Shapes, Not Strengths: Publicity, the Deterrence Feedback, and When a Jump Can Be Destroyed

An administration that publishes its findings chooses how loudly to publish, and that choice does two things at once. It changes how large the sanction looks to those who might incur it, and it changes what the enforcement apparatus subsequently sees, because a violation deterred is a violation nobody observes. Each effect is ordinarily modelled, where it is modelled at all, as a strength — a parameter saying how much publicity moves perceptions, or how much the deterred pool feeds back into what enforcement finds. This paper asks whether those strengths decide anything, and finds that they do not: whether a publicity choice can destroy the jump it is meant to produce at an arbitrarily small feedback is decided by the shapes of both responses and by neither strength. Dividing the administration’s equal-value condition by its first-order condition makes the locus of jump points the critical set of a single scalar function, with the weight-to-floor ratio as its critical value. On that locus a fragility point — a publicity level at which the two jump tightnesses merge, the gap between them having closed to zero — exists exactly when the awareness response’s logarithmic slope exceeds a threshold built from the deterrence technology alone, and an arbitrarily small feedback destroys the jump there exactly when the floor response’s elasticity exceeds one. Neither condition contains a strength. Both say the same thing in different variables: a response that has already turned over cannot be made to matter by making it stronger. A feedback affine in publicity is inert, and affine is the only inert shape, so the second channel is one of curvature rather than of slope. The locus observed over a range of publicity separates the two strengths, and the fragility itself is bounded: the point at which it becomes free lies outside the region in which the administration’s own problem has an interior optimum.