Whether marked positions on a 2D integer-spiral lattice prefer quadratic-polynomial diagonals — the geometric signature Ulam (1963) first observed for primes.
Threshold-marks the signal at the median (mark_density = 0.5), then lays sample indices on a square spiral integer lattice centered at the origin. Asks whether the marks cluster on diagonal lines, fit incremental quadratic curves better than linear ones, or concentrate angular/radial energy in directions a uniform mask would not. Six metrics probe complementary projections of the marked grid: a pairwise direction comparison, an algebraic curve fit, an arm-density coefficient-of-variation, a coarse FFT wedge ratio, a per-shell density spectrum, and a per-angle Radon spectrum. uint8-and-rank-invariant — only the mark mask matters, not the raw values.
Normalized excess of diagonal vs. cardinal mark adjacencies: (#diag − #card) / (#diag + #card + 1). 0 is isotropic, > 0 diagonal-leaning, < 0 cardinal-leaning. Logistic Edge-of-Chaos, Period-2, and Period-4 saturate near +1.0 (every other index gets marked → perfect diagonal stripe), followed by Prime Indicator (+0.998) and von Mangoldt Function (+0.997). Rainfall (−0.49), ECG Fusion (−0.29), and LIGO Hanford (−0.28) sit at the cardinal end. The is-prime indicator reproduces Ulam 1963.
Incremental quadratic R² (R²_quad − R²_linear) on the densest single diagonal of the marked grid. Up to 1.0 for a clean quadratic ridge; near 0 for line-linear or noise. Lotka-Volterra (0.18), Wave Height Buoy (0.17), ETH/BTC Ratio (0.15), GOES X-Ray Flux (0.11), and Ambient Microseism (0.10) score highest — their marked indices on the dominant diagonal fit a quadratic better than a line. Devil's Staircase (0.001), Primes, Partition Function, and Minkowski ?(x) (~0.002) sit at the floor.
Coefficient-of-variation² of per-arm mark density along the square spiral's expanding legs. Devil's Staircase (4.35) dominates because its long flat plateaus produce mark-dense and mark-empty arms in alternation; Minkowski ?(x), Partition Function, and Primes all sit at 2.4 (monotone-trending sources concentrate marks on later arms). Constants and Period-2 score near 0 (uniform per-arm density).
Spectral concentration (max bin / total non-DC power) of the per-shell mark-density sequence, where shell s = positions at Chebyshev distance s from the origin. Ambient Microseism (0.89), Quantum Walk (0.75), Wave Height Buoy (0.68), Hilbert Walk (0.65), and Minkowski ?(x) (0.64) top the list — they re-visit similar value ranges at scales that match the shell length, producing a dominant FFT bin in the shell-density curve. Surfaced by the 2026-05-11 embedding-discovery pilot as visual structure the original four Ulam metrics missed.
Variance of per-angle spectral concentration over four Radon-like projections {0°, 45°, 90°, 135°} of the marked grid, with each projection normalized by its own line length. High when one orientation has a dominant periodic bin (oriented stripes — Chirikov-like diagonals); near zero for rotationally-symmetric patterns. Van der Pol (0.045), Chua's Circuit (0.040), Duffing (0.037), LIGO Hanford (0.037), and Square Wave (0.036) score highest. Period-2, Period-4, Noisy Period-2, DNA SARS-CoV-2, and Collatz Parity sit at ~0.0003. Surfaced by 2026-05-11 residual-PCA mining as PC2 of vision-feature space after the existing five metrics were regressed out — picks up oriented periodicity that a coarse wedge-pooling FFT measure misses.
| Source | Domain | Value |
|---|---|---|
| Devil's Staircase | exotic | 4.3451 |
| BTC Close Price | financial | 1.7896 |
| Rainfall (ORD Hourly) | climate | 1.5509 |
| ··· | ||
| Constant 0x00 | noise | 0.0000 |
| Fibonacci Word | exotic | 0.0074 |
| Logistic r=3.2 (Period-2) | chaos | 0.0096 |
| Source | Domain | Value |
|---|---|---|
| Logistic Edge-of-Chaos | chaos | 0.9999 |
| Logistic r=3.5 (Period-4) | chaos | 0.9999 |
| Logistic r=3.2 (Period-2) | chaos | 0.9999 |
| ··· | ||
| Rainfall (ORD Hourly) | climate | -0.4945 |
| ECG Fusion | medical | -0.2889 |
| LIGO Hanford | astro | -0.2791 |
| Source | Domain | Value |
|---|---|---|
| Lotka-Volterra | bio | 0.1837 |
| GOES X-Ray Flux | astro | 0.1767 |
| ETH/BTC Ratio | financial | 0.1518 |
| ··· | ||
| Devil's Staircase | exotic | 0.0008 |
| Quantum Walk | quantum | 0.0094 |
| Tohoku Aftershock Intervals | geophysics | 0.0208 |
| Source | Domain | Value |
|---|---|---|
| Ambient Microseism | geophysics | 0.8887 |
| Quantum Walk | quantum | 0.7498 |
| GOES X-Ray Flux | astro | 0.6850 |
| ··· | ||
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0313 |
| Logistic r=3.2 (Period-2) | chaos | 0.0313 |
| Source | Domain | Value |
|---|---|---|
| Van der Pol Oscillator | exotic | 0.0454 |
| Chua's Circuit | exotic | 0.0402 |
| Duffing Oscillator | chaos | 0.0374 |
| ··· | ||
| Collatz Parity | number_theory | 0.0002 |
| DNA SARS-CoV-2 | bio | 0.0003 |
| Noisy Period-2 | chaos | 0.0003 |
Ulam Spiral (Square) is the framework's only geometry that detects quadratic-polynomial alignment of integer-indexed events on a 2D lattice. The is-prime indicator scores diagonal_alignment +0.998 — the canonical Ulam phenomenon. polynomial_concentration extends the question from "do marks line up?" to "do they fit a parabola?". The three projection metrics (arm_density_variance, radial_periodicity, radon_anisotropy) capture different orientations of regularity that the line/diagonal metrics miss: radial banding from quasi-periodic-amplitude signals (Ambient Microseism, Wave Height), oriented stripes from continuous-time chaos (Van der Pol, Chua), and arm-scale concentration from monotone trends. A sister metric fft_anisotropy (diagonal-vs-cardinal 2D FFT wedge ratio) is class-only — kept in compute_metrics for direct callers but dropped from the atlas because r = +0.90 with Spectral Analysis:high_freq_ratio makes it a near-duplicate spectral-roughness measure; the geometrically distinct signals (diagonal_alignment for prime diagonals, radon/radial for arm structure) carry the real Ulam-lattice information. The Sacks-parametrization variant of the same geometry is excluded from the atlas — see Ulam Spiral (Sacks) below.