RecognitionDeficit
The universal non-equilibrium cost statement asserts that J-cost is strictly positive for every positive real ratio r not equal to one. Cross-domain researchers in hydrodynamics, medicine, game theory, and cognitive science cite this as the common source of deviation cost. The declaration introduces it as a definition with no internal proof structure.
claimThe recognition deficit is the assertion that $J(r) > 0$ whenever $r > 0$ and $r ≠ 1$, where $J$ denotes the J-cost function: $∀ r ∈ ℝ, 0 < r → r ≠ 1 → 0 < J(r)$.
background
This module presents the J-cost positivity universality claim, labeled C16, as the off-equilibrium counterpart to the equilibrium result C7. J-cost quantifies the cost of deviation from the fixed point at ratio one, where the J function satisfies J(1) = 0. The definition RecognitionDeficit captures this positivity in a form that is definitionally identical across seven domains: turbulent flow cost, disease cost from homeostasis deviation, off-target cost in CRISPR, off-equilibrium game cost, market arbitrage gap, biased reasoning cost, and recognition deficit in neurodevelopment.
proof idea
This declaration is a definition that directly states the quantified positivity condition as a Prop. It contains no proof body and applies no lemmas internally.
why it matters in Recognition Science
The definition supplies the common proposition underlying the equality theorem all_seven_are_one, which shows that the seven domain costs are identical. It realizes the structural claim in the module documentation that a single lemma generates non-equilibrium cost in every Recognition Science domain. This step connects the core J function properties to applications in physics, biology, and cognition without additional assumptions.
scope and limits
- Does not prove J-cost positivity; the proof resides in the lemma Jcost_pos_of_ne_one.
- Does not restrict r to any specific physical interpretation.
- Does not address the case r = 1 or r ≤ 0.
formal statement (Lean)
40def RecognitionDeficit : Prop := ∀ r : ℝ, 0 < r → r ≠ 1 → 0 < Jcost r
proof body
Definition body.
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