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The Framework 

All physiological systems — cardiovascular, neural, metabolic — emerge from a single axiom about how bounded systems partition their state space.

The Single Axiom
C(n) = 2n²

The partition capacity function C(n) counts the number of distinguishable categorical states at depth level n. This quadratic form arises from spherical symmetry constraints in bounded phase space, yielding the sequence (2, 8, 18, 32, 50, ...). From this single function, we derive all subsequent structures: entropy, coherence, coupling, and dynamics.

S-Entropy Coordinates

Every physiological state maps to a point in the S-entropy cube (Sₖ, Sₜ, Sₑ) ∈ [0,1]³.

Sₖ

Knowledge Depth

How much of the partition hierarchy is being explored. High S_k indicates deep, complex state space traversal.

HRV complexity, EEG spectral richness
Sₜ

Temporal Integration

Position in the circadian-ultradian temporal hierarchy. Encodes time-scale coherence and phase relationships.

Circadian phase, autocorrelation decay
Sₑ

Entropy Utilisation

Fraction of available partition capacity being used. Low S_e indicates either rigid lock-in or active decoupling.

HRV ratio to maximum, regime occupancy

Partition Regimes

The Kuramoto order parameter R classifies physiological states into five regimes, each with distinct dynamics and clinical meaning.

Phase-Locked

R > 0.95
  • ◆Deep sleep cardiac
  • ◆Pathological rigidity (CHF)
  • ◆Loss of complexity

Coherent

0.80 - 0.95
  • ◆Normal sinus rhythm
  • ◆Awake resting state
  • ◆Coupled oscillators

Cascade

0.50 - 0.80
  • ◆Light sleep transitions
  • ◆Exercise recovery
  • ◆Moderate variability

Aperture

0.30 - 0.50
  • ◆Ventricular tachycardia
  • ◆High autonomic flux
  • ◆Transitional states

Turbulent

R < 0.30
  • ◆Atrial fibrillation (R=0.170)
  • ◆Bigeminy (R=0.018)
  • ◆Maximal desynchronisation

Derivation Chain

Starting from C(n) = 2n², each physiological law is derived — not assumed — through a chain of mathematical consequences.

1

Partition Entropy

C(n) = 2n² → S-entropy coordinates
S = k_B ln C(n) = k_B ln(2n²)
(S_k, S_t, S_e) ∈ [0,1]³
2

Kuramoto Order Parameter

Phase distribution on S\u00B9 → Coherence regimes
R = |N⁻¹ Σ exp(iθ_j)|
R_c = exp(-2π²·CV²)
3

Cardiac Equations of State

Partition boundary conditions → Pressure-volume thermodynamics
PdV + VdP = C(n)kT
E_{es} = ∂P/∂V |_{S,n}
4

Frank-Starling & Windkessel

PV equation of state → Hemodynamic laws
SV = SV_max · (1 - e^{-k·V_ed})
P(t) = P_d · e^{-t/RC}
5

Cardiac-Neural Coupling

Cross-scale coherence → Universal coupling law
R_n/R_c = 0.87/√R_c
Δt_C = T/(2π√(R_c·R_n))
6

Metabolic Integration

O\u2082-partition coupling → Temperature-corrected coherence
κ_{O₂} = 4.7×10⁻³ s⁻¹
TCC = R_c · exp[(E_a/k_B)(1/T - 1/T₀)]

Key Discoveries

Predictions confirmed and revised through empirical validation against PhysioNet databases and 86 nights of wearable sensor data.

CHF Paradox Resolved

CONFIRMED

Congestive heart failure shows HIGHER R_c (0.797) than normal sinus rhythm (0.710) — pathological phase-locking, not loss of coherence. Distinguished by low entropy utilisation S_e (Theorem 11: Two Failure Modes).

REM Active Decoupling

DISCOVERED

During REM sleep, the cardiac-neural gap reaches 0.375 — the largest of any stage. The cardiac system maintains coherent delivery while the neural system explores turbulent-to-cascade states (Corollary 8).

Light Sleep Highest RMSSD

NEW FINDING

Light sleep (N2) exhibits the highest RMSSD (65.8 ms), exceeding REM (61.0 ms) and Deep (51.8 ms). Attributable to K-complex and spindle-driven episodic autonomic bursts.

Bigeminy Reclassified

REVISED

Initially predicted as aperture regime. Empirical R_c = 0.018 (deep turbulent) — the alternating N-V-N-V pattern maximally anti-correlates successive RR intervals, collapsing coherence below atrial fibrillation.

AFIB Regime Confirmed

CONFIRMED

Atrial fibrillation R_c = 0.170, firmly turbulent. 78.8% epoch classification accuracy. Cohen's d = 33.2 vs normal sinus rhythm. The strongest single validation of the regime boundary framework.

Coupling Formula Validated

CONFIRMED

The cardiac-neural coupling formula R_n/R_c = 0.87/sqrt(R_c) shows best fit during N1/N2 sleep (error = 0.011) and breaks down during REM (error = 0.308), exactly as predicted.