empty_ledger_cost
plain-language theorem explainer
The cost of the empty ledger, defined via summation of event costs over an empty list, equals zero. Researchers deriving the existence of zero-cost balanced ledgers under J-symmetry cite this base case. The proof reduces immediately by simplification on the cost summation and empty-list definitions.
Claim. Let $L_0$ be the ledger containing no recognition events. Its total cost, obtained by summing the costs of events in its list, satisfies $cost(L_0) = 0$.
background
The Ledger Forcing module establishes that J-symmetry forces double-entry ledger structure. The empty ledger is defined as the ledger whose event list is empty and is balanced by the nil-count property. Ledger cost is the foldl summation of individual event costs over that list. This result parallels the empty-ledger cost theorem in the QuantumLedger module and rests on the empty ledger definition from the same file.
proof idea
One-line wrapper that applies simp to unfold ledger cost as a fold over events and empty ledger as the nil list, reducing the sum to zero.
why it matters
It supplies the zero-cost instance required by the ledger forcing principle, which chains J-symmetry to paired events, double-entry bookkeeping, and log-sum cancellation. The principle explicitly asserts existence of a balanced ledger with zero cost. The theorem is invoked in the empty-support case of the finite Euler ledger cost formula and in the QuantumLedger counterpart.
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