Masses Coherence Exponent Coherence Exponent Is Fib 5

The coherence exponent

The Fibonacci sequence begins 1, 1, 2, 3, 5, 8, where each term is the sum of the two before it. The number 5 is the fifth term, usually written F₅. In the Recognition Science framework, a ledger, a discrete record of events, carries a cost of recognition, and from that cost the framework derives physical structure. This page concerns one derived quantity: the coherence exponent.

Within the framework, the dimension D is set to 3, and the octave is defined as 2^D, which equals 8. The coherence exponent is defined as the difference, octave minus D, so 8 minus 3 equals 5. The theorem coherence_exponent_is_fib_5 in the framework's machine-checked library of formal theorems proves that this exponent equals the Fibonacci number F₅. A companion theorem, coherence_exponent_from_fibonacci, shows the same result as F₆ minus F₄, since 8 is F₆ and 3 is F₄.

The framework's library also proves that D = 3 is the only dimension for which both D and 2^D are Fibonacci numbers. The theorems D_2_fails, D_5_fails, and D_8_fails show that 2^2 = 4, 2^5 = 32, and 2^8 = 256 are not Fibonacci numbers. This uniqueness is what makes the exponent 5 structurally determined rather than a free choice.

The coherence exponent feeds into the definition of E_coh, the coherence energy, as φ^(-5), where φ is the golden ratio. The theorem E_coh_eq proves this equality. This energy is not a free parameter in the framework; it follows from the Fibonacci constraint on dimension.

What the theorem does not claim is broader. It does not derive the value of the fine-structure constant α, nor does it prove the Riemann Hypothesis. The Fibonacci constraint is a definitional choice within the framework, not a physical law. The theorem establishes a structural relationship inside the framework's formal system; it does not by itself connect that relationship to any measured physical quantity.

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

/-- The coherence exponent equals F₅ -/
theorem coherence_exponent_is_fib_5 : coherence_exponent = fib 5 := by
  rw [coherence_exponent_eq_5, fib_5_eq]

/-- 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]

/-- 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 theorem does not derive the fine-structure constant α. The theorem does not prove the Riemann Hypothesis. The Fibonacci constraint is a definitional choice within the framework, not an empirical law.

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.

Derived articles

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