Foundation Mode Energy Derivation E Coh At Eq Derived

The coherence energy

In the Recognition Science framework, a ledger is a discrete record of recognition events. The declaration E_coh_at_eq_derived is a theorem in the framework's machine-checked library of formal theorems. It proves that the coherence energy, written E_coh and defined for any dimension d as phi raised to the power -(d+2), equals the energy derived from a product of independent modes. At the framework's forced dimension of three spatial dimensions, this value is phi to the power -5, where phi is the golden ratio, approximately 1.618.

The derivation rests on counting independent parameters of a recognition event. The framework models an event as specified by five coordinates: three spatial, one temporal, and one balance coordinate. The theorem total_modes_eq_5 proves that this count is five, and the theorem E_coh_derived_eq proves that the product of the minimum excitation per mode, phi to the power -1, over five modes yields phi to the power -5. The theorem E_coh_at_eq_derived then establishes that the general definition E_coh_at D, evaluated at D = 3, matches this derived product. This is a statement about the framework's internal definitions, not a measurement of any physical system.

The framework further connects this energy to a consciousness gap, defined as the product of nine parity counts and the five modes, yielding the number 45. The theorem E_coh_derived_matches_gap proves that the derived coherence energy equals the energy associated with this gap. This is a formal identity within the framework's structure, showing that the gap's energy and the mode-derived energy coincide. The theorem E_coh_derived_matches_constant proves that this derived value equals the framework's predefined constant E_coh, closing the loop between the general definition and the specific derivation at three dimensions.

The claim is precise: within the framework, the general coherence energy, when evaluated at the forced dimension, is the same object as the energy built from five independent modes. The declaration does not claim that this energy corresponds to any measured physical quantity, nor does it assert that the consciousness gap is a physical phenomenon. It establishes an internal consistency between two ways of defining the same framework quantity. The proof is axiom-clean, meaning it relies only on the framework's stated postulates and the standard axioms of the underlying type theory.

/-- The parametric form matches the derived value. -/
theorem E_coh_at_eq_derived : E_coh_at D = E_coh_derived := by
  rw [E_coh_at_D3, E_coh_derived_eq]

/-- E_coh as a function of dimension (parametric). -/
def E_coh_at (d : ℕ) : ℝ := phi ^ (-(d + 2 : ℤ))

/-- The coherence energy: product of D+2 independent minimum excitations.

    E_coh = (φ^{-1})^{D+2} = φ^{-(D+2)}

    Independence means: the total cost of exciting all modes simultaneously
    is the product of the individual costs (equivalently, the sum in
    log-domain). This follows from the lattice product structure of
    ℤ^D × ℤ_8 × ℤ_2 (spatial × temporal × balance). -/
def E_coh_derived : ℝ := min_excitation_per_mode ^ total_modes

/-- At D = 3, the parametric form matches. -/
theorem E_coh_at_D3 : E_coh_at D = phi ^ (-5 : ℤ) := by
  unfold E_coh_at 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

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

/-- 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]

/-- 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

What this page does not claim

The coherence energy is a measured physical quantity. The consciousness gap corresponds to any observable phenomenon. The derivation establishes the existence of three spatial dimensions in the physical world.

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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