Money Is on the Arrow: A State-Transition Analysis Separating Payment Amounts, Deposit Balances, and History Aggregates

When a firm pays a household 100, both the amount paid and the household's post-payment deposit balance may equal 100. The former, however, is a value of one payment, whereas the latter is a value of an account at one point in time. This paper separates monetary amounts into amounts of individual payment transitions, deposit balances of states, and period aggregates of histories. By making explicit the step from individual transitions to period aggregation, this threefold distinction is finer than the usual stock/flow distinction. Bookkeeping changes can be written as x_i − x_{i−1} = b_i m_i when they are expressible in terms of one reference amount, and as x_i − x_{i−1} = B_i q_i when several distinct bookkeeping amounts are required. This framework yields a ledger-level underdetermination result: even a complete double-entry ledger cannot by itself decide whether the payment amount or the deposit balance should be called “money.” The result is demonstrated through a comparison of Schmitt and Graziani, a zero-profit benchmark sequence, and a positive-profit sequence. Thermodynamics provides intuition for separating state from process, while the extent of reaction supplies a form for decomposing state change into a direction and an executed magnitude. The contribution of the paper is to specify what the same “100” is a value of and to make visible the rules that connect quantities of different types.企業が家計へ100を支払うと、支払額と支払い後の家計預金はともに100になりうる。しかし、前者は一回の支払い、後者は一時点の口座についての値である。本稿は、通貨額を個別遷移の支払額、状態の預金残高、履歴の期間集計額へ分ける。この三分法は、個別遷移から期間集計への一段階を明示する点で、通常のストック/フロー区別より細かい。記帳変化は、一つの基準額で表せる場合には x_i − x_{i−1} = b_i m_i、複数の基礎記帳額を要する場合には x_i − x_{i−1} = B_i q_i と書ける。この形式から、完全な複式簿記も、支払額と預金残高のどちらを「貨幣」と呼ぶべきかを単独では決定しないという台帳上の不決定性が得られる。シュミットとグラツィアーニの比較、利潤ゼロ系列、正の利潤形成系列によってこの結果を示す。熱力学は状態と過程を分ける意味を直観化し、反応進行度は状態変化を方向と実行量へ分解する形式を与える。本稿の貢献は、同じ「100」が何について測られた値かを明示し、異なる量をつなぐ規則を可視化することにある。