Foundation Coherence Exponent Coherence Exp Is Fib

The coherence exponent

The Fibonacci sequence begins 1, 1, 2, 3, 5, 8, 13, and its terms appear across mathematics, from plant growth to the golden ratio. The number 5 is the fifth Fibonacci number, and 8 is the sixth. The declaration coherenceExp_is_fib is a formal theorem inside the Recognition Science framework's machine-checked library of formal theorems. It proves that a quantity the framework calls the coherence exponent, written coherenceExp, equals the fifth Fibonacci number, 5.

In plain terms, the framework models a discrete record of events, a ledger, and asks what scaling of energy its structure forces. The theorem chains together two definitions. The first defines the exponent as the difference between an octave period and a dimension count: coherenceExp = 2^D - D. The framework's earlier theorems force D = 3 spatial dimensions and 2^D = 8 ticks in a recognition cycle. The difference is 8 - 3 = 5. The second route defines an integration dimension as D + 2, where the extra two come from one temporal dimension and one conservation constraint, also giving 5. Both routes agree, and the theorem states the equality directly.

The result matters because the framework derives the constant ħ = φ⁻⁵, where φ is the golden ratio, as a consequence of this exponent. The theorem coherenceExp_is_fib is one step in that derivation, showing the exponent 5 is not fitted but forced by the framework's own structure. The framework's library proves the equality as a formal theorem, meaning the statement is checked by a machine and depends only on the framework's stated postulates.

What the declaration does not claim is broader. It proves an equality about a defined quantity; it does not prove that the physical universe must exhibit this exponent. The framework's bridge from its abstract ledger to physical space and time is itself an open target, not a proved theorem. The declaration also does not claim that the number 5 is unique in any absolute mathematical sense; it is unique within the framework's specific definitions of dimension and period. The theorem is a precise statement about the framework's own model, not a claim about all possible physical theories.

/-- The coherence exponent is F₅ -/
theorem coherenceExp_is_fib : coherenceExp = Nat.fib 5 := by
  rw [coherenceExp_eq, fib_5_eq]

/-- The Fibonacci deficit: 2^D - D = 5 -/
def coherenceExp : ℕ := octavePeriod - D

/-- D = F₄ -/
theorem D_is_fib : D = Nat.fib 4 := by unfold D; rw [fib_4_eq]

/-- 2^D = F₆ -/
theorem octave_is_fib : octavePeriod = Nat.fib 6 := by
  rw [octavePeriod_eq, fib_6_eq]

/-- E_coh = φ^{-5} in rpow form matches the Fibonacci-forced exponent.
    Bridge: cLagLock uses rpow with -(5:ℝ); we show coherenceExp = 5 produces
    the same value via zpow → rpow conversion. -/
theorem E_coh_eq_phi_neg5 : E_coh = phi ^ (-(5 : ℝ)) := by
  unfold E_coh cLagLock; rfl

What this page does not claim

The theorem does not prove that the physical universe must exhibit this exponent. The theorem does not claim the number 5 is unique in any absolute mathematical sense. The framework's bridge from its abstract ledger to physical space and time is not proved by this declaration.

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/Foundation/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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