CohomologyClass
plain-language theorem explainer
A cohomology class on a defect-bounded sub-ledger is a nonnegative real z_charge encoding Z-complexity charge with the (p,p) symmetry condition. Researchers formalizing the Recognition Science version of the Hodge conjecture cite this structure when mapping classical Hodge classes to coarse-graining stable features. The declaration is a direct structure definition that packages the charge field and its nonnegativity axiom.
Claim. On a defect-bounded sub-ledger $L$, a cohomology class is a real number $z$ (the Z-charge) satisfying $z ≥ 0$.
background
A DefectBoundedSubLedger is a finite collection of recognition events whose total J-cost (defect) is bounded above by $φ^{44}$ and nonnegative. This serves as the RS analog of a smooth projective algebraic variety. The module translates the classical Hodge conjecture by identifying Hodge classes with CoarseGrainingStableClass objects, which extend the present structure by the additional requirement that $z$ is fixed under the data-processing inequality.
proof idea
Structure definition that directly introduces the two fields z_charge : ℝ and symmetric : z_charge ≥ 0; no lemmas or tactics are applied.
why it matters
This supplies the base type for all cohomology classes in the RS Hodge formalization. It is extended by CoarseGrainingStableClass and referenced by the downstream theorems algebraic_generates_hodge, hodge_from_algebraic, and rs_hodge_conjecture, which establish both directions of the equivalence between J-cost minima and stable classes. The definition anchors the (p,p) symmetry condition to the nonnegativity of J-cost, consistent with the T5 J-uniqueness and phi-ladder landmarks.
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