Two Records of One Claim Disagreement, rewriting, and the bystander in networks of bilateral obligations

Models of obligation networks carry one number per link. Real books carry two: the creditor's record of the claim and the debtor's record of the liability, each kept on its own books, with no rule that they coincide. We take that second record as the primitive and ask what it forces. Three results are the core. First, any instrument that reads the network through its exposures matrix (who owes whom how much) or through net positions is blind to a datum that decides who bears the cost of rewriting a cycle of obligations at fixed positions: two networks with identical exposures and positions can differ on which parties the cheapest rewrite touches and on whether a party with nothing outstanding consents to it. Second, for every separable, differentiable, strictly convex cost of open positions that is minimised on empty lines, the cheapest rewrite at fixed positions is unique, is determined by the cut space of the graph of standing relationships, and moves onto a relationship carrying nothing exactly when the marginal costs circulate around a cycle that relationship closes; a party who owed nothing and was owed nothing is then charged. Third, a fixed cost of opening a position (a kink at rest, of which the gross-notional objective used in portfolio compression is the extreme case) suppresses this effect whenever the circulation is smaller than the kink, and a one-sided line suppresses it outright; so the compression literature's finding that fixed-net compression does not reach uninvolved parties is a property of its objective and feasible set, not of the network. Around these we prove what two records force with no cost at all (a disagreement is conserved by every move the two parties make together and is closed only by a revaluation whose two halves sum to the gap; the books contain exactly two figures for a disputed claim, and settlement costs neither party more than the gap exactly on the interval between them), and what they leave free (shares of a cost, the split of a revaluation, outside options, rent schedules, which relationships are kept). An axiomatic section characterises the disagreement index that is additive over relationships, orientation-free and scale-free as J(x) = 1/2 (x + 1/x) − 1 of the ratio of the two records, and shows every objective defined on positions or exposures is cycle-blind. A final section gives a located-unit model in which the two records arise: a transfer is recorded by the two books in different periods, so matched books are kept records with different stamps, and each party's free-account balance is forced to be minus its net position.