Masses Coherence Exponent D 1 Fibonacci Constraint

The Fibonacci constraint

The Fibonacci sequence begins 1, 1, 2, 3, 5, 8, 13, and each number after the second is the sum of the two before it. The declaration D_1_fibonacci_constraint is a machine-checked proof of a simple fact about this sequence: the number 1 and the number 2, which is 2 raised to the power 1, both appear in the list. The proof is a direct computation, checking the short list of Fibonacci numbers shown above and confirming that 1 and 2 are on it.

This fact matters inside the Recognition Science framework, which models physical structure as a forced pattern of discrete recognition events. The framework's library of formal theorems argues that the dimension D of space must satisfy a constraint: both D and 2^D must be Fibonacci numbers. The declaration D_1_fibonacci_constraint verifies the case D = 1, showing that 1 and 2^1 = 2 both satisfy this condition. This is a necessary check, not a conclusion: the constraint alone does not single out D = 3, because D = 1 also passes it.

The larger argument proceeds by elimination. The framework's library proves that D = 2 fails the constraint (4 is not Fibonacci), D = 5 fails (32 is not), and D = 8 fails (256 is not). Among the small candidates, only D = 3 survives, with 3 and 8 both Fibonacci. From there, the framework derives the coherence exponent 5 as the difference 8 - 3, and defines the coherence energy E_coh = φ^(-5), where φ is the golden ratio. The declaration D_1_fibonacci_constraint is a supporting lemma in this chain: it rules out the trivial one-dimensional case as a candidate for the unique solution.

The declaration does not claim that D = 1 is the correct dimension, nor that the Fibonacci constraint is the only condition needed to force D = 3. It establishes only the narrow fact that the pair (1, 2) satisfies the constraint. The framework's full argument for three dimensions relies on additional structure beyond this single check.

/-- D = 1 satisfies the Fibonacci constraint but is degenerate -/
theorem D_1_fibonacci_constraint : is_fibonacci 1 ∧ is_fibonacci (2^1) := by
  constructor <;> native_decide

/-- 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

/-- 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]

/-- 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

D_1_fibonacci_constraint does not claim that D = 1 is the correct or unique dimension for space. The Fibonacci constraint alone does not force D = 3; other framework conditions are needed. The declaration does not establish any physical measurement or empirical prediction about particle masses.

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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