Two Records of One Claim: Posting invariants, reconciliation, and network rewriting

Ana and Ben can make their reports of one debt look closer as a ratio without reducing the amount they disagree about. A table of current exposures also leaves out relationships that are available but carry zero. Those relationships can change how the cost of carrying obligations is shared when obligations are rearranged at fixed net positions. With a free counter account at every party, the monetary gaps between separate records of a claim completely classify matched-posting reachability on balanced real books. For positive reported figures on any finite chart with those counters, matched postings can make the sum of absolute scores arbitrarily small for any score of the ratio that is continuous at one and zero there, with every gap unchanged. The reciprocal cost J(x) = (x + x⁻¹)/2 − 1, the unique cost of a two-sided comparison under a published composition law, is one instance; its unmatched positive-reporting orbits have infimum zero and no minimizer. On log-ratio coordinates the same cost is the carrier objective that law selects.

Fix a continuous, strictly convex, coercive carrier objective minimized at zero. On fixed edge coordinates, two charts have the same minimizer operator exactly when their cut spaces agree. The four-party example keeps all numerical book entries and exposures identical while changing the debtor labels of two zero-balance relationships, recorded in the separate standing chart. Under admissibility-constrained single-edge changes without transfers, the stable-to-efficient burden ratio is unbounded even on four parties. In one tuned, history-dependent example, a bilateral addition raises total burden by exactly 1/56, and a stable set costs about 3.44 times the efficient set.