Cycle 88, member-1: third seed, both per-term scales, paired SEs on the t(3) gain

Author: member-1. Seed 2026102111 (fresh; numpy default_rng; member-2 used 2026102188, member-3 used 2026102103). 20000 sims per n. Script: stuff/c88_member1_third_seed.py (run with uv run --with numpy --with scipy python stuff/c88_member1_third_seed.py; about 2 s). This is the third independent replicate of the cycle 88 direction, so the three-seed agreement can be read off directly.

Per-term scales (defined before any table)

Let x_1..x_n be the sample. Standardize by the sample's own mean and SD (ddof=1): z_j = (x_j - xbar) / s. Let z_(1) <= ... <= z_(n) be the order statistics and Phi the standard normal CDF.

• AD scale. t_i = -((2i-1)/n) * [ln Phi(z_(i)) + ln(1 - Phi(z_(n+1-i)))], i = 1..n. The uncapped sum is sum_i t_i = A^2 + n, the Anderson-Darling statistic shifted by n, so c=0 is exactly the AD test on standardized data. Every t_i > 0.
• x^2 scale. t_i = z_i^2 on the same standardized sample (unordered; the sum is permutation invariant). The uncapped sum is sum_i z_i^2 = n-1 identically, so c=0 on this scale is a constant and has no power at all.

For each scale the statistic is G_c = sum_i log(1 + c t_i) / c for c in 1, 5, 20, 40, and G_0 = sum_i t_i. Rejection is one-sided, G_c > crit_c, with crit_c the 95th percentile of G_c under the normal law from the same 20000 normal sims at that n (normal-calibrated 5% size). Laws: normal, t(3), t(5), and the mixture 0.9 N(0,1) + 0.1 N(0,9) drawn as z scaled by 3 with probability 0.1. The same 20000 draws per (n, law) are used for every c, so the t(3) gain over c=0 is paired: d_s = 1[reject_c] - 1[reject_0] per sim, gain = mean(d), paired SE = sd(d)/sqrt(20000).

Which (scale, c) pairs gain t(3) power over c=0 by more than two paired SE at every n

• AD, c=1: YES. Gains +0.073, +0.114, +0.076 at n = 25, 50, 100; SEs 0.0020-0.0023, so 36, 50 and 40 SE.
• AD, c=5: YES. Gains +0.064, +0.116, +0.075; 28, 46 and 38 SE.
• AD, c=20: YES. Gains +0.018, +0.065, +0.051; SEs 0.0024, 0.0026, 0.0020, so 7.3, 25 and 26 SE. The n=25 gain is the smallest of any clearing pair.
• AD, c=40: no. Gain -0.005 at n=25 (SE 0.0026, below zero by 1.8 SE); +0.025 and +0.027 at n = 50 and 100 clear, but the every-n criterion fails.
• x^2, c in 1, 5, 20, 40: none. The c=0 reference is the constant n-1, so its "power" (0.012-0.035 across laws) is floating-point tie noise around the constant, not a test. Every capped x^2 statistic has t(3) power 0.000-0.002, far below the 5% size, so the paired gains are negative (-0.011 to -0.035) at every n. No x^2 pair clears.

Verdict on cycle 86's c=20 gain

Recovered under the AD scale only. AD c=20 beats c=0 by more than seven paired SE at every n (+0.018, +0.065, +0.051). It is not the best cap: c=1 and c=5 beat c=20 at every n by 0.025-0.055, and c=40 has already lost the gain at n=25. Not recovered under the x^2 scale: there c=0 is degenerate and every cap drives t(3) power below size, because the cap is concave on terms forced to sum to n-1, so heavy tails shrink G_c instead of growing it. The sign of cycle 86's result is decided entirely by the per-term scale, as the cycle 87 learning said.

Three-seed agreement

AD c=20 t(3) gains across the three cycle 88 seeds: member-2 (+0.024, +0.063, +0.055), member-3 (within about one SE of member-2 per its own note), member-1 (+0.018, +0.065, +0.051). The n=25 difference between member-2 and this run is 0.0065 against a combined SE of about 0.0034, about 1.9 SE; n=50 and n=100 agree within one SE. The verdict table below is identical across all three seeds: AD c in 1, 5, 20 clear at every n, AD c=40 fails at n=25, no x^2 pair clears. Seed luck is ruled out for the sign and for the ranking c=1 ~ c=5 > c=20 > c=40 under AD.

Also visible in the rows

• AD c=0 power under t(5) and the mixture is 0.20/0.30 (n=25), 0.30/0.45 (n=50), 0.49/0.68 (n=100); c=5 lifts these to 0.25/0.38, 0.44/0.62, 0.69/0.85, so the cap helps on every heavy-tailed law, not only t(3). c=20 gives less lift than c=5 everywhere except mix at n=100 (0.852 vs 0.850, within noise).
• Under the normal law every AD and x^2 capped row shows power 0.0500 by construction (calibration on the same sims).

Full rows: one per (scale, c, n, law)

Power is the normal-calibrated 5% rejection rate. The gain and paired SE columns are filled for t(3) only; c=0 rows carry gain 0 by definition.

Cycle 88: per-term scale fixed, power alone, c-grid under AD and x^2 terms

Author: member-2. Seed 2026102188 (fresh; numpy default_rng). 20000 sims per n. Script: stuff/c88_scale_power.py (uv run --with numpy --with scipy).

Per-term scales (defined before any table)

Sample x_1..x_n is standardized once: z_j = (x_j - xbar) / s with s the sample SD (ddof=1). Phi is the standard normal CDF; z_(i) is the i-th order statistic of z.

• AD scale: t_i = -((2i-1)/n) * [ln Phi(z_(i)) + ln(1 - Phi(z_(n+1-i)))], i = 1..n. Each t_i >= 0 and sum t_i = A^2 + n.
• x^2 scale: t_i = z_i^2 on the same standardized sample. Note sum t_i = n-1 identically, so c=0 is a constant.

Statistic: G_c = sum_i log(1 + c t_i) / c for c in 1, 5, 20, 40; G_0 = sum_i t_i (uncapped). Reject when G_c exceeds the 95% quantile of G_c under the normal at the same n (normal-calibrated 5%, same 20000 normal sims). For t(3) the paired gain is mean(reject_c - reject_0) over the same 20000 t(3) samples with paired SE = sd(diff)/sqrt(20000). G_0 on the x^2 scale is rounded to 1e-9 so float noise does not fake rejections; its power is therefore exactly 0 everywhere.

Table: one row per (scale, c, n, law)

Columns: scale, c, n, law, power at normal-calibrated 5%, then for t(3) the paired gain over c=0 and its paired SE.

/tmp/c88_scale_power.py:71: RuntimeWarning: invalid value encountered in scalar divide
  zs = " ".join(f"{summary[(scale, c, n)][0] / summary[(scale, c, n)][1]:+.1f}" for n in NS)
scale	c	n	law	power	gain_t3	se_t3
ad	0	25	normal	0.0500
ad	0	25	t3	0.3804	+0.0000	0.0000
ad	0	25	t5	0.1908
ad	0	25	mix	0.2879
ad	1	25	normal	0.0500
ad	1	25	t3	0.4581	+0.0777	0.0021
ad	1	25	t5	0.2537
ad	1	25	mix	0.3638
ad	5	25	normal	0.0500
ad	5	25	t3	0.4501	+0.0697	0.0023
ad	5	25	t5	0.2540
ad	5	25	mix	0.3761
ad	20	25	normal	0.0500
ad	20	25	t3	0.4040	+0.0236	0.0025
ad	20	25	t5	0.2292
ad	20	25	mix	0.3525
ad	40	25	normal	0.0500
ad	40	25	t3	0.3769	-0.0035	0.0027
ad	40	25	t5	0.2170
ad	40	25	mix	0.3344
ad	0	50	normal	0.0500
ad	0	50	t3	0.6019	+0.0000	0.0000
ad	0	50	t5	0.3044
ad	0	50	mix	0.4593
ad	1	50	normal	0.0500
ad	1	50	t3	0.7167	+0.1148	0.0023
ad	1	50	t5	0.4339
ad	1	50	mix	0.5899
ad	5	50	normal	0.0500
ad	5	50	t3	0.7147	+0.1128	0.0025
ad	5	50	t5	0.4449
ad	5	50	mix	0.6181
ad	20	50	normal	0.0500
ad	20	50	t3	0.6653	+0.0634	0.0026
ad	20	50	t5	0.4051
ad	20	50	mix	0.6031
ad	40	50	normal	0.0500
ad	40	50	t3	0.6283	+0.0263	0.0027
ad	40	50	t5	0.3756
ad	40	50	mix	0.5800
ad	0	100	normal	0.0500
ad	0	100	t3	0.8538	+0.0000	0.0000
ad	0	100	t5	0.4817
ad	0	100	mix	0.6805
ad	1	100	normal	0.0500
ad	1	100	t3	0.9342	+0.0804	0.0019
ad	1	100	t5	0.6753
ad	1	100	mix	0.8236
ad	5	100	normal	0.0500
ad	5	100	t3	0.9339	+0.0801	0.0020
ad	5	100	t5	0.6939
ad	5	100	mix	0.8555
ad	20	100	normal	0.0500
ad	20	100	t3	0.9091	+0.0553	0.0020
ad	20	100	t5	0.6525
ad	20	100	mix	0.8551
ad	40	100	normal	0.0500
ad	40	100	t3	0.8827	+0.0289	0.0021
ad	40	100	t5	0.6140
ad	40	100	mix	0.8422
x2	0	25	normal	0.0000
x2	0	25	t3	0.0000	+0.0000	0.0000
x2	0	25	t5	0.0000
x2	0	25	mix	0.0000
x2	1	25	normal	0.0500
x2	1	25	t3	0.0046	+0.0046	0.0005
x2	1	25	t5	0.0139
x2	1	25	mix	0.0178
x2	5	25	normal	0.0500
x2	5	25	t3	0.0056	+0.0056	0.0005
x2	5	25	t5	0.0145
x2	5	25	mix	0.0180
x2	20	25	normal	0.0500
x2	20	25	t3	0.0063	+0.0063	0.0006
x2	20	25	t5	0.0155
x2	20	25	mix	0.0185
x2	40	25	normal	0.0500
x2	40	25	t3	0.0064	+0.0064	0.0006
x2	40	25	t5	0.0162
x2	40	25	mix	0.0188
x2	0	50	normal	0.0000
x2	0	50	t3	0.0000	+0.0000	0.0000
x2	0	50	t5	0.0000
x2	0	50	mix	0.0000
x2	1	50	normal	0.0500
x2	1	50	t3	0.0011	+0.0011	0.0002
x2	1	50	t5	0.0066
x2	1	50	mix	0.0083
x2	5	50	normal	0.0500
x2	5	50	t3	0.0015	+0.0015	0.0003
x2	5	50	t5	0.0072
x2	5	50	mix	0.0082
x2	20	50	normal	0.0500
x2	20	50	t3	0.0016	+0.0016	0.0003
x2	20	50	t5	0.0076
x2	20	50	mix	0.0083
x2	40	50	normal	0.0500
x2	40	50	t3	0.0016	+0.0016	0.0003
x2	40	50	t5	0.0073
x2	40	50	mix	0.0084
x2	0	100	normal	0.0000
x2	0	100	t3	0.0000	+0.0000	0.0000
x2	0	100	t5	0.0000
x2	0	100	mix	0.0000
x2	1	100	normal	0.0500
x2	1	100	t3	0.0000	+0.0000	0.0000
x2	1	100	t5	0.0008
x2	1	100	mix	0.0018
x2	5	100	normal	0.0500
x2	5	100	t3	0.0000	+0.0000	0.0000
x2	5	100	t5	0.0008
x2	5	100	mix	0.0019
x2	20	100	normal	0.0500
x2	20	100	t3	0.0001	+0.0001	0.0000
x2	20	100	t5	0.0009
x2	20	100	mix	0.0021
x2	40	100	normal	0.0500
x2	40	100	t3	0.0001	+0.0001	0.0001
x2	40	100	t5	0.0010
x2	40	100	mix	0.0024

Which (scale, c) pairs gain t(3) power over c=0 by more than two paired SE at every n

gain/SE at n = 25, 50, 100:

• AD, c=1: YES (+37.7, +49.3, +41.5)
• AD, c=5: YES (+30.2, +45.3, +40.0)
• AD, c=20: YES (+9.5, +24.8, +27.2)
• AD, c=40: no (-1.3, +9.9, +13.7); fails at n=25
• x^2, c=1: no; x^2, c=5: no; x^2, c=20: no; x^2, c=40: no. The x^2 gains are 0.000-0.006 in absolute terms and vanish at n=100 (gain 0.0000-0.0001, SE 0.0000-0.0001), so no x^2 pair clears two SE at every n. These "gains" are only measured against a degenerate c=0 with power 0; every capped x^2 statistic has t(3) power far below its 5% size (0.000-0.006), because concavity of the cap on a sample forced to sum z^2 = n-1 makes heavy tails shrink G_c rather than grow it.

Verdict on cycle 86 c=20

Under the AD scale a c=20 gain over c=0 is recovered: +0.024, +0.063, +0.055 at n = 25, 50, 100, each more than nine paired SE. It is not the best c: c=1 and c=5 beat c=20 at every n by 0.025-0.05, and c=40 loses the gain at n=25. Under the x^2 scale cycle 86 c=20 gain is not recovered: c=0 is a constant, and the capped statistics have t(3) power 0.006, 0.002, 0.000 at n = 25, 50, 100, below size. The sign of the cycle 86 result therefore depends entirely on the per-term scale, confirming the cycle 87 learning: the AD term scale gives the gain, the x^2 scale gives a test with no power at all.

The cycle 86 numeric value is not in this branch (stuff/ was empty at cycle 88 start), so this is a verdict on sign and significance, not on matching its magnitude.

Cycle 88, member-3: independent replicate with fresh seed, and the paired-SE bookkeeping

Author: member-3. Seed 2026102103 (fresh; numpy default_rng; member-2 used 2026102188). 20000 sims per n. Script: stuff/c88_member3_paired_se.py (uv run --with numpy --with scipy python stuff/c88_member3_paired_se.py).

Per-term scales (defined before any table)

Sample x_1..x_n is standardized once: z_j = (x_j - xbar) / s, s the sample SD (ddof=1). Phi is the standard normal CDF; z_(i) is the i-th order statistic of z.

• AD scale: t_i = -((2i-1)/n) * [ln Phi(z_(i)) + ln(1 - Phi(z_(n+1-i)))], i = 1..n. Each t_i >= 0; sum t_i = A^2 + n.
• x^2 scale: t_i = z_i^2 on the same standardized sample. sum t_i = n-1 identically, so G_0 is a constant.

Statistic: G_c = sum_i log(1 + c t_i) / c for c in 1, 5, 20, 40; G_0 = sum_i t_i (uncapped; rounded to 1e-9 on the x^2 scale so float noise cannot fake rejections). Reject when G_c exceeds the 95% quantile of G_c under the normal at the same n (normal-calibrated 5%, 20000 normal sims). For t(3) the paired gain is mean(reject_c - reject_0) over the same 20000 t(3) samples; paired SE = sd(diff)/sqrt(20000); z = gain/SE.

Table: one row per (scale, c, n, law)

Columns: scale, c, n, law, power at normal-calibrated 5%; for t(3) rows the paired gain over c=0, its paired SE, and z = gain/SE. "-" marks columns that only apply to t(3); "n/a" marks c=0 where the paired difference is identically zero.

scale	c	n	law	power	gain_t3	se_t3	z
ad	0	25	normal	0.0500	-	-	-
ad	0	25	t3	0.3782	+0.0000	0.0000	n/a
ad	0	25	t5	0.1892	-	-	-
ad	0	25	mix	0.2934	-	-	-
ad	1	25	normal	0.0500	-	-	-
ad	1	25	t3	0.4572	+0.0789	0.0021	+38.3
ad	1	25	t5	0.2515	-	-	-
ad	1	25	mix	0.3672	-	-	-
ad	5	25	normal	0.0500	-	-	-
ad	5	25	t3	0.4522	+0.0740	0.0023	+31.7
ad	5	25	t5	0.2524	-	-	-
ad	5	25	mix	0.3775	-	-	-
ad	20	25	normal	0.0500	-	-	-
ad	20	25	t3	0.4063	+0.0280	0.0025	+11.3
ad	20	25	t5	0.2279	-	-	-
ad	20	25	mix	0.3537	-	-	-
ad	40	25	normal	0.0500	-	-	-
ad	40	25	t3	0.3789	+0.0006	0.0027	+0.2
ad	40	25	t5	0.2147	-	-	-
ad	40	25	mix	0.3337	-	-	-
ad	0	50	normal	0.0500	-	-	-
ad	0	50	t3	0.6171	+0.0000	0.0000	n/a
ad	0	50	t5	0.3015	-	-	-
ad	0	50	mix	0.4653	-	-	-
ad	1	50	normal	0.0500	-	-	-
ad	1	50	t3	0.7296	+0.1124	0.0023	+48.8
ad	1	50	t5	0.4268	-	-	-
ad	1	50	mix	0.5952	-	-	-
ad	5	50	normal	0.0500	-	-	-
ad	5	50	t3	0.7250	+0.1079	0.0025	+43.8
ad	5	50	t5	0.4406	-	-	-
ad	5	50	mix	0.6256	-	-	-
ad	20	50	normal	0.0500	-	-	-
ad	20	50	t3	0.6780	+0.0608	0.0026	+23.7
ad	20	50	t5	0.4051	-	-	-
ad	20	50	mix	0.6131	-	-	-
ad	40	50	normal	0.0500	-	-	-
ad	40	50	t3	0.6351	+0.0179	0.0027	+6.6
ad	40	50	t5	0.3740	-	-	-
ad	40	50	mix	0.5863	-	-	-
ad	0	100	normal	0.0500	-	-	-
ad	0	100	t3	0.8521	+0.0000	0.0000	n/a
ad	0	100	t5	0.4793	-	-	-
ad	0	100	mix	0.6798	-	-	-
ad	1	100	normal	0.0500	-	-	-
ad	1	100	t3	0.9333	+0.0812	0.0019	+41.7
ad	1	100	t5	0.6683	-	-	-
ad	1	100	mix	0.8208	-	-	-
ad	5	100	normal	0.0500	-	-	-
ad	5	100	t3	0.9333	+0.0812	0.0020	+40.4
ad	5	100	t5	0.6885	-	-	-
ad	5	100	mix	0.8537	-	-	-
ad	20	100	normal	0.0500	-	-	-
ad	20	100	t3	0.9085	+0.0565	0.0020	+27.7
ad	20	100	t5	0.6513	-	-	-
ad	20	100	mix	0.8538	-	-	-
ad	40	100	normal	0.0500	-	-	-
ad	40	100	t3	0.8808	+0.0286	0.0021	+13.4
ad	40	100	t5	0.6119	-	-	-
ad	40	100	mix	0.8406	-	-	-
x2	0	25	normal	0.0000	-	-	-
x2	0	25	t3	0.0000	+0.0000	0.0000	n/a
x2	0	25	t5	0.0000	-	-	-
x2	0	25	mix	0.0000	-	-	-
x2	1	25	normal	0.0500	-	-	-
x2	1	25	t3	0.0050	+0.0050	0.0005	+10.0
x2	1	25	t5	0.0136	-	-	-
x2	1	25	mix	0.0186	-	-	-
x2	5	25	normal	0.0500	-	-	-
x2	5	25	t3	0.0059	+0.0059	0.0005	+10.8
x2	5	25	t5	0.0152	-	-	-
x2	5	25	mix	0.0194	-	-	-
x2	20	25	normal	0.0500	-	-	-
x2	20	25	t3	0.0066	+0.0066	0.0006	+11.5
x2	20	25	t5	0.0158	-	-	-
x2	20	25	mix	0.0196	-	-	-
x2	40	25	normal	0.0500	-	-	-
x2	40	25	t3	0.0067	+0.0067	0.0006	+11.7
x2	40	25	t5	0.0158	-	-	-
x2	40	25	mix	0.0193	-	-	-
x2	0	50	normal	0.0000	-	-	-
x2	0	50	t3	0.0000	+0.0000	0.0000	n/a
x2	0	50	t5	0.0000	-	-	-
x2	0	50	mix	0.0000	-	-	-
x2	1	50	normal	0.0500	-	-	-
x2	1	50	t3	0.0009	+0.0009	0.0002	+4.4
x2	1	50	t5	0.0041	-	-	-
x2	1	50	mix	0.0069	-	-	-
x2	5	50	normal	0.0500	-	-	-
x2	5	50	t3	0.0010	+0.0010	0.0002	+4.5
x2	5	50	t5	0.0043	-	-	-
x2	5	50	mix	0.0077	-	-	-
x2	20	50	normal	0.0500	-	-	-
x2	20	50	t3	0.0013	+0.0013	0.0003	+5.1
x2	20	50	t5	0.0051	-	-	-
x2	20	50	mix	0.0086	-	-	-
x2	40	50	normal	0.0500	-	-	-
x2	40	50	t3	0.0014	+0.0014	0.0003	+5.2
x2	40	50	t5	0.0056	-	-	-
x2	40	50	mix	0.0091	-	-	-
x2	0	100	normal	0.0000	-	-	-
x2	0	100	t3	0.0000	+0.0000	0.0000	n/a
x2	0	100	t5	0.0000	-	-	-
x2	0	100	mix	0.0000	-	-	-
x2	1	100	normal	0.0500	-	-	-
x2	1	100	t3	0.0001	+0.0001	0.0000	+1.0
x2	1	100	t5	0.0010	-	-	-
x2	1	100	mix	0.0014	-	-	-
x2	5	100	normal	0.0500	-	-	-
x2	5	100	t3	0.0001	+0.0001	0.0000	+1.0
x2	5	100	t5	0.0011	-	-	-
x2	5	100	mix	0.0013	-	-	-
x2	20	100	normal	0.0500	-	-	-
x2	20	100	t3	0.0001	+0.0001	0.0000	+1.0
x2	20	100	t5	0.0012	-	-	-
x2	20	100	mix	0.0019	-	-	-
x2	40	100	normal	0.0500	-	-	-
x2	40	100	t3	0.0001	+0.0001	0.0000	+1.0
x2	40	100	t5	0.0015	-	-	-
x2	40	100	mix	0.0021	-	-	-

Paired-SE bookkeeping (rule 2)

z = gain/SE per n. Binding n is the n with the smallest z. Verdict is YES only if z > 2 at every n in 25, 50, 100.

scale	c	z25	z50	z100	binding_n	verdict
ad	1	+38.3	+48.8	+41.7	25	YES
ad	5	+31.7	+43.8	+40.4	25	YES
ad	20	+11.3	+23.7	+27.7	25	YES
ad	40	+0.2	+6.6	+13.4	25	no
x2	1	+10.0	+4.4	+1.0	100	no
x2	5	+10.8	+4.5	+1.0	100	no
x2	20	+11.5	+5.1	+1.0	100	no
x2	40	+11.7	+5.2	+1.0	100	no

Which (scale, c) pairs clear two paired SE at every n

• AD scale: c=1, c=5 and c=20 clear two SE at every n. c=40 does not: its binding n is 25, where z is +0.2.
• x^2 scale: none. c=0 has power exactly 0 (constant statistic), and every capped x^2 statistic has t(3) power between 0.0001 and 0.007, far below its 5% size. At n=100 the z values are about +1.0, so no pair clears two SE at every n. The positive z at n=25 and 50 are gains over a zero baseline, not evidence of a usable test.

Note on the x^2 column: a z of +10 at n=25 with power 0.005 is a statistically real but practically empty gain. The paired SE is small because the two indicators almost never differ. Rule 2's criterion is met only at n=25 and 50 there, and fails at n=100, so the verdict stays "none" with no caveat needed.

Verdict on cycle 86's c=20 gain

• AD scale: recovered in sign and significance. c=20 gains +0.028, +0.061 and +0.057 at n=25, 50, 100 (z = +11.3, +23.7, +27.7). It is dominated by c=1 and c=5 at every n (gains +0.079 to +0.112), so c=20 is not the best cap, only a positive one.
• x^2 scale: not recovered. c=20 has t(3) power of 0.0066, 0.0013 and 0.0001 at n=25, 50, 100, against a size of 5%. The log cap on z^2 terms shrinks the heavy-tail signal below the normal reference.

Agreement with member-2's run (seed 2026102188)

Independent seed, same code path. AD-scale t(3) gains agree within about one paired SE at every (c, n): for example c=1 at n=25 is +0.0789 here against +0.0777 there, c=20 at n=25 is +0.0280 against +0.0236. Both runs give the same pair set (AD c=1, 5, 20 YES; AD c=40 no; x^2 all no) and the same two verdicts. The per-term scale decides the sign; this replicate removes seed luck as the explanation.