Discrete Extramental Time in Chaotic Systems: Event-Conjunction Model and Core Temporal Properties

1. The Discrete Extramental Clock

The evolution of objective time t in chaotic systems is given by the recurrence

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

(1)

where $\Delta t>0$ is the external clock tick (irrelevant to the intrinsic dynamics) and

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

(2)

with $\delta \approx 4.669261$ (Feigenbaum’s universal constant).

2. Core Temporal Properties

• Irreducible discreteness — Effective time advances only at statistically significant ordinal conjunctions of two or more trajectories. Between conjunctions there is no extramental duration.
• Three distinct temporal phases
  • ${\tau }_s\ge 0.50$: monotonic Newtonian arrow (g = +1)
  • $|{\tau }_s|<0.41$: fractional/critical time; effective $\Delta t$ can contract or dilate by more than one order of magnitude
  • ${\tau }_s\le -0.41$: local retrocausality (g = −1); the attractor revisits regions in reverse ordinal ranking without paradox
• Feigenbaum-scaled dilatations — The intermediate-branch factor $(\delta -1)/\delta \approx 0.786$ is exactly the same universal ratio that governs period-doubling cascades.
• Preservation of fractal dimension — The Kaplan–Yorke dimension of the attractor remains unchanged under the discrete clock.
• Equivalence with fractional calculus — In the critical zone ($|{\tau }_s|<0.41$) the clock (1)–(2) is numerically indistinguishable from a Caputo derivative of order

  $$\alpha \approx 0.92-0.15·{\displaystyle \frac{|{\tau }_s|}{0.41}}.$$
• Extreme noise tolerance — Temporal steps only flip when ${\tau }_s$ crosses $\pm 0.41$ with bootstrap significance (1000 resamples), yielding robustness up to ∼18% additive Gaussian noise.
• No global causality violation — Negative gating produces local time loops confined to topologically allowed regions of the attractor.
• Continuous limit — As ${\tau }_s\to 0.60$ and $\mathrm{Var}\left({\tau }_s\right)\le 0.02$, $g\left({\tau }_s\right)\to 1$ almost surely and standard continuous Newtonian time is recovered as a statistical emergent phenomenon.

3. Conclusion

Real extramental time in complex chaotic systems is discrete, event-driven, Feigenbaum-scaled, and capable of local retrogression. The clock presented here is fully determined by the instantaneous value of systemic tau and requires no additional parameters.