Foundation Coherence Exponent Coherence Exp Eq

The coherence exponent

The coherence exponent is the number 5 that appears in the expression E_coh = φ⁻⁵, a quantity the Recognition Science framework associates with a coherence energy. The declaration coherenceExp_eq establishes, as a theorem, that this exponent is not an adjustable parameter but a fixed consequence of definitions built from earlier results. In plain terms, the framework proves that the exponent must be 5, and it derives this number through two separate routes that arrive at the same answer.

The first route starts from the number of spatial dimensions. The framework forces three spatial dimensions, and from that derives an eight-tick period. Both 3 and 8 are Fibonacci numbers, and their difference, 8 minus 3, is 5, which is also a Fibonacci number. The framework proves that the dimension 3 is the only nontrivial case where both the dimension and its power of two are Fibonacci numbers, making the deficit 5 a unique structural outcome rather than one option among many.

The second route counts independent integration variables. The framework adds one temporal dimension and one conservation constraint to the three spatial dimensions, giving five total variables. Each variable contributes a factor of φ⁻¹, so the product is φ⁻⁵. The theorem routes_agree proves that the two routes yield the same number, and the theorem coherence_energy_forced packages the full result: the exponent is 5, it is a Fibonacci number, it equals the integration dimension, and the energy is φ⁻⁵.

What the declaration does not claim is equally important. It does not prove that the coherence energy itself has a measured physical value; the framework derives the mathematical form, and any comparison with experiment is a separate empirical check. It does not claim that the number 5 is special outside the framework's own definitions. The theorem is a statement about consequences within the framework, not an independent law of nature.

theorem coherenceExp_eq : coherenceExp = 5 := by
  unfold coherenceExp octavePeriod D; norm_num

/-- Both routes agree: Fibonacci deficit = integration dimension -/
theorem routes_agree : coherenceExp = integrationDimension := by
  rw [coherenceExp_eq, integrationDimension_eq]

/-- D=3: both 3 and 8 ARE Fibonacci -/
theorem D3_works : isFib 3 = true ∧ isFib (2 ^ 3) = true := by
  constructor <;> native_decide

/-- The master theorem: E_coh = φ^{-5} is structurally determined.

    The exponent 5 arises from:
    1. D = 3 forced by T8 (unique non-trivial Fibonacci-compatible dimension)
    2. 2^D - D = 8 - 3 = 5 = F₅ (Fibonacci deficit)
    3. Equivalently: D + 2 = 3 + 2 = 5 (integration measure)
    4. Therefore E_coh = φ^{-5} with zero free parameters -/
theorem coherence_energy_forced :
    coherenceExp = 5 ∧
    coherenceExp = Nat.fib 5 ∧
    coherenceExp = integrationDimension ∧
    E_coh = phi ^ (-(5 : ℝ)) := by
  exact ⟨coherenceExp_eq, coherenceExp_is_fib, routes_agree, E_coh_eq_phi_neg5⟩

What this page does not claim

No measured value for the coherence energy is claimed; the theorem concerns only the mathematical form. No claim that the number 5 is special outside the framework's own definitions and forcing chain. No claim that the two routes are independent at the level of physical assumptions; both rely on the same forced dimension.

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.

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