T0-7: Fibonacci Sequence Necessity Theory (Formal)

Formal Framework

Axiom

A1 (Entropy Increase): ∀S: self_referential(S) ∧ complete(S) → ∀t: H(S,t+1) > H(S,t)

Definitions

D7.1 (Weight Sequence): W = {w_n}_{n=1}^∞ ⊂ ℕ₊ is a weight sequence

D7.2 (No-11 Encoding): E: ℕ → {0,1}^ℕ where E(n) = (b₁, b₂, ...) satisfies:

n = ∑ᵢ bᵢwᵢ
∀i: bᵢ ∈ {0,1}
∀i: bᵢ · bᵢ₊₁ = 0

D7.3 (Complete Coverage): W has complete coverage iff ∀n ∈ ℕ: ∃!b ∈ {0,1}^ℕ: n = ∑ᵢ bᵢwᵢ ∧ no-11(b)

D7.4 (Fibonacci Sequence): F₁ = 1, F₂ = 2, ∀n≥2: F_{n+1} = F_n + F_{n-1}

Main Theorems

Theorem T7.1 (Coverage Necessity)

Statement: Complete coverage under no-11 requires w_{n+1} ≤ w_n + w_{n-1}

Proof:

Let W be a weight sequence with complete coverage
Define V_max(n) = max{∑ᵢ≤n bᵢwᵢ : no-11(b)}
For position n+1 to be useful: w_{n+1} ≤ V_max(n) + 1
V_max(n) achieved by alternating pattern
Adding positions n and n-1: V_max(n) ≥ V_max(n-2) + w_n + w_{n-1}
Continuity requires: w_{n+1} ≤ w_n + w_{n-1} ∎

Theorem T7.2 (Uniqueness Necessity)

Statement: Unique representation under no-11 requires w_{n+1} ≥ w_n + w_{n-1}

Proof:

Assume w_{n+1} < w_n + w_{n-1}
∃v: w_{n+1} ≤ v < w_n + w_{n-1}
v cannot be uniquely represented:
- Cannot use just position n+1 (too small)
- Cannot use n and n-1 together (violates no-11)
- Must use complex combinations → non-uniqueness
Therefore: w_{n+1} ≥ w_n + w_{n-1} ∎

Theorem T7.3 (Fibonacci Recurrence)

Statement: W satisfies complete unique coverage iff w_{n+1} = w_n + w_{n-1}

Proof: From T7.1: w_{n+1} ≤ w_n + w_{n-1} From T7.2: w_{n+1} ≥ w_n + w_{n-1} Therefore: w_{n+1} = w_n + w_{n-1} ∎

Theorem T7.4 (Initial Conditions)

Statement: Complete unique coverage requires w₁ = 1, w₂ = 2

Proof:

w₁ determination:
- Must represent 1: w₁ ≥ 1
- Minimality: w₁ = 1
w₂ determination:
- Must represent 2 uniquely
- If w₂ = 1: redundant with w₁
- If w₂ = 2: exactly represents 2
- If w₂ > 2: cannot represent 2
- Therefore: w₂ = 2
Verification: (1,2) generates Fibonacci satisfying all requirements ∎

Theorem T7.5 (Information Density)

Statement: Fibonacci maximizes information density under no-11 constraint

Proof:

Information capacity: I(n) = log₂(#{valid n-bit strings})
Under no-11: #{valid} = F_{n+1}
Density: ρ = lim_{n→∞} I(n)/n = lim_{n→∞} log₂(F_{n+1})/n
F_n ~ φⁿ/√5 where φ = (1+√5)/2
ρ = log₂(φ) ≈ 0.694
This is maximum for no-11 constraint ∎

Theorem T7.6 (Optimal Coupling)

Statement: Fibonacci spacing minimizes coupling variance in component interactions

Proof:

From T0-6: κᵢⱼ = min(Fᵢ,Fⱼ)/max(Fᵢ,Fⱼ)
For consecutive: κ_{n,n+1} = F_n/F_{n+1}
lim_{n→∞} F_n/F_{n+1} = 1/φ ≈ 0.618
Variance: Var(κ) → 0 as n → ∞
Constant ratio minimizes coupling variance ∎

Theorem T7.7 (Self-Similarity)

Statement: Fibonacci exhibits perfect self-similar structure

Proof:

Define shift operator: T({a_n}) = {a_{n+1}}
For Fibonacci: T²(F) - T(F) - F = 0
Scaling property: {F_{n+k}} = F_k·{F_n} + F_{k-1}·{F_{n-1}}
Proof by induction on k:
- Base k=2: F_{n+2} = F_n + F_{n+1} ✓
- Step: Assume for k, prove for k+1
- F_{n+k+1} = F_{n+k} + F_{n+k-1}
- = (F_k·F_n + F_{k-1}·F_{n-1}) + (F_{k-1}·F_n + F_{k-2}·F_{n-1})
- = F_{k+1}·F_n + F_k·F_{n-1} ✓
Self-similarity established ∎

Theorem T7.8 (Synchronization Optimality)

Statement: Fibonacci minimizes synchronization threshold

Proof:

From T0-6: κ_critical = |Fᵢ - Fⱼ|/(Fᵢ + Fⱼ)
Adjacent: κ_critical^{(n,n+1)} = F_{n-1}/F_{n+2}
lim_{n→∞} κ_critical = 1/φ³ ≈ 0.236
This is minimum stable value for growth
Alternative sequences have higher thresholds ∎

Theorem T7.9 (Error Detection)

Statement: Fibonacci encoding maximizes single-bit error detection

Proof:

Valid encoding has no "11" pattern
Error creating "11" immediately detected
Detection probability: P_d = P(error creates 11)
For random bit flip in valid string:
- P_d approaches 1/φ ≈ 0.618
This is optimal for no-11 constraint ∎

Theorem T7.10 (Uniqueness in Category)

Statement: Fibonacci is the unique initial object in category of no-11 sequences

Proof:

Category C:
- Objects: Sequences with no-11 counting property
- Morphisms: Recurrence-preserving maps
Initial Object Property:
- ∀X ∈ Obj(C): ∃! f: Fib → X
Uniqueness:
- Any two initial objects are isomorphic
- Therefore Fibonacci unique up to isomorphism ∎

Theorem T7.11 (Extremal Characterization)

Statement: Fibonacci minimizes J[a] = ∑n ((a{n+1} - a_n - a_{n-1})/a_n)²

Proof:

Euler-Lagrange: δJ/δa_n = 0
Yields: a_{n+1} = a_n + a_{n-1}
With a₁ = 1, a₂ = 2: unique solution is Fibonacci
Second variation: δ²J > 0 → minimum
Fibonacci is unique minimizer ∎

Theorem T7.12 (Complete Necessity)

Statement: Binary self-referential systems with finite capacity necessarily use Fibonacci weights

Proof:

From Axiom A1: System must increase entropy
From T0-1: Binary representation necessary
From T0-3: No-11 constraint for uniqueness
From T7.3: Recurrence must be a_{n+1} = a_n + a_{n-1}
From T7.4: Initial conditions must be 1, 2
Therefore: Weights must be Fibonacci sequence
No alternatives: Any deviation violates requirements ∎

Formal Properties

Lemma L7.1 (Growth Rate)

F_n = (φⁿ - ψⁿ)/√5 where φ = (1+√5)/2, ψ = (1-√5)/2

Lemma L7.2 (Summation Formula)

∑{i=1}^n F_i = F{n+2} - 1

Lemma L7.3 (GCD Property)

gcd(F_m, F_n) = F_{gcd(m,n)}

Lemma L7.4 (Cassini Identity)

F_{n-1}·F_{n+1} - F_n² = (-1)ⁿ

Conclusion

Main Result: The triple (F, F₁=1, F₂=2) where F satisfies F_{n+1} = F_n + F_{n-1} is the unique solution to the requirements:

Complete coverage under no-11
Unique representation
Optimal information density
Minimal coupling variance
Self-similar structure

Necessity Chain:

Entropy Axiom
    ↓
Binary Representation (T0-1)
    ↓
No-11 Constraint (T0-3)
    ↓
Coverage + Uniqueness
    ↓
Fibonacci Recurrence (T7.3)
    ↓
Initial Conditions (T7.4)
    ↓
Complete Fibonacci Sequence

The Fibonacci sequence emerges as mathematical necessity, not choice.

∎

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