rs_hodge_conjecture
plain-language theorem explainer
In Recognition Science every coarse-graining-stable cohomology class on a defect-bounded sub-ledger is generated by some J-cost minimal sub-ledger, and every such minimum produces a stable class. This is the complete RS translation of the Hodge conjecture. The proof is a one-line term that pairs the two directional lemmas.
Claim. Let $c$ be a cohomology class. If $c$ survives every coarse-graining, then there exists $n$ and a sub-ledger $L$ of size $n$ that is J-cost minimal; conversely, every J-cost minimal sub-ledger generates a coarse-graining-stable class.
background
A cohomology class is an abstract object carrying an integer degree and a sector index. A sub-ledger is a finite collection of voxels equipped with a nonnegative real-valued cost function. J-cost minimal means no proper injective substructure of smaller cardinality has strictly lower total defect; such objects are the recognition-closed subgraphs. Coarse-graining-stable means the class is detected after every resolution reduction, which the data-processing inequality identifies with features that cannot be eliminated by smoothing.
proof idea
The term proof applies exact to the pair consisting of hodge_from_algebraic applied to the given class and stability hypothesis together with the lambda that sends any minimal sub-ledger to the stability of the class via algebraic_generates_hodge.
why it matters
The declaration supplies the full bidirectional RS Hodge conjecture inside the HodgeAlgebraicCycles module. It closes the two directions stated in the module doc-comment and referenced in biggest-questions.md §XIII Q2. The result identifies J-cost minima with algebraic cycles and stable classes with Hodge classes, linking directly to the recognition composition law and the global minima of the J-cost landscape.
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