Suppose two forecasters each have a yes-or-no signal about the same yes-or-no event. Does learning more from one forecaster always reduce the extra value of hearing from the other?

That sounds like a simple question about diminishing returns. But checking only what happens when both signals are revealed completely can miss the answer. One forecaster might reveal a carefully chosen, randomised fragment of their information.

This Evidence Press candidate reduces that whole family of disclosure choices to four exact tests, for one specified quadratic scoring rule. There are two tests for each order of the signals. If all four pass, every admissible partial disclosure satisfies the stated diminishing-returns condition. If one fails and the probabilities are rational, the software constructs a concrete two-report disclosure that demonstrates the failure.

The proof turns the information problem into the shape of a curve. For each order, the relevant curve is concave. Its two endpoint values therefore decide whether it can dip below zero. This is an analytic argument, not a claim that a computer tried every possible disclosure.

The package includes the proof, exact checking software, thousands of finite test cases and explicit counterexamples. A comparison added during review shows that another condition, called projective substitutes, can hold even when this stronger condition fails.

The limits matter. The main classification is for two binary signals and a fixed score. It is not a general theorem about every forecasting system, and it does not test real-world forecast data. Internal checks are not independent external reproduction or formal verification.

The paper and evidence accompany Four tests for binary informational substitutes, released on the eighth of October twenty twenty-six. This briefing uses an AI-generated voice and is not additional mathematical evidence.