Exact in the corner, where nothing was

Coverage of a nominal 95% interval on five designs, at a required half-width of 0.3. The first four are the two-arm field's own and the fifth is its corner — two variances, block sizes that swing by eight, and an allocation that alternates between five to one and one to five — where neither of that field's two conditions holds. The effective-size weights over-cover there at 98.40%; the weights h_b(λ) = (1/m_A + λ/m_B)⁻¹ cover at 94.84%, and at 94.84% when λ is estimated from the within-arm contrasts rather than known. Nothing here is supposed to move.

λ enters only through the weights, so misstating it leaves the estimate unbiased and moves two things — the interval's calibration and its efficiency — both of which are closed forms of the design. Coverage stays at its level over a factor of two in either direction (94.93% at half the truth, 94.27% at twice it) and starts to go at a factor of five. An estimate on hundreds of within-arm degrees of freedom is never wrong by anything like that, which is what makes the feasible rule usable rather than merely definable.

The corner leaves a choice rather than an answer. Equal weights assume nothing about the variance ratio, are correctly scaled at any ratio, and are 118% wider. The weights h_b(λ) assume the ratio, are exact, and are the narrowest interval available. The effective sizes assume one of two conditions that do not hold here, and are both mis-scaled and wide — over-covering at 98.40%, which reads as caution and is not. The first two are a trade; the third is not on the frontier at all.