T0-2 Formal Specification: Entropy Bucket Theory

Type Definitions

type Bit = 0 | 1
type ZeckendorfString = List[Bit] where no adjacent 1s
type FibIndex = ℕ
type Capacity = Fibonacci(FibIndex)

Core Structures

EntropyContainer

structure EntropyContainer:
    level: FibIndex          // Capacity level index
    capacity: Capacity        // F_{level+1}
    state: ZeckendorfString   // Current state, length ≤ level
    entropy: ℕ               // Current entropy value
    
    invariants:
        - capacity = Fibonacci(level + 1)
        - entropy = ZeckendorfDecode(state)
        - entropy < capacity
        - ∀i. state[i] = 1 → state[i+1] ≠ 1

ContainerOperations

function create_container(level: FibIndex) → EntropyContainer:
    return EntropyContainer(
        level = level,
        capacity = Fibonacci(level + 1),
        state = [0] * level,
        entropy = 0
    )

function add_entropy(C: EntropyContainer, ΔE: ℕ) → EntropyContainer:
    new_entropy = C.entropy + ΔE
    if new_entropy < C.capacity:
        return EntropyContainer(
            level = C.level,
            capacity = C.capacity,
            state = ZeckendorfEncode(new_entropy),
            entropy = new_entropy
        )
    else:
        return overflow(C, ΔE)

Formal Axioms

Axiom 1: Finite Capacity Necessity

∀C: SelfReferentialComponent.
    can_encode_self(C) → has_finite_capacity(C)
    
where:
    can_encode_self(C) ≡ ∃φ. φ(C) ⊆ States(C)
    has_finite_capacity(C) ≡ |States(C)| < ∞

Axiom 2: Capacity Quantization

∀C: EntropyContainer.
    C.capacity ∈ {F_1, F_2, F_3, ...}
    
where:
    F_n = Fibonacci sequence

Axiom 3: No Consecutive Ones

∀C: EntropyContainer, ∀i ∈ [0, C.level-1].
    C.state[i] = 1 → C.state[i+1] = 0

Theorems with Formal Proofs

Theorem 1: Capacity Count Formula

theorem capacity_count:
    ∀n: ℕ. count_valid_states(n) = Fibonacci(n + 2)
    
proof:
    Base cases:
        n = 0: {} → count = 1 = F_2 ✓
        n = 1: {0, 1} → count = 2 = F_3 ✓
    
    Inductive step:
        valid(n) = valid(n-1) ending in 0 ∪ valid(n-2) ending in 01
        count(n) = count(n-1) + count(n-2)
        This is Fibonacci recurrence
    ∎

Theorem 2: Maximum Entropy Value

theorem max_entropy:
    ∀C: EntropyContainer.
        max_entropy(C) = C.capacity - 1 = Fibonacci(C.level + 1) - 1
        
proof:
    Maximum Zeckendorf representation with n digits:
        10101...010 or 10101...001 (alternating pattern)
    This sums to F_n + F_{n-2} + ... = F_{n+1} - 1
    ∎

Theorem 3: Overflow Conservation

theorem overflow_conservation:
    ∀C: EntropyContainer, ∀ΔE: ℕ.
        let (C', excess) = overflow_with_excess(C, ΔE)
        C'.entropy + excess = min(C.entropy + ΔE, total_system_capacity)
        
proof:
    By construction of overflow function
    Either rejects (excess = ΔE), collapses (C'.entropy = 0), 
    or cascades (excess flows to next container)
    Total entropy is conserved in all cases
    ∎

Overflow Specifications

Overflow Types

enum OverflowType:
    REJECT    // Return unchanged container
    COLLAPSE  // Reset to ground state
    CASCADE   // Transfer to linked container

function overflow(C: EntropyContainer, ΔE: ℕ) → EntropyContainer:
    match C.overflow_type:
        REJECT: return C
        COLLAPSE: return create_container(C.level)
        CASCADE: return cascade_overflow(C, ΔE)

Cascade Rules

function cascade_overflow(C: EntropyContainer, ΔE: ℕ) → (EntropyContainer, ℕ):
    new_entropy = C.entropy + ΔE
    if new_entropy < C.capacity:
        return (update_entropy(C, new_entropy), 0)
    else:
        filled = update_entropy(C, C.capacity - 1)
        excess = new_entropy - (C.capacity - 1)
        return (filled, excess)

Multi-Container Composition

System Capacity

function system_capacity(containers: List[EntropyContainer]) → ℕ:
    return ∏(C.capacity for C in containers)
    
theorem system_capacity_product:
    ∀C₁, C₂: EntropyContainer.
        capacity(C₁ ⊗ C₂) = capacity(C₁) × capacity(C₂)

Capacity Distribution

function redistribute(system: List[EntropyContainer], weights: List[ℝ]) → List[EntropyContainer]:
    total_entropy = sum(C.entropy for C in system)
    new_system = []
    for C, w in zip(system, weights):
        target = min(floor(total_entropy * w), C.capacity - 1)
        new_system.append(update_entropy(C, target))
    return balance_remainder(new_system, total_entropy)

Verification Checklist

Invariants to Verify

1. ✓ All container states are valid Zeckendorf strings
2. ✓ No container exceeds its Fibonacci capacity
3. ✓ Overflow preserves total system entropy
4. ✓ Capacity levels form Fibonacci sequence
5. ✓ State transitions preserve no-11 constraint

Computational Checks

function verify_container(C: EntropyContainer) → bool:
    return (
        is_valid_zeckendorf(C.state) ∧
        C.entropy = zeckendorf_decode(C.state) ∧
        C.entropy < C.capacity ∧
        C.capacity = fibonacci(C.level + 1)
    )

function verify_system(S: List[EntropyContainer]) → bool:
    return all(verify_container(C) for C in S)

Measurement Functions

Entropy Measurement

function measure_entropy(C: EntropyContainer) → ℕ:
    return sum(C.state[i] * fibonacci(i+1) for i in range(C.level))

Utilization Ratio

function utilization(C: EntropyContainer) → ℝ:
    return C.entropy / C.capacity
    
theorem golden_ratio_utilization:
    ∀ε > 0, ∃N: ℕ. ∀n > N.
        |average_utilization(random_container(n)) - 1/φ| < ε
    where φ = (1 + √5)/2 is golden ratio

Connection Interfaces

From T0-1

import from T0-1:
    - valid_zeckendorf_string()
    - zeckendorf_encode()
    - zeckendorf_decode()
    - no_consecutive_ones()

To T0-3

export to T0-3:
    - EntropyContainer
    - system_capacity()
    - overflow_rules
    - cascade_mechanics

Formal Verification Points

Each point maps to testable assertion:

1. FV1: ∀n. fibonacci(n) is computable and unique
2. FV2: ∀s: ZeckendorfString. no_11(s) = true
3. FV3: ∀C, ΔE. total_entropy_before = total_entropy_after (overflow)
4. FV4: ∀C₁, C₂. capacity(compose(C₁, C₂)) = capacity(C₁) × capacity(C₂)
5. FV5: ∀C. lim(n→∞) utilization(C_n) → 1/φ

Complexity Analysis

Space Complexity

Container storage: O(log_φ(capacity)) = O(level)
System storage: O(num_containers × average_level)

Time Complexity

Add entropy: O(level) for Zeckendorf encoding
Measure entropy: O(level) for decoding
Overflow: O(1) for reject/collapse, O(level) for cascade
System operation: O(num_containers × level)