Normal Distribution
nothing is called 95% until it has been counted
Field

Overlap and complementarity, separated

How much two searches over one sample share is measured as the net of two effects: ground both of them find, and configurations the joint search reaches that neither slice contains. Pin the first search at its own answer and search the second, and the two separate exactly — the pinned supremum cancels, so the split adds back to the original number on every draw. The control the whole scale is anchored on reads zero because its two components are several times larger and cancel, and the split depends on which of the two searches is pinned while their difference does not.
What each pair is made of. Each pair of searches, over 300 draws, split into the two effects its excess is the difference of. The overlap is what the second search loses by having the first already run at its own answer; the interaction is what the joint search finds by moving the first off it. They subtract to the excess exactly, on every draw, because the pinned supremum cancels. Two disjoint dictionaries of independent columns read an excess of 0.000011 and are made of 0.000514 and 0.000503. A break paired with a dictionary of step columns has an interaction of exactly 0 and is all overlap. And a break paired with an independent column has an overlap of -0.004395 against an interaction of 0.002364, which is what puts its excess below zero.

Two effects in one number

How much two searches over one sample share is measured as the net of two things — ground both of them find, and configurations only the joint search reaches. One extra supremum per draw separates them exactly.

 Both halves grow; the difference does not. The control pair's two components and their difference, against how much each of its two searches can find, over 1200 draws at each dictionary size. Two disjoint sets of independent columns are additive at every size — the excess stays inside a standard error or two of zero throughout — and it is not because there is nothing there. The overlap grows from 0.000112 at two columns to 0.000870 at ten, a factor of 7.76, and the interaction grows with it, staying within a factor of two of the overlap at every size. Two searches competing for one residual sum share ground and find configurations neither has alone, in almost equal measure, and their difference is what the earlier field's scale calls zero.

What a zero is made of

Two disjoint dictionaries of independent columns read an excess of 0.000116 and are made of an overlap of 0.000583 and an interaction of 0.000467. The control the whole scale is anchored on reads zero because two effects cancel.

 The split depends on the order. How much two searches share, measured both ways round, over 300 draws. Pinning the first search at its own answer and searching the second gives one overlap; pinning the second and searching the first gives another. A break paired with an independent column reads -0.004395 one way and 0.002163 the other, at 8.70 paired standard errors and on opposite sides of zero. The excess the two components subtract to is the same in both orders by construction, so what changes is only how it is attributed. There is no order-free way to say which of two searches found ground both can reach, and the two orders bracket it.

A split that depends on the order

Run the second search first and pin that instead, and the same draw gives a different overlap and a different interaction — with the same difference. And one pair has no second order at all.

 The overlap climbs and the interaction does not. The over-charge at each quantile level split into overlap — the two searches' own quantiles less the sequential run's — and interaction — the joint search's quantile less the sequential run's — for three pairs over 1200 draws. two independent dictionaries: overlap -0.40, 0.01, 1.55, 2.44, 5.72; interaction 0.07, 0.01, 0.11, 0.06, 0.00. two step dictionaries: overlap -0.73, 8.19, 22.53, 31.02, 47.75; interaction 4.87, 4.55, 4.90, 4.75, 1.10. a break and a window: overlap 17.88, 29.52, 43.01, 53.79, 71.61; interaction 1.57, 1.29, 1.19, 3.22, 2.75, at the 50%, 75%, 90%, 95% and 99% points.

The split at the critical value

Split the over-charge of two searches at the joint statistic's 95% point rather than at its mean, and the two components behave differently. The interaction — what only the joint search can reach — stays within two units of its average at every level up to the 95% point. The overlap climbs from nothing to most of the over-charge, on two searches over independent columns as much as on searches that share ground. At the critical value the split's overlap has stopped meaning shared ground: it is mostly the arithmetic of adding tails, and the identity that defines it hides that.

 One order's interaction climbs under long memory. The break-and-window pair's interaction — the joint search's quantile less the sequential run's — at five quantile levels, in the field's order (break first, solid) and the other (window first, dashed), under three laws, 1200 draws each. first-order autoregression: break first 1.57, 1.29, 1.19, 3.22, 2.75; window first 0.98, 1.26, 1.53, 2.55, 2.14. five-period moving average: break first 2.00, 1.88, 2.29, 2.63, 0.60; window first 0.16, 0.21, 0.26, 0.29, 0.00. long memory: break first 0.53, 0.80, 1.05, 0.46, 1.35; window first 2.01, 1.62, 4.75, 6.07, 7.85, at the 50%, 75%, 90%, 95% and 99% points. Under two of the laws, and in the field's order under all three, the interaction stays within a few units of its median. Under long memory with the window searched first it climbs from 2.01 to 7.85, against an average of 1.99; the same order's averages under the other two laws are 0.94 and 0.17.

A tail the law gives the pair

Split the over-charge of two searches at their critical value and the interaction — the configurations only the joint search reaches — stays near its average while the overlap climbs. That held under one law. Under a five-period moving average and under long memory it holds for every pair in the order the field reads, to within three units. In the other order, one pair acquires a tail of its own: with the window searched first and the break second, the interaction under long memory climbs from 2.0 at the median to 7.9 at the 99th percentile, four times its average.

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