Globally Distinct, Locally Identical Duty, Private Value, and the Trap Boundary

A monitor’s duty — the weight at which he charges himself for excused failures — and his valuation of the quality he produces are often treated as one preference under two names. They are not. Duty enters the payoff through attribution, private value through production; their marginal effects have different observability profiles, and no global identity relates them. But on the boundary of the strategic-inattention trap, where minimal attention ceases to be a rest point, the two are exactly interchangeable. That boundary is a straight line whose slope is an exchange rate pricing duty in private-value units. The rate equals the stake plus the base regime boundary, because duty is exactly private value internalized at the stake plus a blame-compression term that private value cannot supply. It rises with the stake, falls with productivity, and is a shadow price rather than a constant. The equivalence is exact on the line and confined to it. A reviewer’s rejected intuition and its rejection are recovered as propositions over different domains, the intuition capturing a measurable nine-tenths of the mechanism — suggesting that identity claims between parameters are better posed as: on which submanifold, and to what order?