IndisputableMonolith.Mathematics.HodgeHarmonicForms
The HodgeHarmonicForms module defines the J-cost gradient as the variation in total defect under small perturbations of recognition events and proves that every ledger is J-cost critical. It further states harmonic form theorems for zero-charge and minimal ledgers together with a Hodge decomposition. Researchers assembling the hard direction of the RS Hodge conjecture cite these results. The module proceeds by importing LedgerForcing and HodgeConjecture then deriving the listed theorems from those foundations.
claimThe J-cost gradient measures the first-order change in total defect when a recognition event is perturbed. A ledger is J-cost critical when this gradient vanishes. Every ledger is J-cost critical. Globally minimal ledgers determine harmonic forms, and the Hodge decomposition exists for the associated cohomology classes.
background
The module sits inside the Recognition Science translation of the Hodge conjecture. LedgerForcing establishes that J-symmetry forces double-entry ledger structure. HodgeConjecture supplies the RS rendering of the classical statement that every Hodge class on a smooth projective variety is a rational linear combination of algebraic subvarieties. Constants fixes the base time quantum as one tick.
proof idea
The module first introduces the definitions JCostGradient and IsJCostCritical. It then proves all_ledgers_are_jcost_critical directly from the double-entry structure supplied by LedgerForcing. Subsequent results establish harmonic_form_theorem_zero_charge, harmonic_form_theorem_minimal_ledger, HodgeDecomposition and the two hard-direction lemmas via defect-budget arguments.
why it matters in Recognition Science
This module supplies the harmonic-forms layer required by HodgeHardDirection to prove that every CoarseGrainingStableClass is generated by JCostMinimalCycles. It therefore occupies the position in the proof chain that links J-cost minimality to the existence of algebraic cycles in the RS Hodge conjecture.
scope and limits
- Does not prove the full Hodge conjecture.
- Does not address the easy direction of the conjecture.
- Does not compute explicit numerical values for any gradient.
- Does not treat ledgers outside the J-cost minimal class.
used by (1)
depends on (3)
declarations in this module (13)
-
structure
JCostGradient -
def
IsJCostCritical -
theorem
all_ledgers_are_jcost_critical -
def
IsGloballyMinimal -
theorem
globally_minimal_gives_cycle -
theorem
harmonic_form_theorem_zero_charge -
theorem
harmonic_form_theorem_minimal_ledger -
structure
HodgeDecomposition -
theorem
hodge_decomposition_exists -
theorem
hard_direction_via_defect_budget -
theorem
hard_direction_for_zero_limit_ledger -
structure
HarmonicFormsCert -
theorem
harmonicFormsCert