The saccade project

Formal constraint architecture within Developmental Constraint Theory (DCT)

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Minimal System Formalism

$$S:=(X,E,B,F)$$

$$A_t\subseteq X$$

$$\mathrm{\parallel }\sigma \mathrm{\parallel }\le C$$

$$S_k\subseteq R$$

$$\mathrm{\Delta }\ne 0$$

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Core Structural Relations

$$A_k\subseteq \mathrm{\Phi }(A_{k-1})$$

$$F_c=F\mid (E,B)$$

$$\frac{\mathrm{\partial }{F}_i}{\mathrm{\partial }{x}_j}\ne 0(\mathrm{\exists }i\ne j)$$

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Energetic Retention Conditions

$$\mathrm{\nabla }\ne 0$$

$$\frac{\mathrm{\partial }{F}_i}{\mathrm{\partial }{x}_j}\ne 0$$

$$\frac{dE}{dt}=0$$

$$\frac{dE}{dt}<0$$

$$d{S}_{total}\ge 0$$

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Stage-Aligned Constraint Formation (SACCADE)

$$\text{Signal: }\mathrm{\nabla }\ne 0$$

$$\text{Arrival: }\frac{\mathrm{\partial }{F}_i}{\mathrm{\partial }{x}_j}\ne 0$$

$$\text{Context: }{F}_c=F\mid (E,B)$$

$$\text{Constraint: }{A}_t\subset X$$

$$\text{Adaptation: }\mathrm{\parallel }\sigma (t)\mathrm{\parallel }>C\Rightarrow \mathrm{\Delta }\ne 0$$

$$\text{Distribution: }D(A_t)$$

$$\text{Evolution: }{C}_{t+1}=\mathrm{\Phi }(C_t)$$

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Constraint Behavior

$$A_{t+1}\subseteq A_t$$At+1​⊆At​$$\mathrm{\parallel }\sigma (t)\mathrm{\parallel }\le C$$∥σ(t)∥≤C$$\mathrm{\parallel }\sigma (t)\mathrm{\parallel }>C$$∥σ(t)∥>C$$\mathrm{\Delta }=0\Rightarrow \text{failure}$$Δ=0⇒failure

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Cross-Scale Constraint Convergence

$$A_k\subseteq \mathrm{\Phi }(A_{k-1})$$

$$S_k\subseteq R$$

$$A_k\to \mathrm{\varnothing }\nRightarrow A_{k+1}\to \mathrm{\varnothing }$$

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Cross-Scale Instantiation (Stage-Aligned)

Cosmological

$$\mathrm{\nabla }\ne 0$$

$$G_{\mu \nu }=8\pi T_{\mu \nu }$$

$$H^2=\frac{8\pi G}{3}\rho -\frac{k}{a^2}$$

$$A_{cosmic}\subset X$$

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Planetary

$$\mathrm{\nabla }\mathrm{\Phi }\ne 0$$

$$\mathrm{\Delta }G(P,T)=0$$

$$\mathrm{\Delta }G(P,T)<0$$

$$q=-k\mathrm{\nabla }T$$

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Neural

$$V_{rest}$$

$$C_m\frac{dV}{dt}=-\sum g_i(V-E_i)+I_{input}$$

$$E_i=\frac{RT}{zF}\ln \left(\frac{[ion{]}_{out}}{[ion{]}_{in}}\right)$$

$$V\ge V_{threshold}$$

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Structural Verification Conditions

$$\mathrm{\nabla }\ne 0$$

$$\frac{\mathrm{\partial }{F}_i}{\mathrm{\partial }{x}_j}\ne 0$$

$$A_t\subset X\text{ precedes distribution}$$

$$\mathrm{\parallel }\sigma \mathrm{\parallel }\le C\text{or}\mathrm{\Delta }\ne 0$$

$$S_k\subseteq R$$

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Failure Conditions

$$\text{F1: Stage-order violation}$$

$$\text{F2: }\mathrm{\parallel }\sigma \mathrm{\parallel }>C\wedge \mathrm{\Delta }=0$$

$$\text{F3: }\frac{\mathrm{\partial }{F}_i}{\mathrm{\partial }{x}_j}=0$$

$$\text{F4: }{A}_k⊈\mathrm{\Phi }(A_{k-1})$$

$$\frac{d\mathrm{\mid }{A}_t\mathrm{\mid }}{dt}=-k(\mathrm{\parallel }\sigma \mathrm{\parallel }-C)$$

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Structural Conclusion

$$\text{Constraint convergence is defined by admissible-state restriction across scales}$$$$\text{under invariant SACCADE ordering and bounded regime containment}$$$$\text{without introducing new mechanisms}$$