module module high

`IndisputableMonolith.Mathematics.HodgeConjecture`

The module sets up the Recognition Science analog of the Hodge conjecture by introducing DefectBoundedSubLedger as a finite collection of recognition events whose total J-cost is bounded. Researchers assembling algebraic geometry statements inside the RS framework would cite it when building the proof chain for Hodge-type algebraicity. The module organizes definitions via imports from Constants, LedgerForcing, and HodgeConjectureStructure without containing proofs itself.

claimA defect-bounded sub-ledger is a finite set $S$ of recognition events such that the total J-cost satisfies $J(S) < ∞$, serving as the RS analog of a smooth projective algebraic variety.

background

Constants supplies the base time quantum τ₀ = 1 tick. LedgerForcing establishes that J-symmetry forces double-entry ledger structure. HodgeConjectureStructure provides the M-006 scaffold for Hodge-type algebraicity statements. The supplied DOC_COMMENT defines the central object: a DefectBoundedSubLedger is a finite set of recognition events with bounded total J-cost, the RS analog of a smooth projective algebraic variety.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the core objects for the RS Hodge conjecture and feeds the hard direction assembled in HodgeHardDirection, where every CoarseGrainingStableClass is generated by JCostMinimalCycles. It also supports the harmonic forms analog developed in HodgeHarmonicForms, which translates the classical Hodge theorem on unique harmonic representatives minimizing the L² norm.

scope and limits

Does not prove the classical Hodge conjecture over the complex numbers.
Does not treat sub-ledgers with unbounded total J-cost.
Does not compute explicit numerical bounds on J-cost for concrete examples.
Does not address non-finite collections of recognition events.

used by (2)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Mathematics.HodgeHardDirection module_import
IndisputableMonolith.Mathematics.HodgeHarmonicForms module_import

depends on (3)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Foundation.LedgerForcing module_import
IndisputableMonolith.Mathematics.HodgeConjectureStructure module_import

declarations in this module (19)

structure DefectBoundedSubLedger L56
structure CohomologyClass L68
structure CoarseGrainingStableClass L78
structure JCostMinimalCycle L84
theorem j_cost_minimal_is_cgstable L100
theorem j_cost_minimal_is_cgstable' L119
def RSHodgeConjecture L145
theorem hodge_implies_zero_charge L154
theorem rs_pp_condition L167
theorem sub_ledger_exists L172
structure CoarseGrainingFlow L188
def trivialFlow L199
def flowLimit L206
theorem flowLimit_nonneg L211
def IsFlowStable L216
theorem flow_stable_at_zero L221
theorem defect_budget_theorem L229
structure HodgeCert L260
theorem hodgeCert L272