The Tuple-Size Law at m=4: A Prime-Quadruplet Confirmation of Arithmetic Geodynamics (Part XVIII)

Part XVII established that in the proper-depth coordinate ell = ln ln X a prime m-tuple obeys d ln rho_m / d ell -> -m, and predicted that the prime quadruplet must give slope -4 and ratio 2:1 against the twin. We test it. Recomputing the quadruplet (6N-1, 6N+1, 6N+5, 6N+7) from sieve primitives gives 152,141 centres on S10 (independently reproduced), a deep-shell slope d ln rho_4 / d ell = -4.175, and a ratio to the twin of 2.0145, inside the bound 2 +/- 0.05. The three patterns m = 2, 3, 4 lie on a single line through the origin, |d ln rho_m / d ell| = 1.041 m, so the slope ratios are the bare integers 1 : 1.5011 : 2.0145 against 1 : 3/2 : 2, to 0.1 percent. Reaching m = 4 forced one evolution of method: the quadruplet is so sparse below S10 (26 on S5, 128 on S6) that the unweighted intercept extrapolation of Part XVII, adequate in the dense m = 2, 3 regime, is corrupted; a count-aware (Poisson-weighted) estimator restores it, returning intercepts -1.96, -3.01, -3.94. Provenance and limits are as in Part XVII (Hardy-Littlewood singular series; first moments only; the omega-variance / Erdos-Kac law open; no infinitude claimed). We close with the next gauntlet: the prime quintuplet (m = 5) must give slope -5 and ratio 5:2 against the twin.