Exact finite-size corrections for spacelike-volume estimators in causal diamonds: preregistered tests in 1+1 and 2+1 dimensions

We study a counting statistic for a causal set sprinkled at random into a causal diamond, keeping only the causal order. For the diamond's past half R, the separator is the maximal elements of R together with the minimal elements of its complement: the last layer inside the diamond's widest slice (its waist) and the first layer outside it. The order alone defines it; no thickness or cut-off enters. Buck, Dowker, Jubb and Surya showed that such counts estimate the surface's spatial volume (coefficients sqrt(pi) in 1+1, kappa_3 = 1.7587 in 2+1); in 1+1 the exact finite-count expectation is also classical (Bai, Hwang, Liang and Tsai). We specialize the count to a finite diamond's waist and compute its expectation exactly at finite N. In 1+1 the closed form is that classical one, its correction a taper term; in 2+1 the finite-diamond evaluation appears to be new, its correction a rim term of relative order N^(-1/3). Both formulas were tested in campaigns frozen and registered before any confirmatory counting ran (the 1+1 test carries a disclosed pre-freeze qualification). At the largest size the measured coefficient is 1.7563 in 1+1 (finite-size prediction 1.7501; exponent 0.5126) and 1.6603 in 2+1 (prediction 1.6715, -0.7%; exponent 0.6890); per-size agreement is within 0.8% at all four 2+1 sizes. Rival counts on the same sprinklings - causal links and total relations - follow no boundary law. Finally, we state the boundary-capacity premise any gravity-style reading requires: in factorized form it is independent of the axiomatic core it would extend (explicit countermodels; a machine-checked proof); in capacity form it yields only an upper bound on an average conditional capacity, entailment unresolved. The count estimates a codimension-one spacelike volume; no premise adopted here bridges that to the codimension-two area of the black-hole entropy law.