The Tuple-Size Law at m=5: Resolving the Sparse Regime

Part XVIII confirmed the tuple-size law d ln rho_m / d ell -> -m at m=4 and predicted the prime quintuplet (m=5): slope -5, ratio 5:2 against the twin. We test it. At the baseline S9->S10 the quintuplet (only 2,936 members on S9) reads slope -5.43 and ratio 2.62, overshooting the 2.5 +/- 0.06 bound -- the prediction's own caveat that sparsity would pull the estimate up. Descending one shell to S10->S11, where the quintuplet count recomputed from sieve primitives is 102,254 (with the twin counted in the same pass, 196,963,369), the slope falls to -5.065 +/- 0.09 and the ratio to 2.4488 +/- 0.042, inside the bound, on the law. The theory predicted the bias and the cure, and the computation obeyed. The ladder is now ironclad at four rungs m=2,3,4,5, with |d ln rho_m / d ell| = kappa m and ratios 1 : 1.5011 : 2.0145 : 2.4488 against 1 : 3/2 : 2 : 5/2. The count-aware estimator of Part XVIII passed its stress test in the sparsest regime yet reached. Hardy-Littlewood inputs, first moments only, the omega-variance left open, no infinitude claimed. This closes the empirical mining of the series; m=6 (slope -6, ratio 3:1, requiring shells beyond S11) is left as the open challenge for supercomputer centres.