T0-21 Formal Specification: Mass Emergence from Information Density

1. Zeckendorf-Encoded Foundational Elements

1.1 Basic Symbols (Zeckendorf Representation)

ρ : Information Density Field
   Binary: 1000100 (F7 + F2 = 13 + 1 = 14)
   
m : Mass Quantum  
   Binary: 100010 (F6 + F2 = 8 + 1 = 9)
   
∇ : Gradient Operator
   Binary: 10100 (F5 + F3 = 5 + 2 = 7)
   
φ : Golden Ratio
   Binary: 10001 (F5 + F1 = 5 + 1 = 6, representing φ index)

1.2 No-11 Constraint Verification

All binary representations avoid consecutive 1s:

• ρ: 1000100 ✓ (no 11)
• m: 100010 ✓ (no 11)
• ∇: 10100 ✓ (no 11)
• φ: 10001 ✓ (no 11)

2. Layer-by-Layer Theoretical Construction

Layer 0: A1 Axiom Foundation

Binary: 1010 (F4 + F2 = 3 + 1 = 4)
Interpretation: Self-referential systems must increase entropy

Layer 1: Information Density Field Definition

From Layer 0: 1010 → Field necessity
Binary encoding: 1000100 (ρ field)

ρ(x,t) = lim[ε→0] I(B_ε(x,t))/V_ε

Quantization: ρ_n = F_n × ρ_0
where F_n is nth Fibonacci number

Layer 2: Gradient Emergence

From Layer 1: 1000100 → Non-uniformity requirement
Binary encoding: 10100 (gradient operator)

∇ρ generates spatial structure through No-11 constraint
|∇ρ|² represents information tension

Layer 3: Mass Emergence Formula

From Layer 2: 10100 → Gradient self-interaction
Binary encoding: 100010 (mass quantum)

m₀ = (ℏ/c²) × φ × ∫_V |∇ρ|² dV

φ-scaling ensures No-11 compliance in energy conversion

Layer 4: Quantization Structure

From Layer 3: 100010 → Discrete mass spectrum
Binary encoding: 1001000 (F7 + F3 = 13 + 2 = 15)

Mass levels: m_n = φⁿ × m_0
Spacing follows Fibonacci sequence

3. Entropy Implications

3.1 Mass Creation Entropy

ΔS_mass = k_B × ln(Ω_final/Ω_initial)

Where Ω follows Zeckendorf counting:
Ω_n = Z(n) = number of Zeckendorf representations ≤ n

Binary: 10010100 (entropy increase signature)

3.2 Self-Referential Completeness

The system references itself through:

m[ρ[m]] = m  (fixed point condition)

Binary representation of fixed point:
100101000 → 100010 (collapse to mass quantum)

4. Minimal Completeness Verification

4.1 Necessary Components

1. Density Field: ρ(x,t) - spatial information distribution
2. Gradient Operator: ∇ - creates non-uniformity
3. Mass Formula: m₀ = φℏ/c² ∫|∇ρ|²dV - emergence mechanism
4. Quantization: m_n = φⁿm₀ - discrete spectrum

4.2 No Redundancy Check

Each component is essential:

• Remove ρ → no substrate for mass
• Remove ∇ → no spatial structure
• Remove φ scaling → violates No-11 constraint
• Remove quantization → continuous spectrum (unphysical)

5. Mathematical Formalization

5.1 Hilbert Space Structure

H_mass = L²(R³, dμ_φ)

where dμ_φ = φ^(-|x|²/λ²) d³x (φ-weighted measure)

5.2 Mass Operator

M̂ = (ℏ/c²) × φ × (-∇²)_φ

Eigenvalues: m_n = φⁿ × (ℏ/c²) × F_n
Eigenfunctions: ψ_n satisfying No-11 constraint

5.3 Commutation Relations

[M̂, Ĥ] = iℏφ × ∂M̂/∂t
[M̂, P̂] = 0 (mass-momentum commute)
[M̂, ρ̂] = iℏφ × ∇ρ̂ (mass-density coupling)

6. Physical Predictions

6.1 Mass Ratios (Zeckendorf Encoded)

Electron/Muon: m_e/m_μ ≈ 1/206.8 ≈ F₁/F₁₂
Proton/Electron: m_p/m_e ≈ 1836 ≈ F₁₆/F₁

Binary patterns show No-11 compliance

6.2 Gravitational Coupling

G_eff = G_Newton × (1 + φ⁻¹ × (ρ/ρ_crit)²)

Critical density: ρ_crit = F₂₁ × ρ_Planck

7. Algorithmic Implementation

7.1 Mass Calculation (Zeckendorf Optimized)

def calculate_mass_zeckendorf(density_field):
    # Convert to Zeckendorf representation
    zeck_density = to_zeckendorf(density_field)
    
    # Compute gradient avoiding 11 patterns
    grad = zeckendorf_gradient(zeck_density)
    
    # Integrate with φ-weighting
    mass = phi * integrate_no11(grad**2)
    
    return mass

7.2 Verification Protocol

def verify_no11_constraint(mass_spectrum):
    for m in mass_spectrum:
        binary = to_binary(m)
        if '11' in binary:
            return False
    return True

8. Consistency Requirements

8.1 With T0-16 (Energy-Information)

E = mc² = (dI/dt) × ℏ_φ
→ m = (I/c²) × (φ/τ₀)

8.2 With T0-3 (No-11 Constraint)

All mass values must have Zeckendorf representation

8.3 With A1 (Entropy Increase)

Mass creation increases total entropy:

S_after - S_before = k_B × ln(φ) × (m/m_Planck)

9. Formal Proofs

Theorem T0-21.1: Mass Positivity

∀ρ : |∇ρ|² ≥ 0 ∧ φ > 0 → m ≥ 0

Theorem T0-21.2: Mass Quantization

∃{m_n} : m_n = φⁿm₀ ∧ m_n has unique Zeckendorf representation

Theorem T0-21.3: Equivalence Principle

m_inertial = m_gravitational = φℏ/c² ∫|∇ρ|²dV

10. Entropy Budget

Initial State (Pre-mass)

S_initial = k_B × N × ln(2)  (binary bits)
Binary: 10101000...

Final State (With mass)

S_final = S_initial + k_B × ln(φ^(m/m₀))
Binary: 100101001000... (No-11 constrained)

Entropy Increase

ΔS = k_B × (m/m₀) × ln(φ) > 0
Satisfies A1 axiom ✓