empty_balanced
The empty ledger page satisfies the balance condition that the sum of event flows equals zero. Researchers modeling conserved quantities in Recognition Science ledger systems would cite this as the base case for proving balance preservation under event pairing. The proof proceeds by direct simplification of the balance computation on the empty event list.
claimThe ledger page with empty event list and initial value 0 satisfies the balance predicate, i.e., the fold of event flows over the empty list equals zero.
background
In the LedgerAlgebra module a LedgerPage pairs a list of LedgerEvent (each carrying a flow integer) with an initial integer. The function computeBalance reduces the event list by left-folding addition of flows, starting from zero; IsBalanced then asserts that this reduction equals zero. The empty page therefore meets the predicate by the definition of foldl on the nil list.
proof idea
The proof is a one-line term-mode wrapper that invokes simp on the definitions of IsBalanced and computeBalance, reducing the empty-list case of the fold to the identity 0 = 0.
why it matters in Recognition Science
This supplies the base case for inductive arguments that paired events preserve balance, as indicated by the sibling declaration paired_preserves_balance. It anchors the algebraic treatment of neutral configurations (sum of flows zero) that later feed into graded ledgers and window constructions. The result closes the trivial end of the ledger induction without touching open questions about physical embedding.
scope and limits
- Does not establish balance for any non-empty event list.
- Does not incorporate the initial balance field into the computed sum.
- Does not relate the ledger balance to J-cost or the phi-ladder.
- Does not address units, constants, or forcing-chain steps T0-T8.
formal statement (Lean)
97theorem empty_balanced : IsBalanced ⟨[], 0⟩ := by
proof body
Term-mode proof.
98 simp [IsBalanced, computeBalance]
99
100/-- **PROVED: Adding a paired event preserves balance.** -/