PotentialFunction
plain-language theorem explainer
PotentialFunction equips a graded ledger with an integer map on vertices such that every edge weight equals the potential difference between its endpoints. Algebraic topologists and Recognition Science researchers cite it when verifying that cycle sums vanish in conserved ledgers. The declaration is a plain structure definition with two fields and no proof obligations.
Claim. Let $L$ be a graded ledger with $n$ vertices and edge weights $w(u,v) = L.edges(u,v) : Fin n → Fin n → ℤ$. A potential function on $L$ is a map $φ : Fin n → ℤ$ such that $w(u,v) = φ(v) - φ(u)$ for all vertices $u,v$.
background
GradedLedger is the structure with vertex count $n$, edge-weight function edges : Fin n → Fin n → ℤ, and the conservation axiom that inflow equals outflow at every vertex. This supplies the algebraic setting for ledgers in the Recognition framework, where conservation encodes the balance law used in double-entry bookkeeping and discrete homology. The module LedgerAlgebra imports the basic ledger type and develops its exactness properties; upstream results include the GradedLedger definition itself together with auxiliary lists of modes and systems that do not enter the structure fields.
proof idea
This is a structure definition. It introduces PotentialFunction L by declaring the potential map and the gradient condition that edge weights must equal potential differences. No lemmas are invoked and no tactics are used; the two fields are taken as primitive data.
why it matters
The structure supplies the exactness notion for the discrete differential on ledgers. It is invoked by the downstream theorem potential_implies_exact, which shows that existence of such a potential forces every cycle sum to zero. In the Recognition Science framework it realizes the algebraic half of T4 discrete exactness, connecting the conservation law of graded ledgers to the forcing chain where inflow-outflow balance implies potentials unique up to constants on components. It closes the algebraic bookkeeping model before geometric or cohomological extensions.
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