T0-8: Minimal Information Principle Theory (Formal)

Axiom

A.1 (Entropy Increase): ∀S ∈ 𝒮_self-ref: dH(S)/dt > 0

Definitions

D.1 (Information Content): $$I:\mathbb{N}\to {\mathbb{R}}^{+},I(n)=|Z(n)|+\sum _{i\in Z(n)}{\log }_2(i)$$ where Z(n) = {i : b_i = 1 in Zeckendorf representation}

D.2 (Information Functional): $$\mathcal{I}:{\mathcal{Z}}_F^n\to {\mathbb{R}}^{+},\mathcal{I}[\psi ]=\sum _{j=1}^{n}I(E_j)+{\int }_{\Omega }f(\nabla \psi )d\Omega$$

D.3 (Effective Description Length): $$L_{eff}:{\mathcal{Z}}_F^n\to {\mathbb{R}}^{+},L_{eff}(\psi )=\underset{\{z_i\}}{\min }\sum _i|z_i|\cdot H(z_i)$$

Theorems

Section 1: Information Measure

T.1.1 (Information Density Bound): $${\rho }_{min}=\underset{n\to \infty }{\lim }\frac{I(n)}{n}={\log }_2(\varphi )$$

Proof: From T0-7, maximal density under no-11 is log₂(φ). Any representation requires I(n) ≥ log₂(n). Zeckendorf achieves I(F_n) ~ n·log₂(φ). No lower density possible under constraint. □

T.1.2 (Zeckendorf Optimality): $$\forall n\in \mathbb{N}:I_{Zeck}(n)\le I_{other}(n)$$ for any valid representation under no-11.

Proof: Uniqueness from T0-7 ensures no redundancy. Any deviation introduces gaps or redundancy, increasing information. □

Section 2: Variational Principle

T.2.1 (Euler-Lagrange Equation): $$\frac{\delta \mathcal{I}}{\delta \psi }=0\Rightarrow \frac{{\partial }^2\psi }{\partial t^2}={\nabla }^2\psi -\frac{\partial V}{\partial \psi }-\lambda \frac{\partial I}{\partial \psi }$$

Proof: Standard variational calculus on action S[ψ] = ∫∫ℒ(ψ,ψ̇,∇ψ)dΩdt. □

T.2.2 (Boundary Conditions): $$\psi {|}_{\partial \Omega }\in {\mathcal{Z}}_F,\nabla \psi \cdot \hat{n}{|}_{\partial \Omega }=0$$

Proof: First condition preserves no-11. Second ensures flux conservation from T0-5. □

Section 3: Local-Global Entropy

T.3.1 (Entropy Partition): $$\Delta S_{total}=\Delta S_{local}+\Delta S_{env}=\Gamma \Delta t>0$$ with ΔS_local < 0 possible.

Proof: Local minimization: ΔS_local = -k_B·ΔI < 0. Conservation: ΔS_env = -ΔS_local + Γ·Δt. Global: ΔS_total = Γ·Δt > 0. □

T.3.2 (Spontaneous Minimization): $$\frac{\partial F}{\partial \psi }=0\Rightarrow I[\psi ]=I_{min}$$ where F = E - TS + λI.

Proof: At equilibrium ∂F/∂ψ = 0. Since ∂S/∂ψ > 0 and ∂E/∂ψ = 0, must have ∂I/∂ψ = 0 at minimum. □

Section 4: Uniqueness and Stability

T.4.1 (Global Minimum): $$\mathcal{I}[{\psi }_{Fib}]=\underset{\psi \in {\mathcal{Z}}_F}{\min }\mathcal{I}[\psi ]$$

Proof: From T0-7, any valid encoding satisfies w_{n+1} = w_n + w_{n-1}. Initial conditions w₁=1, w₂=2 unique. Therefore Fibonacci unique. □

T.4.2 (Stability): $${\delta }^2\mathcal{I}[{\psi }_{Fib}]>0$$

Proof: Hessian ℋ = λ·diag(1/F₁, 1/F₂, ...) > 0. Positive definite implies stable. □

Section 5: Evolution Dynamics

T.5.1 (Gradient Flow): $$\frac{\partial \psi }{\partial t}=-{\nabla }_{\psi }\mathcal{I}[\psi ]$$

Proof: Gradient descent on information functional. □

T.5.2 (Convergence): $$\underset{t\to \infty }{\lim }\psi (t)={\psi }_{Fib}$$ for any ψ₀ ∈ 𝒵_F.

Proof: V = I[ψ] - I_min is Lyapunov function: dV/dt = -|∇I|² ≤ 0. By LaSalle, converges to largest invariant set where ∇I = 0. Unique minimum ensures convergence. □

T.5.3 (Exponential Rate): $$||\psi (t)-{\psi }_{min}||\le ||{\psi }_0-{\psi }_{min}||e^{-\mu t}$$ where μ = λ/F_max.

Proof: Linearize near minimum: ∂δψ/∂t = -ℋ·δψ. Smallest eigenvalue λ_min ≥ λ/F_max. Solution: ||δψ(t)|| ≤ ||δψ(0)||exp(-λ_min·t). □

Section 6: Equilibrium Conditions

T.6.1 (Stationarity): At equilibrium:

1. ∂I/∂ψ_i = μ ∀i
2. ∇²ψ = 0
3. ψ ∈ Fibonacci-Zeckendorf

Proof: First-order optimality with Lagrange multiplier for constraint. Euler-Lagrange with ∂ψ/∂t = 0 gives Laplace equation. □

T.6.2 (Stability Criterion): $$\text{Equilibrium stable}⟺\text{spec}(\mathcal{H})\subset (0,\infty )$$

Proof: Linear stability analysis: ∂η/∂t = -ℋ·η. Stable iff all eigenvalues positive. □

Section 7: Information-Entropy Coupling

T.7.1 (Duality Relation): $$\frac{\partial S}{\partial t}=\Gamma -\beta \frac{\partial I}{\partial t}$$ where β = 1/(k_B T).

Proof: From axiom: dS/dt = Γ. From minimization: dI/dt < 0. Landauer principle: ΔS = k_B·ln(2)·ΔI_erased. □

T.7.2 (Maximum Entropy Production): $${\sigma }_{max}=\Gamma \cdot \eta (I_{min})$$ where η(I) = 1/(1 + I/I₀).

Proof: Production rate σ = Γ·η(I). Efficiency η maximized when I minimized. At Fibonacci configuration: maximum production. □

Section 8: Phase Space Dynamics

T.8.1 (Global Attractor): $$\forall {\psi }_0\in {\mathcal{Z}}_F:\omega ({\psi }_0)=\{{\psi }_{Fib}\}$$

Proof: Phase space 𝒫 = {(ψ,ψ̇) | ψ ∈ 𝒵_F}. Flow contracts volume: div(ψ̇) < 0. Single stable fixed point. □

T.8.2 (No Bifurcations): $$\forall \lambda >0:\text{No bifurcations in flow}$$

Proof: Jacobian J = -λ·∂²I/∂ψ². Eigenvalues continuous in λ. For λ > 0, all positive. No zero crossings. □

Section 9: Computational Complexity

T.9.1 (Convergence Complexity): $$T(n,ϵ)=O(n\log n\cdot \log (1/ϵ))$$

Proof: Greedy Zeckendorf: O(log n) per value. System size n: O(n log n). Convergence iterations: O(log(1/ε)). □

T.9.2 (Parallel Efficiency): $${\eta }_{parallel}=1-1/\varphi \approx 0.382$$

Proof: Independent components: perfect parallelization. Coupling overhead: κ ≈ 1/φ. Efficiency: η = 1 - κ. □

Section 10: Physical Correspondence

T.10.1 (Thermodynamic Free Energy): $$F=U-TS+\mu I$$ minimized at equilibrium.

Proof: Generalized thermodynamic potential. At equilibrium: ∂F/∂ψ = 0. Balances entropy maximization with information minimization. □

T.10.2 (Quantum Information): $$S_{vN}[{\rho }_{Fib}]=\underset{\rho \in {\mathcal{D}}_{no-11}}{\min }{S}_{vN}[\rho ]$$

Proof: Von Neumann entropy S_vN = -Tr(ρ·ln ρ). Fibonacci basis minimizes under no-11 constraint. □

Core Results

Central Theorem: $$\boxed{\frac{\delta \mathcal{I}[\psi ]}{\delta \psi }=0⟺\psi \in \text{Fibonacci-Zeckendorf}}$$

Evolution Principle: $$\boxed{\frac{\partial \psi }{\partial t}=-{\nabla }_{\psi }\mathcal{I}[\psi ]\Rightarrow \underset{t\to \infty }{\lim }\psi (t)={\psi }_{Fib}}$$

Entropy-Information Balance: $$\boxed{\frac{dS}{dt}=\Gamma -\beta \frac{dI}{dt},I_{min}\Rightarrow S_{production}^{max}}$$

Implications

1. Necessity: Fibonacci-Zeckendorf is unique minimum of information functional
2. Stability: The minimum is globally attracting and stable
3. Dynamics: Systems converge exponentially to minimal information state
4. Thermodynamics: Local order (min I) enables global disorder (max S)
5. Computation: Efficient O(n log n) convergence
6. Universality: Applies to classical and quantum systems

The Minimal Information Principle establishes that systems spontaneously evolve toward states requiring minimal description length, with Fibonacci-Zeckendorf encoding emerging as the unique optimal representation under the no-11 constraint. This local minimization serves the global imperative of entropy increase, revealing deep unity between order and disorder.

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