CohomologyClass
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.
claimOn 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 in Recognition Science
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.
scope and limits
- Does not assert that every such class is generated by a J-cost minimal cycle.
- Does not compute or bound explicit numerical values of z_charge.
- Does not encode the full de Rham cohomology ring structure.
- Does not address the converse direction of the Hodge conjecture.
formal statement (Lean)
68structure CohomologyClass (L : DefectBoundedSubLedger) where
69 /-- The characteristic value of the class (Z-charge) -/
70 z_charge : ℝ
71 /-- The type (p,p) condition in RS: charge is symmetric about 0 -/
72 symmetric : z_charge ≥ 0
73
74/-- A CoarseGrainingStableClass is a cohomology class that survives
75 "zooming out" — the Data Processing Inequality (DPI).
76 A class survives coarse-graining iff its z_charge is anchored to
77 the actual J-cost structure (not just a statistical artifact). -/