knowledge_equilibrium
plain-language theorem explainer
The declaration establishes that the recognition cost vanishes at the unit element, formalizing knowledge as zero-defect recognition in the Recognition Science framework. Researchers in epistemic logic or foundations of physics would reference it when grounding knowledge claims. The proof reduces immediately to the unit lemma for the J-cost function via direct substitution.
Claim. $J(1) = 0$, where $J(x) = (x-1)^2/(2x)$ is the recognition cost function.
background
In Recognition Science the J-cost function measures recognition defect and is given by the squared ratio $J(x) = (x-1)^2/(2x)$. It is zero precisely when the model matches the world exactly. The module EpistemologyFromRS maps five epistemic stances onto configuration dimension D = 5, with knowledge identified as the zero-cost state and justified true belief requiring additional sigma-conservation.
proof idea
The proof is a one-line wrapper that applies the Jcost_unit0 lemma imported from the Cost module. That lemma itself follows by simplification on the definition of Jcost.
why it matters
This theorem supplies the knowledge_zero component of the EpistemologyCert structure, which certifies the five-stance mapping in the philosophy module. It instantiates the principle that knowledge equals recognition at J = 0. The result sits inside the J-uniqueness step (T5) of the forcing chain and supports the module's claim that genuine knowledge demands full conservation rather than isolated zero cost.
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