More budget never removes effort: the measure-order form of Gavish and Katriel's budget-monotonicity conjecture for periodic regulation

Unrefereed research candidate. In Gavish and Katriel's periodic regulation model a stock (a pollutant, a pest or vector population, a pool of susceptible people, a harvested population) is refilled by a seasonal inflow, decays naturally and is reduced by control effort that removes a fixed fraction of what is present, continuously or in instantaneous pulses. With a fixed yearly effort budget the planner minimises the seasonally weighted average stock. Gavish and Katriel (Applied Mathematics & Optimization 93, 2026) conjectured that a larger budget never makes it optimal to do less at any time.

This package proves the measure-order (weak) form of that conjecture: under their uniqueness assumption, and allowing any mixture of continuous treatment and pulses, the optimal effort assigned to every Borel set of times is nondecreasing in the budget, so supports are nested, pulses never shrink and an optimal plan can be built incrementally. The strict inequality displayed in the source is false in general. Monotonicity survives caps on pulses and treatment rates; it can fail under seasonal prices or seasonal efficacy (exact counterexamples, including among all measures with one continuous price) and, with finitely many dates, under a nonlinear state cost; a price-band sufficient condition and an elasticity criterion are proved for finitely many dates. With finitely many dates a dose budget makes the problem linear. The vaccination Conjecture 8 of Gavish and Katriel (Bulletin of Mathematical Biology 88, 2026) follows in measure form and pointwise except where the density formula is undefined or uses a discontinuous derivative representative.

The archive contains the 24-page manuscript (PDF and LaTeX), Python check scripts (exact rational and symbolic certificates plus labelled floating-point explorations), committed deterministic outputs, a one-command byte-for-byte replay, semantic negative controls and SHA-256 sums. Assurance: written proofs; exact certificates for identities, instances and counterexamples; producer-side replay, including a clean replay in a separate copy that reproduced every output byte-identically; internal AI-assisted review, AI-assisted adversarial review (OpenAI gpt-5.6-sol) and a verification re-review. This is not peer review, independent reproduction or formal verification. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.