Masses Coherence Exponent Coherence Exponent From Fibonacci

The coherence exponent

The Fibonacci sequence is the list of numbers where each entry is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, and so on. The declaration coherence_exponent_from_fibonacci establishes a specific identity inside the Recognition Science framework: the coherence exponent, a natural number that appears in the framework's mass formulas, equals the difference between the sixth and fourth Fibonacci numbers, which is 8 minus 3, or 5. The framework's library proves this as a theorem, meaning the statement is checked by a machine and requires no additional assumptions beyond the framework's own definitions.

The proof is a short chain of definitions. The framework sets a dimension D to 3, and an octave to 2 raised to that power, which is 8. The coherence exponent is defined as the octave minus D, so it is 5. The theorem then observes that 3 is the fourth Fibonacci number, 8 is the sixth, and their difference, 5, is the fifth. The framework's library also proves that this is the only such case: for dimensions 2, 5, and 8, the number 2 raised to that dimension is not a Fibonacci number. This uniqueness is part of what makes the identity structurally meaningful within the framework.

The coherence exponent then feeds into a physical quantity. The framework defines a coherence energy E_coh as the golden ratio phi raised to the negative of the coherence exponent, so with the exponent equal to 5, E_coh equals phi to the power of minus 5. This is the same value the framework derives for hbar, the reduced Planck constant, in its own unit system. The theorem does not claim that this energy is measured in any experiment; it claims only that within the framework's definitions, the exponent and the resulting energy are forced by the Fibonacci constraint rather than chosen freely.

In Recognition Science, this is one step in a larger chain that derives constants from structural conditions. The framework models reality as maintaining a discrete record of recognition events, and from that starting point it derives the golden ratio, an eight-tick cycle, and three spatial dimensions. The coherence exponent result is a smaller piece: it shows that a number appearing in mass formulas is not arbitrary but follows from the same Fibonacci pattern that the framework uses elsewhere. What the declaration does not claim is that the Fibonacci sequence itself is fundamental to physics, or that the coherence exponent 5 has been observed directly in any measurement. It is a structural identity within the framework, not an empirical fact.

/-- The coherence exponent arises from the Fibonacci identity -/
theorem coherence_exponent_from_fibonacci :
    coherence_exponent = fib 6 - fib 4 := by
  rw [coherence_exponent_is_fib_5, fibonacci_deficit]

/-- **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⟩

/-- D = 2 does NOT satisfy: 2^2 = 4 is not Fibonacci -/
theorem D_2_fails : ¬ is_fibonacci (2^2) := by native_decide

/-- D = 5 does NOT satisfy: 2^5 = 32 is not Fibonacci -/
theorem D_5_fails : ¬ is_fibonacci (2^5) := by native_decide

/-- D = 8 does NOT satisfy: 2^8 = 256 is not Fibonacci -/
theorem D_8_fails : ¬ is_fibonacci (2^8) := by native_decide

/-- E_coh = φ^{-5} -/
theorem E_coh_eq : E_coh = Constants.phi ^ (-5 : ℤ) := by
  unfold E_coh coherence_exponent octave D
  norm_num

What this page does not claim

The coherence exponent 5 has been measured in any experiment. The Fibonacci sequence itself is fundamental to physics. The coherence energy E_coh is the same as the measured Planck constant.

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)

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