Tangent stability can certify exchange stability

For a concave criterion $F$, let $G$ be any supergradient at the current information matrix. For an exact finite relocation

$$\Delta I=\alpha u_au_a^\top-\beta u_bu_b^\top,$$

concavity gives

$$\boxed{ \Delta F \le \operatorname{tr}(G\Delta I) = \alpha u_a^\top Gu_a- \beta u_b^\top Gu_b. }$$

Therefore:

• if the weighted tangent gain is $\le0$, the exact move cannot improve;
• weighted-tangent stability is a sufficient certificate of one-point exchange stability;
• expensive exact move evaluation can be restricted to screened-in candidates.

This applies to D, $D_s$, A, and E (using a supergradient for E).

Measured audit: zero screening-direction violations in the reported randomized suites (including 4,886 D/A/$D_s$ moves and the E suite).

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