DefectBoundedSubLedger
plain-language theorem explainer
A DefectBoundedSubLedger is a finite list of recognition events equipped with a real-valued total J-cost that is strictly less than phi to the 44 and nonnegative. Researchers formalizing the Hodge conjecture inside Recognition Science cite this structure as the direct analog of a smooth projective variety. The declaration is a plain structure definition that encodes the bounded-defect and nonnegativity conditions with no lemmas or tactics.
Claim. A DefectBoundedSubLedger consists of a finite list of recognition events together with a real number $d$ satisfying $d < phi^{44}$ and $d >= 0$.
background
In the Recognition Science translation of the Hodge conjecture, a smooth projective algebraic variety becomes a DefectBoundedSubLedger: a finite collection of recognition events whose aggregate J-cost (defect) remains finite. The J-cost functional originates in the LedgerFactorization upstream result, which calibrates the multiplicative structure on positive reals and ensures nonnegativity. Coarse-graining flows and stable cohomology classes are subsequently defined on top of such ledgers.
proof idea
This is a structure definition that directly packages a list of recognition events with a real defect value and the two inequalities defect < phi^44 and 0 <= defect. No lemmas are invoked; the definition itself supplies the non-singularity condition required by the module's RS dictionary.
why it matters
The structure supplies the ambient space for every downstream object in the module, including CoarseGrainingFlow, CoarseGrainingStableClass, CohomologyClass, defect_budget_theorem, and HodgeCert. It fills the base case of the RS Hodge dictionary (variety = DefectBoundedSubLedger) and enables the one-direction theorem that every JCostMinimalCycle is a CoarseGrainingStableClass. The bounded-defect hypothesis directly supports the defect-budget argument that stable classes must carry zero charge.
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