Foundation Mode Energy Derivation Min Excitation Pos

The minimum excitation

The golden ratio φ, about 1.618, appears throughout mathematics as the number whose square equals itself plus one. In the Recognition Science framework, it also sets the scale of the smallest possible energy step on the framework's discrete ledger, a record of recognition events. The theorem min_excitation_pos states that this minimum excitation per mode is positive, meaning it is a real number greater than zero. The proof is short: the minimum excitation is defined as φ⁻¹, the reciprocal of the golden ratio, and since φ is positive, so is its reciprocal.

The framework builds its account from a cost function J(x) = (x + 1/x)/2 - 1, which its machine-checked library of formal theorems proves is the unique function satisfying five plain conditions. From that cost function, the framework derives the golden ratio as the minimal non-trivial ratio on the ledger, and then identifies the smallest resolvable energy quantum as φ⁻¹. This is not a free choice; it follows from the structure of the cost function and the ledger's own units, where moving one rung of the φ-ladder is the unit step.

The positivity theorem is a small but necessary piece of a larger derivation. The framework defines a recognition event by D+2 independent parameters: D spatial coordinates, one temporal coordinate, and one balance coordinate. At D = 3, this gives five total modes. The coherence energy is then the product of the minimum excitation per mode raised to the number of modes, so E_coh = (φ⁻¹)⁵ = φ⁻⁵. The theorem min_excitation_pos ensures that this energy is a positive quantity, a prerequisite for the framework's later claims about a consciousness gap and its connection to the parity count.

In Recognition Science, the framework models the minimum excitation as a positive real number, and its library proves the associated theorems. The declaration does not claim that φ⁻¹ is the actual measured energy of any physical particle, nor does it assert that the derivation applies outside the framework's own axioms. It is a statement about the internal consistency of the framework's ledger, not a prediction about laboratory measurements.

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

/-- The minimum excitation energy per mode on the φ-ladder.
    One rung step on the φ-ladder = φ^{-1} in coherence units.

    This is forced by the φ-forcing theorem (T6): φ is the unique
    self-similar ratio on the discrete ledger, so the minimum
    non-trivial energy quantum is 1/φ = φ^{-1}. -/
def min_excitation_per_mode : ℝ := phi ^ (-1 : ℤ)

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 minimum excitation φ⁻¹ is not a measured energy of any known particle. The derivation does not claim to apply outside the framework's own axioms and definitions.

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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