Fig. S1 Information estimates for Gaussian channels and invertibly transformed channels. GC denotes the Gaussian-copula covariance score; KSG denotes the Kraskov–Stögbauer–Grassberger estimator with neighbour order k = 5. Curves show means over 400 independent catalogues per point. Shaded bands show one standard deviation of the estimates, not confidence intervals for their means. Dashed lines mark the analytic information, which is unchanged by the transformation. Each catalogue has four chemical and three kinematic dimensions. Panels (a) and (b) use ρ = 0.4, and (c) and (d) use ρ = 0.6; (a) and (c) are Gaussian, while (b) and (d) apply the curved transformation

Fig. S2 Rejection fractions under independence and the specified linear and quadratic alternatives. The quadratic N = 78 point uses the separate 5,000-catalogue follow-up; all other points use the original 200 catalogues. Each catalogue uses 199 permutations; vertical bars are 95% Wilson intervals. The dotted line marks the nominal significance level 0.05, and panel (a) uses an expanded vertical scale. GC denotes the Gaussian-copula covariance test; KSG denotes the Kraskov–Stögbauer–Grassberger test with k = 5. The intersection requires both tests to reject the same null. The alternatives are not matched in mutual information. Panels (a), (b) and (c) show independence, linear dependence and quadratic dependence, respectively

Fig. S3 Dependence generated by six group-level mean shifts. Global and within-group permutations address marginal and within-group independence, respectively. The simulated domains are dependent marginally and independent within groups. Points and bars show rejection fractions and 95% Wilson intervals over 200 catalogues per sample size. GC denotes the Gaussian-copula covariance test; KSG denotes the Kraskov–Stögbauer–Grassberger test with k = 5. The intersection requires both tests to reject and is especially conservative in this example. Panels (a) and (b) use N = 78 and N = 156, respectively

Fig. S4 Mutual information induced by joint logistic selection from independent standard-normal variables. Panel (a) integrates the selected density at five boundary sharpnesses. Panel (b) compares that integral with the Gaussian score from the raw selected covariance at t = 1.5. All values use order-640 Gauss–Hermite quadrature and describe population quantities. The raw-covariance score does not use marginal rank normalisation and is distinct from the Gaussian-copula statistic used for the other GC analyses. This joint-selection example does not model the catalogue’s speed cut or targeting

Fig. S5 Permutation distributions for the 78-star catalogue. The observed statistic is unchanged when the null changes. Global permutations use 1,000 draws; instrument-by-metallicity permutations use 2,000. Histogram densities are normalised separately. GC denotes the Gaussian-copula covariance score; KSG denotes the Kraskov–Stögbauer–Grassberger estimator with k = 5. Distance correlation is dimensionless, while the information scores are in nats. Coarse stratification does not estimate conditional mutual information. Panels (a), (b) and (c) show GC, KSG5 and distance correlation, respectively

Fig. S6 Permutation p values after 100 rounding perturbations and 100 separate added-noise perturbations. Each point uses 499 instrument-by-metallicity permutations. Horizontal black marks show medians and the dotted line marks 0.05. GC denotes the Gaussian-copula covariance test; KSG denotes the Kraskov–Stögbauer–Grassberger test with k = 5. Original sample membership and stratum labels remain fixed. The draws assess perturbation sensitivity and are not independent replications of the stellar population. Panels (a) and (b) show publication-rounding and added-noise perturbations, respectively

Fig. S7 Prespecified nearest-neighbour sensitivity in the six development designs. Curves show mean estimation error for permutation-centred KSG at k=3,4,5,8 using the same datasets and permutation sequences. Solid and dashed lines denote true information of 0.10 and 0.40 nats. Each point summarises 200 independent parent catalogues. Prefixes within a parent are nested. No k is selected from the displayed performance; full uncertainty summaries are archived. Panels (a)–(f) show L1, L3, T3, Q1, Q1x and Q3, respectively

Fig. S8 Absolute calibration in the reserved H1 and H3 geometries. Mean estimation errors are shown for raw and centred KSG4 and analytically corrected GC. Solid and dashed lines denote true information of 0.10 and 0.40 nats. Bars are two-sided 95% Student-t Monte Carlo intervals over 200 independent parent catalogues. The Gaussian correction is not assumed exact after empirical rank normalisation. Under these symmetric nonlinear alternatives, GC also has population approximation error. Panels (a) and (b) show H1 and H3, respectively

Fig. S9 Equal-information comparisons in the reserved geometries. Curves show mean errors in B-minus-A information differences for centred KSG4; the true difference is zero. Baseline estimates use the indicated sample sizes. P1 uses 78 rows on both sides, averaging five subsets from each larger source sample and reusing the baseline at N=78. Solid and dashed lines denote common information of 0.10 and 0.40 nats. Bars are 95% Student-t Monte Carlo intervals over 200 independent A/B pairs, reused across the specified size comparisons. Panels (a) and (b) show H1 and H3, respectively

Fig. S10 Constant-information profiles in the reserved geometries. Curves show mean errors for raw KSG4, centred KSG4 and centred KSG4 with common-size subsampling. Solid and dashed lines denote true information of 0.10 and 0.40 nats. Every prefix is drawn from the same population; each curve summarises 200 independent parent catalogues. Whole-catalogue bootstrap intervals for the endpoint and mean-profile range are supplied in the archived profile table. A flat mean profile does not establish accurate absolute information. Panels (a) and (b) show H1 and H3, respectively

Fig. S11 Prespecified nearest-neighbour sensitivity in the reserved geometries. Mean estimation errors are shown for centred KSG at k=3,4,5,8. Solid and dashed lines denote true information of 0.10 and 0.40 nats. Each point summarises 200 independent parent catalogues; datasets and permutations are shared across k. These fixed sensitivity analyses assess extension to the absolute-value family without selecting a preferred k from its observed performance. Panels (a) and (b) show H1 and H3, respectively
