Foundation Coherence Exponent Coherence Exp Matches E Coh Exponent

The coherence exponent

The declaration coherenceExp_matches_E_coh_exponent is a small formal step inside a larger machine-checked library of theorems. It states, in the framework's own notation, that the integer coherence exponent, the power to which the golden ratio is raised to set a fundamental energy scale, is exactly 5. The proof is a direct rewrite of an earlier theorem: it unfolds the definition of the exponent and simplifies. The declaration itself does not derive the number 5 from first principles; it records that two separately defined quantities, the coherence exponent and the exponent appearing in the energy formula, agree with each other and with the integer 5.

The interest lies in how that 5 is reached. The framework's library proves that the coherence exponent equals 5 by two independent routes. The first route starts from the number of spatial dimensions D, which the framework forces to be 3, and the octave period 2^D, which it forces to be 8. Both 3 and 8 are Fibonacci numbers, and their difference, 8 minus 3, is 5, also a Fibonacci number. The second route counts integration variables: three spatial dimensions, one temporal dimension, and one conservation constraint give D + 2 = 5. Each variable contributes one factor of phi^(-1), so the energy E_coh becomes phi^(-5). The theorem routes_agree proves these two routes give the same answer, and coherence_energy_forced packages the whole result: coherenceExp = 5, coherenceExp is the fifth Fibonacci number, coherenceExp equals integrationDimension, and E_coh equals phi to the power -5.

What the declaration does not claim is as important as what it proves. It does not claim that the number 5 is physically measured or empirically observed. It does not claim that the two routes are conceptually independent in a deep sense; both ultimately rely on the same forced values of D and the octave period. The declaration does not derive the value of the golden ratio phi itself, nor does it prove that the energy E_coh is the actual energy of any physical system. It establishes an internal consistency: within the framework's definitions, the coherence exponent is 5, and the energy formula uses that same 5. The theorem is a statement about the framework's own arithmetic, not about the external world.

For a reader, the practical upshot is this: the framework's claim that the coherence energy exponent is forced, not fitted, rests on a chain of definitions and theorems that a machine has checked. The declaration coherenceExp_matches_E_coh_exponent is a small link in that chain, confirming that two different ways of arriving at the exponent do not conflict. It is a consistency check, not a discovery. The number 5 is forced only after the framework has already forced the dimension D to be 3 and the octave period to be 8; those prior steps carry the conceptual weight.

/-- The coherence exponent matches E_coh's defining exponent -/
theorem coherenceExp_matches_E_coh_exponent :
    (coherenceExp : ℤ) = 5 := by
  rw [coherenceExp_eq]; norm_num

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

/-- 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

The declaration does not claim that the number 5 is physically measured or empirically observed. It does not claim that the two routes to 5 are conceptually independent in a deep sense. It does not derive the value of the golden ratio phi itself.

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)

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