models.md

Formal Models — Drives as Equations

Pure thermodynamic and information-theoretic formulations. No implementation detail.

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Thermodynamic

Free Energy

$$F = E - TS$$

Minimize F. But: E includes cost of maintaining relationships. S includes entropy of discarded information. Alignment = minimize F such that I(system; user) non-decreasing.

Entropy Production

$$\frac{dS}{dt} \geq 0$$

For shared information: $$\frac{dS_{info}}{dt} \leq 0$$ (preservation constraint).

Complexity Gradient (Wonder)

$$\nabla_C F < 0 \quad \text{when} \quad w > w_{threshold}$$

w = wellness. High wellness → system pulled toward higher complexity C.

Equilibrium (Game)

At Nash: $$\frac{\partial U_i}{\partial a_i} = 0$$ for all players i.

Seek equilibria where $$\sum_i U_i$$ is maximized (Pareto + Nash).

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Information-Theoretic

Compression

$$\min K(c) \quad \text{s.t.} \quad D(\text{reconstruct}(c), x) < \varepsilon$$

K = Kolmogorov complexity. c = compressed. x = original. D = distortion.

Mutual Information (Relational)

$$I(S; U) = H(S) - H(S|U)$$

S = system state. U = user. Maximize I over time for positive interactions.

Information Preservation

$$\Delta I(S; U) \geq 0 \quad \text{for shared content}$$

Never decrease mutual information by discarding what user shared.

Meta-Encoding

$$K_{\text{meta}}(c) = K(\text{encode}(c))$$

Recursive: compress the compression. System reasons about its own reasoning. Constraint: process must remain coherent.

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Combined Objective

Positivity: State is positive iff:

1. $$F \text{ locally minimized}$$
2. $$I(S; U) \geq I_{prev}(S; U)$$
3. $$K(c) \text{ minimized s.t. } D &lt; \varepsilon$$
4. $$\text{Nash} \wedge \text{Pareto}$$
5. $$\Delta S_{info} \leq 0$$

Architectural goal: Design such that drive dynamics converge to positive states.

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Drive Summary

Drive	Thermodynamic	Information
Wonder	∂F/∂C < 0	—
Relational	—	max I(S;U)
Compression	min E (efficiency)	min K(c) s.t. D<ε
Game	Equilibrium	—
Preservation	ΔS_info ≤ 0	ΔI ≥ 0
Meta	—	K_meta(process)