IsFlowStable
plain-language theorem explainer
IsFlowStable encodes the condition that a coarse-graining stable class on a defect-bounded sub-ledger survives a given coarse-graining flow. Researchers formalizing the Recognition Science Hodge conjecture cite it when reducing stable classes to zero charge under flows that converge to the ground state. The definition is the direct inequality between the class z-charge and the limiting defect extracted from the flow's monotone sequence.
Claim. Let $L$ be a defect-bounded sub-ledger, let $F$ be a coarse-graining flow on $L$, and let $C$ be a coarse-graining stable class on $L$. The predicate holds precisely when the z-charge of $C$ is at most the limiting defect of $F$.
background
A DefectBoundedSubLedger is a finite collection of recognition events whose total J-cost (defect) satisfies $0 ≤ defect < φ^{44}$. A CoarseGrainingFlow on such an $L$ consists of a monotone nonincreasing sequence of nonnegative defects with coarsened_defect 0 equal to the ledger defect; it models successive zooming out. A CoarseGrainingStableClass extends a cohomology class by the data-processing inequality z_charge ≤ L.defect, ensuring the class survives arbitrary coarsening.
proof idea
One-line definition that directly sets the predicate equal to the inequality cls.z_charge ≤ flowLimit cgf, where flowLimit is the limit of the coarsened_defect sequence.
why it matters
The definition supplies the hypothesis used by defect_budget_theorem to conclude that any class stable under all flows (including the zero-limit flow) has z_charge = 0. It thereby feeds the hard-direction arguments in HodgeHardDirection.hodge_hard_direction_case_A and HodgeHarmonicForms.hard_direction_via_defect_budget, which establish that every stable class on a zero-limit ledger is generated by a JCostMinimalCycle. This step closes the defect-budget bridge in the RS translation of the Hodge conjecture.
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