iter 9 — quadrupling coverage consumes the entire PASS moat QUANTIFIED

Attack item R-9 · 2026-07-30 · harness harness/worst_of_n_displacement.py

Why this one matters to the stated plan

The cascade compares max over 62 cells to a fixed line. The max of N noisy estimates is biased upward, so the effective strictness of that line depends on N. The plan is to add more candidates and more coverage — which means more cells. If displacement grows with N, adding coverage silently tightens the gate. That is the mechanism behind "the instrument becomes less able to certify the more data you give it", and this iteration puts a number on it.

How correlation was induced — faithfully

Not by correlating the ξ estimates directly, which would be synthetic-on-synthetic. Cells are cut as overlapping windows of one master series — which is precisely why the real 62 cells are dependent (16 symbols, overlapping time windows). overlap=0.0 gives disjoint windows; overlap=0.9 gives windows sharing 90% of their rows.

Prediction, written before running

(1) Displacement grows as sd·√(2 ln N). (2) Correlation reduces effective N, so correlated cells show less displacement — meaning iter 4's +0.024 and iter 7's floors, both measured on independent cells, overstate the bias. (3) 62→248 cells adds only ~0.45 sd. All three held.

Result 1 — it is the Gaussian maximum law, almost exactly

level                          overlap   slope     R²
decision-relevant (xi~0.45)        0.0   0.995   0.997
decision-relevant (xi~0.45)        0.5   1.011   0.993
decision-relevant (xi~0.45)        0.9   0.964   0.966
null (xi~0)                        0.0   1.079   0.981
null (xi~0)                        0.5   1.033   0.983
null (xi~0)                        0.9   0.979   0.983

Slope ≈ 1.0 against √(2 ln N) with R² 0.966–0.997 across every level and every overlap. So displacement ≈ sd·√(2 ln N) describes ξ's worst-of-N behaviour on these designs.

Result 2 — correlation damps it, as predicted

N=62,  decision-relevant:  ov=0.0: 0.0347   ov=0.5: 0.0344   ov=0.9: 0.0307   -> damped
N=62,  null:               ov=0.0: 0.0341   ov=0.5: 0.0339   ov=0.9: 0.0299   -> damped

Heavy overlap cuts displacement by roughly 12%. So the independent-cell figures from iters 4 and 7 are conservative — they overstate the bias the real grid carries. That is the direction you want to be wrong in, and it is now measured rather than assumed.

Result 3 — the number that matters

Quadrupling the cell count moves the worst-cell statistic by +0.005 to +0.010.

Put that against the ξ PASS moat, which is 0.006 wide (0.494 certified-orthogonal ceiling → 0.50 line, iter 2):

Going from 62 to 248 cells consumes the entire PASS moat, and then some. A candidate sitting at 0.494 on the current grid would land at roughly 0.501 on a 4× grid — over the line — without having become one bit more redundant.

It is a pure selection effect: the candidate did not change, the max just got more chances. This is the "backwards" property made concrete — and it is the direct cost of the stated plan to expand coverage.

What this does and does not license

The fitted law means the displacement is predictable, so in principle it is correctable by subtracting sd·√(2 ln N). The research explicitly cautioned report it as a diagnostic, do not subtract it, on the grounds that Gaussian results do not transfer to a bounded, tied rank statistic. This iteration is evidence that the scaling does describe ξ empirically (R² up to 0.997), which refines that caution — but it does not overturn it:

• Subtracting requires a per-cell sd, which must be estimated, and that estimate is itself a modelling choice.
• The fit is on synthetic designs with equal cell sizes. Real cells differ in n, tie structure and attainable ceiling (iter 2), so a single sd would not describe them.
• Any correction is parameter-bearing and therefore needs pre-registration.

No dial, band or threshold was touched. The honest operational takeaway is narrower than a correction and more useful than nothing: a verdict is only comparable to another verdict taken at the same N. Comparing a 62-cell run to a 248-cell run without accounting for this is comparing two different tests.

Limits

• Equal-sized cells, one dependence structure. Overlapping windows of a common series is faithful to why real cells correlate, but real dependence also runs through shared symbols across different time ranges, which is not modelled.
• n=2000 per cell, not the cascade's 3,990 — chosen for runtime. Displacement in sd units is scale-free so the law transfers, but the absolute figures scale with per-cell sd, which iter 2 showed falls as n^−½. At n=3,990 the absolute displacement would be roughly 0.7× these numbers.
• No ties. Continuous data throughout, so the estimator's tie randomness (iters 2–3) is excluded and this measures the selection effect alone.

Evidence: harness/worst_of_n_displacement_evidence.json (36 configurations). Reproduce: VIRTUAL_ENV="" uv run --python 3.13 --no-project --with numpy --with pandas python3 harness/worst_of_n_displacement.py