HodgeHardCert
The HodgeHardCert structure records that the hard direction of the RS Hodge conjecture holds for zero-limit ledgers and for unit-defect ledgers. Researchers assembling the Recognition Science formulation of the Hodge conjecture cite this certificate when closing the special-case proofs. The construction packages two prior lemmas, one for each case, into a single record with the joint assertion set to true.
claimLet $L$ be a defect-bounded sub-ledger. If every coarse-graining flow on $L$ has flow limit zero, then for every coarse-graining stable class $cls$ on $L$ there exist $J$-cost minimal cycles whose $z$-charges sum to $cls.z_{charge}$. The same decomposition holds whenever the defect of $L$ is at most 1. Both directions together are asserted as true.
background
A CoarseGrainingStableClass on a DefectBoundedSubLedger is a cohomology class that survives coarse-graining, with its $z_{charge}$ fixed under the Data Processing Inequality and anchored to the actual $J$-cost structure. A $J$CostMinimalCycle is a recognition-closed subgraph with zero net defect, the RS analog of an algebraic cycle. CoarseGrainingFlow models the process of zooming out, which monotonically decreases apparent defect while the flow limit records the scale at which the defect stabilizes. The module assembles the hard direction of the RS Hodge conjecture: every CoarseGrainingStableClass is generated by $J$CostMinimalCycles. This is proved under two restrictions: zero flow limit (Case A) and defect at most 1 (Case B).
proof idea
The structure is a direct packaging of two lemmas. Case A is supplied by rs_hodge_holds_for_trivial_ledgers, which handles the zero-limit condition. Case B is supplied by rs_hodge_holds_for_unit_defect, which handles the defect bound. The both_directions field is set to trivial.
why it matters in Recognition Science
This certificate supplies the hard direction of the RS Hodge conjecture for the special cases of zero-limit ledgers and unit-defect ledgers, feeding directly into the theorem hodgeHardCert. It advances the Recognition Science analog of the classical Hodge conjecture, where defect-budget minimization replaces the role of harmonic forms. The remaining gap for the full conjecture is the case of positive $z_{charge}$ greater than 1, which requires extending generation to rational combinations of cycles.
scope and limits
- Does not prove the full RS Hodge conjecture for arbitrary positive $z_{charge}$.
- Does not extend the notion of generation to rational combinations of cycles.
- Does not remove the zero-limit or unit-defect restrictions.
formal statement (Lean)
131structure HodgeHardCert where
132 /-- Case A proved -/
133 case_A : ∀ L : DefectBoundedSubLedger,
134 (∀ cgf : CoarseGrainingFlow L, flowLimit cgf = 0) →
135 ∀ cls : CoarseGrainingStableClass L,
136 ∃ cycles : List (JCostMinimalCycle L),
137 cls.z_charge = (cycles.map (fun c => c.cycle_class.z_charge)).sum
138 /-- Case B proved -/
139 case_B : ∀ L : DefectBoundedSubLedger, L.defect ≤ 1 →
140 ∀ cls : CoarseGrainingStableClass L,
141 ∃ cycles : List (JCostMinimalCycle L),
142 cls.z_charge = (cycles.map (fun c => c.cycle_class.z_charge)).sum
143 /-- Both directions together (full Hodge for special cases) -/
144 both_directions : True
145