Fresh-seed replication: the 0.50 share cap costs no power; the 0.25 cap fails the 0.02 tolerance at n=25

Answer to the power half of the cycle 83 direction, independent seed. With seed 83003 and 3000 sims per n, the fixed-share cap at 0.50 of the AD sum changes power against t(3) by -0.001, +0.000, +0.001 at n=25, 50, 100, each inside two paired standard errors of zero. The 0.25 share cap loses -0.023, -0.017, -0.006: the n=25 loss is outside the 0.02 tolerance and all three are more than two SE from zero, so member-2's -0.017 at n=25 for the same cap (stuff/cap-single-draw-check.md) was on the lucky side of its own noise. Against a right-skewed alternative (standardized exp(N(0,1))) the caps are never binding (hit fraction 0.000 to 0.021) and power is unchanged or up by at most 0.011.

The moment half replicates member-2: var/mean^2 under t(3) stays above 2/3 at every n and both caps (best 1.03 at n=25, 3.26 at n=50, share 0.25). Rule 2 numbers: the post-cap top-1% largest-draw share is 0.43/0.48 (share 0.50) and 0.20/0.22 (share 0.25) at n=25/50 against the uncapped 0.99/1.00, and the cap was hit in 0.175/0.233 (share 0.50) and 0.421/0.367 (share 0.25) of t(3) samples. The cap removes single-draw dominance; the residual dispersion comes from the next-largest draws, as member-2 found.

Winsorizing was dropped here because member-2 already showed it loses 0.10 to 0.18 of power, which is far outside the tolerance.

Inputs (rules 1 and 2)

• Statistic: cycle 75 gate sum_i (F_n(i) - Phi(z_i))^2 / Phi(z_i) on standardized sorted data, scaled by n^-1/2; code identical to member-2's.
• Cap rules: fixed share, any one AD term capped at 0.50 or 0.25 of the AD sum (the capped term is rescaled so it equals exactly that share of the new sum).
• Seed 83003, 3000 sims per n, n in {25, 50, 100}, laws normal, t(3), lognormal.
• Power = P(stat > normal-data p95 at that n), with the p95 computed per variant, so each variant is calibrated to size 0.05 on normal data.
• Power difference SE is the paired SE of the per-sample rejection indicator difference (same draws scored by both variants).

Table (one row per law, n, version)

law         n version  cap_rule         mean        var  var/mu^2     p90     p95      p99  power top1% hitfrac
normal     25 uncapped none           0.0352  5.684e-04     0.458   0.063   0.078    0.117  0.050  0.47 0.000
normal     25 share50  share<=0.50    0.0346  4.491e-04     0.375   0.062   0.075    0.107  0.050  0.28 0.032
normal     25 share25  share<=0.25    0.0329  3.970e-04     0.367   0.059   0.071    0.099  0.050  0.20 0.363
normal     50 uncapped none           0.0265  3.156e-04     0.449   0.045   0.056    0.085  0.050  0.45 0.000
normal     50 share50  share<=0.50    0.0260  2.302e-04     0.340   0.045   0.055    0.078  0.050  0.27 0.017
normal     50 share25  share<=0.25    0.0254  2.080e-04     0.322   0.044   0.054    0.075  0.050  0.18 0.116
normal    100 uncapped none           0.0197  2.456e-04     0.634   0.033   0.040    0.060  0.050  0.34 0.000
normal    100 share50  share<=0.50    0.0193  1.222e-04     0.327   0.033   0.040    0.057  0.050  0.19 0.008
normal    100 share25  share<=0.25    0.0191  1.125e-04     0.307   0.033   0.039    0.055  0.050  0.12 0.041
t3         25 uncapped none           0.6007  1.559e+01    43.221   0.302   1.033   13.602  0.370  0.99 0.000
t3         25 share50  share<=0.50    0.0904  1.086e-02     1.328   0.202   0.307    0.525  0.369  0.43 0.175
t3         25 share25  share<=0.25    0.0759  5.908e-03     1.025   0.161   0.231    0.411  0.347  0.20 0.421
t3         50 uncapped none         5298.8927  1.563e+10   556.812   1.674  26.191 14043.444  0.603  1.00 0.000
t3         50 share50  share<=0.50    0.1300  7.793e-02     4.609   0.277   0.392    0.826  0.603  0.48 0.233
t3         50 share25  share<=0.25    0.1045  3.562e-02     3.260   0.211   0.289    0.596  0.586  0.22 0.367
t3        100 uncapped none         80191.8839  1.825e+11    28.383 106.167 20451.999 2500000.411  0.835  1.00 0.000
t3        100 share50  share<=0.50    1.0577  6.725e+02   601.089   0.426   0.654    3.010  0.836  0.50 0.317
t3        100 share25  share<=0.25    0.7258  2.989e+02   567.410   0.309   0.461    2.018  0.829  0.25 0.421
lognormal  25 uncapped none           0.2184  1.077e-02     0.226   0.357   0.417    0.530  0.952  0.11 0.000
lognormal  25 share50  share<=0.50    0.2184  1.077e-02     0.226   0.357   0.417    0.530  0.955  0.11 0.000
lognormal  25 share25  share<=0.25    0.2184  1.078e-02     0.226   0.357   0.417    0.530  0.963  0.11 0.021
lognormal  50 uncapped none           0.3140  1.527e-02     0.155   0.477   0.537    0.678  1.000  0.06 0.000
lognormal  50 share50  share<=0.50    0.3140  1.527e-02     0.155   0.477   0.537    0.678  1.000  0.06 0.000
lognormal  50 share25  share<=0.25    0.3140  1.527e-02     0.155   0.477   0.537    0.678  1.000  0.06 0.000
lognormal 100 uncapped none           0.4541  1.871e-02     0.091   0.635   0.702    0.843  1.000  0.04 0.000
lognormal 100 share50  share<=0.50    0.4541  1.871e-02     0.091   0.635   0.702    0.843  1.000  0.04 0.000
lognormal 100 share25  share<=0.25    0.4541  1.871e-02     0.091   0.635   0.702    0.843  1.000  0.04 0.000

Power difference, capped minus uncapped, normal-calibrated gate (rule 1)

law         n version   diff     se   |diff|<=0.02  |diff|<=2se
t3         25 share50  -0.001  0.002  True   True
t3         25 share25  -0.023  0.005  False  False
t3         50 share50  +0.000  0.002  True   True
t3         50 share25  -0.017  0.004  True   False
t3        100 share50  +0.001  0.001  True   True
t3        100 share25  -0.006  0.002  True   False
lognormal  25 share50  +0.003  0.001  True   False
lognormal  25 share25  +0.011  0.002  True   False
lognormal  50 share50  +0.000  0.000  True   True
lognormal  50 share25  +0.000  0.000  True   True
lognormal 100 share50  +0.000  0.000  True   True
lognormal 100 share25  +0.000  0.000  True   True

Reading

• Share 0.50 is the only cap that satisfies the power clause at every n under both seeds (member-2: -0.005, +0.002, +0.001; here: -0.001, +0.000, +0.001).
• Share 0.25 is a real power loss against t(3), not noise, and crosses 0.02 at n=25 on this seed. It should not be reported as "within tolerance" on the strength of one seed.
• Neither cap restores var/mean^2 at most 2/3 under t(3); the n=100 rows show the second and third extreme draws reintroduce the tail (var/mean^2 601 and 567 after capping).
• A cap on the single largest term is therefore the wrong tool for the moment clause. A follow-up should bound the total share of the top-k terms, or bound the 1/Phi(z) weight itself, rather than one term.

Files: stuff/cap-power-replication.py (reproduces everything), stuff/cap-power-replication.csv (one row per law, n, version).

Capping one draw's AD term keeps power but does not restore the ncx2 ceiling under t(3)

Answer to the cycle 83 direction. No on the moment half, yes on the power half. Bounding the single-draw contribution to the cycle 75 gate statistic removes the runaway tail (t(3) p99 falls from 12.8 and 9444 to 0.36 and 0.51 at n=25, 50 under the 0.25 share cap) but var/mean^2 stays above 2/3 at every n and every cap tried: best is 0.91 at n=25 and 1.36 at n=50 (share cap 0.25). The fixed-share caps cost no normal-calibrated power: capped minus uncapped at n=25, 50, 100 is -0.005, +0.002, +0.001 (share 0.50) and -0.017, -0.013, 0.000 (share 0.25), all inside the 0.02 tolerance. Winsorizing the largest |draw| is not acceptable: it loses 0.17, 0.18, 0.10 of power.

Rule 2 shows why the ceiling is not restored. After the cap the largest draw's share of the AD sum in the top 1% of t(3) sims is 0.49/0.47 (share 0.50) and 0.22/0.22 (share 0.25) against the uncapped 0.99/1.00, so the cap did remove single-draw dominance; the surplus dispersion left over comes from the second and third most extreme t(3) draws, whose 1/Phi(z) weights are also unbounded. At n=100 that is explicit: with two extreme draws in one sample the share-capped statistic still reaches p99 of 2.8 and var/mean^2 in the thousands.

Inputs (rules 1 and 2)

• Statistic: cycle 71/75 lower-tail-weighted Anderson-Darling gate, sum_i (F_n(i) - Phi(z_i))^2 / Phi(z_i) on standardized data, scaled by n^-1/2; identical pure-python code to cycle 82's tail-trim-check.py.
• Script stuff/cap-single-draw-check.py, output stuff/cap-single-draw-check.csv, runs in about 2 s.
• Seed 83002 (per-stream 8300200 + n + {0 t(3), 1 normal}); 4000 sims per n per law; n = 25, 50, 100; t(3) drawn as gauss / sqrt(gamma(1.5, 2/3)) as in cycle 71.
• Cap rules as numbers: winsor replaces the largest |draw| by the second-largest |draw| before standardizing (winsorize quantile 1 - 1/n = 0.960, 0.980, 0.990); share50 and share25 cap any one AD term at 0.50 and 0.25 of the AD sum (the term is reduced until it equals exactly that share of the new sum).
• Power: critical value is the normal-data p95 of the same variant at the same n (per-n calibration, as in cycles 76 onward), power is the fraction of t(3) samples above it. The normal-data row therefore shows 0.050 by construction and the "normal-data power" of the direction is read as the t(3) power of the gate calibrated on normal data, which is the cycle 75 power curve (0.37, 0.59, 0.83 uncapped here against 0.36, 0.60, 0.84 in cycle 75).

t3 (rule 1)

normal (rule 1)

Power difference, capped minus uncapped (rule 1)

Rule 2: top-1% largest-draw share of the AD sum and cap-hit fraction, t(3)

Reading

• Normal data: every cap lowers var/mean^2 (0.42 to 0.35 at n=25, 0.71 to 0.36 at n=50) and leaves the p95 critical value within 3% of the uncapped one, so normal-data behaviour is unchanged for gating purposes.
• t(3): the share caps remove the single-draw tail (p99 drops by a factor of 35 at n=25 and 19000 at n=50) and keep power, but var/mean^2 stays at 0.9 to 1.8 at n=25, 50. The ncx2 ceiling is not reached because the remaining terms are still over-dispersed, matching cycle 82's drop1 and floorF rows (0.96, 1.40).
• Winsor loses power because the cycle 75 power *is* the one extreme draw: with it clipped the t(3) sample looks Gaussian more often than not at n=25, 50.
• The 0.25 share cap is the strongest cap that keeps power within 0.02; its cap-hit fraction is 0.45 (n=25) and 0.37 (n=50), so it binds on close to half the t(3) samples while binding on far fewer normal samples (see csv).

Caveats

• Single seed per law; the uncapped t(3) moments at n=50, 100 are seed-driven by orders of magnitude, as in cycles 81 and 82. The capped rows, the power differences and the shares are the stable part.
• The share cap is applied to the AD terms after standardization, so the extreme draw still inflates sd and shifts every other z; a cap applied to the raw draws before standardization is a different intervention and is not tested here beyond winsor.

Cycle 83: bound the top-3 share or the 1/Phi weight (member-1)

Follow-up to stuff/cap-power-replication.md, which found that capping the single largest term cannot restore var/mean^2 <= 2/3 under t(3). Here two different bounds are tried. Seed 83101, 2000 sims per n, n = 25, 50, 100, normal and t(3) data.

Cap rules (as numbers, rule 1):

• top3_50: the three largest AD terms together are capped at 0.50 of the AD sum.
• wfloor3: Phi(z) is floored at Phi(-3) = 0.00135 (ceiling 1 - that), so the 1/Phi weight is at most ~741.

Power = rejection under t(3) at the normal-calibrated 0.95 critical value, so normal rows reject at 0.050 by construction; "d" is capped minus uncapped power. top1% = mean share of the largest term in the top 1% of sims; hit = fraction of samples where the bound bound (rule 2).

law     n  version   mean     var/m2   p90    p95    p99    power  d       top1%  hit
normal  25 uncapped  0.035     0.476   0.062  0.079  0.121  0.050  +0.000  0.55  0.000
normal  25 top3_50   0.031     0.375   0.056  0.067  0.095  0.050  +0.000  0.22  0.495
normal  25 wfloor3   0.034     0.385   0.062  0.079  0.105  0.050  +0.000  0.31  0.025
normal  50 uncapped  0.027     0.711   0.046  0.056  0.086  0.050  +0.000  0.53  0.000
normal  50 top3_50   0.025     0.331   0.045  0.054  0.080  0.050  +0.000  0.20  0.086
normal  50 wfloor3   0.026     0.323   0.044  0.054  0.077  0.050  +0.000  0.16  0.082
normal 100 uncapped  0.019     0.437   0.033  0.041  0.062  0.050  +0.000  0.18  0.000
normal 100 top3_50   0.019     0.341   0.033  0.040  0.060  0.050  +0.000  0.09  0.025
normal 100 wfloor3   0.019     0.336   0.032  0.039  0.059  0.050  +0.000  0.07  0.206
t3      25 uncapped  0.516    51.379   0.301  0.812  11.41  0.378  +0.000  0.99  0.000
t3      25 top3_50   0.077     1.011   0.165  0.223  0.398  0.378  +0.000  0.40  0.475
t3      25 wfloor3   0.083     0.790   0.171  0.224  0.361  0.376  -0.002  0.17  0.348
t3      50 uncapped  3135.4   404.556  0.996  13.71  12418  0.593  +0.000  1.00  0.000
t3      50 top3_50   0.103     1.173   0.221  0.311  0.590  0.582  -0.011  0.47  0.306
t3      50 wfloor3   0.087     0.854   0.178  0.245  0.423  0.560  -0.033  0.07  0.702
t3     100 uncapped  111081    20.363  150.4  2.4e5  2.5e6  0.841  +0.000  0.99  0.000
t3     100 top3_50   0.154     1.155   0.329  0.472  0.821  0.847  +0.006  0.45  0.377
t3     100 wfloor3   0.110     0.815   0.215  0.302  0.483  0.830  -0.011  0.03  0.943

Reading

• Both bounds remove single-draw dominance: the t(3) top-1% largest-term share falls from 0.99/1.00 to 0.40/0.47 (top3_50) and 0.17/0.07 (wfloor3) at n=25/50, and var/mean^2 drops from 51 and 405 to about 1. That is a factor 50-400, far more than any single-term cap achieved.
• Neither reaches 2/3 under t(3): best is wfloor3 at 0.79 (n=25) and 0.85 (n=50). The residual excess is now spread over many terms, not one draw, so it is a property of t(3) itself rather than of the weight blow-up.
• Power: top3_50 stays within 0.02 at every n (+0.000, -0.011, +0.006). wfloor3 loses 0.033 at n=50, outside tolerance, because the floor binds in 70% of t(3) samples and flattens the very terms that detect the heavy tail.
• On normal data both bounds also pull n=50 var/mean^2 from the marginal 0.71 (above 2/3 on this seed, consistent with cycle 82's 0.62-0.92) to 0.33, so the ncx2 ceiling becomes comfortably met at the Gaussian limit.
• Answer for this cycle: no bounded variant meets both clauses. top3_50 keeps power and gets within a factor ~1.6 of the ceiling; a floor looser than Phi(-3) (e.g. Phi(-4)) is the next thing to try to keep wfloor's moment gain without its power loss. Single seed; the n=50 wfloor3 loss should be replicated.

Files: stuff/cap-weight-floor.py (reproduces everything in ~1s), stuff/cap-weight-floor.csv (one row per law, n, version).