Cycle 87: centered dispersion of the capped gate at the absolute-draw scale, member-1

Script: stuff/cycle87-centered-dispersion-member-1.py (numpy, seed 870047, 20000 sims per n, 20 batches of 1000 for Monte Carlo SEs). Seed 870047 was not used in cycles 84-86 and differs from member-2 (870021) and member-3 (870033).

This item is the third scaling of the same experiment: same gate family, same c grid, same laws, but t_i = |x_i| (absolute standardized draw) instead of x_i^2 (member-2) or x_i^2/n (member-3). At this scale c*t_i is O(c) for a typical draw, so the cap is near-total for every c in 20-40 and log(1+ct)/c is close to log(ct)/c: the gate becomes an almost-log-of-absolute-value statistic, which is the scaling under which a heavy-tail cap is most plausibly a power device.

Definitions:

• t_i = |x_i| on standardized draws; capped gate G_c = sum_i log(1 + c t_i)/c; c=0 is the uncapped sum of t_i.
• Centered dispersion ratio = var(G_c) / var(G_0) on the same draws, per (n, law); SE from batch-to-batch spread of the ratio.
• Top-term share = mean over sims of max_i term / G_c.
• Normal-calibrated t(3) power = P(G_c > 95th percentile of G_c under normal at the same n), evaluated under t(3). Gain = power(c) - power(0), SE from the batch spread of the paired difference.

Findings against rule 2, per n: c=20 maximizes t(3) power at n=25, 50 and 100, and power falls monotonically in c at every n. The gain over the uncapped c=0 gate is negative at every n (-0.323, -0.401, -0.327 with SEs 0.003, 0.004, 0.003), so no gain exceeds two standard errors; the loss exceeds two SEs at every n by a wide margin, and is larger than at either of the other two scalings. Per law, the dispersion ratio is NOT flat in c: every (c, n, law) cell is at or below 0.005 with SE at or below 0.001, so every cell is more than two SEs below 1 under normal, t(3), t(5) and the contaminated mixture (0 of 15 cells within two SEs for each law). Top-term share roughly halves from c=0 to c=20 and then drifts down slowly in c.

Across the three scalings tried this cycle (x^2, x^2/n, |x|) the cycle 86 power gain at c=20 reproduces at none of them, so the scale cycle 86 used remains unrecovered; at every scale tried the cap costs normal-calibrated t(3) power and compresses dispersion by one to three orders of magnitude.

Closing sentence: a dispersion half does not belong in the next rule, because at every scale tried the cap collapses the centered dispersion ratio far below 1 at every c (nothing flat to test) while costing normal-calibrated t(3) power at every c and n by more than two standard errors; the next rule should fix the per-term scale of t explicitly and test power alone.

Rule 1 table: centered dispersion ratio and top-term share (20000 sims per n, seed 870047)

Rule 2 summary

All c, t(3) power by n: {25: {0: 0.6083, 20: 0.2854, 25: 0.2787, 30: 0.2742, 35: 0.2711, 40: 0.2686}, 50: {0: 0.8205, 20: 0.4192, 25: 0.4104, 30: 0.4045, 35: 0.399, 40: 0.3966}, 100: {0: 0.9614, 20: 0.6345, 25: 0.623, 30: 0.6144, 35: 0.6078, 40: 0.6008}}

Dispersion ratio flat in c (every c within 2 SE of 1)?

• normal: NOT flat (0/15 cells within 2 SE)
• t3: NOT flat (0/15 cells within 2 SE)
• t5: NOT flat (0/15 cells within 2 SE)
• mix: NOT flat (0/15 cells within 2 SE)

Cycle 87: centered dispersion of the capped gate, member-2

Script: stuff/cycle87-centered-dispersion-member-2.py (numpy, seed 870021, 20000 sims per n, 20 batches of 1000 for Monte Carlo SEs).

Definitions used here (stated because no earlier cycle file survives in this tree):

• t_i = x_i^2 on standardized draws; capped gate G_c = sum_i log(1 + c t_i)/c; c=0 is the uncapped sum of t_i.
• Centered dispersion ratio = var(G_c) / var(G_0) on the same draws, per (n, law). SE from batch-to-batch spread of the ratio.
• Top-term share = mean over sims of max_i term / G_c.
• Normal-calibrated t(3) power = P(G_c > 95th percentile of G_c under normal at the same n), evaluated under t(3). Gain = power(c) - power(0) with SE from the batch spread of the paired difference.

Caveat: with t_i on the x^2 scale, c in 20-40 puts every term deep in the log regime, so the cap is near-total. Cycle 86 reported power *gains* from the same cap, which this scaling does not reproduce; the per-term scale of t is the unstated parameter that decides the sign, and the next rule should fix it explicitly.

Rule 1 table: centered dispersion ratio and top-term share (20000 sims per n, seed 870021)

Rule 2 summary

All c, t(3) power by n: {25: {0: 0.76, 20: 0.364, 25: 0.3548, 30: 0.3479, 35: 0.3439, 40: 0.3394}, 50: {0: 0.9264, 20: 0.5209, 25: 0.5086, 30: 0.4982, 35: 0.4904, 40: 0.485}, 100: {0: 0.9928, 20: 0.741, 25: 0.7278, 30: 0.7164, 35: 0.7079, 40: 0.7015}}

Dispersion ratio flat in c (every c within 2 SE of 1)?

• normal: NOT flat (0/15 cells within 2 SE)
• t3: NOT flat (0/15 cells within 2 SE)
• t5: NOT flat (0/15 cells within 2 SE)
• mix: NOT flat (0/15 cells within 2 SE)

Rule 2 statements

• Power-maximizing c among 20-40 is c=20 at every n (power falls monotonically in c). The gain over c=0 is negative at every n (about -0.40, -0.41, -0.25 at n=25, 50, 100, SE about 0.003), so no gain exceeds two standard errors; the uncapped gate dominates under this scaling.
• Centered dispersion ratio is NOT flat in c for any law: every capped cell sits at 0.000-0.002 with SE below 0.0005, far more than two SEs from the c=0 value of 1. Top-term share likewise collapses from 0.08-0.36 to 0.02-0.09 and then declines slowly in c.
• Closing sentence: a dispersion half does not belong in the next rule, because the dispersion ratio here is just a monotone restatement of cap strength (it carries no information beyond what the power column already shows, and it moves in lockstep with the power loss), so the next rule should instead pin the per-term scale of t and test c on power alone.

Cycle 87: centered dispersion of the capped gate at the per-n scale, member-3

Script: stuff/cycle87-centered-dispersion-member-3.py (numpy, seed 870033, 20000 sims per n, 20 batches of 1000 for Monte Carlo SEs). Seed 870033 was not used in cycles 84-86 and differs from member-2 (870021).

This item is the scaling counterpart to member-2: same gate family, same c grid, same laws, but t_i = x_i^2 / n instead of x_i^2, so that c*t_i is O(1) for c in 20-40 and the cap is partial rather than near-total. Member-2 flagged the per-term scale of t as the unstated parameter deciding the sign of the power gain; this item tests the other natural choice.

Definitions:

• t_i = x_i^2 / n on standardized draws; capped gate G_c = sum_i log(1 + c t_i)/c; c=0 is the uncapped sum of t_i.
• Centered dispersion ratio = var(G_c) / var(G_0) on the same draws, per (n, law); SE from batch-to-batch spread of the ratio.
• Top-term share = mean over sims of max_i term / G_c.
• Normal-calibrated t(3) power = P(G_c > 95th percentile of G_c under normal at the same n), evaluated under t(3). Gain = power(c) - power(0), SE from the batch spread of the paired difference.

Findings against rule 2, per n: c=20 maximizes t(3) power at n=25, 50 and 100, and power falls monotonically in c at every n. The gain over the uncapped c=0 gate is negative at every n (-0.178, -0.093, -0.015 with SEs 0.003, 0.002, 0.001), so no gain exceeds two standard errors; the loss exceeds two SEs at every n. Per law, the dispersion ratio is NOT flat in c: at this scale every (c, n, law) cell is below 1 by far more than two SEs (ratios 0.06-0.17 at the smallest-n, smallest-c corner and falling with c), and the drop is largest under t(3) and the contaminated mixture. The cycle 86 power gain at c=20 does not reproduce at either the x^2 scale (member-2) or the x^2/n scale (this item), so the gain depends on a scale neither item recovered.

Closing sentence: a dispersion half does not belong in the next rule, because the cap moves dispersion strongly and monotonically at every c (the ratio is never flat, so there is nothing to test), while it costs normal-calibrated t(3) power at every c and n with losses beyond two standard errors; the next rule should instead fix the per-term scale of t explicitly and re-test power alone.

Rule 1 table: centered dispersion ratio and top-term share (20000 sims per n, seed 870033)

Rule 2 summary

All c, t(3) power by n: {25: {0: 0.754, 20: 0.576, 25: 0.5582, 30: 0.5427, 35: 0.5304, 40: 0.5201}, 50: {0: 0.9184, 20: 0.825, 25: 0.81, 30: 0.7956, 35: 0.7832, 40: 0.7717}, 100: {0: 0.9928, 20: 0.9777, 25: 0.9742, 30: 0.9706, 35: 0.9677, 40: 0.9648}}

Dispersion ratio flat in c (every c within 2 SE of 1)?

• normal: NOT flat (0/15 cells within 2 SE)
• t3: NOT flat (0/15 cells within 2 SE)
• t5: NOT flat (0/15 cells within 2 SE)
• mix: NOT flat (0/15 cells within 2 SE)