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).
| Source | Domain | Value |
|---|---|---|
| LIGO Hanford | astro | 1.0453 |
| Thue-Morse | exotic | 1.0398 |
| Random Steps | exotic | 1.0202 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Quantum Walk | quantum | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Sine Wave | waveform | 0.9947 |
| Exponential Chirp | exotic | 0.9937 |
| Van der Pol Oscillator | exotic | 0.9929 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Quantum Walk | quantum | 1.0000 |
| Square Wave | waveform | 0.9642 |
| Pulse-Width Modulation | waveform | 0.9456 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Ikeda Map | chaos | 0.0475 |
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.