Governing Equations of Human Sleep Macro-Structure:Symbolic Discovery from 1,106 Nights
Abstract
Human sleep is organized into a small number of cycles per night: deep sleep dominates early cycles, REM lengthens toward morning, and light sleep clusters at cycle boundaries. These regularities have long been described qualitatively but never distilled into closed-form laws from large-scale data. A total of 1,106 nights (Sleep-EDF 153 + SHHS 953) are segmented into REM-anchored cycles, and four cycle-indexed statistics follow simple closed-form equations: cycle length declines nearly linearly (R^2=0.996), REM duration follows an inverted U (R^2=0.873), deep-sleep (N3) run length decays exponentially (R^2=0.987) with a time constant of 2.94 cycles, consistent with the homeostatic Process S of the two-process model, and light-sleep (N1) run length follows a U shape (R^2=0.956), confirming a "long-short-long" pattern across the night. The laws are reproduced by per-night differentiable fitting (142/142 and 773/773 nights converge), by unconstrained symbolic regression (PySR), and by a 500-night generative synthesizer built from them. Their robustness is established against the two main methodological threats: bootstrap CIs computed by resampling whole nights (1,000 replications) exclude zero for every shape term, and a five-level scan of the REM-merge segmentation threshold preserves every qualitative form. These appear to be the first closed-form laws governing the macro-structure of sleep, distilled and tested across an age-mixed sample (healthy young adults and community-based older adults).
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BibTeX
@misc{gao2026governing,
title = {{Governing Equations of Human Sleep Macro-Structure:Symbolic Discovery from 1,106 Nights}},
author = {Junhui Gao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00550},
eprint = {2611.00550},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00550},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Governing Equations of Human Sleep Macro-Structure:Symbolic Discovery from 1,106 Nights
AU - Junhui Gao
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00550
UR - https://arxiv.org/abs/2611.00550
AB - Human sleep is organized into a small number of cycles per night: deep sleep dominates early cycles, REM lengthens toward morning, and light sleep clusters at cycle boundaries. These regularities have long been described qualitatively but never distilled into closed-form laws from large-scale data. A total of 1,106 nights (Sleep-EDF 153 + SHHS 953) are segmented into REM-anchored cycles, and four cycle-indexed statistics follow simple closed-form equations: cycle length declines nearly linearly (R^2=0.996), REM duration follows an inverted U (R^2=0.873), deep-sleep (N3) run length decays exponentially (R^2=0.987) with a time constant of 2.94 cycles, consistent with the homeostatic Process S of the two-process model, and light-sleep (N1) run length follows a U shape (R^2=0.956), confirming a "long-short-long" pattern across the night. The laws are reproduced by per-night differentiable fitting (142/142 and 773/773 nights converge), by unconstrained symbolic regression (PySR), and by a 500-night generative synthesizer built from them. Their robustness is established against the two main methodological threats: bootstrap CIs computed by resampling whole nights (1,000 replications) exclude zero for every shape term, and a five-level scan of the REM-merge segmentation threshold preserves every qualitative form. These appear to be the first closed-form laws governing the macro-structure of sleep, distilled and tested across an age-mixed sample (healthy young adults and community-based older adults).
ER -