The declaration hodgeHardCert in module IndisputableMonolith.Mathematics.HodgeHardDirection assembles proofs for special cases of the RS Hodge conjecture.
(1) In plain English: It constructs a certificate showing that every CoarseGrainingStableClass is generated by a list of JCostMinimalCycle objects (with matching total z_charge) when the sub-ledger is either asymptotically trivial (coarse-graining flow limit equals 0) or has defect at most 1.
(2) Why it matters in Recognition Science: Recognition Science frames the Hodge conjecture as a statement about cost-minimal cycles in defect-bounded ledgers; this theorem supplies the hard direction (stable classes arise from minimal cycles) for the ground-state and low-defect regimes that correspond to physical ground states and unit-charge excitations.
(3) How to read the formal statement: hodgeHardCert is a theorem that returns a value of type HodgeHardCert. The structure HodgeHardCert has three fields: case_A (a quantified statement that the conjecture holds for all trivial-limit ledgers), case_B (the same for all ledgers with defect ≤ 1), and both_directions (a trivial witness that both cases are covered). The proof simply packages the already-proved lemmas rs_hodge_holds_for_trivial_ledgers and rs_hodge_holds_for_unit_defect.
(4) Visible dependencies or certificates in the supplied source: The module imports HodgeConjecture and HodgeHarmonicForms; it directly uses defect_budget_theorem, harmonic_form_theorem_zero_charge, and harmonic_form_theorem_minimal_ledger to build hodge_hard_direction_case_A and hodge_hard_direction_case_B, which are then assigned to the certificate fields. The summary theorem hodge_hard_direction_summary bundles the two cases.
(5) What this declaration does not prove: It does not establish the full RSHodgeConjecture for arbitrary positive z_charge without the defect ≤ 1 or zero-limit assumptions; the general case requiring rational combinations of cycles remains open and is explicitly deferred to HodgeConjecture.lean §RSHodgeConjectureGeneral.