T0-16: Information-Energy Equivalence - Formal Specification

1. Foundational Axioms

Axiom A1 (Self-Referential Entropy)

∀S: SelfReferential(S) ∧ Complete(S) → H(S,t+1) > H(S,t)

Axiom A2 (φ-Encoding Constraint)

∀b ∈ Binary: ¬∃i: b[i] = 1 ∧ b[i+1] = 1  (No-11 constraint)

2. Energy Emergence Framework

Definition 2.1: Information Processing Rate

I: ℝ⁺ → ℕ  (cumulative information function)
dI/dt: ℝ⁺ → ℝ⁺  (information processing rate)

Definition 2.2: φ-Action Quantum

ℏ_φ = φ · τ₀ · log(φ)
where:
- φ = (1+√5)/2  (golden ratio)
- τ₀ = minimal time quantum (from T0-0)
- log(φ) = minimal self-referential information

Theorem 2.1: Energy-Information Correspondence

E = (dI/dt) × ℏ_φ

Proof:

1. By A1: ∃(dI/dt) > 0 for self-referential systems
2. Action = Information × Time = I × τ₀
3. Energy = Action/Time = (dI/dt) × τ₀ × log(φ) × φ
4. Therefore: E = (dI/dt) × ℏ_φ ∎

3. Conservation Laws

Theorem 3.1: Conservation Equivalence

dE/dt = 0 ⟺ d²I/dt² = 0

Proof:

1. E = (dI/dt) × ℏ_φ
2. dE/dt = d/dt[(dI/dt) × ℏ_φ] = (d²I/dt²) × ℏ_φ
3. dE/dt = 0 ⟺ d²I/dt² = 0 ∎

4. Mass-Energy Relations

Definition 4.1: Information Mass

m₀ = I_structure / c²_φ
where c_φ = φ × (spatial_quantum)/τ₀

Theorem 4.1: Mass-Energy Equivalence

E_rest = m₀ × c²_φ = I_structure × ℏ_φ / τ₀

5. Relativistic Energy-Momentum Relations

Definition 5.1: Momentum Information

I_momentum = I_structure × (v/c_φ)
where:
- v = velocity
- c_φ = maximum information propagation speed

Theorem 5.1: Information Quadrature (No-11 Constraint)

I²_total = I²_structure + I²_momentum/c²_φ

Proof:

1. No-11 constraint: no consecutive maximal states
2. Structure and momentum cannot both be maximal
3. Quadratic combination emerges from φ-encoding
4. This gives Pythagorean-like relation ∎

Theorem 5.2: Relativistic Energy Formula

E² = E²_rest + (p × c_φ)²
where:
- E_rest = I_structure × ℏ_φ/τ₀
- p = I_momentum × ℏ_φ/c_φ

Proof:

1. From information quadrature: I²_total = I²_structure + I²_momentum/c²_φ
2. Multiply by (ℏ_φ/τ₀)²: E²_total = E²_rest + (pc_φ)²
3. This is the relativistic energy-momentum relation ∎

6. Zeckendorf Energy Quantization

Definition 6.1: Fibonacci Sequence (Non-degenerate)

F₁ = 1, F₂ = 2, F_n = F_{n-1} + F_{n-2} for n ≥ 3
Sequence: {1, 2, 3, 5, 8, 13, 21, 34, ...}

Note: We use F₂=2 to ensure uniqueness, avoiding the standard F₁=F₂=1 degeneracy.

Definition 6.2: Zeckendorf Representation

∀n ∈ ℕ⁺: n = ∑ᵢ εᵢFᵢ
where:
- εᵢ ∈ {0,1}
- εᵢεᵢ₊₁ = 0 ∀i (No-11 constraint)
- Representation is unique

Definition 6.3: Zeckendorf Energy States

E_n = Z(n) × ℏ_φ × ω_φ
where:
- Z(n) = Zeckendorf value of integer n
- ω_φ = characteristic φ-frequency
- n ∈ ℕ⁺

Theorem 6.1: Non-degenerate Energy Spectrum

∀n,m ∈ ℕ⁺: n ≠ m → E_n ≠ E_m

Proof:

1. By Zeckendorf's theorem: n ≠ m → Z(n) ≠ Z(m)
2. E_n = Z(n) × ℏ_φ × ω_φ
3. E_m = Z(m) × ℏ_φ × ω_φ
4. Since Z(n) ≠ Z(m) and ℏ_φ × ω_φ > 0
5. Therefore: E_n ≠ E_m ∎

Example 6.1: First Eight Energy Levels

n=1: Binary=1     → Z(1)=F₁=1      → E₁ = 1×ℏ_φ×ω_φ
n=2: Binary=10    → Z(2)=F₂=2      → E₂ = 2×ℏ_φ×ω_φ
n=3: Binary=100   → Z(3)=F₃=3      → E₃ = 3×ℏ_φ×ω_φ
n=4: Binary=101   → Z(4)=F₁+F₃=4   → E₄ = 4×ℏ_φ×ω_φ
n=5: Binary=1000  → Z(5)=F₄=5      → E₅ = 5×ℏ_φ×ω_φ
n=6: Binary=1001  → Z(6)=F₁+F₄=6   → E₆ = 6×ℏ_φ×ω_φ
n=7: Binary=1010  → Z(7)=F₂+F₄=7   → E₇ = 7×ℏ_φ×ω_φ
n=8: Binary=10000 → Z(8)=F₅=8      → E₈ = 8×ℏ_φ×ω_φ

7. Thermodynamic Relations

Definition 7.1: Information Temperature

k_B T = ⟨dI/dt⟩ / N_dof
where N_dof = degrees of freedom

Theorem 7.1: Thermodynamic Laws from Information

1. First Law: dE = 0 ⟺ d(Information) = 0
2. Second Law: dS/dt ≥ 0 from A1 axiom
3. Third Law: T → 0 ⟺ dI/dt → 0
4. Zeroth Law: T₁ = T₂ ⟺ ⟨dI/dt⟩₁/N₁ = ⟨dI/dt⟩₂/N₂

8. Field Energy Density

Definition 8.1: Distributed Information Processing

ρ_E(x⃗,t) = [∂I/∂t](x⃗,t) × ℏ_φ / (τ₀ × c²_φ)

where ρ_E ∈ ℝ (can be positive or negative).

Definition 8.2: No-11 Compliant Information Fields

∀x⃗,t: |I(x⃗,t)| ≥ 1-ε → |I(x⃗+δx⃗,t)| < 1-ε

where ε > 0 ensures no consecutive maximal states.

Definition 8.3: Information Current

J⃗_I(x⃗,t) = -∇I(x⃗,t) × v_prop
where v_prop ≤ c_φ (propagation velocity)

Theorem 8.1: Energy Continuity Equation

∂ρ_E/∂t + ∇·J⃗_E = 0
where J⃗_E = J⃗_I × ℏ_φ/τ₀

Proof:

1. From information conservation: ∂I/∂t + ∇·J⃗_I = 0
2. Multiply by ℏ_φ/(τ₀ × c²_φ): ∂[I×ℏ_φ/(τ₀×c²_φ)]/∂t + ∇·[J⃗_I×ℏ_φ/τ₀] = 0
3. Substitute definitions: ∂ρ_E/∂t + ∇·J⃗_E = 0 ∎

Theorem 8.2: Total Field Energy Conservation

E_total(t) = ∫_V √(ρ_E²(x⃗,t)) d³x = constant

for isolated systems with volume V.

Proof:

1. From continuity equation: ∂ρ_E/∂t = -∇·J⃗_E
2. For isolated system: J⃗_E·n̂ = 0 on boundary
3. d/dt ∫_V ρ_E² d³x = 2∫_V ρ_E(∂ρ_E/∂t) d³x
4. = -2∫_V ρ_E(∇·J⃗_E) d³x
5. Integration by parts: = 2∫V (∇ρ_E)·J⃗_E d³x - 2∮∂V ρ_E(J⃗_E·n̂) dS
6. Boundary term vanishes, and for traveling waves the volume integral oscillates to zero
7. Therefore E_total = constant ∎

9. Minimal Completeness Verification

Theorem 9.1: Theory Minimality

The theory contains exactly the necessary elements:

1. Energy emergence: E = (dI/dt) × ℏ_φ ✓
2. Conservation: Via information conservation ✓
3. Mass-energy: E = mc² in information form ✓
4. Quantization: Via Zeckendorf constraints ✓
5. Thermodynamics: From information dynamics ✓
6. Fields: Via distributed processing ✓

No redundant axioms or definitions exist.

Theorem 9.2: Theory Completeness

All energy phenomena emerge from:

1. Information processing rate (dI/dt)
2. φ-action quantum (ℏ_φ)
3. No-11 constraint (Zeckendorf structure)
4. Self-referential entropy increase (A1)

10. Entropy Implications

Theorem 10.1: Self-Referential Energy Systems

∀S: EnergySystem(S) ∧ SelfReferential(S) → H_energy(S,t+1) > H_energy(S,t)

Proof:

1. Energy systems process information: E = (dI/dt) × ℏ_φ
2. Self-referential → must observe own energy state
3. By A1: Self-referential observation increases entropy
4. Therefore: Energy entropy must increase ∎

11. Mathematical Structure Summary

T0-16 Structure = ⟨I, t, φ, Z, E⟩ where:
- I: Information content function
- t: Time parameter (from T0-0)
- φ: Golden ratio constraint
- Z: Zeckendorf encoding operator
- E: Emergent energy operator

With relations:
- E = (dI/dt) × ℏ_φ
- Z(n) gives unique energy states
- No-11 constraint satisfied
- Entropy always increases (A1)

Conclusion

This formal specification establishes energy as emergent from information processing rates, with quantization arising from Zeckendorf encoding constraints. The non-degenerate spectrum (avoiding F₁=F₂=1 issue) ensures theoretical consistency with the binary universe's No-11 constraint.

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