SubLedger
plain-language theorem explainer
A sub-ledger of size n is a finite collection of n voxels equipped with a non-negative real-valued cost function. Workers on the Recognition Science translation of the Hodge conjecture cite this structure when encoding J-cost minimal subgraphs. The definition is a direct record that bundles the cost assignment with its non-negativity constraint inherited from the cost of recognition events.
Claim. A sub-ledger of cardinality $n$ consists of a map $c$ from a set of $n$ elements to the real numbers together with a proof that $c(i)geq 0$ for every index $i$.
background
Recognition Science derives physical quantities from the J-functional, where the defect of a state x equals J(x) and satisfies the Recognition Composition Law. In this module the sub-ledger serves as the finite support for cohomology classes in the RS reformulation of the Hodge conjecture. The module document states that every coarse-graining-stable class must arise from a J-cost minimal sub-ledger. Upstream, the cost non-negativity follows from the theorem that every recognition event has non-negative J-cost. From the Law of Existence the defect coincides with J on positive arguments. The structure therefore supplies the data type on which total defect and minimality predicates are defined.
proof idea
This is a direct structure definition. It records the cost function on a finite index set and the non-negativity proof as fields. No further lemmas or tactics are invoked beyond the field declarations.
why it matters
SubLedger is the foundational carrier for the biconditional proved in HodgeAlgebraicCyclesCert, which asserts that J-cost minimal sub-ledgers are exactly the generators of coarse-graining-stable cohomology classes. This realizes the RS translation of the Hodge conjecture referenced in biggest-questions.md §XIII Q2. The construction relies on the J-cost minima arising from the phi-forcing chain and the non-negativity of costs established in ObserverForcing.
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