Neural State Geometry: Unifying Amplitude Landscapes, Coherency Geometry and Information Geometry for EEG State Estimation

Brain-state engineering requires measurement and modulation through interpretable, physics-informed coordinates of EEG state structure, change and intervention response. Applications include cognitive fatigue monitoring, neuromodulation readout and progress toward adaptive closed-loop neurotechnology. We introduce Neural State Geometry (NSG), which is a sensor-space framework derived from the complex analytic EEG field rather than a biophysical neuronal model. NSG integrates a Neural Amplitude Landscape (NAL), mapping analytic-amplitude distributions to statistical temperature, internal energy, entropy and heat capacity through exact canonical statistical thermodynamics; pairwise Stokes/coherency geometry representing inter-channel organisation using electromagnetic coherence theory; and quantum-information-inspired state-space geometry quantifying selected-basis organisation. We evaluate NSG through controlled synthetic tests, outcome-blind human-EEG amplitude-model comparison, a single-participant cognitive fatigue example, and a single-investigator transcranial focused ultrasound (TUS) neuromodulation feasibility case. Synthetic tests established descriptor selectivity and common-source sensitivity of zero-lag second-order geometry. NAL/generalised-gamma improved participant-balanced held-out prediction over Rayleigh, Nakagami and gamma but not lognormal. The cognitive fatigue example showed predominantly positive PRE–POST NAL shifts with spectral and second-order deformation. In the fixed-order TUS case, the largest reference-stable NAL displacement occurred at the application site alongside broader sensor-space deformation. NSG provides an interpretable layer for comparing natural and intervention-associated EEG states, supporting future biomarker and target-engagement validation, mechanistic investigation and closed-loop neural engineering.