Explanation of globally_minimal_gives_cycle

(1) In plain English, the declaration states that if a defect-bounded sub-ledger is globally minimal (its defect is exactly 0 or at most 1), then there exists a J-cost minimal cycle associated to it whose z-charge is at most the ledger's defect. The proof constructs such a cycle explicitly by taking the ledger's events and assigning a zero z-charge class with the zero-defect property.

(2) In Recognition Science this matters because it supplies a concrete link between global cost minima (the RS analog of harmonic forms) and the cycle generators of cohomology classes. It forms part of the bridge from J-cost criticality to the Hodge-harmonic-form theorem, showing that minimal ledgers produce the required minimal cycles for stable classes.

(3) The formal statement is read as follows:

theorem globally_minimal_gives_cycle (L : DefectBoundedSubLedger)
    (h : IsGloballyMinimal L) :
    ∃ cyc : JCostMinimalCycle L,
      cyc.cycle_class.z_charge ≤ L.defect

It takes a sub-ledger L and a hypothesis h that L satisfies IsGloballyMinimal (i.e., L.defect = 0 ∨ L.defect ≤ 1), then asserts existence of a JCostMinimalCycle whose z-charge obeys the inequality. The proof uses L.defect_nonneg to discharge the bound.

(4) Visible dependencies in the supplied source include the definition of IsGloballyMinimal in the same module and the construction of JCostMinimalCycle instances (with fields cycle_events, cycle_class, zero_defect). It is used inside the larger harmonicFormsCert structure and appears alongside harmonic_form_theorem_zero_charge and hodge_decomposition_exists. No external axioms are invoked; the proof is a direct construction.

(5) This declaration does not prove the general Hodge conjecture, nor does it show that every CoarseGrainingStableClass (without the global-minimality assumption on L) admits a minimal cycle. It is restricted to the zero-or-unit defect case and leaves the full defect-budget argument as a hypothesis in the module documentation.