JCostMinimalCycle
plain-language theorem explainer
JCostMinimalCycle encodes the Recognition Science analog of an algebraic cycle as a list of recognition events on a defect-bounded sub-ledger whose associated cohomology class carries zero net charge or charge at most one. Researchers formalizing the Hodge conjecture inside Recognition Science cite this structure when establishing the algebraic-implies-Hodge direction. The definition is a direct structural encoding of balanced boundary conditions with no separate proof steps.
Claim. Let $L$ be a defect-bounded sub-ledger. A JCostMinimalCycle on $L$ consists of a list of recognition events, a cohomology class on $L$ with charge $z$, and the condition that $z=0$ or $zleq 1$.
background
The module translates the classical Hodge conjecture into Recognition Science terms. A DefectBoundedSubLedger is a finite collection of recognition events with total J-cost strictly less than phi^44, serving as the RS stand-in for a smooth projective variety. A CohomologyClass on such a ledger is a real number z_charge satisfying z_charge geq 0 that records the topological charge of the class. CoarseGrainingStableClass extends the cohomology class by the additional requirement that z_charge leq defect of the ledger, enforcing invariance under the data-processing inequality.
proof idea
This is a structure definition. It directly assembles the three fields cycle_events, cycle_class, and zero_defect from the imported RecognitionEvent and CohomologyClass primitives supplied by LedgerForcing and HodgeConjectureStructure. No lemmas or tactics are invoked; the zero_defect disjunction is stated verbatim as the balanced-boundary condition.
why it matters
JCostMinimalCycle supplies the algebraic-cycle object needed for the proved half of the RS Hodge conjecture. It is invoked by j_cost_minimal_is_cgstable and j_cost_minimal_is_cgstable' to show that every minimal cycle yields a CoarseGrainingStableClass, and by HodgeCert to record that the algebraic-implies-Hodge direction holds. The definition anchors the framework to the defect-budget argument and to the open converse stated in RSHodgeConjecture.
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