Sleep Cycle (NREM/REM)
The sleep cycle is the τ-categorical nested ultradian lemniscate inside the rest lobe of the circadian temporal lemniscate: a closed traversal S¹_NREM ∨ S¹_REM with period ≈90 min. It is the consumer-sector specialization of the temporal lemniscate to neural dynamics, mandated by the topological requirement that the τ¹ trajectory be periodic.
τ-Definition
The sleep cycle is the τ-categorical nested ultradian lemniscate inside the rest lobe of the circadian temporal lemniscate: a closed traversal S¹_NREM ∨ S¹_REM with period ≈90 min. It is the consumer-sector specialization of the temporal lemniscate to neural dynamics, mandated by the topological requirement that the τ¹ trajectory be periodic.
Categorical invariant. Nested ultradian lemniscate L_sleep = S¹_NREM ∨ S¹_REM (VI.P19) within the rest lobe of L_T (VI.D27); the wedge point is the bistable VLPO/orexin flip-flop transition between sleep stages.
Primary registry anchor:
VI.P19
τ-Derivation Chain
Empirical Correlate
Biomarker: EEG signatures: N1 (theta), N2 (sleep spindles 12–14 Hz, K-complexes), N3 (delta 0.5–4 Hz), REM (low-voltage mixed-frequency, sawtooth waves); EOG (REMs); EMG (atonia in REM); orexin/hypocretin and VLPO firing rates.
Measurable range: Cycle period ≈ 90 min; 5–6 cycles per night (total ~7–9h adult); N1 ~5%, N2 ~50%, N3 ~20%, REM ~25% of total sleep; deep sleep predominates in early cycles, REM in later cycles.
Observation method: Polysomnography (EEG + EOG + EMG), actigraphy, intracranial recordings of hippocampal sharp-wave ripples and cortical slow oscillations, glymphatic flow imaging via contrast-enhanced MRI.
Calibration anchor: LG-Y02-kinetic-pseudoscalar-channel
Anchor chain:
- VI.L18 chirality channel
- stereospecificity of orexin/hypocretin and adenosine receptor binding (L-amino acid receptors, D-ribose nucleotide)
- homochirality of melatonin and serotonin biosynthetic precursors
Manuscript reference: manuscript-sources/book-06/part06/ch41-learning-sleep.tex
Lean Coverage
Status: Planned