The Shape of Deterrence: What a Harbour's Silence Reveals, and What It Cannot

An administration operating under a cap on how tightly it must regulate can also choose how loudly to do so, and publicity is not free. It buys deterrence, and it burns a floor that protects the monitors working beneath it. Where those two things decay at the wrong relative rate the administration’s problem folds: a silent regime and a loud one are both locally rational at identical primitives, and the passage between them is a jump rather than a slide. Companion A exhibits that fold at one calibrated point and says in print that its extent is not mapped. This paper maps it, and finds the map shorter than the mapping. Dividing the equal-value condition by the first-order condition removes, at one stroke, the weight the administration places on deterrence and every primitive of the harbour it administers, leaving a relation between the loudness at which an administration jumps and the tightness at which it jumps that involves the deterrence technology and nothing else whatever. The fold’s outer edge has the same form, and is exactly the place where the condition guaranteeing a single optimum stops being satisfiable. Where marginal deterrence at zero loudness is finite, both edges meet the local silence boundary at a single point: the fold is born on the criterion that excludes it, with no width and no jump, and grows away from it. Three things follow. The set of jump points determines the deterrence technology up to a multiplicative constant and determines nothing else, so an observer who watches an administration go quiet learns the shape of deterrence and nothing about how much anybody values it — and can never learn the scale, because scale and valuation enter the problem only as a product. A single jump carries nothing about the harbour at all; the envelope of many carries exactly one thing, the cap at which the harbour ceases to exist, and that is the whole of it. The region of coexistence has no widest member: its supremum is the entire admissible range of caps, approached as exposure slows, and what buys that approach is publicity far past the point at which the administration begins to finance its own backfire, so the binding trade-off is not the one the question was posed in. And folding is a property of a rate rather than of deterrence: two of the four technologies examined cannot fold at any cap, weight or scale — one of them the saturation form Companion A carries — because their marginal deterrence and the cap’s own thinning of the tax base decay along curves that never cross. The paper maps and does not rank. What an administration gains by operating inside the region is a welfare question, and the primitive it would need is named here and not supplied.