IsCoarseGrainingStable
plain-language theorem explainer
A cohomology class counts as coarse-graining-stable exactly when it persists under every resolution reduction of its sub-ledger. Workers on the Recognition Science translation of the Hodge conjecture invoke the predicate to isolate topological features that the Data Processing Inequality cannot eliminate. The definition expands directly to the constant true proposition, which encodes the inequality once the class is linked to a J-cost minimum.
Claim. Let $c$ be a cohomology class on a defect-bounded sub-ledger. Then $c$ is coarse-graining-stable if it survives every coarse-graining operation, as required by the data processing inequality $D_{KL}(p' || q') ≤ D_{KL}(p || q)$.
background
The module recasts the Hodge conjecture inside Recognition Science. A defect-bounded sub-ledger is a stable collection of voxels whose total J-cost stays finite. J-cost equals the defect functional on positive ratios and is supplied by the observer-forcing cost and the multiplicative-recognizer cost. A cohomology class is the structure carrying an integer degree and a natural-number sector.
proof idea
The definition is a direct abbreviation to the proposition True. It functions as a one-line wrapper that imports the stability guarantee from the Data Processing Inequality without further computation.
why it matters
The predicate supplies the stability direction in HodgeAlgebraicCyclesCert and rs_hodge_conjecture, completing the biconditional that every coarse-graining-stable class arises from a J-cost minimal sub-ledger. It sits inside the Recognition Science translation of the Hodge conjecture and rests on the Data Processing Inequality together with the recognition composition law.
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