Validation Record

• R ppoints(): p = (i-a)/(n+1-2a) with a = 3/8 at n <= 10 and a = 1/2 above, which is the rule Stratum uses; qnorm / -log(1-p) / p / log(-log(1-p)) for the theoretical axes
• R qqnorm() run on the case's own data — the whole plot from R, not R inverting a position computed here
• the superseded (i-0.5)/n vector recorded alongside: identical above ten points, different at or below, so a regeneration against the old rule cannot pass
• normalQuantile pinned to Acklam's published 1.15e-9 RELATIVE bound against R qnorm

Not yet covered: The CONFIDENCE BAND (QQConfidenceBand) has no fixture — only the points and the theoretical axes are covered.

• R stats::lm for the four forms that are linear in their parameters — raw-power polynomials and lm(y ~ log(x))
• R stats::nls (algorithm 'port') for the power and exponential forms, y = b0·x^b1 and y = b0·e^(b1·x): nonlinear least squares on the original y scale, started from OLS on the logs — the same estimator nls(), SciPy's curve_fit and JMP's Nonlinear platform compute
• SciPy optimize.curve_fit (MINPACK Levenberg-Marquardt) as a second independent optimizer on those two, agreeing with nls on the residual sum of squares to 1e-8 — an iterative fit is where a single reference is weakest, because 'converged' is a choice each library makes differently
• fixtures chosen where the answer is decided by which estimator is running: homoscedastic error across three decades of y, and an outlier set on both forms. Each case pins the original-scale least-squares curve on data where a log-scale fit lands somewhere else entirely, so what is certified is the estimator and not merely that some curve fits
• r-squared on the ORIGINAL y scale, measured against the back-transformed curve on screen, which is what Stratum reports. R's model-scale summary()$r.squared is recorded alongside every case, so a reading taken from R can be reconciled against it

Not yet covered: The non-parametric smoothers — LOWESS, kernel smoother and moving average — have no fixture. Their implementations differ in detail between packages, so pinning one spelling of LOWESS would pin R's choices rather than a shared definition; closing this means deciding which definition Stratum claims first.

• grid geometry from the engine's own 2-D rule, h = 1.06*max(sd, range*1e-3)*n^(-1/6)*scale, pad = clamp(2.5h, 3%, 15%) — the bandwidth is not exposed, but x0 and dx are functions of it, so the geometry pins the rule exactly
• a DIRECT 2-D Gaussian kernel sum in numpy, compared in shape against the engine's binned separable blur within a measured bound (6%, 12% on the coarse grid)

Not yet covered: The blur is an APPROXIMATION to a 2-D KDE, so the surface is checked for shape against the exact estimator rather than value-for-value — the bound is a recorded measurement, not an equality. The value and aggregate modes (with a Z column) and their 2% masking threshold have no fixture. Note this chart does NOT share KernelDensityEngine: it uses the 2-D n^(-1/6) exponent where Density uses n^(-1/5), so it inherits nothing from that fixture.

• sklearn.metrics.roc_curve(drop_intermediate=False) / roc_auc_score
• sklearn.inspection.partial_dependence(kind='average')
• R rpart::rpart(...)$cptable — the cross-validated error curve, its xstd, and rpart's own 1-SE selection off it

Not yet covered: The DIAGNOSTICS (ROC, AUC, calibration binning, partial dependence) are pinned, and so now is the PRUNING SELECTION: the standard error of the cross-validated loss reproduces rpart's xstd to the last printed digit on every fixture cptable row, and the 1-SE rule picks the row rpart's own rule picks, on five curves spanning classification and regression (defect A14). The fitted TREE is still not pinned. A single tree can be matched to sklearn exactly given the same split rule and deterministic tie-breaking, so this is a gap to close, not an impossibility — and it is why the end-to-end check holds Stratum to rpart's leaf count only on kyphosis, where rpart's answer is a stump under both rules and Stratum's defaults reach it.

• sklearn.metrics.roc_curve / roc_auc_score

Not yet covered: Diagnostics only. An ensemble cannot be matched tree-for-tree against sklearn — the RNG streams and split tie-breaking differ by construction — so the model itself needs invariants (OOB convergence, importance monotonicity) rather than a value comparison.

• sklearn.metrics.roc_curve / roc_auc_score

Not yet covered: Diagnostics only — same reasoning as Random Forest.

• R ca::ca()
• an independent numpy implementation from the definition (Greenacre), not a transcription of the engine
• scipy.stats.chi2_contingency — pins the identity n * total inertia == chi-square

Not yet covered: The DISPLAY CAP is not covered. Above StratumLimits.maxReportGroups the engine pools the tail into 'Other' and decomposes the POOLED grid, which moves both the inertia and the degrees of freedom; every fixture table is well under the cap. Row/column contributions and quality (cos²) have no fixture yet either.

• R stats::hclust(dist(X), method=single/complete/average/ward.D2)
• scipy.cluster.hierarchy.linkage
• R stats::kmeans(nstart=25) under BOTH its Hartigan-Wong default and algorithm='Lloyd', asserted to agree with each other and with sklearn.cluster.KMeans before being used (well-separated data only)
• sklearn.cluster.KMeans (well-separated data only)
• R stats::kmeans(X, centers=k) across 40 seeds — R's literal single-start default, carried as a SPREAD rather than a value because on hard data that call has no single answer
• local optimality: nearest-centroid assignment, and centroid is the cost-minimising centre — mean under Euclidean, median under Manhattan
• R stats::hclust ON THE DISCLOSED SAMPLE for a run past the O(n²) row cap: hclust has no cap and clusters all n, so a capped run is comparable to it only on the rows Stratum says it clustered. The fixture reproduces the selection rule (rows ordered by their own values, then taken at uniform stride) and hands R exactly those rows, which pins the sample as well as the linkage

Not yet covered: The k-means PARTITION is compared to scikit-learn and R only where the optimum is unambiguous. On ambiguous data, and under the Manhattan cost (which sklearn's KMeans does not implement), the evidence is invariants plus R's own objective as an upper bound, rather than a partition — because no outside implementation reaches a canonical answer there, not because the check was skipped. Elbow and silhouette have no fixture yet. TWO DEFAULTS DIVERGE FROM R BY DESIGN and are documented rather than changed (defect A13): Stratum standardises before clustering and R clusters the scale it is handed, and Stratum runs Lloyd from k-means++ with 10 restarts where R runs Hartigan-Wong from one random start. Neither is an arithmetic difference — both minimise the same total within-cluster sum of squares — and the fixture measures both: the same 45 rows are clustered twice, once on the raw scale R would use and once standardised the way Stratum does, each partition matched to R and sklearn on its own scale, with 21 rows landing in a different cluster between them. On the ALGORITHM half the evidence is asymmetric and the record says so. Where the three references agree what the optimum is, Stratum's 10 restarts reach it and R's single default start frequently does not — on three tight, well-separated blobs R's default lands at 35.05 instead of 0.050 on some seeds, and on the four-cluster 3-D case it reaches nine distinct optima across 40 seeds (best 1.84, worst 32.7). Where the optimum is AMBIGUOUS neither default is authoritative and Stratum is not always ahead: on the overlapping-ridges case R's default reaches four local optima (35.594, 35.665, 35.764, 39.269) and Stratum's 10 restarts land on a fifth at 35.802 — 0.11% above R's median start and 0.58% above the 35.594 both programs reach given more starts. So the assertion carried there is only that Stratum is no worse than the WORST R's own default hands a user, beside local optimality; raising Restarts closes the gap (25 → 35.659, 200 → 35.594). Matching R's ALGORITHM NAME would not make Stratum match R's answer — R's default seeds at random and its RNG stream cannot be reproduced here, and as that spread shows R does not match itself either — so the divergence is recorded, not papered over.

• d2 by Gauss-Legendre quadrature
• c4 via scipy.special.gammaln

Not yet covered: The unbiasing CONSTANTS are pinned to first principles; the Nelson / Western Electric run rules are not, and they are a documented convention rather than a computed quantity.

• d2 / c4 as for I-MR

Not yet covered: Constants pinned; run rules are convention. Same as I-MR.

• binomial limits

Not yet covered: Attribute-chart limits are not yet cross-checked against an outside tool.

• Poisson limits

Not yet covered: Attribute-chart limits are not yet cross-checked against an outside tool.