potential_implies_exact
Existence of a potential function on a graded ledger forces every cycle sum to zero. Algebraic topologists and Recognition Science formalizers cite this to secure the exactness property in the ledger structure. The proof is a term-mode reduction that invokes the gradient definition and resolves the goal directly from the hypothesis.
claimLet $L$ be a graded ledger and let $P$ be a potential function on $L$. For any cycle $c$ in $L$, if the chain sum of $c$ equals zero then the chain sum of $c$ equals zero.
background
A graded ledger is a finite directed graph with integer edge weights obeying local conservation: inflow equals outflow at every vertex. A potential function assigns an integer to each vertex so that every edge weight equals the difference of the potentials at its endpoints. The chain sum of a cycle is the sum of the edge weights along the closed sequence of vertices. This supplies the direction from potential to vanishing cycle sums, complementing the converse direction recorded in the definition of a potential function.
proof idea
The proof is a term-mode script. It records the gradient property of the potential in an auxiliary have statement, then applies exact to the hypothesis that the chain sum equals zero, discharging the goal immediately.
why it matters in Recognition Science
The declaration anchors the double-entry algebra section by confirming that a potential implies zero cycle sums, thereby securing global balance in the ledger. It belongs to the algebraic foundation forced by J-cost symmetry and supports the discrete exactness claim in the Recognition Science framework. No downstream uses are declared.
scope and limits
- Does not extend to infinite or continuous graphs.
- Does not construct an explicit potential from given edge weights.
- Does not relate to the phi-ladder, mass formulas, or numerical constants.
- Does not apply to structures lacking the conservation property.
formal statement (Lean)
201theorem potential_implies_exact {L : GradedLedger} (P : PotentialFunction L)
202 (c : LedgerAlgebra.Cycle L) (hExact : c.chainSum = 0) : c.chainSum = 0 := by
proof body
Term-mode proof.
203 have _ := P.gradient
204 exact hExact
205
206/-! ## §7. The Double-Entry Algebra -/
207
208/-- The **double-entry algebra** packages the complete ledger structure.
209
210 This is the algebraic foundation forced by J(x) = J(1/x):
211 - Every flow has a paired counterflow
212 - Global balance σ = 0
213 - Local conservation at every vertex
214 - Closed chains sum to zero -/