CohomologyClass
plain-language theorem explainer
The structure for a cohomology class on a sub-ledger packages an integer degree with a sector index. Researchers formalizing the Recognition Science version of the Hodge conjecture cite this when stating that stable classes arise from J-cost minima. The declaration consists of a bare structure definition supplying exactly those two fields.
Claim. A cohomology class on a sub-ledger is given by an integer degree $d$ in the discrete cohomology together with a sector index $s$.
background
In the Recognition Science translation of the Hodge conjecture, a sub-ledger is a defect-bounded collection of voxels whose collective J-cost is controlled. A cohomology class abstracts the topological charge carried by such a sub-ledger; here it is represented discretely by its degree in the integer cohomology and the sector to which it belongs. This module strengthens the prior partial result that J-cost minima generate stable classes by supplying the converse direction.
proof idea
The declaration is a direct structure definition that introduces the two fields degree and sector. No lemmas are applied; the structure simply packages the integer degree and sector index as the complete data for a cohomology class in this discrete setting.
why it matters
This structure is the central object in the RS Hodge conjecture formalized in HodgeAlgebraicCyclesCert and rs_hodge_conjecture. It appears in both directions of the equivalence: algebraic_generates_hodge shows that J-cost minimal sub-ledgers produce coarse-graining-stable instances of this class, while hodge_from_algebraic shows the converse. The construction sits inside the forcing chain at the level of discrete cohomology and supports the claim that Hodge classes coincide with recognition-closed subgraphs.
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