PartialMap Ledger: When Mathematical Agents May Reuse Tool Receipts

Mathematical agents often cache outputs from external tools and reuse them in later reasoning. If this history is compressed, however, retaining individually correct answers need not preserve what the agent is justified in doing next. We study a transparent setting in which a tool implements one fixed but unknown linear map $M$ on $V=(\mathbb{Z}/n\mathbb{Z})^d$, and each call returns a receipt $(x,b)$ with $b=Mx$. A history determines a partial input-to-output map $\phi_H$ on the span $X_H$ of previously queried inputs. We show that $(X_H,\phi_H)$ is exact for future receipt acceptance: any two unequal ledger states are separated by one admissible receipt. The same object also yields exact algebraic tests for whether two histories admit a shared completion and whether every completion is invertible. Importantly, stronger logical conclusions do not necessarily require richer memory states: the universal invertibility decision depends only on the socle restriction, even though exact receipt acceptance requires the full ledger state. A controlled study of 3,120 held-out actions serves as an illustration: replay and the ledger agree with the exact labels, while the declared lossy policy--decoder pairs make both unsafe approvals and unnecessary refusals. The result isolates a distinction between preserving true outputs and preserving the relational authority needed for later actions.