defect_energy
Defect energy extracts the J-cost from any recognition defect state. Workers on RS error correction and thermodynamics cite this when assigning an energy scale to ledger deviations from the J=0 ground state. The definition is a direct projection onto the J-cost functional.
claimLet $d$ be a recognition defect for state function $X$. The defect energy is defined by $E(d) = J(X(d.state))$, where $J$ is the cost functional.
background
The module develops an error-correction perspective on Recognition Science thermodynamics. Ledger dynamics achieve stability through fault tolerance, with recognition defects defined as states carrying positive J-cost. Defect energy is the J-cost itself, providing the scale for deviations from the J=0 ground state.
proof idea
The definition is a one-line extraction that applies the J-cost functional directly to the state field of the input defect.
why it matters in Recognition Science
This definition is invoked by the theorem establishing positivity of defect energy. It fills the role of quantifying defect cost in the error-correction module, connecting to the eight-tick cycle and phi-ladder code distance in the Recognition framework. The construction supports viewing physical laws as stable under ledger fault tolerance.
scope and limits
- Does not prove that defect energy is positive.
- Does not provide a numerical evaluation or units for the energy.
- Does not implement correction protocols or detect syndromes.
- Does not reference specific values of phi or spatial dimensions.
Lean usage
theorem defect_energy_pos {X : Ω → ℝ} (d : RecognitionDefect X) : 0 < defect_energy d := d.is_defectformal statement (Lean)
45noncomputable def defect_energy {X : Ω → ℝ} (d : RecognitionDefect X) : ℝ :=
proof body
Definition body.
46 Jcost (X d.state)
47
48/-- Defect energy is positive by definition. -/