Masses Coherence Exponent Fib Recurrence At 6

The Fibonacci step at 6

The Fibonacci numbers are the sequence where each term is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, and so on. The declaration fib_recurrence_at_6 is a machine-checked proof of one specific instance of this rule: the sixth Fibonacci number, 8, equals the fifth plus the fourth, 5 plus 3. In isolation, this is a trivial arithmetic fact, the kind a student verifies in seconds. Its significance comes entirely from the framework in which it is embedded.

In Recognition Science, the framework models physical structure as a discrete record of events, a ledger, where each entry records a recognition event at a cost. The framework's library, a machine-checked collection of formal theorems, builds a chain of results from a proved cost function. Within this chain, the framework defines a dimension D as 3 and an octave as 2 to the power D, which is 8. The theorems in the pack show that 3 is the fourth Fibonacci number, 8 is the sixth, and that their difference, 8 minus 3, is 5, the fifth Fibonacci number. The declaration fib_recurrence_at_6 is the formal step that connects the sixth Fibonacci number to its predecessors, a link in this arithmetic chain.

The framework then defines a coherence exponent as the difference between the octave and the dimension, which is 5. This exponent is not a free parameter; it is derived from the definitions. The coherence energy is defined as the golden ratio phi raised to the negative of this exponent, giving phi to the power minus 5. The theorem coherence_exponent_unique assembles these pieces, proving that the exponent is 5 and that this value is forced by the Fibonacci constraint that both D and 2 to the power D be Fibonacci numbers. The declaration fib_recurrence_at_6 is a component of this larger proof, not the proof itself.

What fib_recurrence_at_6 does not claim is just as important. It does not, by itself, prove that the coherence exponent is 5, nor that the energy is phi to the minus 5. Those are separate theorems in the library, built on the definitions of D, octave, and coherence_exponent. The declaration only proves the arithmetic identity fib 6 = fib 5 + fib 4. It is a lemma, a single verified step. The framework's larger claim, that this exponent is structurally determined, rests on the entire chain of definitions and theorems, not on this one recurrence. The declaration also makes no empirical claim about measured particle masses; it is a statement about the framework's internal arithmetic, not about experimental data. The comparison to measured values is a separate empirical check, not part of this theorem.

/-- The Fibonacci recurrence: F₆ = F₅ + F₄ -/
theorem fib_recurrence_at_6 : fib 6 = fib 5 + fib 4 := by
  rw [fib_6_eq, fib_5_eq, fib_4_eq]

/-- The Fibonacci deficit: 2^D - D = 5 -/
def coherence_exponent : ℕ := octave - D

/-- **Main Theorem**: The coherence exponent 5 is uniquely determined.

The number 5 arises from:
1. D = 3 is the unique non-trivial dimension where both D and 2^D are Fibonacci
2. The Fibonacci identity F₆ - F₄ = F₅ gives 8 - 3 = 5
3. Therefore E_coh = φ^{-5} is structurally determined, not a free parameter.
-/
theorem coherence_exponent_unique :
    D = fib 4 ∧
    octave = fib 6 ∧
    coherence_exponent = fib 5 ∧
    coherence_exponent = 5 := by
  exact ⟨D_is_fib_4, octave_is_fib_6, coherence_exponent_is_fib_5, coherence_exponent_eq_5⟩

What this page does not claim

The declaration fib_recurrence_at_6 alone does not prove the coherence exponent is 5. The declaration makes no empirical claim about measured particle masses. The declaration does not establish the physical recognition-to-linking bridge.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Masses/CoherenceExponent.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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