Actual Economics: Separate records, reconciliation, and the standing network

Arbitrarily close relative agreement need not imply a small monetary gap, and an exposures table cannot identify a network rewrite. 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 is one instance; its unmatched positive-reporting orbits have infimum zero and no minimizer.

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. Identical numerical book entries and exposures can coexist with different debtor labels on two zero-balance relationships in the separate standing chart, giving different optimal carriers and individual charges. Exact triangle and four-party examples locate this missing information; finite network certificates show an unstable efficient tree, an inefficient stable cycle, and dependence on the order of deletions.

Recognition Science supplies the composition law whose product term, with the stated normalization, uniquely selects the hyperbolic kernel (Ground 4.1). Recognition Science also supplies the recording clock (Section 12) and the discount weight φ⁻ᴺ (Sections 9 and 12). On log-ratios, cosh t − 1 = J(eᵗ) is the unit-free carrier objective; its triangle minimum differs from the quadratic one. Every theorem of Sections 2 through 7 and Appendix A holds on classical grounds alone. The remaining framework locates prices through reservations, distinguishes cash payments from accruals, and states the further data needed for coalitions, splitting, genesis, and the economic dictionary.