cert_inhabited

The module treats algorithmic probability in the Recognition Science sense: the Solomonoff prior $P(x)=\sum_{U(p)=x}2^{-|p|}$ is tied to J-cost, so that at an RS-optimal encoding the shortest program length scales as $J(\varphi)\cdot|x|$ and the universal prior peaks at the $J(\varphi)$ compression rate. Status is a structural theorem (zero sorry, zero axiom).

Domain cost is the local cost functional on positive reals used in place of raw Kolmogorov complexity for the RS encoding. The certificate structure simply bundles three elementary facts about that cost: it is zero when measure equals encoding scale (nonzero), it is nonnegative for positive arguments, and the module's canonical threshold is positive. Those three props are the only data the structure carries.

One-line term proof. Nonempty is introduced by the anonymous constructor applied to the already-defined value cert, which is the concrete inhabitant of the certificate structure assembled from the three component lemmas (diagonal vanishing, nonnegativity, threshold positivity). No further rewriting or case analysis.

Closes the existence half of the structural package in AlgorithmicProb3FromJCost: once the certificate type is inhabited, downstream arguments may assume the three cost axioms without re-proving them. The module frames this as the bridge from J-cost (T5 uniqueness $J(x)=(x+x^{-1})/2-1$) to a Solomonoff-style prior peaked at the $J(\varphi)$ rate, with $\varphi$ the self-similar fixed point from the forcing chain. No further used-by edges are recorded in-tree yet; the declaration is the terminal existence stamp for the certificate rather than a computational lemma. It does not itself derive the prior sum or the compression-rate identity; it only guarantees the hypothesis bundle is realizable.