Masses Coherence Exponent Coherence Exponent Unique

The coherence exponent

The coherence exponent is the whole number 5. In the Recognition Science framework, it appears as the power of the golden ratio that sets a basic energy scale. The framework's machine-checked library of formal theorems contains a declaration, coherence_exponent_unique, that pins this number down. It proves that the exponent is 5, and that this value is forced by a Fibonacci constraint rather than chosen by hand.

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 constraint in question asks that both a dimension D and its octave, 2^D, be Fibonacci numbers. The theorem shows that D = 3 satisfies this, since 3 is the fourth Fibonacci number and 8, which is 2^3, is the sixth. The octave minus the dimension gives 8 - 3 = 5, and 5 is the fifth Fibonacci number. So the coherence exponent is 5, and the energy scale becomes E_coh = φ^(-5), where φ is the golden ratio, about 1.618.

The declaration does not claim that D = 3 is the only dimension that satisfies the Fibonacci constraint. The library itself records that D = 1 also works, since 1 and 2^1 = 2 are both Fibonacci numbers. The uniqueness claim is narrower: it establishes that the coherence exponent, defined as octave minus D, equals 5 for the specific case where D = 3. It does not rule out other dimensions satisfying the constraint, nor does it claim that the Fibonacci constraint itself is derived from more basic principles within the framework.

In Recognition Science, this result matters because it turns a seemingly arbitrary number, the exponent -5, into a structural consequence. The framework models the cost of recognition events as forced by a proved functional equation, and from that equation a chain of theorems leads to the golden ratio and to dimensions. The coherence exponent is one link in that chain: it shows that the energy scale is not a free parameter but follows from the Fibonacci pattern. The theorem is a formal result within the framework's library, and its force is conditional on the framework's definitions and axioms.

/-- The coherence exponent equals 5 -/
theorem coherence_exponent_eq_5 : coherence_exponent = 5 := by
  unfold coherence_exponent octave D
  norm_num

/-- D = 3 satisfies the Fibonacci constraint -/
theorem D_3_fibonacci_constraint : is_fibonacci 3 ∧ is_fibonacci (2^3) := by
  constructor <;> native_decide

/-- Key identity: 8 - 3 = 5, or F₆ - F₄ = F₅ -/
theorem fibonacci_deficit : fib 6 - fib 4 = fib 5 := by
  rw [fib_6_eq, fib_5_eq, fib_4_eq]

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

/-- 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

What this page does not claim

The declaration does not claim that D = 3 is the only dimension satisfying the Fibonacci constraint, since D = 1 also works. The declaration does not claim that the Fibonacci constraint itself is derived from more basic principles within the framework. The declaration does not claim that the coherence exponent is unique across all possible definitions, only for the specific definition given.

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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