Structure Atlas — Exotic Geometry Framework

297 data sources from 16 domains, analyzed through 318 metrics across 73 exotic geometries.

Astro (11 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: Seismic Noise (ANMO), Temperature, BTC Volatility)
• Kepler Exoplanet — Kepler confirmed transiting-exoplanet light curve. Long-cadence (~30 min) PDCSAP flux with periodic transit dips. (structural neighbors: Gut Motility, BTC Volume)
• Kepler Non-planet — Kepler long-cadence flux for a target without a confirmed planet. Stellar variability + instrumental noise, no transit signal. (structural neighbors: BTC Volume, BTC Range)
• LIGO H1 Whitened (GW150914) — LIGO H1 strain, 4096 Hz, 32s around GW150914 --- ASD-whitened + 30-400 Hz bandpassed. Chirp (~0.2s) of first-detected binary-black-hole merger embedded in stationary detector noise. Distinct from raw 'LIGO Hanford' (dominated by 1-10 Hz seismic). ~12% trial-window hit rate on the chirp (public domain, GWOSC) (structural neighbors: Bearing Outer, EEG Seizure, Bearing Normal)
• LIGO Hanford — LIGO Hanford H1 strain around the GW150914 binary black-hole merger. 4096 Hz, broadband detector noise with the chirp embedded. (structural neighbors: Langevin Double-Well, Stochastic Resonance)
• LIGO Livingston — LIGO Livingston L1 strain around GW150914. 4096 Hz; the same chirp arrives ~7 ms earlier than Hanford, opposite-sign. (structural neighbors: Kuramoto Oscillators, Langevin Double-Well)
• Solar Flares Daily Peak — GOES daily peak X-ray flux 1975-2016 (NGDC NOAA, 15098 days). 22.7% quiescent at the 1e-9 W/m^2 baseline, sparse B/C/M/X-class events to max X28 (Nov 2003). Forest-Fire-style phenotype: metastable baseline with rare extreme excursions. Tests whether the atlas detects 'slow accumulation + sudden discharge' as a universality class spanning exotic+astro domains (structural neighbors: Geomagnetic ap Index, Rainfall (ORD Hourly), Aubry-André Critical)
• Solar Wind IMF — OMNI hourly interplanetary magnetic-field magnitude at L1. Carries the Parker-spiral structure and CME-driven enhancements. (structural neighbors: Wind Speed, Wave Height (Buoy))
• Solar Wind Speed — OMNI hourly solar-wind bulk velocity at L1. Highly intermittent: long slow-wind background punctuated by fast streams. (structural neighbors: Wave Height (Buoy), Ocean Wind (Buoy))
• Sunspot Number — SIDC daily international sunspot number. Strong ~11-year solar cycle on top of intermittent active-region bursts. (structural neighbors: Wind Speed, Ocean Wind (Buoy))
• Voyager 1 X-band — Breakthrough Listen Voyager 1 fine-res spectrum at X-band (8419 MHz, 2.79 Hz/chan, ~1M channels) --- known artificial residual carrier + telemetry sidebands at +/-22.5 kHz, ~3x noise median. DC filterbank artifact masked. Most random trial windows see GBT noise floor; rare windows capture the narrowband signal (public, Breakthrough Listen) (structural neighbors: VLF Radio (Eclipse), VLF Radio (Baseline), Noisy Sine (SNR 3 dB))

Bearing (11 sources)

• Bearing Ball — CWRU bearing vibration, ball fault. Drive-end accelerometer at 12 kHz with a 0.014 in diameter ball-element defect. (structural neighbors: Earthquake P-wave, EEG Seizure)
• Bearing Inner — CWRU bearing vibration, inner-race fault. Drive-end accelerometer at 12 kHz; periodic impulses ride a broadband resonance. (structural neighbors: EEG Seizure)
• Bearing Normal — CWRU bearing vibration, normal baseline. Drive-end accelerometer at 12 kHz, ~1797 RPM, healthy ball bearing. (structural neighbors: Ocean Swell, ARMA(2,1))
• Bearing Outer — CWRU bearing vibration, outer-race fault. Drive-end accelerometer at 12 kHz; cleaner impulse train than inner-race because the defect is stationary in the load zone. (structural neighbors: LIGO H1 Whitened (GW150914))
• IMS Bearing Degraded — NASA IMS test 2, day 6 snapshot, bearing 1. Same rig as healthy but mid-degradation; outer-race spalling has begun, periodic impulse component starts to emerge above the broadband floor. (structural neighbors: ARMA(2,1))
• IMS Bearing Failed — NASA IMS test 2, peak-fault snapshot 2004-02-19 05:02 (file 975 of 984), bearing 1. Outer-race fully failed; RMS ~9x baseline, kurtosis ~17, impulses hitting the ±5 g accelerometer rail. Paired with healthy/degraded sources to expose a 3-point degradation trajectory.
• IMS Bearing Healthy — NASA IMS run-to-failure test 2, day 1 snapshot, bearing 1. 20 kHz accelerometer, 2000 RPM, 6000 lb radial load; baseline before any fault develops. (structural neighbors: ARMA(2,1))
• MFPT Inner Loaded — MFPT bearing vibration, inner-race fault under 300 lb radial load. 48.828 kHz; pairs with MFPT Inner Unloaded to expose load-zone amplitude modulation. (structural neighbors: EEG Resting, NASDAQ Returns)
• MFPT Inner Unloaded — MFPT bearing vibration, inner-race fault under zero radial load. 48.828 kHz; impulse train still present but with reduced load-zone modulation. (structural neighbors: BTC Returns, NASDAQ Returns)
• MFPT Normal — MFPT bearing vibration, healthy baseline. NICE rig, 97.656 kHz accelerometer, ~1500 RPM (25 Hz shaft), 270 lb radial load. (structural neighbors: ARMA(2,1))
• MFPT Outer Race — MFPT bearing vibration, outer-race fault. 97.656 kHz, 270 lb load; matches CWRU outer-race fault topology but at higher RPM/higher sample-rate and on a different bearing model. (structural neighbors: ARMA(2,1), Gaussian Noise)

Binary (22 sources)

• AES Encrypted — AES-256-CTR ciphertext of structured data --- the gold standard for pseudorandom streams, indistinguishable from random without the key (structural neighbors: Euler-Mascheroni γ Digits, Golden Ratio Digits, ln(2) 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: ln(2) Digits)
• 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 (structural neighbors: Catalan G Digits)
• ChaCha20 — Bernstein's ChaCha20 stream cipher (2008) --- cryptographic-strength PRNG, the keystream of TLS 1.3 / QUIC / WireGuard. Detection-ceiling source: the framework should find ChaCha20 keystream geometrically indistinguishable from White Noise --- a stronger null than MT19937 (cryptographic vs statistical PRNG) (structural neighbors: BSL Residues)
• 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: Uniform Chaos (Logistic Scramble), English Literature, Fibonacci Quasicrystal)
• Gzip (level 1) — Gzip-compressed data (1.4 MB, level 1) --- fast DEFLATE with more residual structure (structural neighbors: Catalan G Digits, White Noise)
• Gzip (level 9) — Gzip-compressed data (1.3 MB, level 9) --- maximum DEFLATE compression, near-entropy stream (structural neighbors: Catalan G Digits, White Noise)
• 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
• MT19937 (Mersenne Twister) — The Mersenne Twister (Matsumoto & Nishimura, 1998) --- period 2^19937-1, default PRNG of numpy's legacy RandomState, CPython's random module, and many other languages. Detection-ceiling source: well-mixed enough that the framework should find it indistinguishable from White Noise on size-8192 windows --- a positive null calibrating where the weak-PRNG/strong-PRNG boundary actually sits (structural neighbors: White Noise)
• 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.
• Network Packet Sizes — Simulated network-packet length sequence. Moderate entropy, smooth spectrum, bursty arrivals. (structural neighbors: Shuffled Blocks, Random Telegraph, Kepler Non-planet)
• Neural Net (Dense) — Synthetic trained dense-layer weights --- Xavier init plus 50 momentum-smoothed gradient updates, L1-like shrinkage of small weights, and heavy-tailed outliers. An *untrained* layer is i.i.d. Gaussian (indistinguishable from noise); training imprints local correlation. (structural neighbors: Gaussian Noise, Entanglement Entropy, Poisson Counts)
• 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: Poker Hands, Rainfall (ORD Hourly), 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 (structural neighbors: ECG Beat Conformity)
• 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: White Noise)
• Wichmann-Hill — Wichmann & Hill (1982) --- three combined short-period LCGs. Python 2's random() engine before Mersenne Twister. Periods ~6.95×10¹²
• 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: White Noise)
• macOS Mach-O (dyld) — macOS dyld universal binary (2.6 MB) --- i386 + x86-64 + arm64e __text sections, pure machine code
• 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: Markov Chain (10-state))

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 DNA (AC017075). Tandem 171bp repeats from centromeric heterochromatin; strongly periodic at the repeat unit.
• DNA Chimp — Chimpanzee chr1 genomic DNA (Clint_PTRv2). 500kb contiguous, GC ~41%; near-identical compositional profile to the human contig.
• DNA Dog — Dog chr1 genomic DNA (ROS_Cfam_1.0). 500kb contiguous, GC ~41%; mammalian outgroup to the great-ape genomes.
• DNA Human — Human chr1 genomic DNA (GRCh38). 500kb contiguous block, GC ~41%, 4-symbol alphabet (A/C/G/T) as ASCII bytes.
• DNA Phage Lambda — Bacteriophage lambda complete genome (NC_001416). ~48kb dsDNA virus with balanced GC 50%; canonical genetics-lab reference organism.
• DNA Plasmodium — P. falciparum chr1 (NC_004325). Malaria parasite with extreme AT bias (GC ~21%); 640kb contig.
• DNA SARS-CoV-2 — SARS-CoV-2 complete genome (NC_045512). Compact RNA virus, ~30kb, GC 38%; AT-biased relative to mammalian genomes.
• DNA Thermus — Thermus thermophilus complete genome (NC_006461). Hyperthermophile bacterium with extreme GC content (~69%), opposite bias to Plasmodium.
• Hodgkin-Huxley — Biophysical neuron --- action potentials with Na⁺/K⁺ channel kinetics, refractory periods (structural neighbors: Blood Pressure Waveform, BTC Volatility, GOES X-Ray Flux)
• 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. (structural neighbors: Bzip2 (level 9), Bzip2 (level 1), Gzip (level 1))
• Lotka-Volterra — Predator-prey oscillations --- nonlinear limit cycles with phase-shifted population waves (structural neighbors: Rossler Attractor, Damped Pendulum, Duffing Oscillator)
• 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, Critical Circle Map (Silver Mean), Critical Circle Map)
• SIR Epidemic — Stochastic SIR model --- infection waves with exponential rise, peak overshoot, and power-law decay (structural neighbors: Barometric Pressure (Buoy), Potomac River Flow, Pressure)

Chaos (49 sources)

• 2-Torus Quasiperiodic — 1D observable from dense quasiperiodic orbit on T^2 (frequencies sqrt 2, sqrt 3). Ground-truth intrinsic dimension d=2. Ergodic but not mixing.
• 3-Torus Quasiperiodic — 1D observable from dense quasiperiodic orbit on T^3 (frequencies sqrt 2, sqrt 3, sqrt 5). Ground-truth intrinsic dimension d=3.
• 4-Torus Quasiperiodic — 1D observable from dense quasiperiodic orbit on T^4 (frequencies sqrt 2,3,5,7). Ground-truth intrinsic dimension d=4 --- past typical GP correlation-dim horizon.
• 5-Torus Quasiperiodic — 1D observable from dense quasiperiodic orbit on T^5 (frequencies sqrt 2,3,5,7,11). Ground-truth intrinsic dimension d=5.
• Aizawa Attractor — Aizawa 3D continuous flow with torus-like topology and two polar spikes --- structurally distinct from Lorenz butterfly, Rossler spiral, and Halvorsen cyclic. Multistable. Output: x-coordinate. (structural neighbors: Berry Random Wave)
• 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: glibc LCG, ChaCha20, MINSTD (Park-Miller))
• 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 (structural neighbors: LFSR (16-bit))
• 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)
• Clifford Attractor — Clifford 2D iterated map x_{n+1}=sin(a*y)+c*cos(a*x), y_{n+1}=sin(b*x)+d*cos(b*y) at (a,b,c,d)=(-1.4,1.6,1.0,0.7). Bounded sin/cos-coupled discrete chaos with rich fractal fine structure, distinct from Hénon's quadratic folding. Output: x-coordinate.
• Coupled Map Lattice — Spatiotemporal chaos in coupled logistic maps --- chimera states and coherence-incoherence patterns (structural neighbors: Blue Noise, GUE Spacings, Entanglement Entropy)
• 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
• Critical Circle Map (Bronze Mean) — Sine circle map at K=1 critical coupling tuned to the bronze-mean winding number (sqrt(13)-3)/2 (continued fraction [0;3,3,3,...]) --- a third distinct winding-number universality class for the quasiperiodic route to chaos
• Critical Circle Map (Silver Mean) — Sine circle map at K=1 critical coupling tuned to the silver-mean winding number sqrt(2)-1 (continued fraction [0;2,2,2,...]) --- a renormalization fixed point distinct from the golden mean's, with its own universal critical-scaling constants (structural neighbors: Phyllotaxis)
• Critical Transition (Fold) — Saddle-node (fold) normal form dx/dt = x - x^3 + mu(t) driven by a slowly ramped control parameter across the tipping point at mu_c = 2/(3*sqrt(3)). As the occupied well flattens, the system shows critical slowing down --- rising autocorrelation and variance --- then transitions abruptly to the surviving stable branch (the canonical early-warning-signal model; Scheffer 2009, Dakos 2012). (structural neighbors: BTC Close Price, Stochastic Resetting Walk, Geometric Brownian Motion)
• Duffing Oscillator — Driven damped nonlinear oscillator --- strange attractor with broad amplitude visits (structural neighbors: Sine Wave, Magnetic Pendulum (3-Magnet))
• FPUT N=16 — Fermi-Pasta-Ulam-Tsingou anharmonic-spring chain (β-model), N=16 masses, fixed boundaries. Conservative (Hamiltonian) high-DOF chaos --- direct Hamiltonian analogue of Lorenz-96 N=8 (dissipative) at matched dimensionality. Famously fails to thermalize for low amplitudes (FPUT recurrences); above the energy threshold the chain mixes ergodically. Velocity-Verlet integration. Observable: q_{N/2}(t). (structural neighbors: Berry Random Wave, Magnetic Pendulum (3-Magnet), Triple Pendulum)
• Halvorsen Attractor — Halvorsen x-coordinate --- 3-fold cyclically symmetric chaotic flow: each axis obeys the same equation with cyclic substitution. Topologically a 'cyclic transport' attractor, distinct from Lorenz switching or Rossler spiral.
• 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: Spring Pendulum, Magnetic Pendulum (3-Magnet))
• 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: AES Encrypted, BSL Residues)
• Intermittency Type-II — Type-II intermittency near a subcritical Neimark-Sacker (Hopf) bifurcation. The laminar phase is a slow spiral: a rotation whose amplitude grows smoothly until it leaves the channel, bursts, and is reinjected near zero. Distinct from Pomeau-Manneville type-I (tangent bifurcation) and type-III (period-doubling); laminar length scales as epsilon^-1 (structural neighbors: IMS Bearing Failed, MFPT Inner Unloaded)
• Intermittency Type-III — Type-III intermittency near a subcritical period-doubling (flip) bifurcation. The laminar phase is a growing period-2 subharmonic: the sign alternates every step while the amplitude grows, until it bursts and is reinjected near zero. Lag-1 autocorrelation is strongly negative -- the clean signature distinguishing it from type-I and type-II (structural neighbors: BTC Returns)
• 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 (structural neighbors: Tribonacci Word, Pell Word)
• 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 (structural neighbors: Pell Word)
• 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)
• Lorenz-96 N=36 — Lorenz-96 with N=36 variables at standard forcing F=8 (canonical testbed). Observable is x_0(t). Fully developed spatiotemporal chaos. (structural neighbors: Double Pendulum)
• Lorenz-96 N=4 F=16 — Lorenz-96 atmospheric-dynamics ODE with N=4 variables and high forcing F=16 (required for chaos at small N). Observable is x_0(t). Attractor fractal dim ~2.5-3. (structural neighbors: Double Pendulum)
• Lorenz-96 N=8 — Lorenz-96 with N=8 variables at standard forcing F=8. Observable is x_0(t). Weakly turbulent regime, attractor dim ~3. (structural neighbors: Berry Random Wave, Double Pendulum)
• Newton-Leipnik Attractor — Newton-Leipnik 3D continuous flow --- two-disc strange attractor with coexisting attractors at standard parameters. Distinct multistable topology not present in Lorenz/Rossler/Halvorsen. Output: x-coordinate.
• 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, Aliquot Orbit Lengths)
• 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: Minkowski Question Mark, Hawkes Process, ECG Normal)
• Quartic Map (Feigenbaum) — Quartic map x(n+1) = 1 - r*x^4 at r=1.5949 --- accumulation of the period-doubling cascade for the z=4 family. DIFFERENT Feigenbaum universality class from logistic/sine (quartic-tip, delta=7.2847). Tests whether the atlas distinguishes universality classes: should be near Logistic Edge-of-Chaos and Sine Map (Feigenbaum) but discernibly separate
• Rossler Attractor — Rossler system x-coordinate --- the simplest 3D continuous chaotic flow, a single folded band with period-doubling route to chaos (structural neighbors: Lotka-Volterra)
• Ruelle-Takens Cascade — Quasiperiodic route to chaos (Ruelle-Takens-Newhouse): a ring of coupled Hopf oscillators (complex Ginzburg-Landau) driven Benjamin-Feir unstable, so the multi-frequency quasiperiodic motion breaks down into a strange attractor. Chaos without a period-doubling cascade --- distinct from Lorenz/Rossler and from the intact quasiperiodic tori
• 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, EEG Healthy, EEG Eyes Closed)
• Sine Map (Feigenbaum) — Sine map x(n+1) = r*sin(pi*x) at r=0.8640 --- empirically-determined accumulation of the period-doubling cascade for this functional form (bisected between period-16 at r<=0.8639 and wide chaos at r>=0.8641). Functionally distinct from the logistic map but in the SAME Feigenbaum universality class (quadratic-tip, z=2, delta=4.6692). Tests whether the atlas rediscovers period-doubling universality unsupervised: should cluster tightly with Logistic Edge-of-Chaos despite the different functional form
• 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: EEG Seizure, EEG Eyes Closed, EEG Resting)
• 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
• Tent Map — Piecewise-linear chaos --- a V-shaped map near slope 2, producing uniform invariant density with maximal topological entropy
• Uniform Chaos (Logistic Scramble) — Logistic map at r ≈ 3.9 pushed through its invariant CDF. Chaotic dynamics with a flat histogram; preserves temporal correlations while erasing the arcsine marginal.

Climate (9 sources)

• Barometric Pressure (Buoy) — NOAA buoy 46042 atmospheric pressure --- real synoptic weather patterns off Monterey Bay 2020-2023, 10-min intervals (CC0) (structural neighbors: Potomac River Flow, Regime Switching)
• Humidity — Max Planck Jena station relative humidity. 10-min sampling; strong inverse coupling to temperature on the diurnal cycle. (structural neighbors: Seismic Noise (ANMO))
• Ocean Wind (Buoy) — NOAA buoy 46042 wind speed --- real 10-min observations off Monterey Bay 2020-2023, capturing marine boundary layer turbulence and diurnal sea breeze cycles (CC0) (structural neighbors: Langevin Double-Well, Solar Wind Speed, Stochastic Resonance)
• Pressure — Max Planck Jena station barometric pressure. 10-min sampling; very smooth, dominated by synoptic-scale (multi-day) variation. (structural neighbors: Potomac River Flow)
• 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: Solar Flares Daily Peak, Devil's Staircase, Neural Net (Pruned 90%))
• 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: Wave Height (Buoy))
• Temperature — Max Planck Jena weather station air temperature. 10-min sampling; strong diurnal cycle on top of synoptic weather variability. (structural neighbors: Seismic Noise (ANMO))
• Temperature Drift — Slow room temperature with HVAC cycles. Very smooth, high Hurst, low bandwidth. (structural neighbors: Brownian Walk, Langevin Double-Well, Regime Switching)
• Wind Speed — Max Planck Jena station wind speed. 10-min sampling; intermittent, heavy-tailed (Weibull-like), with diurnal modulation. (structural neighbors: Solar Wind IMF, Wave Height (Buoy), Solar Wind Speed)

Exotic (51 sources)

• Categorical Sensor — Random draws from 6 to 12 non-uniform categories. Peaked, unpredictable, low entropy.
• Chladni Plate Mode — Square plate vibration eigenmode with free edges, Z_{n,m}(x,y) = cos(nπx/L)cos(mπy/L) − cos(mπx/L)cos(nπy/L), evaluated on a √size × √size grid and raster-scanned to 1D. Mode (n,m) drawn per call from {(3,1),(5,2),(4,3),(7,2),(6,5)} --- non-trivial Chladni figures with intersecting nodal lines. The 2D nodal pattern is the canonical plate-vibration positive control: sand falls to nodal lines where Z=0. Raster encoding produces row-periodicity at stride √size + slower across-row modulation, mirroring the row/column lattice that Chladni geometry's nodal_* and plate_low_mode_fraction metrics target. (structural neighbors: Ruelle-Takens Cascade, Tidal Gauge (SF))
• Chua's Circuit — Double-scroll electronic chaos --- bimodal attractor with different topology from Lorenz (structural neighbors: FPUT N=16, Magnetic Pendulum (3-Magnet), Double Pendulum)
• Devil's Staircase — Numerical derivative of the Cantor function C(x), sampled on a uniform grid in a random sub-window of (0,1). C(x) is continuous, monotone, and singular --- derivative zero almost everywhere, with all of its increase concentrated on the ternary Cantor set (a fractal of measure zero). Emitting forward differences C(x_{i+1}) - C(x_i) exposes that Cantor measure: exactly zero on the deleted middle-thirds, sharp positive bursts on the Cantor set. Different pathology from Minkowski's ?'(x) --- ternary-Cantor geometry vs continued-fraction arithmetic. Raw C(x) emission was retired 2026-05-20: the monotone values collapsed to a near-linear ramp under [0,1] normalization (pixel-identical to Primes / Partition / Minkowski-? in rank-based geometries) and a stray '% 1.0' wrap left a spurious 1->0 cliff at the tail. Mirrors the Minkowski ?(x) -> ?'(x) fix. (structural neighbors: Rainfall (ORD Hourly), Neural Net (Pruned 90%))
• 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: Euler-Mascheroni γ Digits, AES Encrypted)
• Exponential Chirp — Exponential frequency sweep --- instantaneous frequency grows geometrically, creating equal time per octave. Used in acoustic measurements and radar (structural neighbors: Magnetic Pendulum (3-Magnet), Triple Pendulum)
• Fibonacci Word — Sturmian sequence --- the simplest aperiodic word, with golden-ratio quasiperiodicity and complexity function p(n) = n + 1 (minimal for non-periodic sequences)
• Forest Fire — Drossel-Schwabl forest fire model --- tree density fluctuates as growth and burns compete (structural neighbors: ECG Fusion, BTC Volatility, Rainfall (ORD Hourly))
• Free Group F₂ Walk — Non-backtracking random walk on the Cayley graph of the free group F₂ = ⟨a, b⟩ --- the 4-regular tree, canonical δ-hyperbolic (δ=0) space. At each step the next generator is drawn uniformly from {a, a⁻¹, b, b⁻¹} excluding the inverse of the previous step, so the walk strictly extends a reduced word. Emits step labels as a 4-level uint8 sequence {0, 85, 170, 255}. Word length grows linearly in step count (tree metric) while ball volume grows like 4·3^{n-1} (exponential) --- the textbook Gromov-hyperbolic positive control for Cayley:delta_hyp_norm, growth_exponent, saturation_radius, spectral_gap. (structural neighbors: Codon Usage, DNA SARS-CoV-2)
• Geometric Waiting Times — Geometric(p) inter-arrival times with p ~ Uniform(0.05, 0.2). Strongly peaked at low values with an exponential tail; memoryless by construction, no temporal correlations. (structural neighbors: Poisson Spacings, Prime Gaps, Earthquake Magnitudes)
• Gray Code Counter — 8-bit Gray-code counter with variable step. Low entropy, high structure, complex bit transitions. (structural neighbors: Triangle Wave, Clipped Sine, De Bruijn Sequence)
• Hawkes Process — Self-exciting point process --- events trigger more events, creating clustered bursts (structural neighbors: Solar Wind IMF, Collatz Trajectory, Wind Speed)
• Heisenberg Walk — Random walk on the discrete Heisenberg group H(ℤ) with generators drawn uniformly from {(±1,0,0), (0,±1,0)} (the standard symmetric Carnot generating set). Emits the (x,y) coordinates interleaved as a byte stream (out[2i]=x_i mod 256, out[2i+1]=y_i mod 256), which is exactly the encoding the framework's Heisenberg geometry consumes — its embed() builds the path from consecutive byte pairs and accumulates the Levy area z_{n+1} = z_n + x_n·y_{n+1} itself. Pure Heisenberg-walk reference: x,y are independent ±1 random walks, so the Levy area grows like Var(z_n) ~ n²/8 (faster than the n^{3/2} that an independent (x,y) byte pair would give) — the defining Carnot–Carathéodory anisotropy of nilpotent geometry. (structural neighbors: Copeland-Erdős, Windows PE x86-64, Champernowne)
• Hilbert Walk — 1D trace of Hilbert space-filling curve --- preserves 2D locality, creating structured self-similar fluctuations at every scale (structural neighbors: Mertens Function, Rudin-Shapiro, Geometric Brownian Motion)
• Hofstadter Q — Hofstadter Q-sequence (A005185): Q(1)=Q(2)=1, Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2)). Quasi-chaotic nested recurrence with no closed form; not proven well-defined for all n, though empirically holds far past the ranges sampled here. Q(n) ~ n/2 with Mallows-conjectured O(√n) fluctuations, so the emitted first differences Q[n]-Q[n-1] form an integer stream with AC1 ~ -0.5 (oscillatory: big drops followed by big rises) and a slowly-ramping variance over the window --- nonstationary by design, not a generator artifact. Trending metrics (variance_trend, vol_of_vol) respond to the ramp; that is faithful detection of Q's mathematical structure, not noise. (structural neighbors: Blue Noise, GUE Spacings, GOE Spacings)
• Intermittent Silence — Signal with long silent stretches punctuated by active bursts --- on/off intermittency (structural neighbors: Wind Speed)
• Kolakoski Sequence — Kolakoski K = 1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,... --- the unique infinite sequence over {1,2} starting with 1 whose run-length encoding equals itself. Density of 1s conjectured to be exactly 1/2 (Keane 1991, still open in 2026). Aperiodic, self-similar in run-length structure, complexity class unknown.
• Kuramoto Oscillators — Order parameter of coupled oscillators near synchronization transition --- partial coherence (structural neighbors: Kilauea Tremor, Kicked Rotor, Ornstein-Uhlenbeck)
• 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
• LFSR (16-bit) — 16-bit linear feedback shift register (LFSR). Low entropy, deterministic, complex bit transitions. (structural neighbors: Bernoulli Shift, Logistic Chaos, Standard Map K=0.5 (Mixed))
• Langton's Ant — Classic 2-state Langton's Ant (rule RL) on a 256x256 grid --- chaotic build-up then emergent diagonal 'highway' (structural neighbors: Brownian Walk, Mertens Function, Perlin Noise)
• Levy Flight — Random walk with Cauchy-distributed jumps --- infinite variance, occasional extreme leaps create fractal clustering of visited sites (structural neighbors: Regime Switching, ETH/BTC Ratio, BTC Close Price)
• Markov Chain (10-state) — Ergodic 10-state Markov chain. Moderate entropy, smooth spectrum, diverse transitions. (structural neighbors: Codon Usage)
• Minkowski Question Mark — Numerical derivative ?'(x) of Minkowski's question-mark function, sampled on a uniform grid in a random sub-window of (0,1). ?(x) is continuous, strictly increasing, and singular (derivative zero almost everywhere): its derivative is a Cantor-like measure concentrated on rationals. Emitting d?/dx via finite differences exposes that structure as a heavy-tailed signal --- mostly tiny values where ?(x) is flat, with sharp bursts where ?(x) jumps at rational breakpoints. Bijects quadratic irrationals with rationals via the continued-fraction expansion ?(x) = 2 * sum_{k>=1} (-1)^(k+1) / 2^(a_1 + ... + a_k) where [a_0; a_1, a_2, ...] is the continued fraction of x. Different pathology from Cantor's Devil's Staircase --- singularity rooted in continued-fraction arithmetic rather than ternary-Cantor geometry. Raw ?(x) emission was retired 2026-05-18: monotone-increasing values collapsed to a near-linear ramp under [0,1] normalization, becoming pixel-identical to Primes / Partition Function in every rank-based geometry. (structural neighbors: Geomagnetic ap Index, Wind Speed)
• Multiplicative Cascade — Random multiplicative process on dyadic tree --- multifractal burstiness at all scales (structural neighbors: Wind Speed)
• PID Controller — PID controller tracking a step-changing setpoint. Deterministic, narrowband, strongly correlated. (structural neighbors: Triple Pendulum, FPUT N=16, Magnetic Pendulum (3-Magnet))
• Padovan Word — Smallest-Pisot substitution: a→b, b→c, c→ab. Substitution matrix [[0,0,1],[1,0,1],[0,1,0]] has eigenvalue ρ ≈ 1.3247 (plastic number, real root of x³=x+1, smallest Pisot number greater than 1). Ternary alphabet, cubic algebraic ratio at the LOW end of the Pisot spectrum --- pairs with Tribonacci (cubic, ≈1.839) to disentangle 'cubic-Pisot' from 'ρ-magnitude' effects when probing algebraic-substitution-targeted metrics. (structural neighbors: Moebius Function)
• Pell Word — Silver-mean Sturmian: substitution a→aab, b→a. Substitution matrix [[2,1],[1,0]] has eigenvalue 1+√2 ≈ 2.414 (silver mean). Algebraic-symbolic sibling of Fibonacci Word but with silver-mean rather than golden-mean rotation; serves as a silver-mean contrast class against the golden-quasi anchors.
• Penrose Substitution — Robinson-triangle 1D inflation: substitution a→aba, b→ab. Substitution matrix [[2,1],[1,1]] has eigenvalue (3+√5)/2 = φ² ≈ 2.618 (golden squared). Canonical 1D cross-section through the rhombic Penrose tiling along the golden direction --- distinct from Fibonacci Word's φ-scale substitution; same golden family but at a different golden-power resonance.
• Poisson Counts — Poisson-distributed event counts (λ ≈ 3 to 8). Strongly peaked histogram, no temporal order. (structural neighbors: Neural Net (Dense), Entanglement Entropy, Gaussian Noise)
• 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)
• Random Steps — Random piecewise-constant signal --- flat plateaus of random height with geometrically-distributed durations, creating a sparse derivative and blocky texture (structural neighbors: Triple Pendulum, ECG Ventricular)
• Random Telegraph — Two-state Markov process with exponential holding times --- binary correlation structure (structural neighbors: Network Packet Sizes, Gut Motility)
• Riemann-Hardy-Littlewood — Riemann's 1859 'monster' R(t) = sum_{n>=1} sin(pi n^2 t)/(pi n^2). Continuous everywhere, differentiable only at rationals (2p+1)/(2q+1) where R' = -1/2 (Gerver 1970), nowhere else (Hardy 1916). Quadratic frequencies n^2 with 1/n^2 amplitude decay --- the steepest known cosine fractal: ~92% of energy in the fundamental, non-differentiability emerging from the long thin Hausdorff-dimensional tail. (structural neighbors: fBm (Persistent), Perlin Noise, Brownian Walk)
• 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: Morse Code)
• Rule 30 — Wolfram Rule 30 center column --- produces output indistinguishable from random despite simple local rules. Used as Mathematica's default random generator (structural neighbors: Liouville Function)
• 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)
• 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%), Collatz Gap Lengths)
• Shuffled Blocks — Structured blocks shuffled in random order. Uniform marginals, moderate block entropy, low recurrence. (structural neighbors: Network Packet Sizes, Kicked Rotor)
• Spike Train — Integrate-and-fire neuron model --- quiescent charging with sudden discharge spikes (structural neighbors: Network Packet Sizes, Kicked Rotor)
• Stochastic Resetting Walk — Evans-Majumdar process: x(t+1) = x(t) + drift + Gauss noise, with Poisson-rate full reset to zero. Canonical model in non-equilibrium statistical physics for accumulation-with-catastrophe dynamics. Tests whether Forest Fire phenotype (ramp + reset) is detected without spatial-lattice coupling (structural neighbors: BTC Close Price, Regime Switching)
• Stochastic Resonance — Weak periodic signal amplified by noise in a bistable potential --- intermittent well-hopping (structural neighbors: Langevin Double-Well, fBm (Antipersistent), Geometric Brownian Motion)
• 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 (structural neighbors: Liouville Function, Collatz Parity)
• 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: Liouville Function, Morse Code)
• Takagi Function — Takagi (blancmange) function T(x) = sum_{n>=0} s(2^n x)/2^n with s(x) the distance to the nearest integer. Continuous everywhere, differentiable nowhere; range [0, 2/3]. Sister to Weierstrass/Riemann-H-L but built from triangle waves on dyadic doublings instead of trigonometric series --- a fundamentally different self-similar mechanism with graph Hausdorff dimension 2 (filling the plane up to log corrections). (structural neighbors: fBm (Persistent), Magnetic Pendulum (3-Magnet))
• Tank Drain Cascade — Filling tank with multiplicative partial drains: x grows at constant rate to saturation K, at Poisson rate drains to x*Uniform(0, 0.5). Random drain magnitude mirrors Forest Fire's variable fire-cluster sizes. Positive control for slow-build + sudden-drop phenotype with non-uniform drop magnitudes (structural neighbors: Sawtooth Wave, Pressure)
• Thue-Morse — Binary substitution sequence (0→01, 1→10) --- perfectly balanced yet aperiodic, with flat Fourier spectrum like random noise despite being entirely deterministic
• Tribonacci Word — Cubic-Pisot Sturmian-like: substitution a→ab, b→ac, c→a. Substitution matrix [[1,1,0],[1,0,1],[1,0,0]] has eigenvalue Tribonacci constant ≈1.839 (real root of x³=x²+x+1). Ternary alphabet, cubic algebraic ratio --- a degree-3 Pisot contrast class against degree-2 Fibonacci/Pell.
• 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, Duffing Oscillator, Magnetic Pendulum (3-Magnet))
• Weierstrass — Weierstrass function --- continuous everywhere but differentiable nowhere, the first known mathematical 'monster'. Self-similar roughness at all scales (structural neighbors: Langevin Double-Well, fBm (Antipersistent))
• Zipf Distribution — IID draws from Zipf/power-law distribution --- steep rank-frequency, heavy right tail (structural neighbors: Benford's Law, Poisson Spacings, Entanglement Entropy)

Financial (9 sources)

• BTC Close Price — Bitcoin hourly closing prices --- contiguous 16KB block sampled from 96K bars (2015-2026) (structural neighbors: Regime Switching, Levy Flight, Stochastic Resetting Walk)
• BTC Range — Bitcoin hourly high-low range --- intrabar volatility proxy with characteristic clustering (structural neighbors: VLF Radio (Baseline), VLF Radio (Eclipse))
• BTC Returns — Bitcoin hourly log-returns --- heavy tails (kurtosis ~18), volatility clustering, and leverage effects. The canonical non-Gaussian financial time series
• BTC Volatility — Bitcoin realized volatility --- rolling 24h std of log-returns, exhibits long memory (structural neighbors: Solar Wind IMF, Solar Wind Speed, Wave Height (Buoy))
• 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 (Eclipse))
• ETH/BTC Ratio — Ethereum-to-Bitcoin price ratio --- cross-asset relative strength with regime shifts (structural neighbors: Regime Switching, Levy Flight, Brownian Walk)
• NASDAQ Returns — NASDAQ Composite daily close-to-close log returns (^IXIC, 1971+). Tech-tilted U.S. index; higher volatility regime than the S&P. Block-bootstrap padded to DATA_SIZE since native history is ~14k bars.
• Nikkei Returns — Nikkei 225 daily close-to-close log returns (FRED NIKKEI225, 1949+). Japan-market analog of the S&P series; similar heavy-tailed structure.
• S&P 500 Returns — S&P 500 daily close-to-close log returns (^GSPC, 1928+). The canonical U.S. equity benchmark; heavy tails, volatility clustering, leverage effects.

Geophysics (19 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: EEG Resting, Accel Walk)
• 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: Lorenz-96 N=36, Berry Random Wave, Double Pendulum)
• Earthquake Depths — Real USGS earthquake depths --- bimodal: shallow crustal (<30km) and deep subduction (>100km), ~77k M4.0+ events 2020-2024 (USGS ANSS ComCat) (structural neighbors: Sandpile, Multiplicative Cascade)
• Earthquake Intervals — Real USGS interevent times --- Omori-law clustering after mainshocks, Poisson background, ~77k M4.0+ events 2020-2024 (USGS ANSS ComCat) (structural neighbors: Voyager 1 X-band)
• Earthquake Magnitudes — Real USGS earthquake magnitudes --- Gutenberg-Richter distribution, ~77k M4.0+ events 2020-2024 (USGS ANSS ComCat) (structural neighbors: Geometric Waiting Times, Poisson Spacings, Prime Gaps)
• 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: EEG Tumor, Bearing Ball, Speech "Nine")
• El Centro 1940 — 1940 Imperial Valley NS strong motion --- the most-studied earthquake record in structural engineering (structural neighbors: EEG Seizure, EEG Eyes Closed, Sprott-B)
• 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 Flares Daily Peak, Wind Speed)
• 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, Newton-Leipnik Attractor, EEG Healthy)
• Ocean Swell — Real sea-surface displacement from CDIP buoy 067 (Harvest, CA) --- 1.28 Hz vertical heave dominated by long-period Pacific swell: smooth, narrowband, strongly autocorrelated (public domain, CDIP/Scripps). Falls back to a synthetic swell if the data file is not staged (tools/fetch_cdip_swell.py) (structural neighbors: ARMA(2,1), Bearing Normal, MFPT Normal)
• Potomac River Flow — USGS gauge 01646500 --- real 15-min discharge, Potomac near DC, 2024 (public domain) (structural neighbors: ETH/BTC Ratio, Regime Switching, Levy Flight)
• 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, GOES X-Ray Flux, Barometric Pressure (Buoy))
• 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: Langevin Double-Well, Barometric Pressure (Buoy), Ornstein-Uhlenbeck)
• Seismograph (ANMO) — IRIS IU.ANMO broadband vertical --- real 40Hz seismograph, Albuquerque NM, 2024-01-01 (CC0) (structural neighbors: Kuramoto Oscillators)
• Tidal Gauge (SF) — NOAA CO-OPS water-level series at San Francisco, 6-min sampling. Dominated by the semidiurnal M2 lunar tide (~12.4 h) plus diurnal components. (structural neighbors: Blood Pressure Waveform, 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: Voyager 1 X-band)
• 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: Voyager 1 X-band, Pink Noise)
• 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: Voyager 1 X-band, Pink Noise)
• Wave Height (Buoy) — NOAA buoy 46042 significant wave height --- real ocean swell measurements off Monterey Bay 2020-2023, 10-min intervals (CC0) (structural neighbors: Wind Speed, Solar Wind Speed, Solar Wind IMF)

Medical (15 sources)

• Blood Pressure Waveform — MGH/Marquette ART channel. Real ICU arterial blood pressure (mmHg) at 360 Hz, 3 recordings concatenated. Sharp systolic upstroke + dicrotic notch on each cardiac cycle. (structural neighbors: Tidal Gauge (SF), Lorenz Attractor, Temperature)
• ECG Beat Conformity — Per-beat correlation against a record-specific normal-beat template, MIT-BIH Long-Term ECG (7 records, 14-22 h each, 128 Hz). One scalar per heartbeat: normal beats saturate near 1.0; ectopic beats (V, A, F, S) drop proportionally to morphology deviation and recover on the next normal beat. Stays-at-max + episodic-downward-spike phenotype. (structural neighbors: Earthquake Magnitudes, Multiplicative Cascade, Earthquake Depths)
• ECG Fusion — MIT-BIH Arrhythmia DB, fusion beats. 360 Hz; QRS morphology that is a hybrid of normal-conduction and ventricular-ectopic activation.
• ECG Normal — MIT-BIH Arrhythmia DB, normal sinus rhythm (NSR). 360 Hz single-channel ECG; classic P-QRS-T morphology, ~1 Hz heart-rate with respiratory sinus arrhythmia.
• ECG Supraventr. — MIT-BIH Arrhythmia DB, supraventricular premature beats (SVPB). 360 Hz; ectopic atrial beats interleaved with normal sinus QRS.
• ECG Ventricular — MIT-BIH Arrhythmia DB, ventricular premature beats (VPB). 360 Hz; wide aberrant QRS complexes punctuating sinus rhythm.
• EEG Eyes Closed — Bonn EEG set Z, healthy subjects with eyes closed. 173.61 Hz single-channel surface EEG; prominent posterior alpha rhythm (~10 Hz).
• EEG Eyes Open — Bonn EEG set O, healthy subjects with eyes open. 173.61 Hz single-channel; suppressed alpha relative to the eyes-closed set. (structural neighbors: Ornstein-Uhlenbeck)
• EEG Healthy — Bonn EEG set D, interictal recordings from hippocampal contralateral (non-epileptogenic) zone in epilepsy patients. 173.61 Hz.
• EEG Resting — PhysioNet resting-state EEG, 109 subjects, contiguous blocks at 160 Hz across 64 channels (first channel sampled here). Broadband noise with intermittent alpha bursts. (structural neighbors: Bearing Ball, ARMA(2,1))
• EEG Seizure — Bonn EEG set E, ictal activity. 173.61 Hz single-channel, recorded intracranially during epileptic seizure; large-amplitude rhythmic spike-and-wave complexes. (structural neighbors: Bearing Ball, LIGO H1 Whitened (GW150914))
• EEG Tumor — Bonn EEG set F, interictal recordings from epileptogenic zone outside seizure. 173.61 Hz single-channel; subtler abnormality than the ictal set. (structural neighbors: Earthquake P-wave, Bearing Ball)
• Gut Motility — Zenodo 3878435 three-channel surface electrogastrogram. 20 subjects (fasting + postprandial), 2 Hz band-passed 0.03 to 0.25 Hz; gastric slow-wave activity around 3 cpm. (structural neighbors: Solar Wind IMF, Wind Speed, Kepler Exoplanet)
• Mackey-Glass — Delay-differential equation modeling blood cell regulation --- high-dimensional chaos (structural neighbors: 4-Torus Quasiperiodic, Ruelle-Takens Cascade, 5-Torus Quasiperiodic)
• Respiration Waveform — PhysioNet Fantasia RESP channel. Real chest/abdomen respiratory effort at 250 Hz, 3 subjects (~5.4M samples). Smooth quasiperiodic with breath-to-breath variability. (structural neighbors: Langevin Double-Well, Temperature Drift, Ocean Wind (Buoy))

Motion (11 sources)

• Accel Jog — MotionSense smartphone accelerometer magnitude during jogging. 50 Hz; faster, larger-amplitude gait cycle than walking. (structural neighbors: EEG Resting)
• Accel Sit — MotionSense smartphone accelerometer magnitude while sitting. 50 Hz; near-DC with low-amplitude postural sway and breathing. (structural neighbors: Speech "Nine", Speech "Five")
• Accel Stairs — MotionSense smartphone accelerometer magnitude on stairs. 50 Hz; asymmetric step impulses, larger vertical excursions than walking. (structural neighbors: EEG Eyes Closed)
• Accel Walk — MotionSense smartphone accelerometer magnitude during walking. 50 Hz triaxial IMU collapsed to vector norm; ~2 Hz gait cycle. (structural neighbors: EEG Eyes Closed)
• Damped Pendulum — Nonlinear pendulum with friction --- transient chaos at high amplitude, decay to limit cycle or fixed point (structural neighbors: Sine Wave, Duffing Oscillator, Lotka-Volterra)
• Double Pendulum — Angular velocity of second arm --- mechanical chaos with mixed regular/chaotic regions (structural neighbors: Lorenz-96 N=36, Lorenz-96 N=8, FPUT N=16)
• Langevin Double-Well — Brownian particle in double-well potential --- noise-driven switching between metastable states (Kramers problem) (structural neighbors: Stochastic Resonance, fBm (Antipersistent), Brownian Walk)
• Magnetic Pendulum (3-Magnet) — Pendulum tip in 2D plane above three magnets at triangle vertices, with linear gravity-restoring force --- conservative (no damping) 2-DOF Hamiltonian chaos. The damped version has the famous fractal basin boundaries; the undamped version sweeps the energy surface chaotically. Velocity-Verlet (symplectic) integration. Output: x(t). (structural neighbors: Hénon-Heiles, FPUT N=16)
• Projectile with Drag — Ballistic arcs with quadratic air resistance --- asymmetric trajectories, terminal velocity approach (structural neighbors: FPUT N=16, Chua's Circuit)
• Spring Pendulum — Elastic pendulum (mass on stretchable spring swinging in a vertical plane). 2-DOF Hamiltonian with parameters tuned to 1:2 resonance (k = 4g/L₀), where energy swaps chaotically between radial (spring) and angular (pendulum) modes --- canonical internal-resonance chaos. RK4 with dt=0.002. Output: spring length r(t). (structural neighbors: Hénon-Heiles, FPUT N=16)
• Triple Pendulum — 3-segment Lagrangian pendulum chain, equal m=L=1 --- Hamiltonian 3-DOF conservative chaos with sin-coupling. Direct extension of Double Pendulum by one link; meant to populate the multi-DOF pendulum family the way Lorenz-96 {N=4,8,36} populates the dissipative-chain family. RK4 with dt=0.002. Output: angular velocity of bottom segment. (structural neighbors: FPUT N=16, Lorenz-96 N=8)

Noise (13 sources)

• ARMA(2,1) — Autoregressive moving-average process --- linear short-memory dynamics, the workhorse model of classical time-series analysis (structural neighbors: MFPT Normal, Ocean Swell, MFPT Outer Race)
• 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: MT19937 (Mersenne Twister), XorShift32, AES Encrypted)
• 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: GUE Spacings, GOE Spacings)
• Brownian Walk — Cumulative sum of Gaussian increments --- the canonical random walk, self-affine with Hurst exponent H=0.5 and PSD ~ 1/f²
• Gaussian Noise — IID standard Gaussian noise --- bell-curve density centered at zero, the baseline null model for continuous-valued signals (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
• 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 Open)
• 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: Levy Flight, ETH/BTC Ratio)
• White Noise — IID uniform random floats in [0,1) --- the null model, maximum entropy with zero sequential correlation (structural neighbors: glibc LCG, XorShift32, RANDU)
• fBm (Antipersistent) — Fractional Brownian motion with H=0.3 --- antipersistent (trend-reversing), rougher than standard Brownian motion (structural neighbors: Langevin Double-Well)
• fBm (Persistent) — Fractional Brownian motion with H=0.7 --- persistent (trend-following), smoother trajectories with long-range dependence

Number_Theory (37 sources)

• A5 Artin a_p — Hecke traces a_p for 100 weight-1 cuspidal newforms with projective Galois image A_5 (icosahedral), pooled across LMFDB labels for primes p ≤ 1999 (303 a_p per form, 30,300 pooled). Each a_p is the integer trace of Frobenius in a 2-dimensional Artin representation; the icosahedral case is the Buhler 1978 modularity result, completed for all 2-dim odd Artin reps by Khare–Wintenberger / Kisin. Values lie in the finite set {-32, ±16, ±12, ±8, ±6, ±4, ±2, 0} (all even, since each is the trace down to ℤ of an even-degree coefficient field); the pooled distribution follows the Chebotarev density theorem over A_5 conjugacy classes, heavily zero-biased (~64%) and symmetric. Distinct universality class from continuous Sato-Tate semicircle: a discrete, finite-support Galois-distributed integer stream — the standard 'arithmetic trace' calibration anchor for metrics responding to discrete-categorical structure with deep number-theoretic provenance. (structural neighbors: Categorical Sensor, Poker Hands)
• Aliquot Orbit Lengths — Aliquot sequence orbit lengths --- number of steps before s(n)=sigma(n)-n iteration terminates (reaches 1, enters a cycle, or escapes the sieve). Distinct erratic arithmetic from Collatz Stopping Times: divisor-sum dynamics instead of 3n+1 parity. Catalan-Dickson conjecture (some orbits unbounded, e.g. n=276) means a few entries cap at the divergence ceiling --- treated as MAX_STEPS, not specially flagged
• 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: ChaCha20, Wichmann-Hill, MT19937 (Mersenne Twister))
• 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, Beta Noise)
• Catalan G Digits — Base-256 digits of Catalan's constant G = Σ(−1)ⁿ/(2n+1)². Irrationality and transcendence both unproven; normality unknown. (structural neighbors: glibc LCG)
• Champernowne — Provably normal number in base 256 --- uniform digit distribution but deterministic concatenation (structural neighbors: Linux ELF x86-64)
• Collatz Flights — Peak altitude of Collatz orbits --- how high consecutive integers fly before crashing to 1
• 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, Geometric Waiting Times, Categorical Sensor)
• 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, Penrose Substitution)
• 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: Earthquake Magnitudes, Surface Wind (ORD 5-min), Windows PE x86-64)
• Collatz Trajectory — Collatz orbit values log1p(n) along the trajectory, restarting from consecutive integers to avoid 4-2-1 cycle collapse. Preserves the multiplicative walk structure (3n+1 then halve) on a compressed scale. (structural neighbors: Hawkes Process, Gut Motility, Solar Wind Speed)
• Continued Fractions — CF digits of random quadratic irrationals --- bounded partial quotients with hidden periodicity (structural neighbors: Sandpile, Markov Chain (10-state))
• Copeland-Erdős — Provably normal number formed by concatenating primes in base 256 (Copeland-Erdős 1946). Same headline normality as Champernowne but driven by the prime sequence rather than the natural numbers --- bridges Champernowne's structured-byte-stream class into the prime-density-flavored corner of the atlas (structural neighbors: Middle-Square (von Neumann), Windows PE x86-64)
• De Bruijn Sequence — B(4,4) De Bruijn cycle --- every 4-symbol window over alphabet 0-3 appears exactly once (structural neighbors: LFSR (16-bit), Gray Code Counter, Middle-Square (von Neumann))
• Divisor Count — Divisor count d(n) --- erratic arithmetic function averaging ~ln(n), spikes at highly composite numbers, modulated by prime factorization structure
• 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))
• Euler-Mascheroni γ Digits — Base-256 digits of γ = lim(Hₙ − ln n)'s fractional part. Even irrationality is an open question. Atlas-novelty here would be a striking empirical hint about γ's structure. (structural neighbors: AES Encrypted, MINSTD (Park-Miller))
• Fibonacci Quasicrystal — 1D Fibonacci quasicrystal --- aperiodic tiling with long-range order, discrete diffraction (structural neighbors: Uniform Chaos (Logistic Scramble), Standard Map K=0.5 (Mixed), Critical Circle Map (Bronze Mean))
• Gaussian Collatz Orbit — Collatz map over Gaussian integers Z[i] with divisor pi=1+i and multiplier 3. Signal is split into 4 segments: 3 seeded from the proven period-16 cycle starts (2-8i, -48+4i, -43-35i) and 1 from random init showing the divergent regime. Captures both the proven cycle dynamics and the surrounding chaotic divergence (structural neighbors: Accel Walk, Linux ELF x86-64, Gut Motility)
• Goldbach r(2n) — Goldbach r(2n) --- number of unordered representations of 2n as p+q with both p,q prime. Highly erratic (Hardy-Littlewood: mean ~2*C_2*S(n)*2n/log^2(2n)) with strong primorial bias (r(2n) jumps when 2n is divisible by many small primes). Distinct mechanism from Divisor Count d(n): additive prime structure vs multiplicative divisor structure
• Golden Ratio Digits — Base-256 digits of φ = (1+√5)/2's fractional part. Algebraic of degree 2 (root of x²−x−1); normality unknown. Contrast vs Pi/e/ln2 tests the transcendental-vs-algebraic split. (structural neighbors: ChaCha20, AES Encrypted)
• Liouville Function — Liouville lambda(n) = (-1)^Omega(n) --- parity of prime-factor count with multiplicity. Pure +/-1 sequence with P(+1) -> 1/2 (Landau). Sum lambda(n) = O(n^(1/2+eps)) iff the Riemann Hypothesis; the Polya conjecture (false at n=906_150_257) probed the same balance. (structural neighbors: Rule 30, Symbolic Lorenz, Symbolic Henon)
• 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: Hilbert Walk, Brownian Walk)
• Mian-Chowla — Greedy Sidon (B₂) sequence: a₁=1, a_{n+1} = smallest integer strictly greater than a_n such that all pairwise sums {a_i + a_j : i ≤ j} remain distinct. First terms 1, 2, 5, 10, 11, 13, 37, 42, 75, 96, ... (Mian–Chowla 1944). Emitted as length-N binary indicator (255 at positions a_n < N, 0 elsewhere) cyclically shifted by an rng-chosen offset --- cyclic shift preserves the Sidon property on ℤ/N. Density ~1.1% at N=16384. Canonical positive control for sumset-isolation metrics (Erdős #152 family): by construction every pairwise sum is uniquely realized. (structural neighbors: Rainfall (ORD Hourly), Neural Net (Pruned 90%), Devil's Staircase)
• Moebius Function — Moebius mu(n) in {-1, 0, +1} --- 0 on non-squarefree n (~39.2% by density), (-1)^omega(n) on squarefree n. Generator of multiplicative number theory via the identity sum_{d|n} mu(d) = [n=1]; stationary ternary sequence whose partial sum is Mertens. (structural neighbors: Free Group F₂ Walk, Categorical Sensor, Codon Usage)
• Partition Function — p(n) mod 5, the number of integer partitions of n reduced modulo 5. Captures the first Ramanujan congruence p(5k+4) ≡ 0 (mod 5) (Ramanujan 1919, proved Watson 1938) as a stationary integer stream over {0,1,2,3,4}: every 5th position is deterministically zero, the other four positions look pseudo-random in {0,1,2,3,4}. Computed via Euler's pentagonal number theorem with arithmetic carried in Z/5, so the exponential growth of p(n) (Hardy-Ramanujan: p(n) ~ exp(pi sqrt(2n/3))) never appears directly --- the source carries the algebraic-periodicity content rather than the smooth envelope. Raw log p(n) emission was retired 2026-05-18: it was monotone-increasing and collapsed to the same rank sequence as Primes / Minkowski ?(x) in every rank-based geometry. (structural neighbors: Codon Usage, DNA Centromere, DNA SARS-CoV-2)
• Pi Digits — Digits of pi in base 256 --- conjectured normal (equidistributed), but entirely deterministic. Passes most statistical randomness tests (structural neighbors: MT19937 (Mersenne Twister), AES Encrypted)
• 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: Geometric Waiting Times, Poisson Spacings, Zipf Distribution)
• Prime Indicator — Stanisław Ulam's 1963 reference signal: 1 at prime indices, 0 elsewhere, over the integers starting from a random offset. Density ≈ 1/ln(n) (PNT). Fed through the Ulam Spiral geometry this reproduces the classical prime-diagonal plot --- diagonal_alignment should approach 1.0. (structural neighbors: Neural Net (Pruned 90%))
• Ramanujan Tau — Normalized Ramanujan tau τ̃(p) = τ(p)·p^(-11/2) for consecutive primes, where τ(n) is the q-coefficient of Δ(q) = η(q)^24 — the unique weight-12 cusp form on SL(2,ℤ). By Deligne (1974, formerly Ramanujan-Petersson), |τ̃(p)| ≤ 2 for all p; by Barnet-Lamb–Geraghty–Harris–Taylor (2011), {τ̃(p)} equidistributes to the Sato-Tate semicircle (2/π)√(1-x²/4) on [-2,2] — a non-Gaussian, non-uniform universal distribution distinct from GUE/Wigner spacing. Calibration anchor for metrics responding to semicircle-distributed sequences with deep automorphic structure. (structural neighbors: ChaCha20, Wichmann-Hill, MT19937 (Mersenne Twister))
• Riemann Zeta Zeros — Unfolded nearest-neighbour spacings of the first 100,000 Riemann zeta nontrivial zeros (Odlyzko) --- Montgomery-Odlyzko: GUE-distributed (quadratic level repulsion), the canonical real-data exemplar of random-matrix statistics (structural neighbors: GUE Spacings, GOE Spacings, Blue Noise)
• Rudin-Shapiro — Partial sums of Rudin-Shapiro --- deterministic random walk with flat spectral density (structural neighbors: Brownian Walk, Hilbert Walk)
• Stern-Brocot Walk — Stern diatomic sequence --- every positive rational visited once, fractal self-similar structure
• e Digits — Base-256 digits of Euler's number e's fractional part. Transcendental (Hermite 1873); conjectured normal but unproven. (structural neighbors: AES Encrypted)
• ln(2) Digits — Base-256 digits of ln(2)'s fractional part. Transcendental (Hermite-Lindemann); conjectured normal but unproven. (structural neighbors: glibc LCG, AES Encrypted)
• von Mangoldt Function — von Mangoldt Lambda(n) = log(p) if n = p^k for some prime p and k >= 1, else 0. Prime-power-supported sparse signal whose cumulative sum is the Chebyshev psi function. By Riemann's explicit formula psi(x) = x - sum_rho x^rho/rho - log(2 pi) - (1/2) log(1 - x^-2), so Lambda encodes the nontrivial zeros of zeta directly via psi --- the closest one gets to the zeta zeros in pure number-theoretic form. (structural neighbors: Neural Net (Pruned 90%))
• √2 Digits — Base-256 digits of √2's fractional part. Algebraic of degree 2; irrational (Theaetetus / Pythagoreans, ~5th c. BC); normality unknown. (structural neighbors: AES Encrypted, ChaCha20)

Quantum (13 sources)

• Anderson 1D Localized — 1D tight-binding chain with random on-site disorder W=2 --- exponentially localized eigenstates from K independent disorder realizations, re-centered and concatenated (structural neighbors: Solar Flares Daily Peak, Minkowski Question Mark)
• Aubry-André Critical — Almost-Mathieu operator at critical coupling λ=2 with golden-ratio incommensurate potential --- multifractal eigenstate intensity (deterministic multifractality, canonical exemplar from quasiperiodic origin) (structural neighbors: Solar Flares Daily Peak)
• Berry Random Wave — Berry's random plane wave --- superposition model for chaotic eigenstate statistics, Gaussian amplitude distribution (structural neighbors: FPUT N=16, Lorenz-96 N=8, Double Pendulum)
• Entanglement Entropy — Page curve entanglement entropy of random bipartite states --- ⟨S⟩ ≈ log(d_A) - d_A/(2d_B) (structural neighbors: Gaussian Noise, Neural Net (Dense), Zipf Distribution)
• Fibonacci Tight-Binding — 1D tight-binding chain with on-site potential modulated by Fibonacci substitution rule A->AB, B->A --- Cantor-set spectrum, multifractal eigenstates (deterministic multifractality from substitution-aperiodic mechanism) (structural neighbors: Minkowski Question Mark, Multiplicative Cascade)
• GOE Spacings — Gaussian Orthogonal Ensemble spacings --- linear repulsion P(s)~s exp(-πs²/4), time-reversal symmetric (structural neighbors: Riemann Zeta Zeros, Blue Noise)
• GUE Spacings — Gaussian Unitary Ensemble spacings --- level repulsion P(s)~s² exp(-4s²/π), broken time-reversal symmetry (structural neighbors: Riemann Zeta Zeros, Blue Noise)
• Kicked Rotor — Quantum kicked rotor --- dynamical localization freezes classical chaos, Anderson localization in momentum space (structural neighbors: Kuramoto Oscillators, Ornstein-Uhlenbeck, Bearing Normal)
• OTOC Growth — Out-of-time-order correlator d(log C)/dt for a chaotic mixed-field Ising chain (L=8, g=0.9, h=0.8, J=π/4) --- quantum Lyapunov scrambling rate; flat plateau at 2λ during exponential growth, decays toward zero at saturation (structural neighbors: PID Controller, SIR Epidemic, Magnetic Pendulum (3-Magnet))
• Poisson Spacings — Uncorrelated Poisson spacings P(s)=exp(-s) --- integrable quantum systems, no level repulsion (structural neighbors: Geometric Waiting Times, Prime Gaps, Zipf Distribution)
• Quantum Walk — Hadamard walk on a 257-site ring --- coin polarization ⟨σ_z(t)⟩ with ongoing wraparound dynamics and quantum recurrences (structural neighbors: Hofstadter Q, IMS Bearing Degraded, Riemann Zeta Zeros)
• Spectral Form Factor — Ensemble-averaged spectral form factor log K(t) for a GUE Hamiltonian --- dip-ramp-plateau curve over log-spaced times; captures long-range eigenvalue correlations not seen by local spacings (structural neighbors: Potomac River Flow, Regime Switching, ETH/BTC Ratio)
• Wigner Semicircle — Eigenvalue density from large random matrices --- converges to Wigner semicircle law ρ(x)=√(4-x²)/(2π) (structural neighbors: ChaCha20, glibc LCG, BSL Residues)

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: Poisson Counts, Categorical Sensor, Neural Net (Dense))
• Speech "Five" — FSDD spoken digit 'five', 8 kHz waveform. Fricative-vowel-fricative structure; mid-band energy peak. (structural neighbors: Earthquake P-wave)
• Speech "Nine" — FSDD spoken digit 'nine', 8 kHz waveform. Nasal onset, diphthong, sustained voicing throughout. (structural neighbors: Earthquake P-wave)
• Speech "Zero" — FSDD spoken digit 'zero', 8 kHz waveform. Sustained voicing on the vowel, broadband fricative onset. (structural neighbors: Accel Walk)

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)
• 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, Rule 110)
• 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: Voyager 1 X-band, VLF Radio (Eclipse), VLF Radio (Baseline))
• 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: Symbolic Lorenz)
• Sawtooth Wave — Sawtooth wave --- linear ramp with discontinuous reset, rich in harmonics (1/n Fourier decay) (structural neighbors: Tank Drain Cascade, Duffing Oscillator, Rossler Attractor)
• Sine Wave — Pure sinusoid --- perfectly periodic, single-frequency, minimal complexity. The baseline for all oscillatory comparisons (structural neighbors: Damped Pendulum, Duffing Oscillator, Lotka-Volterra)
• Square Wave — Square wave --- binary oscillation (0 or 255), maximal harmonic content with 1/n odd-harmonic Fourier series
• Triangle Wave — Triangle wave --- linear ramps up and down, smooth peaks with 1/n² Fourier decay (only odd harmonics) (structural neighbors: Damped Pendulum, 2-Torus Quasiperiodic)
• 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, Lotka-Volterra)

Geometric Views

• Biological: How does the framework distinguish biological sequence structure from biological process dynamics and physiological signals? (69 geometries)
• Chaotic: How does the framework distinguish types of deterministic chaos? (69 geometries)
• Distributional: What does the value distribution look like? (11 geometries)
• Dynamical: What is the character of the signal's dynamics? (12 geometries)
• Empirical: How does the real world organize itself, ignoring synthetic constructs? (69 geometries)
• Geophysical: How does the framework organize the natural world — seismic, atmospheric, solar, and gravitational signals? (69 geometries)
• High Entropy: Can the framework distinguish sources that all look like noise? (69 geometries)
• Optimized: What structure remains after removing noise and redundancy? (65 geometries)
• Ordinal: What structure survives re-encoding? (13 geometries)
• Quasicrystal: Does the data have aperiodic order? (5 geometries)
• Reference: What do standard time-series features see? (1 geometries)
• Scale: How does structure change across scales? (5 geometries)
• Surrogate-Contrast: What structure survives shuffling? What is sequence-dependent? (59 geometries)
• Symmetry: What algebraic structure underlies the data? (15 geometries)
• Temporal: How does the signal behave under time-domain operations? (4 geometries)
• Topological: What is the shape of the data's phase space? (12 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.
• AutoRegressive — Box-Jenkins AR(10) coefficient vector via Yule-Walker. 10 coefficients (clipped |coef|≤5) plus ar_residual_frac and ar_bic_order (BIC-selected order in 0..10). Detects: AR-process structure, linear short-memory dependence, white-noise residuality.
• Bispectrum — Third-order spectral statistics: bicoherence over the principal triangle measures phase coupling between frequency triplets (f1, f2, f1+f2). Zero for linear Gaussian processes; ↑ for quadratic nonlinearity. Detects: Phase coupling between frequencies; quadratic nonlinearity; non-Gaussian higher-order structure; harmonic vs cross-mode interaction.
• 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.
• Catch24 — catch22/catch24 canonical time-series features (Lubba et al. 2019) embedded as a calibration lens — the standard comparator placed inside the atlas's own redundancy structure. Detects: Standard time-series features: autocorrelation timescale, distribution shape, automutual information, fluctuation/DFA scaling, Welch spectral summaries, symbolic motifs.
• 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.
• Chladni — Nodal structure analysis inspired by Chladni plate vibration patterns: zero-crossing spacing statistics (level repulsion), nodal clustering (burstiness), and multi-scale modal cascade. Detects: Nodal regularity, zero-crossing clustering, multi-scale modal complexity.
• Continued Fraction — Continued-fraction partial quotients of each sample. Probes Khintchine-Lévy norm and Gauss-Kuzmin deviation on continuous data; gated to inputs with ≥64 distinct values where the underlying measure-theoretic statements apply. Detects: Khintchine-Lévy deviation and Liouville-class partial-quotient outliers on continuous signals.
• D4 Triality — Projects 4-byte windows onto the 24 roots of D4 and measures structural probes — edge graph, spectral coherence, lattice closure. Detects: D4 lattice structure, 4-byte symmetry constraints.
• 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.
• 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: deep diverse edge walks on the 600-cell with diff-vector closure on H4 roots; gated against degenerate-plateau and pingpong saturators.
• 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.
• Hodge–Laplacian — Hodge-Helmholtz decomposition of a vector field built from split-and-reshaped signal halves, plus 2D scalar Laplacian / biharmonic / Dirichlet energy. Compressible vs solenoidal fractions separate gradient-like from rotational data. Detects: Compressible/solenoidal energy split, harmonic 1-forms, curl/div ratio, Laplacian energy, source/sink density.
• 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 multi-scale structure functions S(q, ℓ) = mean(|x(t+ℓ) − x(t)|^q). The regularity spectrum (multifractal formalism) measures local smoothness variation. (Wavelet-leaders is an equivalent classical alternative not used here.) 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.
• Isochronicity — Per-cycle amplitude→frequency coupling (KAM twist, dω/dA). Bandpass-Hilbert cycle extraction; |ρ(A_cycle, 1/T_cycle)| measures anisochronicity. Validated orthogonal to the atlas + the 4-axis dynamical fingerprint. Detects: Anisochronous oscillation — amplitude-dependent frequency (Van der Pol, Duffing, Stuart-Landau shear, Ruelle-Takens cascade); also separates AM (high amp spread, zero shear) from FM (zero amp spread, time-varying frequency).
• Julia Set — Fixed Julia set parameter c acts as a tunable sensor. A Halton-bank of K starting points z0 is iterated with data-modulated perturbations z ← z² + c + ε·(d[i]−⟨d⟩); escape-time statistics depend on the data sequence, not just its value histogram. Detects: Sequence-dependent dynamical trapping, basin structure, Julia boundary sensitivity.
• 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.
• Laplacian — Discrete Laplacian analysis: biharmonic energy cascade, Poisson recovery error, gradient sign persistence, curvature spectral ratio. Detects: Curvature cascade, non-periodic boundary content, monotone runs.
• Level Statistics — Random-matrix-theory level-spacing statistics: KS-distances from the input value distribution to Poisson (integrable) and Wigner-Dyson GOE/GUE surmises. Probes Berry's conjecture. Detects: Wigner-Dyson vs Poisson level statistics; integrable vs quantum-chaotic spectral signatures.
• 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.
• Marginal — Marginal distribution shape via tail occupancy beyond 1.5·σ. Operates on z-scored data to preserve tail-shape information that per-geometry [0,1] normalization destroys. Detects: shoulder weight / marginal-distribution shape.
• Modular Residue — Cyclic mod-7 residue transition dynamics: cycle/drift flow decomposition, spectral mixing rate, stationary occupancy entropy. Detects: Discrete cyclic transport, modular winding bias, recurrent backbone strength on integer-quantized sequences.
• Moiré
• 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.
• Möbius-S³ — Binary icosahedral (2I) quotient structure on S³. Measures topological sector transition entropy on the Poincaré homology sphere S³/2I. Detects: Topological sector transitions on S³/2I.
• Navier-Stokes — Tests turbulence scaling predictions: She-Leveque vs K41 structure function exponents (ESS), signed energy cascade direction, intermittency flatness growth rate, and the exact 4/5 third-order law. Detects: Turbulence cascade, intermittency, K41/She-Leveque scaling.
• 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.
• Piecewise-Linear — Piecewise-linear regime detection via second-derivative thresholding (slope_changes), running-minimum envelope area, and windowed slope-bucket diversity. Originally motivated by tropical (min-plus) algebra but uses standard numerical differencing throughout. Detects: Piecewise-linear regimes, slope diversity, envelope structure.
• 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.
• Sidon — Isolated-point fraction in the sumset A+A of the top-5% quantile mask. Lights up on signals whose high-amplitude positions sit at structured-sparse locations (modular arithmetic, periodic chaotic windows, Phyllotaxis); silent on noise and smooth signals. Inspired by Erdős #152. Detects: Sparse modular / periodic structure in the top-quantile positions — e.g. Goldbach r(2n), von Mangoldt Λ(n), Euler totient ratio, periodic windows in chaotic maps.
• 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.
• Spirograph — Epicyclic decomposition: measures arithmetic structure of frequency ratios (gear rationality, closure, petal symmetry). Detects: Frequency ratio arithmetic, nested periodicities, spirograph closure.
• Symplectic — Constructs a phase portrait (position vs. momentum) from the time series. Metrics quantify how densely the trajectory fills phase space, whether the portrait is time-reversal symmetric (Hamiltonian-conservative hallmark), sweep-rate stationarity, orbital coherence, and recurrence. Detects: Phase space filling, reflection symmetry, trajectory stationarity, orbital coherence, 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.
• Time Reversibility — Direction-of-time asymmetry: ordinal-pattern, increment-skewness, and directional-run asymmetries. Near 0 for linear Gaussian processes, ↑ for dissipative chaos and nonlinear non-Gaussian signals. Detects: Time-reversal symmetry breaking; conservative-vs-dissipative dynamics; nonlinear/non-Gaussian asymmetry.
• 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.
• Ulam Spiral (Sacks) — Threshold-marks samples on a 2D spiral integer lattice (square or Sacks). Measures diagonal/quadratic alignment of marked positions, after Ulam (1963). Detects: Quadratic-polynomial alignment of marked positions on a 2D lattice.
• Ulam Spiral (Square) — Threshold-marks samples on a 2D spiral integer lattice (square or Sacks). Measures diagonal/quadratic alignment of marked positions, after Ulam (1963). Detects: Quadratic-polynomial alignment of marked positions on a 2D lattice.
• 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.
• WaveformAsymmetry — Per-cycle rise/fall time asymmetry. Positive-going zero crossings mark cycles; |t_rise - t_fall| / T per cycle; aggregated by median. Sibling to Isochronicity in the oscillator lens. Detects: Within-cycle positional asymmetry of argmax/argmin. On smooth waveforms reads rise/fall time (sawtooth, ECG QRS, relaxation oscillators). On binary/sparse-spike signals reads duty-cycle skew (skewed PWM, sparse spike trains, Prime Indicator / von Mangoldt). Insensitive to anisochronicity, amplitude variation, and harmonic-symmetric distortion (square, clipped sine, triangle)..
• Wavelet Cascade — Haar wavelet cascade analysis: cross-scale energy coupling, intermittency, directional flow, and criticality. Detects: Cross-scale energy cascades, intermittency, self-organized criticality.
• 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.