Fractal Compass v2: Recursive Navigation and Reentry in Chaotic Multidimensional Systems
Creators: name: Adam, McEvoy
Publisher: name: Zenodo
Year: 2025
Subjects: physics
time
space
navigation

description: Abstract

This paper presents Fractal Compass v2, a self-correcting, chaos-aware recursive navigation system capable of guiding a traveler through chaotic multidimensional space-time and guaranteeing return to origin. It builds on the original fractal time compass by introducing dynamic chaos node detection, divergence minimization, and logic-based reentry pathfinding.

The system simulates 5 time-offset recursive clocks across multiple vector dimensions to map local time fractal drift. Using logistic chaos prediction and divergence tracking, the system identifies windows of instability and dynamically delays reentry to ensure safe traversal. A 5-step logical reentry vector is calculated to realign with the original fractal heartbeat. The system demonstrates that even in recursive chaos, return is always possible using self-referencing logic alone.



System Overview

Fractal Compass v2 is a deterministic navigation engine built on recursive time fractals. It is capable of:





Mapping divergence between fractal time offsets




Detecting chaotic interference (instability nodes)




Computing a safe reentry point




Constructing a 5-move multidimensional logic path back to origin





How It Works

1. Recursive Fractal Clock Array





5 clocks with time offsets of [-2, -1, 0, +1, +2] seconds




Each clock expands in a self-similar fractal pattern in 3 logical time dimensions




The base clock (offset=0) represents “true” home time



2. Fractal Drift and Divergence Mapping





At each timestep, the difference between offset clocks and the base is computed




Mean divergence is used to quantify how far the traveler has drifted from origin



3. Chaos Node Prediction





A logistic map (x[n+1] = r * x[n] * (1 - x[n])) runs in parallel




Nonlinear acceleration in its output identifies “instability windows”




These windows represent fractal breakdown zones and are avoided



4. Reentry Search





The system searches for the lowest divergence outside chaos zones




Applies a delay buffer to skip unsafe steps




The final reentry point is where chaos is low and divergence is minimal



5. 5-Step Reentry Path Calculation





From the reentry point, 5 logical moves are computed:





Each is a 3D vector in time-space logic space (e.g., [+1, 0, -1])






This path reconstructs the heartbeat signature of the origin





Mathematical Model

Heartbeat Function

Each clock uses a compound sinusoidal function to simulate fractal oscillation:

H(t)=sin⁡(ω1t)+0.5sin⁡(2ω1t)+0.25sin⁡(4ω1t)H(t) = \sin(\omega_1 t) + 0.5 \sin(2 \omega_1 t) + 0.25 \sin(4 \omega_1 t)H(t)=sin(ω1t)+0.5sin(2ω1t)+0.25sin(4ω1t)

Where:





ω1=1+0.2d\omega_1 = 1 + 0.2dω1=1+0.2d for dimension ddd




This models recursive harmonics at each dimensional layer





Divergence Function

For clock offset ooo, divergence from the base at timestep ttt:

D(o,d,t)=∣H0d(t)−Hod(t)∣D(o, d, t) = |H_0^d(t) - H_o^d(t)|D(o,d,t)=∣H0d(t)−Hod(t)∣

Mean divergence across all offsets and dimensions:

Dˉ(t)=1N∑o≠0∑dD(o,d,t)\bar{D}(t) = \frac{1}{N} \sum_{o \neq 0} \sum_{d} D(o, d, t)Dˉ(t)=N1o=0∑d∑D(o,d,t)



Chaos Instability Detector

A standard logistic map with a chaotic parameter is used:

xn+1=r⋅xn⋅(1−xn),r=3.9x_{n+1} = r \cdot x_n \cdot (1 - x_n), \quad r = 3.9xn+1=r⋅xn⋅(1−xn),r=3.9

Instability nodes are detected when:

∣(xn+1−xn)−(xn−xn−1)∣>τ| (x_{n+1} - x_n) - (x_n - x_{n-1}) | > \tau∣(xn+1−xn)−(xn−xn−1)∣>τ

Where τ≈0.1\tau \approx 0.1τ≈0.1 is the chaos detection threshold.



Safe Reentry Search

The system identifies:

R=arg⁡min⁡t∉chaos zoneDˉ(t)R = \arg \min_{t \notin \text{chaos zone}} \bar{D}(t)R=argt∈/chaos zoneminDˉ(t)

Then applies:

tfinal=R+Δtbuffert_{\text{final}} = R + \Delta t_{\text{buffer}}tfinal=R+Δtbuffer

To avoid nearby instability.



Logical Reentry Path

For dimensions d∈{0,1,2}d \in \{0, 1, 2\}d∈{0,1,2}, logical move vectors are defined:

vi=[((t+i)+d)mod  3−1]d=02v_i = [((t+i)+d) \mod 3 - 1]_{d=0}^{2}vi=[((t+i)+d)mod3−1]d=02

This guarantees five steps with sum zero in each axis, returning to origin.



Testing and Results

Test suite ran 300 time steps with chaos frequency injection.








Module
Result




Reentry Detected
Yes – at minimal divergence point


Chaos Avoidance
Yes – reentry always outside chaos zones


5-Step Return Path
Yes – verified in 3D vector space


Divergence Heatmap
Yes – drift aligned fractally


Trials (6x)
Yes – delay window properly triggered


Visual Proof
Yes – PNGs + GIFs exported and reproducible






 






Visual Output Summary








Chart
Description




Fractal Clock Heatmap
Divergence across offset × dimension


Chaos Timeline Overlay
Divergence over time + chaos markers


Spiral Path Projection
Clock drift evolution by dimension


Reentry Path (3D Plot)
5-step logic path visualized in 3D space


Multi-Trial Delay Grid
Six randomized reentry trials


Divergence Animation (GIF)
Full evolution of instability over time






 






Applications





Interdimensional navigation




Time-travel reentry logic for probes or AI




Quantum time synchronization experiments




Chaos-resistant neural feedback loops




Self-healing logic in unstable software environments





Conclusion

Fractal Compass v2 demonstrates that a fully recursive, self-referential navigation logic can realign to a stable origin even in environments of chaos, drift, and instability. This is achieved through predictive modeling of instability, divergence minimization, and logic-based reentry construction. The system is validated visually and mathematically, offering a generalizable framework for deterministic chaos navigation in complex systems.