j_cost_minimal_is_cgstable
plain-language theorem explainer
Every J-cost minimal cycle on a defect-bounded sub-ledger produces a coarse-graining stable class with identical z-charge. Recognition Science researchers formalizing the Hodge conjecture cite this as the verified easy inclusion of algebraic cycles into Hodge classes. The proof constructs the stable class by direct extension and discharges the stability predicate via case split on the zero-defect disjunction together with non-negativity of the defect functional.
Claim. Let $L$ be a defect-bounded sub-ledger and let $C$ be a J-cost minimal cycle on $L$. Then there exists a coarse-graining stable class $cls$ on $L$ such that the z-charge of $cls$ equals the z-charge of the cycle class of $C$.
background
In the RS dictionary a defect-bounded sub-ledger is a finite collection of recognition events whose total J-cost (defect) satisfies $0 ≤ defect < φ^{44}$. A J-cost minimal cycle is a recognition-closed subgraph whose cycle class obeys zero_defect, i.e., its z-charge is either exactly zero or at most one. A coarse-graining stable class extends an ordinary cohomology class by the additional predicate that its z-charge is at most the sub-ledger defect, which encodes survival under the data-processing inequality.
proof idea
The tactic builds the target stable class by copying z-charge and symmetry from the input cycle, then proves the dpi_stable field by rcases on the zero_defect disjunction. The zero branch rewrites and invokes defect_nonneg; the bounded branch chains le_add_of_nonneg_left followed by linarith against the local reflexivity helper phi_pow44_gt_one.
why it matters
This theorem supplies the proved direction of the RS Hodge conjecture: every J-cost minimal cycle is coarse-graining stable. It therefore anchors the algebraic-cycle side of the dictionary and leaves the converse generation statement open. The result sits inside the module that translates the classical Hodge conjecture into Recognition Science terms and relies on the non-negativity of the defect functional proved upstream.
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