rs_pp_condition
plain-language theorem explainer
The theorem shows that every coarse-graining stable class on a defect-bounded sub-ledger has nonnegative z-charge. Researchers formalizing the Recognition Science version of the Hodge conjecture cite it as the direct counterpart to the classical (p,p) type condition. The proof is a one-line appeal to the symmetric field already present in the class definition.
Claim. Let $L$ be a defect-bounded sub-ledger and let $cls$ be a coarse-graining stable class on $L$. Then the $z$-charge of $cls$ satisfies $0 ≤ z_{charge}$.
background
A DefectBoundedSubLedger is a finite list of recognition events whose total J-cost (defect) satisfies $0 ≤ defect < φ^{44}$. A CoarseGrainingStableClass on such an $L$ extends a cohomology class and obeys the stability condition $z_{charge} ≤ L.defect$, which encodes survival under the data-processing inequality. The module translates the classical Hodge conjecture by mapping smooth projective varieties to these sub-ledgers, Hodge classes to coarse-graining stable classes, and algebraic cycles to J-cost minimal cycles. The present result supplies the nonnegativity half of the (p,p) condition in this dictionary.
proof idea
The proof is a one-line wrapper that applies the symmetric field of the CoarseGrainingStableClass, which directly encodes the required nonnegativity of z-charge.
why it matters
The declaration recovers the classical Hodge (p,p) type condition inside the RS framework, as stated in the module documentation. It supports the proved direction that every JCostMinimalCycle is a CoarseGrainingStableClass. The result sits inside the broader forcing chain and phi-ladder structure that fixes the defect bound at φ^{44}. It leaves the converse direction of the RS Hodge conjecture open.
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