Foundation Mode Energy Derivation E Coh Derived Matches Gap

The coherence energy identity

The theorem E_coh_derived_matches_gap is a formal statement inside the Recognition Science framework's machine-checked library of formal theorems. It proves that a derived quantity, the coherence energy E_coh_derived, is numerically identical to another quantity, the gap energy E_coh_gap. In plain terms, it closes a loop: the energy associated with the framework's fundamental recognition events is the same number as the energy that defines a certain gap in the framework's structure.

The derivation works by counting independent modes. The framework models a recognition event as having five independent parameters: three spatial coordinates, one temporal coordinate, and one balance coordinate. Each mode has a minimum excitation of φ⁻¹, where φ is the golden ratio. The coherence energy is the product of these per-mode minimum excitations, so E_coh_derived = (φ⁻¹)⁵ = φ⁻⁵. The theorem proves this equals the gap energy, which is defined separately in the framework's GapDerivation module. The proof is axiom-clean, meaning it relies only on the standard logical axioms of the proof assistant, with zero framework-specific axioms.

What the theorem does not claim is broader significance. It does not claim that φ⁻⁵ is a measured physical constant; it is a derived quantity within the framework. It does not claim that the gap energy has any empirical counterpart. The theorem is a statement of internal consistency: two different paths within the framework lead to the same number. It is a check that the framework's definitions cohere, not a prediction about the external world.

/-- The derived value matches GapDerivation's E_coh_gap. -/
theorem E_coh_derived_matches_gap :
    E_coh_derived = GapDerivation.E_coh_gap := by
  rw [E_coh_derived_eq]
  rw [GapDerivation.E_coh_gap_eq]

theorem total_modes_eq_5 : total_modes = 5 := by
  rw [total_modes_eq]; unfold D; norm_num

/-- E_coh_derived = φ^{-5} at D = 3. -/
theorem E_coh_derived_eq : E_coh_derived = phi ^ (-5 : ℤ) := by
  unfold E_coh_derived min_excitation_per_mode
  rw [total_modes_eq_5]
  rw [← zpow_natCast (phi ^ (-(1 : ℤ))) 5, ← zpow_mul]
  norm_num

What this page does not claim

The theorem does not claim φ⁻⁵ is a measured physical constant. The theorem does not claim the gap energy has empirical significance outside the framework.

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/ModeEnergyDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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