How sharply the signal's distribution changes when you perturb the histogram.
Treats the 16-bin histogram as a point on a statistical manifold — a curved space where each point represents a probability distribution. The Fisher information matrix measures the curvature at that point: high curvature means a small change in the data would radically shift the distribution (informationally sensitive). Low curvature means the distribution is robust to perturbation.
How many independent directions matter in the Fisher matrix? Computed as the participation ratio of eigenvalues. De Bruijn, phyllotaxis, and circle map quasiperiodic all score 16.0 (the maximum — all 16 bins are equally important, so the statistical manifold is fully 16-dimensional). Seismic b-value scores 2.27 (its distribution has only 2-3 effective degrees of freedom, despite occupying many bins). This measures the intrinsic dimensionality of the signal's distributional footprint.
Logarithm of the determinant of the Fisher matrix. This is the log-volume element of the statistical manifold at the data's location. Collatz parity (137.4), Symbolic Henon (137.4), and Fibonacci word (137.3) score highest — their sparse, peaked distributions create enormous Fisher curvature (tiny probabilities in many bins produce large 1/p terms). De Bruijn scores 44.4 (the minimum for a 16-bin uniform — all probabilities equal 1/16). The 93-unit range across the atlas spans 40 orders of magnitude in actual determinant value.
Sum of diagonal Fisher matrix entries (sum of 1/p_i for each bin). Collatz parity (229,605), Symbolic Henon (229,604), and Fibonacci word (229,604) score highest for the same reason as log_det: near-empty bins dominate the trace. De Bruijn scores 256 (16 bins, each with probability 1/16, so 16 * 16 = 256). Trace and log_det are correlated but not identical: log_det captures the product of per-bin information (sensitive to the emptiest bin), while trace captures the sum (sensitive to the total information budget).
| Source | Domain | Value |
|---|---|---|
| De Bruijn Sequence | number_theory | 16.0000 |
| Phyllotaxis | bio | 15.9959 |
| Circle Map Quasiperiodic | chaos | 15.9959 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Seismic b-value (SoCal) | geophysics | 2.2681 |
| Source | Domain | Value |
|---|---|---|
| Collatz Parity | number_theory | 137.3795 |
| Symbolic Henon | exotic | 137.3695 |
| Fibonacci Word | exotic | 137.3158 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| De Bruijn Sequence | number_theory | 44.3614 |
| Source | Domain | Value |
|---|---|---|
| Collatz Parity | number_theory | 229604.5187 |
| Symbolic Henon | exotic | 229604.4734 |
| Fibonacci Word | exotic | 229604.2396 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| De Bruijn Sequence | number_theory | 256.0000 |
Fisher Information geometry detects a specific distributional property: how many bins are nearly empty. Signals that use only a few of 16 bins — binary sequences, periodic orbits, sparse event streams — create Fisher matrices with explosive curvature because the nearly-empty bins contribute 1/p terms near infinity (Laplace smoothing prevents actual infinity but preserves the relative ranking). In the atlas, effective_dimension separates "genuinely multi-valued" signals (dimension near 16) from "effectively binary" signals (dimension 2-4), providing a distributional complexity measure independent of entropy.