Foundation Mode Energy Derivation Gap Uses Coherence Exponent

The gap and its factors

The declaration gap_uses_coherence_exponent is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It states a relationship between two quantities that the framework derives separately: the consciousness gap and the total number of independent modes in a recognition event. In plain terms, the theorem proves that the consciousness gap equals the product of two factors: the parity count (the number of independent binary distinctions the framework assigns to a spatial configuration) and the total number of modes (the count of independent coordinates needed to specify one recognition event).

The framework models a recognition event, a discrete record of a single act of distinguishing, as requiring five independent coordinates: three spatial coordinates (which voxel on a three-dimensional lattice), one temporal coordinate (which tick in an eight-tick cycle), and one balance coordinate (a ledger neutrality constraint). The theorem total_modes_eq_5 proves this count is five. The theorem gap_uses_coherence_exponent then shows that the consciousness gap, a quantity the framework derives from the parity count and the coherence exponent, is exactly the parity count (nine) multiplied by the total number of modes (five), giving forty-five.

The theorem does not claim that the consciousness gap is an input or a free parameter. It proves that the gap follows from the number of spatial dimensions alone, once the framework's other derivations are in place. The theorem also does not claim that the coherence energy, the minimum excitation energy per mode raised to the power of the total number of modes, is a measured physical quantity. It is a derived quantity within the framework, matching the framework's own constant E_coh but not compared against any external measurement in this declaration.

What the theorem changes is the status of the gap: it is not an independent postulate but a consequence of the mode count. A reader can now see that the framework's consciousness gap is not a separate assumption but a derived product, and that the coherence exponent (the number five) is not a separate input but the count of independent coordinates itself.

/-- The consciousness gap uses E_coh's exponent as one factor.
    gap = D² × (D+2) = parityCount × configDim = 9 × 5 = 45.
    The (D+2) factor IS the coherence exponent derived here. -/
theorem gap_uses_coherence_exponent :
    GapDerivation.consciousnessGap GapDerivation.D =
    GapDerivation.parityCount GapDerivation.D * total_modes := by
  rw [GapDerivation.gap_factors, GapDerivation.parityCount_at_D3, total_modes_eq_5]

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

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

/-- The consciousness gap uses E_coh's exponent as one factor.
    gap = D² × (D+2) = parityCount × configDim = 9 × 5 = 45.
    The (D+2) factor IS the coherence exponent derived here. -/
theorem gap_uses_coherence_exponent :
    GapDerivation.consciousnessGap GapDerivation.D =
    GapDerivation.parityCount GapDerivation.D * total_modes := by
  rw [GapDerivation.gap_factors, GapDerivation.parityCount_at_D3, total_modes_eq_5]

What this page does not claim

The consciousness gap is not claimed to be a free parameter or an input to the framework. The coherence energy is not claimed to be a measured physical quantity in this declaration. The theorem does not claim that the mode count of five is derived from anything other than the framework's own definitions and prior theorems.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: