrational angles it thins to isolated points.

Connection to the series: The dm³ fold radius r* ≈ 0.776 corresponds to a 7th-order Chebyshev node (2cos(3π/7)). The Ulam prime diagonals are degree-2 polynomial sequences; their intersection under lattice rotation is a discrete analogue of a Whitney A₁ caustic — the same fold that appears in galactic lensing (→ WP58, WP59).