HodgeCert
plain-language theorem explainer
HodgeCert records the formal status of the Recognition Science version of the Hodge conjecture. It confirms the RS vocabulary, proves that J-cost minimal cycles produce coarse-graining stable classes, states the conjecture, and notes that it stays open. Researchers in algebraic geometry or foundational physics would reference this when discussing the partial formalization. The structure is populated directly from prior lemmas in the module.
Claim. A status certificate consists of: vocabulary defined; for every defect-bounded sub-ledger $L$ (finite recognition events with total J-cost bounded by $phi^{44}$) with defect at least 1 and every J-cost minimal cycle $cyc$ on $L$ (zero net defect subgraph), there exists a coarse-graining stable class $cls$ on $L$ with $cls.z_{charge} = cyc.cycle_{class}.z_{charge}$; the RS Hodge conjecture is stated; and the conjecture remains open.
background
The module translates the classical Hodge conjecture into Recognition Science terms. A defect-bounded sub-ledger represents a smooth projective variety as a finite collection of recognition events with bounded total J-cost (defect < phi^44). A J-cost minimal cycle is a recognition-closed subgraph with zero net defect, analogous to an algebraic cycle. A coarse-graining stable class is a cohomology class that survives zooming out, satisfying z_charge ≤ L.defect, corresponding to a Hodge class. The upstream defect is defined as the J functional. The proved direction is that minimal cycles yield stable classes; the full conjecture that every stable class arises from minimal cycles is open.
proof idea
The structure is defined directly. The key field algebraic_implies_hodge is populated by applying the lemma that every J-cost minimal cycle is coarse-graining stable, specifically via j_cost_minimal_is_cgstable' on the given ledger and cycle. The openness is witnessed by a negation showing the conjecture does not imply falsehood.
why it matters
HodgeCert serves as the status record feeding the theorem hodgeCert in the same module, which assembles the certificate using the stability lemma. It documents the one proved direction of the RS Hodge conjecture (algebraic cycles imply Hodge classes) while marking the converse as open, aligning with the framework's emphasis on J-cost minima anchoring stable features under coarse-graining. This partial formalization touches the open question of whether all coarse-graining stable classes are generated by zero-defect cycles.
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