module module moderate

`IndisputableMonolith.Mathematics.HodgeAlgebraicCycles`

The module Mathematics.HodgeAlgebraicCycles introduces sub-ledgers as finite voxel sets equipped with cost functions to support algebraic cycle analysis in Recognition Science. It supplies supporting objects including totalDefect and IsJCostMinimal for defect bounds and stability. Researchers addressing the RS formulation of the Hodge conjecture would cite these definitions as the starting point for cycle generation and coarse-graining. The module consists entirely of declarations and basic properties with no external lemmas required.

claimA sub-ledger is a finite set $S$ of voxels equipped with a cost function $c: S → ℝ$ such that defect measures and J-cost minimality can be defined on $S$.

background

Recognition Science builds all structures from the J-cost function obeying the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). This module defines SubLedger as the basic carrier object for finite voxel collections and introduces totalDefect as the aggregate defect measure together with IsDefectBounded and IsJCostMinimal as stability predicates. The local setting assumes the phi-ladder and eight-tick octave from the upstream forcing chain while remaining inside standard Mathlib type theory.

proof idea

this is a definition module, no proofs

why it matters in Recognition Science

The module supplies the primitives that feed into rs_hodge_conjecture, algebraic_generates_hodge, and hodge_from_algebraic. It establishes the sub-ledger as the concrete object linking voxel recognition costs to Hodge cycles, closing the first layer of scaffolding for the algebraic-to-Hodge direction in the Recognition framework.

scope and limits

Does not prove any form of the Hodge conjecture.
Does not specify the explicit functional form of the cost function beyond the sub-ledger interface.
Does not contain numerical examples or explicit phi-ladder embeddings.
Does not address higher-dimensional or infinite voxel sets.

declarations in this module (14)

structure SubLedger L46
def totalDefect L51
theorem totalDefect_nonneg L54
def IsDefectBounded L58
structure CohomologyClass L65
structure CoarseGraining L70
def IsCoarseGrainingStable L78
def IsJCostMinimal L85
theorem minimal_defect_concentrated L93
theorem algebraic_generates_hodge L107
theorem hodge_from_algebraic L124
theorem rs_hodge_conjecture L140
structure HodgeAlgebraicCyclesCert L149
def hodgeAlgebraicCyclesCert L158