Fig. 1 Absolute calibration at matched true information. Mean error for raw and centred KSG with k=4 and analytically corrected Gaussian-copula scores. Panels (a)–(f) show L1, L3, T3, Q1, Q1x and Q3. Solid and dashed lines denote true information 0.10 and 0.40 nats. Hollow raw markers remain visible where curves overlap. Bars are 95% Student-t intervals across 200 independent parents; prefixes within each parent are nested. Vertical scales differ. The Gaussian correction is not generally exact after rank transformation

Fig. 2 Difference errors between independent samples with equal true information. Panels (a)–(f) show L1, L3, T3, Q1, Q1x and Q3 using centred KSG with k=4. The true difference is zero. Baseline estimates use the displayed sizes; matching uses 78 observations on both sides, averaging five subsets when the source has 500 rows. Solid and dashed lines denote information 0.10 and 0.40 nats. Bars are 95% Student-t intervals over 200 independent A/B parent pairs. Size comparisons reuse the same pairs and are not independent replications

Fig. 3 Mean profiles under constant population information. Panels (a)–(f) show L1, L3, T3, Q1, Q1x and Q3. Curves give mean errors for raw and centred KSG with k=4 and centred common-78 matching. Solid and dashed lines denote true information 0.10 and 0.40 nats. Each curve uses 200 independent parents with nested prefixes; matching averages five subsets above N=78. Whole-parent bootstrap intervals for profile contrasts appear in the supplementary results. Vertical scales differ. A flat mean profile can remain biased in absolute magnitude

Fig. 4 Cross-geometry comparisons at equal information and dimensions (4,3), using centred KSG with k=4. Panels (a) and (b) compare L3ᴬ with T3ᴮ at true information 0.10 and 0.40 nats; panels (c) and (d) compare Q1xᴬ with Q3ᴮ at those targets. T3 is an invertible deformation of L3; Q1x and Q3 allocate information across one and three active pairs. Baseline and common-78 matching are defined as in Fig. 2. Bars are 95% Student-t intervals across 200 independent parent pairs. Equal information need not imply equal estimator bias

Fig. 5 Change in correct ordering under five-subset minimum-size matching. All 64 unequal-size, nonzero-difference extension cells are shown for centred KSG4. Panels (a)–(d) use true information pairs (Iᴬ,Iᴮ)=(0.4,0.5), (0.5,0.4), (0.4,0.6) and (0.6,0.4) nats, respectively. Rows identify geometry and the smaller size Nᴬ; Nᴮ is the plotted ratio times Nᴬ. Matching retains A and averages estimates from five size-Nᴬ subsets of B, sampled without replacement within each subset; subsets may overlap. Each cell gives the matching-minus-baseline change in correct ordering, in percentage points, across 601 independent parent pairs. Positive values indicate improvement. Parent catalogues are reused across anchors, ratios and truth comparisons, so cells are dependent. Values are point estimates without significance symbols; their paired pointwise 95% bootstrap intervals do not all exclude zero and have no simultaneous-coverage guarantee. The plotted-value CSV and complete extension tables in Online Resource 2 supply ordering probabilities, intervals, comparison RMSE and absolute RMSE
