Foundation Mode Energy Derivation E Coh Derived Matches Constant

Coherence energy

In the Recognition Science framework, the coherence energy is a proposed fundamental quantity, a measure of the minimum energy needed to sustain a coherent recognition event. The framework's machine-checked library of formal theorems derives its value from first principles, rather than setting it as a free parameter. The declaration E_coh_derived_matches_constant is the final step in this derivation: it establishes, as a proved theorem, that the value of the coherence energy derived from the mode-counting argument is exactly equal to the value of the coherence energy constant as it is defined elsewhere in the library.

The derivation itself is a count of independent modes. The framework proves that a recognition event is specified by five independent parameters: three spatial coordinates, one temporal coordinate, and one balance coordinate. Each of these five modes contributes a minimum excitation energy of φ⁻¹, where φ is the golden ratio. Since the modes are independent, the total coherence energy is the product of these per-mode excitations, giving E_coh = (φ⁻¹)⁵ = φ⁻⁵. The theorem E_coh_derived_matches_constant then states that this derived value, φ⁻⁵, is identical to the constant E_coh defined in the CoherenceExponent module. The proof is a direct computation, a formal verification that the two definitions are the same number.

This is a statement of internal consistency, not a prediction of a measured physical quantity. The theorem does not claim that φ⁻⁵ is the measured value of any energy in the physical world. It shows that within the framework, the derived coherence energy is not an arbitrary input but a necessary consequence of the framework's own axioms and definitions. The framework's library proves this equality with no unproven assumptions, relying only on the standard axioms of the underlying type theory.

The significance of this result is that it closes a potential gap in the framework's structure. It confirms that the coherence energy, a quantity that appears in other parts of the framework, is not a separate postulate but is determined by the more basic principles of mode independence and the golden ratio. This is a step toward the framework's goal of deriving all its constants from a single source.

/-- Bridge: the derived E_coh matches the constant in Constants.lean. -/
theorem E_coh_derived_matches_constant :
    E_coh_derived = E_coh := by
  rw [E_coh_derived_eq, CoherenceExponent.E_coh_eq_phi_neg5, ← Real.rpow_intCast phi (-5 : ℤ)]
  norm_num

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

/-- The minimum excitation equals 1/φ. -/
theorem min_excitation_eq_inv_phi :
    min_excitation_per_mode = phi⁻¹ := by
  unfold min_excitation_per_mode
  exact zpow_neg_one phi

/-- 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 that φ⁻⁵ is a measured value of any physical energy. The theorem does not claim that the coherence energy is the same as any conventional physics energy. The theorem does not claim that the mode-counting argument is a physical derivation, only a formal one within 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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