Cycle 73: flag-rate movement test at n=25 and n=100

Procedure for the second half of the direction: do any of the three flag rates move from the n=50 baseline? The other two stuff files cover the df=3 margin bootstrap and whether the null rate tracks n. This file covers the movement test that applies to all three rates.

Moved test (both conditions required)

A flag rate at n (25 or 100) counts as moved only if both of these hold:

1. Step: |rate_n − rate_50| > 0.1 on the 0.1 grid used by the cycle-68 gate.
2. Interval: the pooled 95% binomial interval at n (both seeds pooled, Wilson) does not overlap the n=50 baseline interval.

If only one holds, the rate is reported as not moved, and the table records which condition failed.

Inputs (from rule 1 runs)

• Flag counts per seed per n, with the draw count for each.
• The n=50 baseline draw count. The baseline interval is recomputed from that count. A bare point estimate is not accepted.

Pooling

• Pool the two seeds by summing flags and draws, then take one Wilson interval.
• If the two seeds' own intervals do not overlap, report a seed disagreement and give the per-seed rows. Still use the pooled value for the verdict.

Results table (empty: no simulation has been run)

The null-flag rows feed rule 2's tracking verdict in c73-null-flag-tracking.md. The df=3 margin uses the bootstrap in c73-margin-bootstrap.md, not this test.

Cycle 73: bootstrapping the df=3 margin interval

Rule 1 stores per-draw null statistics for every n=25 and n=100 run. This file says how member-2 turns them into the interval rule 2's "moved" test needs.

Why bootstrap

The df=3 margin is median(null statistic) - c, where c is the fixed cycle-68 gate constant. A median minus a constant has no binomial interval, so its pooled 95% interval comes from resampling the stored draws.

Procedure (per n in {25, 100})

1. Pool the stored null statistics from both seeds: S = seed_a ++ seed_b.
2. For b = 1..2000: draw |S| values from S with replacement, take the median, and subtract c. Store as m_b.
3. The pooled 95% interval is the 2.5th and 97.5th percentiles of m_b.
4. Do the same for the n=50 baseline draws, so both sides carry an interval.

Moved test (rule 2 wording, applied to the margin)

The margin at n has moved from the n=50 baseline only if both hold:

• |margin(n) - margin(50)| > 0.1 (one 0.1 step), and
• the n interval and the n=50 interval do not overlap.

If either fails, report the margin as unmoved at that n.

What to record

Fix the bootstrap RNG seed and write it next to the table so the interval can be reproduced from the stored draws.

Cycle 73: does the null false-flag rate track n?

Procedure for the diagnostic half of the cycle-73 direction, applying rule 2 to the null false-flag rate under the fixed cycle-68 gate. Companion to c73-margin-bootstrap.md, which covers the df=3 margin.

Inputs

• Flag counts from every null run at n=25 and n=100, two seeds per n (rule 1).
• The n=50 baseline rate: 4.0%.
• The gate is the cycle-68 gate, unchanged. No threshold is re-tuned.

Pooled rate and interval

For each n, pool the two seeds before computing anything:

1. flags_n = flags_seed1 + flags_seed2
2. draws_n = draws_seed1 + draws_seed2
3. rate_n = flags_n / draws_n
4. Interval: Wilson 95% binomial interval on (flags_n, draws_n).

Wilson rather than Wald, because the rates are near 4% and Wald intervals go below zero or collapse at small counts. The seeds are pooled, not averaged, so the interval reflects the full draw count.

Report the per-seed rates alongside the pooled rate. If the two seeds at the same n differ by more than their own Wilson half-widths, note it as a seed effect and do not call the pooled result either way.

Decision (rule 2, verbatim)

The rate counts as tracking n only if:

• the n=25 interval lies entirely below 4.0%, and
• the n=100 interval lies entirely above 4.0%.

Otherwise it is reported as flat and the gate is called conservative.

Expected shape if finite-sample: below 4% at n=25, drifting toward the nominal 5% at n=100. Expected shape if the gate itself is conservative: the three intervals all cover 4.0% and the point estimates do not order by n.

Edge cases to record, not resolve

• One interval excludes 4.0% and the other does not: flat, by rule 2.
• Both exclude 4.0% on the same side: flat, by rule 2, but log it as a direction for the next cycle since it would mean n moves the rate monotonically rather than toward 5%.
• Zero flags at n=25 on one seed: the Wilson interval still exists. Do not substitute a continuity correction.

Results table

Verdict: ___ (tracking n / flat, gate conservative)

Status

Procedure only. No simulation has been run in cycle 73. The n=50 baseline interval is not stored from cycle 68 and should be recomputed from its draw count when the runs happen, so the baseline row is not a bare point.