Whether your signal has hidden linear structure over GF(2) -- the telltale signature of XorShift generators, LFSRs, and bit-level linear algebra.
The Klein bottle's first homology group contains a Z/2Z torsion factor -- algebraically, that is GF(2), the two-element field. This geometry exploits that correspondence. It runs Berlekamp-Massey on each bit channel to find the shortest linear feedback shift register that reproduces the sequence, forms binary matrices and computes their rank over GF(2) via Gaussian elimination, and maps byte pairs onto the Klein bottle's twisted surface to measure orientation coherence. Together, these three metrics detect whether a signal was generated by XOR-based operations.
Berlekamp-Massey LFSR length, averaged across 8 bit channels, normalized by n/2. True random bits need an LFSR of length ~n/2 (score ~1.0). LIGO Hanford (1.045) and Thue-Morse (1.04) score just above 1.0 -- they are maximally complex, requiring more bits than expected. Random Steps (1.02) is similarly complex. Quantum Walk (0.0) has zero complexity -- its encoding produces trivially reproducible bit patterns. XorShift generators score low (~0.06) because their 32-bit state fully determines the sequence.
How consistently does the trajectory respect local orientation on the Klein bottle's surface? Sine Wave (0.995) is nearly perfectly coherent -- its smooth, monotonic segments maintain consistent orientation. Exponential Chirp (0.994) and Van der Pol (0.993) follow. Logistic Period-4 (0.00003) is nearly incoherent -- its jumps between four values constantly reverse the local orientation, mirroring the non-orientability of the Klein bottle itself.
Normalized rank deficit of 16x16 binary matrices over GF(2). Quantum Walk (1.0) is maximally rank-deficient -- its bit patterns are heavily dependent. Square Wave (0.96) and Pulse-Width Mod (0.95) are nearly so, because their binary-valued signals create matrices with many identical rows. Ikeda Map (0.047) has almost full rank -- its chaotic bit patterns are nearly independent over GF(2).
The same rank-deficit calculation run at multiple matrix sizes m ∈ {8, 11, 12, 13, 16, 17, 23, 32}, taking the maximum across scales. The fixed-16 metric only sees structure whose period divides 16; a source with period 17, 23, or 233 phase-slips and looks full-rank even when deeply structured. The multi-scale max covers non-dyadic periods — m=12 catches period-3 byte streams (24-bit cycle aligns with 12×12=144 bits), and primes 11/13/17/23 cover the remaining alias gaps. Quantum Walk and Constant 0x00 (1.0) top out, followed by Minkowski ?(x) (0.984) and Square Wave (0.976). PRNGs and compressed streams sit near the random-matrix floor (~0.10). Strongly correlated with rank_deficit when the period is dyadic; lights up independently on prime-period and Sturmian sources. Added after the Klein frame-aliasing audit caught dyadic-only blind spots.
Kurtosis of the Walsh-Hadamard transform spectrum. Constants and logistic period-2 score 9.7 (maximum — the WHT concentrates into a few coefficients). Poker Hands (1.38) and Poisson Spacings (1.38) score lowest (flat WHT spectrum — no dominant binary frequency). The Walsh-Hadamard transform is the natural spectral analysis for GF(2)-structured data: high kurtosis means a few binary frequency components dominate, indicating LFSR-like structure. Evolved via ShinkaEvolve.
| Source | Origin | Value |
|---|---|---|
| MFPT Inner Unloaded | machine-vibration | 1.0626 |
| LIGO Hanford | astrophysical | 1.0453 |
| Thue-Morse | symbolic-dynamics | 1.0398 |
| ··· | ||
| OTOC Growth | random-matrix-quantum | 0.0000 |
| Spectral Form Factor | random-matrix-quantum | 0.0039 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0078 |
| Source | Origin | Value |
|---|---|---|
| Solar Flares Daily Peak | astrophysical | 0.9938 |
| Aubry-André Critical | random-matrix-quantum | 0.9919 |
| Mian-Chowla | number-theory | 0.9875 |
| ··· | ||
| Critical Circle Map | iterated-maps | 0.0471 |
| Ikeda Map | iterated-maps | 0.0475 |
| Gzip (level 9) | algorithmic-bytes | 0.0488 |
| Source | Origin | Value |
|---|---|---|
| OTOC Growth | random-matrix-quantum | 1.0000 |
| Solar Flares Daily Peak | astrophysical | 1.0000 |
| Aubry-André Critical | random-matrix-quantum | 0.9996 |
| ··· | ||
| Clifford Attractor | iterated-maps | 0.0950 |
| MT19937 (Mersenne Twister) | algorithmic-bytes | 0.0967 |
| XorShift32 | algorithmic-bytes | 0.1017 |
| Source | Origin | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | iterated-maps | 9.7041 |
| Logistic r=3.5 (Period-4) | iterated-maps | 9.4294 |
| Prime Indicator | number-theory | 9.1420 |
| ··· | ||
| Mian-Chowla | number-theory | 1.3812 |
| Poker Hands | stochastic-process | 1.3816 |
| Liouville Function | number-theory | 1.3823 |
Klein Bottle is the atlas's only geometry targeting GF(2) algebraic structure. The linear_complexity metric is the definitive LFSR detector -- XorShift32 scores 0.06 while true random scores 1.0, a 16x separation with no overlap. The rank_deficit metric uniquely identifies signals with degenerate binary structure (Quantum Walk, Square Wave) that other topological geometries miss entirely. The orientation_coherence metric separates smooth waveforms from jump processes, complementing the algebraic metrics with a genuinely topological perspective.