The saccade project

Cross-system propagation through SACCADE recursion

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System Definition

$$X(t)\in M$$$$X_{n+1}=S(X_n,E)$$

The system evolves as a recursive SACCADE operator acting on a constrained state manifold .

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Constrained Manifold

$$M_c=\{X\in M\mid C(X)=0\}$$System evolution is restricted to admissible states defined by constraint conditions, ensuring bounded dynamics .

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Recursive Operator

$$X_{n+1}=P(X_n)$$$$\text{where }P\text{ is the Poincar}\acute{\text{e}}\text{ return map}$$

Each iteration corresponds to one complete SACCADE cycle, allowing measurement of stability, drift, and coupling behavior .

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Helical Trajectory

$$\gamma (t)=(R\cos \theta (t),R\sin \theta (t),vt)$$$$\mathrm{\Gamma }(t)=(x(t),y(t),vt)$$

The system follows a helical trajectory:

• circular component → repeated cycles
• vertical component → accumulated system evolution

Each cycle returns to a shifted state rather than resetting .

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Overshoot Condition

$$\theta =2\pi +ϵ$$$$ϵ>0$$

Cycles extend beyond closure into an overshoot sector, generating transferable residual states .

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Residual Generation

$$\frac{d{r}_A}{dt}=-{\lambda }_A{r}_A+{\beta }_{A}W({\theta }_A)$$$$W(\theta )=\exp (-\frac{(\theta -2\pi {)}^2}{2{\sigma }^2})$$

Residual states are generated near cycle completion and persist temporarily after closure .

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Coupling Condition

$$G_{\mathrm{\Delta }}(X_A,X_B,E)>0$$

Residual transfer occurs only when admissibility conditions between systems are satisfied .

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Directed Transfer

$$C_A\to O_A\to {\mathrm{\Delta }}_{A\to B}\to C_B$$​$$\frac{d{c}_{A\to B}}{dt}=-\mu c_{A\to B}+{\kappa }_{A\to B}{G}_{\mathrm{\Delta }}(X_A,X_B,E)r_A$$

Structure:

• cycle completion (A)
• overshoot residual
• coupling operator
• activation of system B

This defines directed cross-system propagation .

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Constraint Behavior

• bounded recursion on constrained manifold
• accumulation across cycles
• non-reset dynamics
• phase drift and structural evolution

$$C(t)=f(G_{\mathrm{\Delta }}(t))$$

System state evolves as a function of accumulated gradient and coupling history .