Navigator's Log R&D — Christopher Head — Fresno CA

The Geometric
Encoder

A single mathematical instrument that detects the moment any physical system departs from its current attractor state — confirmed across 11 independent domains spanning 28 orders of magnitude in physical scale, from gravitational wave strain at 10⁻¹⁹ m to solar wind at 10⁹ m.

11Confirmed domains
28Orders of magnitude
10⁻¹¹Calibration law p-value
01 — Mechanism

Eight geometric features.
One decision.

Every physical process traces a path through state space. The encoder takes a window of signal samples, embeds them in delay-coordinate phase space, and computes eight topological features of the resulting trajectory. When the trajectory departs from its baseline attractor — in any domain, at any scale — the encoder fires.

No labeled training data. No domain-specific feature engineering. The only input is a time-series signal. The only prior is the signal's own recent history.

01
menger_curv
Local curvature of the phase-space trajectory at each step.
02
intr_dim
How many independent directions the trajectory uses. Low = rigid orbit. High = complex organized dynamics.
03
density_var
Uniformity of trajectory point distribution across the embedding space.
04
accel_ps
Phase-space acceleration. Rate at which the trajectory changes direction. Dominant at every confirmed state transition.
05
autocorr
Temporal self-similarity. Separates organized oscillation, rigid monotone, and chaotic states.
06
energy_slope
Direction of signal energy flow during the current window.
07
recurrence
How often the trajectory revisits the same region of phase space.
08
lyapunov_proxy
Local divergence rate of nearby trajectories in the embedding.
"Phase-space acceleration ranks first or second in 18 of 18 transition-detection events across 11 independent physical domains spanning 28 orders of magnitude. The dominance is scale-invariant."
18 / 18Transitions — accel_ps rank ≤ 2
p = 1.6 × 10⁻¹¹Binomial vs chance (1/8)
r = −0.31, p = 0.22Spearman: scale-invariant
02 — Evidence

Confirmed across domains

Eleven independent physical domains. Identical encoder code and detection threshold across all. The top trace in each card is the simulated signal. The bottom bar is the encoder anomaly ratio — amber when elevated, red when GENUINE (>10×).

03 — Two Operating Modes

Transition detection vs
state characterization

The calibration law has two sides. Which feature dominates depends on what the encoder is being asked to do — not on the physical domain.

accel_ps dominant
Transition Mode
When detecting departure from the current attractor — a fault onset, a wave arrival, a state change — phase-space acceleration dominates. The trajectory is actively leaving its prior orbit.
GW chirp arrival at LIGO Bearing fault onset (CWRU) Convective cap break (ERA5) Solar wind shock arrival EEG sleep stage transition
intr_dim dominant
State Characterization Mode
When characterizing the attractor itself — how complex is this state, how constrained is this orbit — intrinsic dimension dominates. Lower intr_dim signals fault-toward-rigidity.
Seismic bulk wave phases (P, S, SKS) CHF cardiac autonomic rigidity Parkinson's gait rigidity Diurnal atmospheric channels
04 — Domain Boundary

Where it works.
Where it doesn't.

The encoder is a change-detector, not a state-classifier. It detects transitions between different attractor types. Two experiments confirmed the boundary condition precisely — and both falsifications were scientifically productive.

Confirmed — transitions between attractor types
Gravitational wave chirp. Two merging masses sweep a characteristic phase-space trajectory. Physically distinct from detector noise.
Atmospheric cap break. The temperature profile reorganizes over hours before the column erupts. Distinct attractor from stable stratification.
Cardiac RR intervals. Inter-beat interval series is aperiodic. CHF shows fault-toward-rigidity (lower intr_dim). AF shows negative autocorr. NSR between both.
Bearing fault onset. Healthy rotating orbit to fault-constrained orbit — two distinct attractor types.
Any aperiodic signal with multi-scale structure where the transition is between physically distinct regimes.
Boundary — same attractor type, different parameters
Cardiac ECG waveform. Sinus rhythm produces the same closed-loop phase portrait each beat, before and after AF onset. Two independent experiments confirmed null. Correct behavior — the encoder cannot distinguish instances of the same attractor.
Purely diurnal channels. T925-T850 lapse rate repeats every 24 hours. A 24-hour baseline already saturates — no information distinguishing convective from quiet days.
Long static baselines. A longer history does not improve detection. The encoder needs the current local attractor state, not a long-run statistical average.
Static classification without a baseline. The encoder requires a reference state to detect departure from. No baseline = no "normal" to depart from.
05 — Applications

Where this matters

Any system that can fail, transition, or change state — and where early detection has operational value — is a candidate. The encoder requires no labeled training data and no domain-specific feature engineering.

Autonomous systems
Multi-drone environmental sensing
IMU, WiFi CSI, barometric, ToF, and acoustic sensors feed a single encoder pipeline. Material classification, obstacle detection, and anomaly flagging without separate models per sensor.
Astrophysics
Gravitational wave classification
91.7% LOO accuracy on 31 GWOSC events vs 5 detector glitch types. BBH, BNS, and NSBH mergers produce geometrically distinct chirp trajectories without matched filtering.
Clinical cardiology
Cardiac autonomic state monitoring
RR interval series from wearables. Two-axis classifier (intr_dim + autocorr) separates NSR, pure AFIB, NSR-in-AF, and CHF from 16-beat windows. No ECG morphology required.
Industrial
Rotating machinery fault onset
CWRU benchmark confirmed. Fault-toward-rigidity: the damaged system settles into a lower-dimensional orbit before mechanical failure — detection before damage propagates.
Meteorology
Convective initiation warning
ERA5 multi-level profiles. Three-class atmospheric system: surface-driven, upper-forced, heat dome. Omega magnitude separates 6/6 in real data with a single threshold.
Grid operations
Power grid pre-failure detection
2021 European grid separation confirmed. Frequency near instability shows increasing accel_ps before the excursion — the calibration law operating at continental scale.
Neuroscience
Seizure onset pre-detection
Pre-ictal EEG shows fault-toward-rigidity: cortical dynamics shift from complex organized activity toward pathological synchrony. Normalized envelope encodes this without raw waveform analysis.
Structural engineering
Civil infrastructure health
Healthy structure in wind: high-dimensional vibration. Cracked beam: simpler orbit at the damage mode. Same fault-toward-rigidity as rotating machinery — different substrate, identical geometry.