Every window
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An eighteen-window search gave a false alarm probability of about 5.42 per cent when each window was tested at a five per cent level. Bonferroni correction lowered it to about 0.075 per cent. Those are exact probabilities for a small model with twelve independent binary positions. They give no measured error rate for human writing or a production watermark.
Chances
Each position has a one-in-four chance of a mark. We enumerated all 4,096 possible sequences, calculated the exact binomial upper-tail probability for every tested window, and recorded both decisions for each sequence. A document alarm means at least one window crossed its threshold.
The sequences have different probabilities. An all-marked sequence has probability (1/4)^12, while an all-unmarked sequence has probability (3/4)^12. Counting rejecting sequences and dividing by 4,096 would use the wrong distribution. The calculation instead adds their exact rational weights, which sum to one across the complete enumeration.
Three searches
The document test uses one window of length twelve. The sliding search uses nine windows of length four at stride one. The dyadic search uses lengths two, four and eight at half-length strides, giving eighteen windows. Each schedule includes the final possible start. Overlapping windows are tested on the same complete sequence, so their dependence is retained.
The uncorrected rule tests every window at 1/20. Bonferroni divides that level by the number of windows in the family, giving 1/20, 1/180 and 1/360. Correction is separate for each family. We made no combined decision across all three searches.
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| Family | Rule | Rejecting sequences | Exact false alarm probability |
|---|---|---|---|
| Document | Uncorrected | 1,586 | 119561/8388608 |
| Document | Bonferroni | 1,586 | 119561/8388608 |
| Sliding | Uncorrected | 1,224 | 14285/524288 |
| Sliding | Bonferroni | 1,224 | 14285/524288 |
| Dyadic | Uncorrected | 2,167 | 909559/16777216 |
| Dyadic | Bonferroni | 259 | 12619/16777216 |
The document and sliding probabilities are about 1.43 and 2.72 per cent under both rules. The uncorrected dyadic search exceeds five per cent, while every corrected family stays below it. These are complete probabilities within the finite model, with no sampling interval to estimate.
Discrete steps
Correction changes no sliding-search decision here. A four-position window has upper-tail probability 1/256 when all four positions are marked, and 13/256 when three are marked. The first is below both 1/20 and 1/180, while the second is above both. Both rules therefore reject the same sequences. A correction can be conservative, and it need not change a result when the attainable p-values are discrete.
The dyadic family has different widths and eighteen tests. Its correction changes which sequences reject. The comparison concerns these complete window families. It does not establish that every larger search exceeds its nominal level.
A known null
Independent binary positions make each fixed-window binomial p-value valid in this model. Bonferroni then bounds the chance of any false alarm without needing the overlapping windows to be independent. Validity of the individual p-values is the requirement.
Real token contexts and keyed word pairs can repeat or depend on each other, and an editor can choose passages after seeing scores. This enumeration supplies no binomial null for those cases. It also contains no marked alternative, so it measures no detection power. The separate mixed-passage study reports its human controls, insertions, misses and location results under a different procedure.
Before outputs
The plan, configuration, runner, independent verifier and tests were committed and pushed before the exhaustive run. Registration commit 99a9ba57 is preserved by the permanent window-family-v1-registration tag. The procedure was reviewed before the outputs were calculated. No random seed or draw was needed.
The complete outcomes and reproduction guide retain every sequence, both rules and all three families. The independent verifier constructs the binomial distribution by integer polynomial expansion and reconstructs the windows separately. It checked every weight, decision and summary. Eight helper and fault tests passed, including checks against uniform weighting, counting only widths, multiplying the correction and changing a decision.
The site serves the complete outcomes, exact summary, registration and file hashes. Its build independently checks every sequence weight, search decision and family probability.