Foundation Mode Energy Derivation Min Excitation Lt One

The smallest energy step

The declaration min_excitation_lt_one is a machine-checked theorem in the framework's library of formal theorems. It states that the minimum excitation per mode, written min_excitation_per_mode, is a positive number less than one. In plain terms: the smallest possible energy step that a single independent mode can take is smaller than the unit step of the ledger, the framework's discrete record of events. The proof derives this from the golden ratio phi, the unique self-similar scaling forced by the framework's cost function. Since phi is greater than one, its reciprocal phi^-1 is less than one, and the theorem establishes exactly that.

The number phi^-1 is not arbitrary. The framework proves that the minimal non-trivial ratio on the ledger is phi, so consecutive rungs of the phi-ladder stand at ratio phi. Moving one rung up that ladder costs J(phi) = phi - 3/2 in the cost function's own units, but in the ledger's native coordinates, where the ladder itself is the coordinate system, that same step is the unit step. The minimum excitation per mode is therefore phi^-1, the inverse of the ladder ratio. The theorem min_excitation_lt_one captures the consequence: this smallest resolvable energy quantum is a fraction, not a whole number, of the ledger's unit.

This single fact feeds a larger derivation. The framework models a recognition event as specified by D+2 independent parameters: D spatial coordinates, one temporal coordinate, and one balance coordinate. At D = 3, that is five modes. Because the modes are independent, their minimum excitations multiply in the log domain, giving E_coh = phi^-(D+2), which at D = 3 equals phi^-5. The theorem min_excitation_lt_one is the per-mode ingredient in that product. It does not by itself establish the total coherence energy, the five-mode count, or the connection to the consciousness gap; those are separate theorems in the same module.

What the declaration does not claim is equally precise. It does not claim that any physical system actually achieves this minimum excitation, only that it is the smallest possible value. It does not claim that the minimum excitation is universal across all possible ledgers; it is a consequence of the specific phi-ladder structure the framework forces. And it does not claim that phi^-1 is an energy in any conventional physical unit, since the framework's units are native to the ledger, not SI units. The theorem is a statement about the framework's internal mathematics, not a measurement of any external system.

theorem min_excitation_lt_one : min_excitation_per_mode < 1 := by
  unfold min_excitation_per_mode
  rw [zpow_neg_one]
  simpa using (inv_lt_one_of_one_lt₀ (a := phi) one_lt_phi)

theorem min_excitation_pos : 0 < min_excitation_per_mode :=
  zpow_pos phi_pos (-1)

theorem min_excitation_lt_one : min_excitation_per_mode < 1 := by
  unfold min_excitation_per_mode
  rw [zpow_neg_one]
  simpa using (inv_lt_one_of_one_lt₀ (a := phi) one_lt_phi)

/-- 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 any physical system actually achieves the minimum excitation. The theorem does not claim that phi^-1 is an energy in SI units. The theorem does not claim that the minimum excitation is universal across all possible ledgers.

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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