A Mathematical Framework for Balance-Aware Federated Analytics of Confidential Multi-Entity Accounting Data

Confidential ledgers cannot usually be pooled across entities, yet generic private federated learning does not preserve the identities that make accounting data meaningful. This paper develops Balance-Aware Federated Analytics (BAFA), a constrained federated framework that combines voucher-level differential privacy, debit–credit and period-continuity regularization, account-hierarchy smoothing, secure aggregation, and a one-sided balance-aware update correction. The formulation defines neighboring ledgers by replacement of one complete voucher, bounds the sensitivity of the released representation, model update, and compressed balance sketch, and composes one cached representation release and all round-level aggregate releases with a Rényi differential-privacy accountant that explicitly models the minimum number of non-colluding noise contributors. It also specifies period-complete aggregation for multi-line vouchers, derives the one-sided correction from a half-space projection, and gives first-order balance-safety, hierarchy-stability, convergence, and complexity results under non-IID data, clipping, privacy noise, and sketch error. Our experiments use PaySim, IEEE-CIS Fraud Detection, and UCI Online Retail transformed into accounting-style multi-entity ledgers. The reported points indicate that BAFA improves predictive utility and normalized balance consistency relative to private federated baselines while keeping membership-inference attack AUC near random guessing. The transformed-ledger evaluation is intended as controlled evidence; validation on native enterprise ledgers remains necessary.