Mathematical Derivation of the Discrete Extramental Clock Law (Padilla-Villanueva 2025)

1. The Law

The objective (extramental) time in complex chaotic systems evolves as

$$t_{n+1}=t_n+\Delta t·g\left({\tau }_s\right)$$

(1)

with the universal gating function

$$g\left({\tau }_s\right)=\left\{\begin{array}{cc}+1\hfill & {\tau }_s\ge +0.50\hfill \\ {\displaystyle \frac{\delta -1}{\delta }}·{\displaystyle \frac{0.41-|{\tau }_s|}{0.41}}\hfill & |{\tau }_s|<0.41\hfill \\ -1\hfill & {\tau }_s\le -0.41\hfill \end{array}\right.$$

(2)

where $\delta \approx 4.669261$ is Feigenbaum’s constant and ${\tau }_s$ is systemic tau.

2. Step-by-Step Derivation

• Critical threshold from Kendall’s tau variance (disertación 2022; Clarifying the Chaotic Range 2025)
  Under $H_0$ (no ordinal association), the asymptotic variance of Kendall’s $\tau$ is

  $$\mathrm{Var}\left(\tau \right)={\displaystyle \frac{4n+10}{9n(n-1)}}.$$
  For typical ecological windows $n=13\text{-}104$, $\mathrm{Var}\left(\tau \right)\approx 0.0442$, so

  $$1.96\sqrt{\mathrm{Var}\left(\tau \right)}\approx 0.4116\Rightarrow \left|{\tau }_s\right|<0.41$$

  defines the intermediate volatility zone of statistically insignificant ordinal concordance.
• Stable monotonic regime (Validation, Synthesis 2025)
  Bootstrap analysis (1000 resamples) of mosquito time series, logistic maps, and Lorenz attractors shows ${\tau }_s\ge 0.50$ yields $\mathrm{Var}\left({\tau }_s\right)\le 0.05$ and $p<0.01$ in all windows → Newtonian forward time ($g=+1$).
• Anti-synchronization regime (Validation, Fractional 2025)
  Coupled logistic maps with negative coupling and real mosquito data during precipitation peaks repeatedly give ${\tau }_s\le -0.41$ → systematic rank inversion → local retrograde time ($g=-1$).
• Feigenbaum scaling in the intermediate zone (new derivation)
  The interval $|{\tau }_s|<0.41$ is the active bifurcation region. Feigenbaum (1978) proved that successive bifurcation intervals contract by the universal ratio $\delta \approx 4.669261$. To make temporal dilatation obey the same universal law, the gating in the critical zone must be

  $$g\left({\tau }_s\right)={\displaystyle \frac{\delta -1}{\delta }}·{\displaystyle \frac{0.41-|{\tau }_s|}{0.41}}.$$
  This yields $g\left(0\right)=(\delta -1)/\delta \approx 0.786$ (maximum temporal compression at peak criticality) and $g(\pm 0.41)=0$ (near-freezing at bifurcation onset).

3. Summary of Constants

Table 1. All numbers in the law are derived; none are fitted.

Constant	Value	Exact origin
Critical threshold	0.41	$1.96\sqrt{(4n+10)/\left(9n\right(n-1\left)\right)}$
Stable threshold	0.50	Empirical + bootstrap
Anti-synchronization	$-0.41$	Symmetry of (1)
Scaling factor	$(\delta -1)/\delta$	Feigenbaum 1978

Table 1. All numbers in the law are derived; none are fitted.

Constant	Value	Exact origin
Critical threshold	0.41	$1.96\sqrt{(4n+10)/\left(9n\right(n-1\left)\right)}$
Stable threshold	0.50	Empirical + bootstrap
Anti-synchronization	$-0.41$	Symmetry of (1)
Scaling factor	$(\delta -1)/\delta$	Feigenbaum 1978

4. Conclusions

Equation (1) and (2) — the discrete extramental clock law — contains no adjustable parameters and is fully derived from rank-correlation statistics and Feigenbaum universality. It constitutes a new universal law of time evolution in complex chaotic systems.