Cyclic-transition dynamics of the signal's mod-m integer residues — recurrent backbone strength, modular winding bias, and stationary occupancy spread.
Quantizes the input to integers r_n ∈ {0, ..., m−1} (default m=7; uint8 input takes byte mod m, float input takes floor(x·m) mod m after min-max normalization). Builds the m×m transition matrix P[i, j] = P(r_{n+1} = j | r_n = i) and decomposes the flow into a cycle component (mass on edges that lie in any short directed cycle of the top-k sparsified transition graph) and a residual drift component. Motivated by the NEXAH PRIME_MODULAR_RESONANCE 2026 finding that prime residues mod m exhibit a structured transition operator with a non-trivial cycle backbone and a net forward winding bias; the geometry generalizes that lens to arbitrary integer-quantized sequences. Not encoding-invariant — modular reduction depends on byte/integer value, not rank.
Σ_{(i,j) ∈ any short cycle} P[i,j] / Σ P. Recurrent-backbone strength: how much of the transition mass lives on edges that participate in a short directed cycle of the top-k sparsified mod-7 graph. Logistic Period-2, Period-4, Period-3, Period-5, and Circle Map Quasiperiodic all saturate at 1.0 — their orbits are exact cycles in any residue. Minkowski ?(x), Partition Function, Primes, Forest Fire, and Temperature sit at 0 (no recurrent cycles in their residue stream; the trajectory drifts past every state without returning).
|⟨(j − i) mod m⟩_P − (m−1)/2| / ((m−1)/2). Absolute deviation of the mean modular step from the uniform-baseline (m−1)/2. Measures forward/backward winding bias. Constants saturate at 1.0 (only one residue, no step), followed by Primes (1.000), Minkowski ?(x), and Partition Function (0.9998) — all monotone-trending sources have residue streams that march in one direction. Period-5 Window, Bernoulli Shift, ln(2) digits, golden-ratio digits, and XorShift32 sit at ~0.003 (no net winding — the chain mixes symmetrically through residues).
Normalized Shannon entropy H(π) / log₂(m) of the stationary distribution π of P. 1.0 means π is uniform across all m residues (well-mixed); 0 means the chain concentrates on a single residue. Circle Map QP, Phyllotaxis, Triangle Wave, Sawtooth Wave, and De Bruijn Sequence all hit 1.0 (perfectly balanced residue occupancy). Constants (0.0), Rainfall (0.009), Poker Hands (0.07), and Geomagnetic ap Index (0.09) are degenerate or strongly concentrated.
| Source | Domain | Value |
|---|---|---|
| Circle Map Quasiperiodic | chaos | 1.0000 |
| Phyllotaxis | bio | 1.0000 |
| Critical Circle Map | chaos | 0.9867 |
| ··· | ||
| Sawtooth Wave | waveform | 0.0000 |
| Kolakoski Sequence | exotic | 0.0000 |
| Fibonacci Word | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Fibonacci Word | exotic | 1.0000 |
| Thue-Morse | exotic | 1.0000 |
| Rule 110 | exotic | 1.0000 |
| ··· | ||
| Bernoulli Shift | chaos | 0.0029 |
| ln(2) Digits | number_theory | 0.0030 |
| Golden Ratio Digits | number_theory | 0.0032 |
| Source | Domain | Value |
|---|---|---|
| Circle Map Quasiperiodic | chaos | 1.0000 |
| Phyllotaxis | bio | 1.0000 |
| Triangle Wave | waveform | 1.0000 |
| ··· | ||
| Kolakoski Sequence | exotic | 0.0000 |
| Thue-Morse | exotic | 0.0000 |
| Fibonacci Word | exotic | 0.0000 |
Modular Residue carves a dynamical axis the rest of the framework cannot see: which integer residues the signal visits and in what cyclic pattern. cycle_fraction separates strictly cyclic dynamics (every logistic periodic orbit) from drifting-without-recurrence dynamics (Primes, Partition Function, ?(x)) — the same monotone-number-theoretic axis that the Log Spiral renderer surfaces visually. drift_anomaly cleanly identifies monotone-trending integer sequences via their residue winding bias. occupancy_entropy is a stationarity check on the residue chain itself. A residual_fraction candidate (= 1 − cycle_fraction, trivially redundant) and spectral_gap (r = 0.90 with OrdinalPartition:markov_mixing across 120 trials) were both dropped at the intra-geometry redundancy check before integration.