How to read this:
The Ulam spiral places integers in a square outward coil, centre = 1.
Primes are denser along the diagonals (on the 101 × 101 grid, 25.1% of the cells on the two main diagonals are prime, against 12.0% elsewhere).
Euler's n² + n + 41 lies on the two main diagonals only when the spiral is started at 41, not at 1 as here.
Spiral B is the same spiral as A, rotated as a picture, not a second lattice.
At 0° it lies exactly on A: every prime is a coincidence (1,252 of 1,252 on the 101 × 101 grid).
Cell positions coincide exactly only at rotations of 90°: 359 of 1,252 primes coincide at 90° and 270°, 292 at 180°, against about 154 if the two were independent.
At every other angle on the slider (steps of 0.5°) the rotated cells only partly overlap the grid cells; only −180°, −90°, 0°, 90° and 180° are exact.
Connection to the series:
The dm³ fold radius r* ≈ 0.776 is not a 7th-order Chebyshev node: 2cos(3π/7) = 0.445, and the nearest node of T₇ is 0.7818.
Whether the prime diagonals are a discrete analogue of a Whitney A₁ caustic, and the link to galactic lensing (→ WP58, WP59), are MODEL statements, not shown here.
Verification note (2026-09-29):
ulam-dual-verify.py ports the page's spiral and sieve and checks the statements above.
It replaced four sentences that the script contradicted: “at 0° the coincidence set is dense along the main diagonals”, “at irrational angles it thins to isolated points” (the slider has no irrational angles), “Euler's polynomial lies on diagonal lines” (not in a spiral centred on 1) and the Chebyshev-node line.
Not checked: the visual “interference bands”, the “contact-geometric fingerprint” wording, and the sources (Ulam, Euler), which are cited from general knowledge.
Also noted: the drawing loops stop at N²−1, so the last integer is not drawn (harmless for primes).
The page stays
unfinished.