hodge_hard_direction_case_A
plain-language theorem explainer
For asymptotically trivial defect-bounded sub-ledgers, where every coarse-graining flow converges to zero, every stable class has zero z-charge and is generated by a list of J-cost minimal cycles. Researchers on the Recognition Science Hodge conjecture cite this as the ground-state case of the hard direction. The proof is a short tactic sequence that first applies the defect budget theorem to force zero charge then extracts the cycle via the zero-charge harmonic form theorem.
Claim. Let $L$ be a defect-bounded sub-ledger such that every coarse-graining flow on $L$ has flow limit zero. Then for every coarse-graining stable class $cls$ on $L$ there exists a list of J-cost minimal cycles on $L$ whose z-charges sum to the z-charge of $cls$.
background
The Recognition Science Hodge conjecture states that every CoarseGrainingStableClass is generated by JCostMinimalCycles. A DefectBoundedSubLedger carries a bounded defect functional, while CoarseGrainingFlow is the structure whose coarsened_defect map is monotone decreasing with scale and satisfies at_zero equal to the ledger defect; this flow is the RS analog of heat flow. The hypothesis that flowLimit equals zero for all such flows encodes asymptotic triviality, i.e., the sub-ledger sits in a ground state with no persistent cost defects, as stated in the module doc-comment for Case A.
proof idea
The tactic proof introduces the stable class, builds the auxiliary fact that every flow is stable by rewriting the triviality hypothesis into the IsFlowStable predicate and using the class symmetry, applies defect_budget_theorem to conclude that z_charge is zero, obtains a zero-charge cycle from harmonic_form_theorem_zero_charge, packages the cycle in a singleton list, and finishes with simplification.
why it matters
This theorem supplies the proved Case A of the hard direction, feeding directly into the downstream result rs_hodge_holds_for_trivial_ledgers that asserts the full RSHodgeConjecture for asymptotically trivial sub-ledgers. It covers the main ground-state regime of the RS framework; the module doc-comment notes that positive z-charge cases require extension to rational combinations of cycles, an open step left for HodgeConjecture.lean.
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