Structure Atlas — Exotic Geometry Framework

211 data sources from 16 domains, analyzed through 200 metrics across 54 exotic geometries.

Astro (8 sources)

• GOES X-Ray Flux — GOES satellite 1-min X-ray flux --- solar flare monitoring, log-scale A/B/C/M/X classes (NOAA) (structural neighbors: Temperature, Seismic Noise (ANMO), Barometric Pressure (Buoy))
• Kepler Exoplanet — Kepler confirmed exoplanet light curve (structural neighbors: BTC Volume, BTC Range)
• Kepler Non-planet — Kepler non-planet light curve (structural neighbors: VLF Radio (Baseline), Pink Noise)
• LIGO Hanford — GW150914 Hanford strain (structural neighbors: Langevin Double-Well, Stochastic Resonance)
• LIGO Livingston — GW150914 Livingston strain (structural neighbors: Kuramoto Oscillators, Weierstrass)
• Solar Wind IMF — OMNI hourly interplanetary magnetic field magnitude (structural neighbors: Wind Speed, Wave Height (Buoy))
• Solar Wind Speed — OMNI hourly solar wind bulk velocity (structural neighbors: Ocean Wind (Buoy), Langevin Double-Well)
• Sunspot Number — SIDC daily international sunspot number (structural neighbors: Wind Speed)

Bearing (4 sources)

• Bearing Ball — CWRU ball fault vibration (structural neighbors: EEG Healthy)
• Bearing Inner — CWRU inner race fault vibration (structural neighbors: EEG Seizure)
• Bearing Normal — CWRU normal bearing vibration (structural neighbors: ARMA(2,1), Ornstein-Uhlenbeck, EEG Eyes Open)
• Bearing Outer — CWRU outer race fault vibration (structural neighbors: EEG Eyes Closed, EEG Eyes Open)

Binary (19 sources)

• AES Encrypted — AES-256-CTR ciphertext of structured data --- the gold standard for pseudorandom streams, indistinguishable from random without the key (structural neighbors: White Noise, Pi Digits)
• Bzip2 (level 1) — Bzip2-compressed data (1.2 MB, level 1, 100k blocks) --- Burrows-Wheeler transform with small block size preserves more local structure from the original data (structural neighbors: BSL Residues)
• Bzip2 (level 9) — Bzip2-compressed data (1.1 MB, level 9, 900k blocks) --- BWT with maximum block size, approaching the entropy floor of the source data
• Classical MIDI — Real MIDI files --- Bach (chorale, Brandenburg 3), Beethoven (5th), Mozart (Figaro). Raw Standard MIDI format 1 bytes: delta-time VLQs, note-on/off events, multi-track. 365KB. (structural neighbors: Collatz Flights, Chirikov Standard Map)
• Gzip (level 1) — Gzip-compressed data (1.4 MB, level 1) --- fast DEFLATE with more residual structure
• Gzip (level 9) — Gzip-compressed data (1.3 MB, level 9) --- maximum DEFLATE compression, near-entropy stream (structural neighbors: Pi Digits)
• Linux ELF x86-64 — Stripped Linux x86-64 PIE executable (3.7 MB) --- ELF with .text, .rodata, relocation tables, and x86-64 instruction byte patterns
• MINSTD (Park-Miller) — The 'minimum standard' LCG from Park & Miller (1988 CACM) --- a=16807, m=2³¹-1. Widely adopted as a baseline, but exhibits lattice structure in dimensions ≥6 (structural neighbors: Pi Digits)
• Middle-Square (von Neumann) — Von Neumann's 1949 middle-square PRNG --- famously degenerates to short cycles. Square the state, extract middle digits. Exhibits visible lattice structure. (structural neighbors: ECG Fusion)
• Neural Net (Dense) — Synthetic neural network weights --- Gaussian-initialized dense layer, resembling the near-random parameters of an untrained network (structural neighbors: Blue Noise, Gaussian Noise, Entanglement Entropy)
• Neural Net (Pruned 90%) — Pruned neural network weights (90% zero) --- extreme sparsity typical of compressed models. Zeros encode as byte 0; non-zero weights scaled to [1, 255] preserving sign via abs (structural neighbors: Rainfall (ORD Hourly), Poker Hands, Sensor Event Stream)
• OpenBSD ELF x86-64 — Stripped OpenBSD x86-64 PIE executable (1.1 MB) --- ELF with W^X enforcement, subtly different instruction mix from Linux due to compiler and ABI differences
• RANDU — IBM's infamous RANDU (1968) --- the worst widely-deployed RNG in history. Points fall on just 15 parallel planes in 3D. Multiplier 65539 = 2¹⁶+3 (structural neighbors: BSL Residues, White Noise)
• Wichmann-Hill — Wichmann & Hill (1982) --- three combined short-period LCGs. Python 2's random() engine before Mersenne Twister. Periods ~6.95×10¹² (structural neighbors: BSL Residues)
• Windows PE x86-64 — Windows PE32+ console executable (2.3 MB) --- x86-64 with PE headers, import tables, and different code generation patterns from Unix compilers
• XorShift32 — Marsaglia's XorShift (2003) --- three shift-xor operations per step. Fast and passes most tests, but fails binary rank and linear complexity (structural neighbors: White Noise)
• glibc LCG — The C standard library LCG --- used by glibc rand() for decades. Low bits cycle with short period, high bits have lattice structure in dimensions >5 (structural neighbors: BSL Residues)
• macOS Mach-O (dyld) — macOS dyld universal binary (2.6 MB) --- i386 + x86-64 + arm64e __text sections, pure machine code (structural neighbors: Beta Noise)
• x86-64 Machine Code — Pure .text section from Linux x86-64 ELF --- REX-prefixed instruction stream, 6.3-bit entropy, strong opcode bigram structure, function prologue/epilogue periodicity (structural neighbors: Beta Noise)

Bio (14 sources)

• Codon Usage — Synthetic coding DNA --- nucleotide triplets (codons) with Zipf-like usage bias and weak Markov transitions, emitted as raw ASCII bases (A=65, C=67, G=71, T=84)
• DNA Centromere — Human alpha satellite (AC017075) — tandem 171bp repeats, centromeric
• DNA Chimp — Chimpanzee chr1 genomic DNA (Clint_PTRv2) — 500kb contiguous, GC ~41%
• DNA Dog — Dog chr1 genomic DNA (ROS_Cfam_1.0) — 500kb contiguous, GC ~41%
• DNA Human — Human chr1 genomic DNA (GRCh38) — 500kb contiguous block, GC ~41%
• DNA Phage Lambda — Bacteriophage lambda genome (NC_001416) — balanced GC 50%, 48kb
• DNA Plasmodium — P. falciparum chr1 (NC_004325) — extreme AT bias, GC 21%, 640kb
• DNA SARS-CoV-2 — SARS-CoV-2 complete genome (NC_045512) — RNA virus, 30kb, GC 38%
• DNA Thermus — Thermus thermophilus (NC_006461) — extreme GC 69%, thermophile
• Hodgkin-Huxley — Biophysical neuron --- action potentials with Na⁺/K⁺ channel kinetics, refractory periods (structural neighbors: BTC Volatility, Pomeau-Manneville)
• Human Proteome — Real human proteome --- 20,431 reviewed Swiss-Prot sequences (UniProt 2025), raw ASCII bytes (20 amino acid letters in the 65-89 range). 11.4M residues.
• Lotka-Volterra — Predator-prey oscillations --- nonlinear limit cycles with phase-shifted population waves (structural neighbors: Pomeau-Manneville, BTC Volatility)
• Phyllotaxis — Golden-angle sunflower spiral --- successive seeds placed at 137.508° (= 360°/phi²), the most irrational angle. Quantized angular positions carry genuine quasicrystalline spectral structure with peaks at golden-ratio frequencies (structural neighbors: Circle Map Quasiperiodic, Standard Map K=0.5 (Mixed), Critical Circle Map)
• SIR Epidemic — Stochastic SIR model --- infection waves with exponential rise, peak overshoot, and power-law decay (structural neighbors: Potomac River Flow, Barometric Pressure (Buoy), Pressure)

Chaos (28 sources)

• Arnold Cat Map — Hyperbolic toral automorphism [[2,1],[1,1]] on the 2-torus --- uniformly hyperbolic, Anosov diffeomorphism, mixing with Lyapunov exponent ln((3+√5)/2) (structural neighbors: RANDU, glibc LCG, Wichmann-Hill)
• Baker Map — Baker's map --- stretching and folding like kneading dough, the canonical model of chaotic mixing. Re-seeded every 40 iterations to avoid float collapse
• Bernoulli Shift — Simplest exactly solvable chaotic map: x(n+1) = 2x mod 1. Maximal entropy h=log2, uniform invariant measure, Bernoulli process on binary digits
• Chirikov Standard Map — Chirikov standard map at K=0.9716 --- mixed phase space with coexisting regular islands and chaotic sea, the paradigm of Hamiltonian chaos
• Circle Map Quasiperiodic — Sine circle map at K=0 with golden-mean rotation --- pure quasiperiodic motion on a torus, deterministic but never periodic, flat spectrum with dense peaks at golden-mean harmonics (structural neighbors: Phyllotaxis)
• Coupled Map Lattice — Spatiotemporal chaos in coupled logistic maps --- chimera states and coherence-incoherence patterns (structural neighbors: Blue Noise, Entanglement Entropy, Neural Net (Dense))
• Critical Circle Map — Sine circle map at K=1 critical coupling with golden-mean bare frequency --- sits on the boundary of mode-locking, a different universality class from Feigenbaum's period-doubling (structural neighbors: Phyllotaxis)
• Duffing Oscillator — Driven damped nonlinear oscillator --- strange attractor with broad amplitude visits (structural neighbors: Chua's Circuit, Sine Wave)
• Henon Map — Classic 2D chaotic map --- the strange attractor has fractal cross-sections and correlation dimension ~1.25, a benchmark for testing dimension estimators
• Henon Near-Crisis (a=1.2) — Henon map at a=1.2, b=0.3 --- near the boundary crisis where the attractor collides with its basin boundary, producing intermittent bursts of transient chaos
• Hénon-Heiles — Hamiltonian chaos in a 2D potential: V = ½(x²+y²) + x²y - y³/3. Conservative (no attractor), energy-surface confinement. At E=1/8 mixed regular/chaotic phase space (structural neighbors: Chua's Circuit)
• Ikeda Map — Laser cavity chaos: x(n+1) = 1 + u(x cos t - y sin t), y(n+1) = u(x sin t + y cos t) where t = 0.4 - 6/(1+x²+y²). Multistable at u=0.9, different attractor topology from Hénon (structural neighbors: RANDU, Gzip (level 9), Pi Digits)
• Logistic Chaos — The simplest chaotic system --- one-line recurrence x(n+1) = 4x(1-x) at r=4, filling the unit interval ergodically with a beta(½,½) distribution
• Logistic Edge-of-Chaos — Logistic map at the Feigenbaum point r=3.5699 --- the boundary between order and chaos, where the period-doubling cascade accumulates
• Logistic r=3.2 (Period-2) — Logistic map in stable period-2 --- output alternates between two fixed values, the simplest oscillation after the first bifurcation
• Logistic r=3.5 (Period-4) — Logistic map in period-4 --- four-value cycle after the second bifurcation, output concentrates on four narrow bands
• Logistic r=3.68 (Banded Chaos) — Logistic map in banded chaos --- chaotic within disjoint bands that the orbit cycles through, mixing local unpredictability with global periodicity
• Logistic r=3.74 (Period-5 Window) — Logistic map inside the period-5 stability window at r=3.74 --- an island of order embedded in the chaotic regime, intermittent near its boundary
• Logistic r=3.83 (Period-3 Window) — Logistic map in the period-3 window --- a stable island inside the chaotic sea. By Sharkovskii's theorem, period-3 implies all periods exist nearby
• Logistic r=3.9 (Near-Full Chaos) — Logistic map at r=3.9 --- nearly full chaos but with thin gaps in the attractor revealing remnants of periodic windows
• Lorenz Attractor — Lorenz '63 system x-coordinate --- the butterfly attractor, sensitive dependence on initial conditions. Two-lobe structure with unpredictable switching (structural neighbors: Double Pendulum, Berry Random Wave)
• Noisy Period-2 — Period-2 oscillation corrupted by additive noise --- deterministic alternation partially washed out by Gaussian perturbation, bridging periodic and stochastic regimes (structural neighbors: Euler Totient Ratio)
• Pomeau-Manneville — Pomeau-Manneville type-I intermittency --- near a tangent bifurcation, the orbit lingers in long laminar phases before erupting into chaotic bursts. Power-law laminar length distribution (structural neighbors: ECG Fusion, Lotka-Volterra, Hodgkin-Huxley)
• Rossler Attractor — Rossler system x-coordinate --- the simplest 3D continuous chaotic flow, a single folded band with period-doubling route to chaos (structural neighbors: Projectile with Drag)
• Rössler Hyperchaos — 4D Rössler hyperchaotic system with two positive Lyapunov exponents --- qualitatively different from standard chaos, simultaneous expansion in two directions (structural neighbors: EEG Seizure, Humidity)
• Sprott-B — Sprott case B --- simplest known dissipative chaotic flow with quadratic nonlinearity: dx/dt=yz, dy/dt=x-y, dz/dt=1-xy. Strange attractor with Kaplan-Yorke dimension ≈ 2.01 (structural neighbors: Kicked Rotor, Kuramoto Oscillators, Spike Train)
• Standard Map K=0.5 (Mixed) — Chirikov standard map at K=0.5 --- mixed phase space where large KAM islands coexist with thin chaotic layers, the transitional regime between integrable and fully chaotic (structural neighbors: Phyllotaxis, Fibonacci Quasicrystal)
• Tent Map — Piecewise-linear chaos --- a V-shaped map near slope 2, producing uniform invariant density with maximal topological entropy

Climate (8 sources)

• Barometric Pressure (Buoy) — NOAA buoy 46042 atmospheric pressure --- real synoptic weather patterns, 10-min intervals (CC0) (structural neighbors: Deep Earthquake P-wave)
• Humidity — Jena relative humidity (structural neighbors: Solar Wind Speed)
• Ocean Wind (Buoy) — NOAA buoy 46042 wind speed --- real 10-min observations off Monterey Bay 2023, capturing marine boundary layer turbulence and diurnal sea breeze cycles (CC0) (structural neighbors: Langevin Double-Well, Solar Wind Speed, Stochastic Resonance)
• Pressure — Jena atmospheric pressure (structural neighbors: Seismic Noise (ANMO))
• Rainfall (ORD Hourly) — Real hourly precipitation at Chicago O'Hare 2020--2023 (IEM/ASOS). 35k observations, ~85% zero (dry hours), heavy-tailed wet spells. Public domain. (structural neighbors: Neural Net (Pruned 90%), Forest Fire, Poker Hands)
• Surface Wind (ORD 5-min) — Real 5-minute surface wind speed at Chicago O'Hare 2023 (IEM/ASOS). 114k observations, quantized to ~1-knot steps (31 levels). Captures gusts, diurnal cycles, and frontal passages. Public domain. (structural neighbors: Solar Wind IMF, BTC Range)
• Temperature — Jena temperature time series (structural neighbors: Seismic Noise (ANMO))
• Wind Speed — Jena wind speed (structural neighbors: Solar Wind IMF, Wave Height (Buoy), Multiplicative Cascade)

Exotic (29 sources)

• Chua's Circuit — Double-scroll electronic chaos --- bimodal attractor with different topology from Lorenz (structural neighbors: Duffing Oscillator, Hénon-Heiles, Sine Wave)
• Devil's Staircase — Cantor function --- continuous and monotone but with derivative zero almost everywhere. A singular measure concentrated on a fractal set of measure zero (structural neighbors: Regime Switching, BTC Close Price)
• Dice Rolls — Simulated dice rolls --- uniform over just 6 levels (0, 51, 102, 153, 204, 255), creating a maximally discrete distribution with IID independence (structural neighbors: Beta Noise, RANDU, Gzip (level 1))
• Exponential Chirp — Exponential frequency sweep --- instantaneous frequency grows geometrically, creating equal time per octave. Used in acoustic measurements and radar (structural neighbors: Projectile with Drag)
• Fibonacci Word — Sturmian sequence --- the simplest aperiodic word, with golden-ratio quasiperiodicity and complexity function p(n) = n + 1 (minimal for non-periodic sequences) (structural neighbors: Collatz Parity)
• Forest Fire — Drossel-Schwabl forest fire model --- tree density fluctuates as growth and burns compete (structural neighbors: Rainfall (ORD Hourly), Pomeau-Manneville, Lotka-Volterra)
• Hawkes Process — Self-exciting point process --- events trigger more events, creating clustered bursts (structural neighbors: Solar Wind Speed, Solar Wind IMF)
• Hilbert Walk — 1D trace of Hilbert space-filling curve --- preserves 2D locality, creating structured self-similar fluctuations at every scale (structural neighbors: Rudin-Shapiro, Mertens Function, fBm (Persistent))
• Intermittent Silence — Signal with long silent stretches punctuated by active bursts --- on/off intermittency (structural neighbors: Wind Speed, Geomagnetic ap Index)
• Kuramoto Oscillators — Order parameter of coupled oscillators near synchronization transition --- partial coherence (structural neighbors: Kilauea Tremor, LIGO Livingston)
• L-System (Dragon Curve) — Dragon curve L-system (F→F+G, G→F-G) --- the turn sequence is the regular paper-folding sequence, a non-periodic deterministic binary sequence with fractal spectral measure (structural neighbors: Morse Code)
• Langton's Ant — Multi-state turmite on small torus --- complex emergent trajectories from simple rules (structural neighbors: fBm (Antipersistent), Kilauea Tremor)
• Levy Flight — Random walk with Cauchy-distributed jumps --- infinite variance, occasional extreme leaps create fractal clustering of visited sites (structural neighbors: fBm (Persistent), Regime Switching, BTC Close Price)
• Multiplicative Cascade — Random multiplicative process on dyadic tree --- multifractal burstiness at all scales (structural neighbors: Geomagnetic ap Index, Wind Speed, Solar Wind IMF)
• Poker Hands — Stream of poker hand ranks (High Card through Royal Flush) --- extremely skewed: ~50% High Card, ~42% Pair, <0.002% Straight Flush or better (structural neighbors: Collatz Gap Lengths, Neural Net (Pruned 90%))
• Random Steps — Random piecewise-constant signal --- flat plateaus of random height with geometrically-distributed durations, creating a sparse derivative and blocky texture (structural neighbors: ECG Ventricular, Seismic Noise (ANMO))
• Random Telegraph — Two-state Markov process with exponential holding times --- binary correlation structure (structural neighbors: Wind Speed, BTC Range, Solar Wind IMF)
• Rule 110 — Wolfram Rule 110 center column --- proven Turing-complete, the simplest known universal computer. Supports gliders and localized structures like a 1D Game of Life (structural neighbors: Collatz Parity)
• Rule 30 — Wolfram Rule 30 center column --- produces output indistinguishable from random despite simple local rules. Used as Mathematica's default random generator
• Sandpile — Bak-Tang-Wiesenfeld sandpile --- self-organized criticality produces power-law avalanche sizes without parameter tuning. Slow buildup with sudden cascading collapses (structural neighbors: Continued Fractions, macOS Mach-O (dyld))
• Sensor Event Stream — Categorical sensor event stream --- door open/close, motion, temperature alerts with state-dependent transitions. Mostly quiet (value 0) with coupled event bursts (structural neighbors: Neural Net (Pruned 90%), OpenBSD ELF x86-64)
• Spike Train — Integrate-and-fire neuron model --- quiescent charging with sudden discharge spikes (structural neighbors: Kicked Rotor)
• Stochastic Resonance — Weak periodic signal amplified by noise in a bistable potential --- intermittent well-hopping (structural neighbors: Langevin Double-Well, fBm (Antipersistent))
• Symbolic Henon — Henon map reduced to a binary symbol stream (x < 0 or x > 0) --- the symbolic itinerary of a 2D strange attractor, encoding its topological horseshoe structure
• Symbolic Lorenz — Lorenz attractor reduced to a binary symbol stream (left lobe vs right lobe), one symbol per half-orbit. The symbolic itinerary encodes the topological structure of chaos (structural neighbors: Morse Code)
• Thue-Morse — Binary substitution sequence (0→01, 1→10) --- perfectly balanced yet aperiodic, with flat Fourier spectrum like random noise despite being entirely deterministic (structural neighbors: Morse Code)
• Van der Pol Oscillator — Van der Pol relaxation oscillations --- slow drift along cubic nullcline punctuated by fast jumps, the canonical model of nonlinear self-sustained oscillation (structural neighbors: μ-law Sine)
• Weierstrass — Weierstrass function --- continuous everywhere but differentiable nowhere, the first known mathematical 'monster'. Self-similar roughness at all scales (structural neighbors: Langevin Double-Well)
• Zipf Distribution — IID draws from Zipf/power-law distribution --- steep rank-frequency, heavy right tail (structural neighbors: Poisson Spacings, Benford's Law, GOE Spacings)

Financial (9 sources)

• BTC Close Price — Bitcoin hourly closing prices --- contiguous 16KB block sampled from 96K bars (2015-2026) (structural neighbors: Levy Flight, Regime Switching, Geometric Brownian Motion)
• BTC Range — Bitcoin hourly high-low range --- intrabar volatility proxy with characteristic clustering (structural neighbors: Pink Noise, VLF Radio (Baseline))
• BTC Returns — Bitcoin hourly log-returns --- heavy tails (kurtosis ~18), volatility clustering, and leverage effects. The canonical non-Gaussian financial time series (structural neighbors: Accel Jog, Accel Stairs, Speech "Five")
• BTC Volatility — Bitcoin realized volatility --- rolling 24h std of log-returns, exhibits long memory (structural neighbors: EEG Tumor, Solar Wind IMF, Hodgkin-Huxley)
• BTC Volume — Bitcoin hourly trading volume (log-transformed) --- extreme burstiness with power-law spikes, strong autocorrelation, and intraday seasonality (structural neighbors: Pink Noise, VLF Radio (Baseline))
• ETH/BTC Ratio — Ethereum-to-Bitcoin price ratio --- cross-asset relative strength with regime shifts (structural neighbors: Regime Switching, Brownian Walk, Geometric Brownian Motion)
• NASDAQ Returns — NASDAQ Composite log returns (structural neighbors: Speech "Zero")
• NYSE Returns — NYSE Composite log returns (structural neighbors: Speech "Zero")
• Nikkei Returns — Nikkei 225 log returns

Geophysics (18 sources)

• Ambient Microseism — IU.ANMO long-period vertical (1 sps) --- ocean-generated microseism noise at 0.1-0.3 Hz, 6-hour quiet period with seasonal/storm modulation (IRIS CC0) (structural neighbors: Accel Walk, Accel Jog, Accel Stairs)
• Deep Earthquake P-wave — 2013 Okhotsk M8.3 (depth 598 km) P-wave at ANMO --- sharp impulsive onset, minimal coda, 20 sps broadband vertical (IRIS CC0) (structural neighbors: Barometric Pressure (Buoy), Pressure, Double Pendulum)
• Earthquake Depths — Real USGS earthquake depths --- bimodal: shallow crustal (<30km) and deep subduction (>100km) (structural neighbors: Multiplicative Cascade)
• Earthquake Intervals — Real USGS interevent times --- Omori-law clustering after mainshocks, Poisson background (CORGIS/USGS) (structural neighbors: Entanglement Entropy, Gaussian Noise)
• Earthquake Magnitudes — Real USGS earthquake magnitudes --- Gutenberg-Richter distribution, 8395 events (CORGIS/USGS) (structural neighbors: GOE Spacings, Zipf Distribution, GUE Spacings)
• Earthquake P-wave — P-wave arrivals from 80 earthquakes (M3.0-5.4, 111-498 km) at IU.ANMO --- 40 sps broadband vertical --- impulsive, anti-persistent (low Hurst) (structural neighbors: Accel Sit, EEG Tumor, Speech "Five")
• El Centro 1940 — 1940 Imperial Valley NS strong motion --- the most-studied earthquake record in structural engineering (structural neighbors: EEG Seizure, EEG Resting, Bearing Ball)
• Geomagnetic ap Index — Real 3-hourly geomagnetic ap index 1932-present --- solar-driven magnetic storms, 93 years (CC BY 4.0, GFZ) (structural neighbors: Multiplicative Cascade, Solar Wind IMF)
• Kilauea Tremor — HV.DEVL volcanic tremor during 2018 Kilauea eruption --- harmonic tremor at 100 sps, quasi-periodic with complex amplitude modulation, narrowband (IRIS CC0) (structural neighbors: Kuramoto Oscillators, Lorenz Attractor)
• Potomac River Flow — USGS gauge 01646500 --- real 15-min discharge, Potomac near DC, 2024 (public domain) (structural neighbors: Regime Switching, ETH/BTC Ratio, fBm (Persistent))
• Seismic Noise (ANMO) — Verified-quiet ambient seismic noise at IU.ANMO (no earthquakes) --- 40 sps broadband vertical --- persistent microseismic hum (high Hurst) (structural neighbors: Temperature, Pressure, Double Pendulum)
• Seismic b-value (SoCal) — Sliding-window Gutenberg-Richter b-value over ~20k SoCal M≥1.5 earthquakes 2015-2023 --- low dynamic range (b ≈ 0.5-1.5), strongly autocorrelated, stress-sensitive (USGS FDSN) (structural neighbors: EEG Healthy, fBm (Antipersistent))
• Seismograph (ANMO) — IRIS IU.ANMO broadband vertical --- real 40Hz seismograph, Albuquerque NM, 2024-01-01 (CC0) (structural neighbors: Double Pendulum, Temperature)
• Tidal Gauge (SF) — NOAA CO-OPS 6-min water level, San Francisco (structural neighbors: Mackey-Glass)
• Tohoku Aftershock Intervals — Log inter-event times of ~35k Tohoku 2011 M≥2 aftershocks --- Omori-law decay with temporal clustering, 3 months post-mainshock (ISC/JMA catalog, public domain) (structural neighbors: Pink Noise)
• VLF Radio (Baseline) — VLF radio RMS envelope from normal day (2024-04-10) --- same sensor ET0001 Cleveland OH, 20 kHz, no eclipse, standard D-layer daytime absorption conditions (Eclipse Research Group) (structural neighbors: Pink Noise, BTC Range)
• VLF Radio (Eclipse) — VLF radio RMS envelope during 2024-04-08 total solar eclipse --- sensor ET0001 Cleveland OH (path of totality), 20 kHz sample rate, D-layer collapse creates higher predictability (Eclipse Research Group) (structural neighbors: Pink Noise, BTC Range)
• Wave Height (Buoy) — NOAA buoy 46042 significant wave height --- real ocean swell measurements, 10-min intervals (CC0) (structural neighbors: Solar Wind IMF, Wind Speed, Solar Wind Speed)

Medical (11 sources)

• ECG Fusion — MIT-BIH fusion beat
• ECG Normal — MIT-BIH normal heartbeat
• ECG Supraventr. — MIT-BIH supraventricular premature beat
• ECG Ventricular — MIT-BIH ventricular premature beat
• EEG Eyes Closed — Bonn EEG eyes closed
• EEG Eyes Open — Bonn EEG eyes open (healthy)
• EEG Healthy — Bonn EEG healthy area
• EEG Resting — PhysioNet resting EEG, contiguous blocks, 109 subjects (structural neighbors: Accel Walk)
• EEG Seizure — Bonn EEG seizure activity (structural neighbors: Bearing Ball)
• EEG Tumor — Bonn EEG tumor area (structural neighbors: Earthquake P-wave, Accel Sit)
• Mackey-Glass — Delay-differential equation modeling blood cell regulation --- high-dimensional chaos (structural neighbors: Tidal Gauge (SF), Kuramoto Oscillators, Kilauea Tremor)

Motion (8 sources)

• Accel Jog — MotionSense jogging accelerometer (structural neighbors: EEG Resting)
• Accel Sit — MotionSense sitting accelerometer (structural neighbors: Earthquake P-wave, Speech "Five", EEG Tumor)
• Accel Stairs — MotionSense stairs accelerometer (structural neighbors: EEG Resting)
• Accel Walk — MotionSense walking accelerometer (structural neighbors: EEG Resting)
• Damped Pendulum — Nonlinear pendulum with friction --- transient chaos at high amplitude, decay to limit cycle or fixed point (structural neighbors: Sine Wave, Duffing Oscillator, Triangle Wave)
• Double Pendulum — Angular velocity of second arm --- mechanical chaos with mixed regular/chaotic regions (structural neighbors: Lorenz Attractor, Berry Random Wave, Hénon-Heiles)
• Langevin Double-Well — Brownian particle in double-well potential --- noise-driven switching between metastable states (Kramers problem) (structural neighbors: Stochastic Resonance, Ocean Wind (Buoy), fBm (Antipersistent))
• Projectile with Drag — Ballistic arcs with quadratic air resistance --- asymmetric trajectories, terminal velocity approach (structural neighbors: Rossler Attractor, Berry Random Wave, Chua's Circuit)

Noise (15 sources)

• ARMA(2,1) — Autoregressive moving-average process --- linear short-memory dynamics, the workhorse model of classical time-series analysis (structural neighbors: Bearing Normal)
• Beta Noise — IID Beta(0.3, 0.3) noise --- U-shaped distribution concentrating mass at both extremes with a valley in the middle, structurally opposite to Gaussian (structural neighbors: Gzip (level 1), MINSTD (Park-Miller), Gzip (level 9))
• Blue Noise — High-pass filtered noise (PSD ~ f) --- anti-correlated increments that suppress low frequencies, common in error-diffusion dithering and spatial statistics (structural neighbors: Neural Net (Dense), Coupled Map Lattice)
• Brownian Walk — Cumulative sum of Gaussian increments --- the canonical random walk, self-affine with Hurst exponent H=0.5 and PSD ~ 1/f² (structural neighbors: ETH/BTC Ratio)
• Constant 0x00 — All zero bytes --- degenerate constant signal, minimum entropy, zero variance (structural neighbors: Logistic r=3.2 (Period-2), Logistic r=3.5 (Period-4))
• Constant 0xFF — All 0xFF bytes --- degenerate constant signal, minimum entropy, zero variance (structural neighbors: Logistic r=3.2 (Period-2), Logistic r=3.5 (Period-4))
• Gaussian Noise — Gaussian noise scaled to fill uint8 --- bell-curve density with 3σ mapped to [0,255], concentrating samples around the midpoint (structural neighbors: Neural Net (Dense), Entanglement Entropy)
• Geometric Brownian Motion — Geometric Brownian motion --- multiplicative random walk (dS/S = μdt + σdW), the basis of Black-Scholes option pricing. Log-normal distribution with positive skew (structural neighbors: Mertens Function)
• Ornstein-Uhlenbeck — Ornstein-Uhlenbeck process --- mean-reverting diffusion with stationary Gaussian distribution, the continuous-time analog of an AR(1) process. Bounded excursions unlike Brownian motion (structural neighbors: EEG Eyes Closed)
• Perlin Noise — Spectral Perlin-like noise --- coherent smooth fluctuations with 1/f amplitude falloff, widely used for procedural terrain and texture generation
• Pink Noise — 1/f noise (PSD ~ 1/f) --- equal energy per octave, ubiquitous in nature from heartbeat intervals to semiconductor flicker noise. Between white noise and Brownian motion (structural neighbors: VLF Radio (Eclipse), VLF Radio (Baseline))
• Regime Switching — Two-state hidden Markov process --- alternates between calm (σ=0.5) and volatile (σ=3.0) regimes with asymmetric transition rates (structural neighbors: ETH/BTC Ratio)
• White Noise — IID uniform random bytes --- the null model, maximum entropy with zero sequential correlation (structural neighbors: AES Encrypted, Wichmann-Hill, glibc LCG)
• fBm (Antipersistent) — Fractional Brownian motion with H=0.3 --- antipersistent (trend-reversing), rougher than standard Brownian motion (structural neighbors: Stochastic Resonance, Langevin Double-Well)
• fBm (Persistent) — Fractional Brownian motion with H=0.7 --- persistent (trend-following), smoother trajectories with long-range dependence (structural neighbors: Levy Flight, Mertens Function)

Number_Theory (19 sources)

• BSL Residues — BSL numerator rho(v) mod D for random gap vectors --- probes equidistribution of the Collatz cycle equation. If residues are uniform, no algebraic structure prevents cycles; zero-avoidance is then a 'near miss' phenomenon. Uses (p=10, q=19, D=465239), a verified zero-avoidance case (structural neighbors: Wichmann-Hill, glibc LCG, RANDU)
• Benford's Law — Benford-distributed significands --- leading-digit law P(d) = log₁₀(1+1/d), the distribution of first digits in real-world datasets spanning multiple orders of magnitude (structural neighbors: Zipf Distribution, White Noise, AES Encrypted)
• Champernowne — Provably normal number in base 256 --- uniform digit distribution but deterministic concatenation (structural neighbors: Chirikov Standard Map, Standard Map K=0.5 (Mixed), Linux ELF x86-64)
• Collatz Flights — Peak altitude of Collatz orbits --- how high consecutive integers fly before crashing to 1 (structural neighbors: Linux ELF x86-64, GOE Spacings)
• Collatz Gap Lengths — BSL gap vector from Collatz orbits --- number of halvings between consecutive odd steps (v_i in the Böhm-Sontacchi-Lagarias equation). Distribution approximately geometric(1/2) with correlations encoding the multiplicative walk structure of 3n+1 (structural neighbors: Poker Hands, DNA Thermus, Geomagnetic ap Index)
• Collatz Parity — Collatz parity sequence --- binary trace (even/odd) of the 3n+1 orbit, conjectured to behave like a biased coin flip with P(odd) ≈ log₂(3)/(1+log₂(3)). Restarts from consecutive integers on cycle collapse (structural neighbors: Symbolic Henon, Rule 110, Symbolic Lorenz)
• Collatz Stopping Times — Collatz stopping times --- how many steps each integer takes to reach 1. Wildly erratic, with typical values around 3.5·log₂(n) but enormous outliers (structural neighbors: Surface Wind (ORD 5-min), Windows PE x86-64, Kepler Non-planet)
• Collatz Trajectory — Collatz trajectory mod 256, restarting from consecutive integers to avoid 4-2-1 cycle collapse (structural neighbors: Bernoulli Shift, Henon Map, macOS Mach-O (dyld))
• Continued Fractions — CF digits of random quadratic irrationals --- bounded partial quotients with hidden periodicity (structural neighbors: Sandpile, macOS Mach-O (dyld), x86-64 Machine Code)
• De Bruijn Sequence — B(4,4) De Bruijn cycle --- every 4-symbol window over alphabet 0-3 appears exactly once (structural neighbors: Bernoulli Shift, Henon Map)
• Divisor Count — Divisor count d(n) --- erratic arithmetic function averaging ~ln(n), spikes at highly composite numbers, modulated by prime factorization structure (structural neighbors: Poisson Spacings)
• Euler Totient Ratio — Euler totient ratio φ(n)/n --- the fraction of integers below n that are coprime to n, dense in (0,1) with a fractal structure governed by prime factorization (structural neighbors: Noisy Period-2, Henon Near-Crisis (a=1.2), Chirikov Standard Map)
• Fibonacci Quasicrystal — 1D Fibonacci quasicrystal --- aperiodic tiling with long-range order, discrete diffraction (structural neighbors: Standard Map K=0.5 (Mixed), Chirikov Standard Map, Phyllotaxis)
• Gaussian Collatz Orbit — Collatz map over Gaussian integers Z[i] with divisor pi=1+i and multiplier 3. Unlike integer Collatz, non-trivial cycles are proven to exist: three period-16 orbits at D=13. The only atlas source with proven non-trivial dynamical cycles (structural neighbors: macOS Mach-O (dyld), Beta Noise, x86-64 Machine Code)
• Mertens Function — Cumulative sum of the Moebius function --- a random-walk-like sequence whose growth rate is equivalent to the Riemann Hypothesis (O(n^(½+ε)) iff RH) (structural neighbors: Geometric Brownian Motion, Hilbert Walk)
• Pi Digits — Digits of pi in base 256 --- conjectured normal (equidistributed), but entirely deterministic. Passes most statistical randomness tests (structural neighbors: MINSTD (Park-Miller), AES Encrypted, Gzip (level 9))
• Prime Gaps — Gaps between consecutive primes --- locally erratic but statistically governed by the prime number theorem, with conjectured connections to random matrix theory (structural neighbors: Poisson Spacings, GOE Spacings)
• Rudin-Shapiro — Partial sums of Rudin-Shapiro --- deterministic random walk with flat spectral density (structural neighbors: Hilbert Walk, fBm (Persistent))
• Stern-Brocot Walk — Stern diatomic sequence --- every positive rational visited once, fractal self-similar structure (structural neighbors: Accel Walk, Accel Jog)

Quantum (8 sources)

• Berry Random Wave — Berry's random plane wave --- superposition model for chaotic eigenstate statistics, Gaussian amplitude distribution (structural neighbors: Double Pendulum, Projectile with Drag, Seismic Noise (ANMO))
• Entanglement Entropy — Page curve entanglement entropy of random bipartite states --- ⟨S⟩ ≈ log(d_A) - d_A/(2d_B) (structural neighbors: Neural Net (Dense), Gaussian Noise, Blue Noise)
• GOE Spacings — Gaussian Orthogonal Ensemble spacings --- linear repulsion P(s)~s exp(-πs²/4), time-reversal symmetric (structural neighbors: Zipf Distribution)
• GUE Spacings — Gaussian Unitary Ensemble spacings --- level repulsion P(s)~s² exp(-4s²/π), broken time-reversal symmetry (structural neighbors: Zipf Distribution, Earthquake Magnitudes)
• Kicked Rotor — Quantum kicked rotor --- dynamical localization freezes classical chaos, Anderson localization in momentum space (structural neighbors: Kuramoto Oscillators, Sprott-B, Weierstrass)
• Poisson Spacings — Uncorrelated Poisson spacings P(s)=exp(-s) --- integrable quantum systems, no level repulsion (structural neighbors: Zipf Distribution, Prime Gaps)
• Quantum Walk — Hadamard walk on Z --- ballistic spreading (σ~t not √t), interference creates asymmetric peaks (structural neighbors: Pomeau-Manneville, ECG Fusion, BTC Volatility)
• Wigner Semicircle — Eigenvalue density from large random matrices --- converges to Wigner semicircle law ρ(x)=√(4-x²)/(2π) (structural neighbors: Sawtooth Wave, Rossler Attractor, Triangle Wave)

Speech (4 sources)

• English Literature — Real Project Gutenberg text --- Austen, Carroll, Doyle, Melville (2.8MB). Raw ASCII bytes with natural English letter frequencies and word structure. (structural neighbors: Accel Jog, Poisson Spacings, BTC Returns)
• Speech "Five" — FSDD spoken digit 5 (structural neighbors: Earthquake P-wave)
• Speech "Nine" — FSDD spoken digit 9 (structural neighbors: Earthquake P-wave)
• Speech "Zero" — FSDD spoken digit 0 (structural neighbors: Earthquake P-wave)

Waveform (9 sources)

• Clipped Sine — Sine wave hard-clipped at 70% --- flat tops and bottoms create a hybrid of smooth oscillation and constant segments, concentrating the distribution at extremes (structural neighbors: Damped Pendulum, Duffing Oscillator)
• Morse Code — Synthetic Morse code --- dots (1 unit), dashes (3 units), and hierarchical gaps (1/3/7 units). Creates multi-scale on/off structure encoding random letter sequences (structural neighbors: Symbolic Lorenz, Thue-Morse)
• Noisy Sine (SNR 3 dB) — Sine wave buried in equal-power Gaussian noise (SNR=3 dB) --- the perceptual boundary where periodicity is barely detectable, transitional between C1 waveform and C5 noise regimes (structural neighbors: Tohoku Aftershock Intervals, Gaussian Noise, VLF Radio (Eclipse))
• Pulse-Width Modulation — Pulse-width modulation --- binary carrier (0/255) whose duty cycle sweeps smoothly via sine modulation, encoding an analog signal in digital on/off timing (structural neighbors: Rule 110)
• Sawtooth Wave — Sawtooth wave --- linear ramp with discontinuous reset, rich in harmonics (1/n Fourier decay) (structural neighbors: Rossler Attractor, Wigner Semicircle)
• Sine Wave — Pure sinusoid --- perfectly periodic, single-frequency, minimal complexity. The baseline for all oscillatory comparisons (structural neighbors: Damped Pendulum, Duffing Oscillator, Chua's Circuit)
• Square Wave — Square wave --- binary oscillation (0 or 255), maximal harmonic content with 1/n odd-harmonic Fourier series (structural neighbors: Symbolic Lorenz)
• Triangle Wave — Triangle wave --- linear ramps up and down, smooth peaks with 1/n² Fourier decay (only odd harmonics) (structural neighbors: Damped Pendulum, Hénon-Heiles, Rossler Attractor)
• μ-law Sine — Sine wave with non-linear mu-law quantization (G.711). Creates a structural bridge between analogue waves and discrete logic. (structural neighbors: Van der Pol Oscillator)

Geometric Views

• Biological: How does the framework distinguish biological sequence structure from biological process dynamics? (54 geometries)
• Chaotic: How does the framework distinguish types of deterministic chaos? (54 geometries)
• Distributional: What does the value distribution look like? (9 geometries)
• Dynamical: How does the signal evolve in time? (10 geometries)
• Empirical: How does the real world organize itself, ignoring synthetic constructs? (54 geometries)
• Geophysical: How does the framework organize the natural world — seismic, atmospheric, solar, and gravitational signals? (54 geometries)
• High Entropy: Can the framework distinguish sources that all look like noise? (54 geometries)
• Optimized: What structure remains after removing noise and redundancy? (43 geometries)
• Ordinal: What structure survives re-encoding? (8 geometries)
• Quasicrystal: Does the data have aperiodic order? (7 geometries)
• Scale: How does structure change across scales? (4 geometries)
• Surrogate-Contrast: What structure survives shuffling? What is sequence-dependent? (38 geometries)
• Symmetry: What algebraic structure underlies the data? (14 geometries)
• Topological: What is the shape of the data's phase space? (10 geometries)

Exotic Geometries

• 2-adic — Measures distances via 2-adic valuation: two bytes are close if they agree on many trailing bits. Ultrametric violations reveal non-hierarchical structure. Detects: p-adic clustering, divisibility depth, modular structure.
• Ammann-Beenker (Octagonal) — Tests for eightfold quasicrystalline order (the Ammann-Beenker tiling). Measures cardinal vs. diagonal anisotropy in the diffraction pattern. Detects: Eightfold diffraction symmetry, Bragg peak contrast, cardinal-diagonal anisotropy.
• Attractor Reconstruction — Delay-embedding reconstruction of the phase space attractor. Grassberger-Procaccia correlation dimension and maximum Lyapunov exponent distinguish chaos from noise. Detects: Correlation dimension, Lyapunov exponent, attractor filling.
• Boltzmann — Pairwise Ising model on binary windows: coupling strength, geometric frustration, and spectral gap of the interaction matrix. Detects: Interaction strength, spin-glass frustration, criticality.
• Cantor Set — Interprets bits as a base-3 address into the Cantor set. Gap structure and dust fraction measure self-similarity in the ternary digit stream. Detects: Gaps, dust fraction, ternary self-similarity.
• Cayley — Geometric group theory invariants on k-NN graphs: Gromov hyperbolicity, polynomial growth exponent, and spectral gap (Cheeger constant). Detects: Large-scale curvature, intrinsic dimension, graph expansion.
• D4 Triality — Projects 4-byte windows onto the 24 roots of D4 and measures triality invariance — the unique order-3 symmetry of Spin(8). Detects: Triality symmetry, 4-byte structural constraint.
• Decagonal (Al-Ni-Co) — Tests for 10-fold quasicrystalline order via Golden Ratio spacing. Decagonal quasicrystals (Al-Ni-Co) are periodic in one axis, aperiodic in the other two. Detects: Tenfold diffraction symmetry, Golden Mean scaling.
• Dodecagonal (Stampfli) — Tests for 12-fold quasicrystalline order (Stampfli square-triangle tiling, ratio 2+sqrt(3)). Found in Ta-Te and V-Ni-Si alloys. Detects: Twelvefold diffraction symmetry, square-triangle tiling order.
• E8 Lattice — Projects 8-byte windows onto the 240 roots of E8, the densest lattice sphere-packing in 8 dimensions. Root usage diversity and alignment measure algebraic constraint. Detects: Lattice alignment, algebraic constraint.
• Einstein (Hat Monotile) — Correlates the signal with the Hat monotile boundary kernel (Smith et al., 2023). Detects the aperiodic tiling's distinctive chiral hexagonal structure. Detects: Hat motif correlation, tortuosity self-similarity, chirality.
• Fisher Information — Treats windowed histograms as points on a statistical manifold. The Fisher metric measures how sharply the distribution changes — high curvature means the data is informationally rich. Detects: Information gradient, statistical curvature, parameter sensitivity.
• Fractal (Mandelbrot) — Uses byte pairs as starting points z0 for Mandelbrot iteration z -> z^2 + c. Escape times and orbit statistics measure fractal boundary structure. Detects: Escape rate, fractal dimension, boundary complexity.
• G2 Root System — Projects byte pairs onto the 12 roots of G2. Detects hexagonal symmetry in consecutive-byte correlations. Detects: Hexagonal byte-pair symmetry.
• Gottwald-Melbourne — 0-1 test for chaos: K≈1 for chaotic dynamics, K≈0 for regular. No embedding required. Detects: Chaos vs regular dynamics, classification confidence.
• H3 Icosahedral — Projects 3-byte windows onto the 30 roots of H3 (icosidodecahedron vertices). Detects non-crystallographic 5-fold symmetry. Detects: Icosahedral symmetry, 5-fold rotational order.
• H4 600-Cell — Projects 4-byte windows onto the 120 roots of H4 (600-cell vertices). Detects non-crystallographic symmetry in 4D. Detects: 600-cell alignment, non-crystallographic 4D symmetry.
• Heisenberg (Nil) (centered) — Lifts byte pairs to the 3D Heisenberg group, where the z-coordinate accumulates signed area (xy cross-products). Correlated data twists the path; uncorrelated data stays flat. Detects: Correlation twist, phase coupling, area accumulation.
• Higher-Order Statistics — Computes windowed skewness, kurtosis, permutation entropy, and bispectral coherence — the statistical fingerprint beyond mean and variance. Detects: Skewness, kurtosis, tail asymmetry, non-Gaussianity.
• Hyperbolic (Poincaré) — Embeds data in the Poincare disk, where distances grow exponentially near the boundary. Hierarchical or tree-like data clusters near the edge. Detects: Hierarchy depth, branching, boundary clustering.
• H² × ℝ (Thurston) — Thurston geometry: product of the hyperbolic plane and the real line. Combines hierarchical depth (hyperbolic) with vertical drift (Euclidean). Detects: Hyperbolic layering, vertical drift.
• Hölder Regularity — Estimates pointwise Holder exponents via wavelet leaders. The regularity spectrum (multifractal formalism) measures local smoothness variation. Detects: Local roughness, regularity spectrum, singularity strength.
• Inflation (Substitution) — Detects substitution-rule structure via linear subword complexity, zero topological entropy, bounded discrepancy, concentrated return times, and geometric ACF peaks. Lights up on Fibonacci, Thue-Morse, and L-system sequences. Detects: Substitution inflation symmetry, linear complexity, bounded discrepancy.
• Information Theory — Block entropies at multiple depths (1, 2, 4 bytes), entropy rate, compression ratio, and mutual information at lag 1 and 8. Measures sequential redundancy. Detects: Shannon entropy, complexity, redundancy.
• Julia Set — Fixed Julia set parameter c acts as a tunable sensor. Data values become starting points z0; escape time and orbit stability detect dynamical trapping. Detects: Dynamical trapping, basin structure, Julia dimension.
• Klein Bottle — Detects GF(2) linear structure via Klein bottle topology: LFSR complexity, binary rank over GF(2), and non-orientable trajectory coherence. The Klein bottle's Z/2Z torsion is the topological manifestation of GF(2) algebra. Detects: GF(2) linear recurrences, LFSRs, XorShift generators, bit-level linear dependencies.
• Logarithmic Spiral — Maps the time series to a logarithmic spiral in polar coordinates. Growth rate, winding number, and angular uniformity measure multiplicative structure. Detects: Growth rate, spiral tightness, radial regularity.
• Lorentzian — Treats consecutive (time, value) pairs as events in 1+1 Minkowski spacetime. Classifies intervals as timelike, spacelike, or lightlike to detect causal structure. Detects: Causal ordering, lightcone structure, timelike fraction.
• Mostow Rigidity — 3D Poincaré ball embedding measuring geometric rigidity — how determined the hyperbolic metric is by combinatorial structure Detects: Geometric rigidity, volume invariance, Margulis thickness.
• Multi-Scale Wasserstein — Computes Wasserstein distances between histograms at multiple coarsening scales. Scale coherence reveals distributional self-similarity. Detects: Scale coherence, distributional drift, self-similarity.
• Multifractal Spectrum — Estimates the multifractal singularity spectrum f(alpha) via structure functions. Spectrum width measures scaling heterogeneity — monofractal vs. rich multiscale structure. Detects: Spectrum width, dominant singularity, scaling nonlinearity.
• Nonstationarity — Measures how local geometric character changes over time: speed of change, burstiness, regime duration, and trajectory dimensionality in a 5D local descriptor space. Detects: heteroskedasticity, regime switches, geometric non-stationarity.
• Ordinal Partition — Ordinal pattern transition dynamics: conditional entropy, time irreversibility, and statistical complexity. Detects: Transition predictability, time irreversibility, edge-of-chaos complexity.
• Penrose (Quasicrystal) — Projects the 1D signal into a 2D diffraction pattern and tests for fivefold Bragg peaks, the hallmark of Penrose quasicrystalline order. Detects: Fivefold diffraction symmetry, Bragg peak contrast, aperiodic order.
• Persistent Homology — Builds a Vietoris-Rips filtration on delay-embedded points. Persistent features (connected components H0, loops H1) that survive across scales indicate robust topological structure. Detects: Holes, loops, connected components, topological persistence.
• Predictability — Measures conditional entropy at increasing history depths (1, 2, 4, 8 bytes). The decay rate quantifies memory — fast decay means unpredictable, slow decay means structured. Detects: Conditional entropy, memory depth, sample entropy.
• Projective ℙ² — Lifts byte triples to projective 2-space (homogeneous coordinates). Cross-ratio variance measures departure from projective invariance. Detects: Scale invariance, cross-ratio stability, projective curvature.
• Recurrence Quantification — Recurrence quantification analysis (RQA) on a delay-embedded phase portrait. Determinism, laminarity, and trapping time distinguish chaos from noise. Detects: Recurrence rate, determinism, laminarity, trapping time.
• SL(2,ℝ) (Thurston) — Thurston geometry: the universal cover of the unit tangent bundle of H^2. Accumulates 2x2 matrices and classifies the path as elliptic, parabolic, or hyperbolic by trace. Detects: Shear flow, geodesic divergence, rotation-shear coupling.
• Septagonal (Danzer) — Tests for 7-fold quasicrystalline order (Danzer tiling). Sevenfold symmetry is crystallographically forbidden, arising only in aperiodic structures. Detects: Sevenfold diffraction symmetry, Danzer tiling order.
• Sol (Thurston) — One of Thurston's eight 3D geometries. The Sol metric stretches one direction exponentially while contracting the other, detecting anisotropic scaling. Detects: Hyperbolic splitting, exponential divergence.
• Spectral Analysis — FFT power spectrum analysis: spectral slope (1/f^beta exponent), entropy, flatness, centroid, and bandwidth characterize the frequency content. Detects: Dominant frequency, spectral slope, bandwidth, periodicity.
• Spectral Graph — Heat kernel spectral invariants on epsilon-neighborhood graphs: spectral dimension, Weyl exponent, eigenvalue gap ratio. Detects: Diffusion dimensionality, spectral shape, cluster structure.
• Spherical S² — Maps byte triples to the 2-sphere via spherical coordinates. Measures directional concentration, hemisphere balance, and angular spread. Detects: Directional clustering, angular uniformity.
• Symplectic — Constructs a phase portrait (position vs. momentum) from the time series. The symplectic area form measures trajectory stationarity and return structure. Detects: Phase space area, trajectory stationarity, recurrence.
• S² × ℝ (Thurston) — Thurston geometry: product of the 2-sphere and the real line. Detects data with layered spherical structure — directional concentration that drifts over time. Detects: Spherical layering, radial drift.
• Torus T^2 — Maps consecutive byte pairs to a flat torus (quotient R^2/Z^2). Coverage and nearest-neighbor distance reveal periodic or constrained structure. Detects: Periodicity, cyclic coverage, uniformity.
• Tropical — Tropical algebra replaces (add,multiply) with (min,add). The resulting piecewise-linear envelope reveals slope transitions and regime boundaries. Detects: Piecewise-linear regimes, slope diversity, envelope structure.
• Visibility Graph — Converts the time series to a graph: two points are connected if no intermediate value blocks the line of sight. Degree distribution and clustering reveal dynamical class. Detects: Graph degree distribution, clustering, small-worldness.
• Wasserstein — Computes optimal transport (earth mover's) distance between windowed histograms. Self-similarity and concentration measure distributional stability. Detects: Distribution shape, transport cost, self-similarity.
• Zariski — Non-Hausdorff algebraic structure via Zariski topology. Detects polynomial recurrences via Vandermonde SVD and Heyting lattice gaps in pattern complement structure. Detects: Polynomial recurrences, algebraic varieties, non-Boolean pattern lattice.
• Zipf–Mandelbrot (16-bit) — Treats N-bit sequences as symbols and fits Zipf-Mandelbrot frequency decay f(r) ~ (r+q)^-alpha. Vocabulary richness, hapax ratio, and Gini coefficient measure symbolic diversity. Detects: Zipf exponent, vocabulary richness, frequency decay.
• Zipf–Mandelbrot (8-bit) — Treats N-bit sequences as symbols and fits Zipf-Mandelbrot frequency decay f(r) ~ (r+q)^-alpha. Vocabulary richness, hapax ratio, and Gini coefficient measure symbolic diversity. Detects: Zipf exponent, vocabulary richness, frequency decay.
• p-Variation — Computes the p-th variation (sum of |increments|^p) for multiple p. The critical p where variation transitions from finite to infinite characterizes path roughness. Detects: Path roughness, variation index, regularity.