Cheap and Powerful Tests for Supervised Subspaces: Per-Component Inference for PLS

1.1 Setting; 1.2 The gap

• Supervised low-rank regression asks whether a K-dimensional outcome-aligned subspace predicts better than chance, but existing tests do not combine good calibration with low cost.The same question arises for PLS, supervised PCA, sparse linear discriminant analysis, linear probes, and sparse-autoencoder concept directions.Source: §1.1 · p. 1, §1.2 · p. 2

1.3 Contribution

• The corrected asymptotic test uses held-out OLS-refit predictions, while a pre-run check based on n and stable rank routes flagged designs to an exact permutation test.The stated asymptotic condition is n≥25 and stable rank of standardized X at least 3; for p≥20,000, the exact test costs about 1.15× the asymptotic test at p=20,000, n=200, and less as p grows.Source: §1.3 · p. 2
• NB has matched or better power than CV-permutation-Q2 at substantially lower cost.The introduction also describes a pre-run check for when the asymptotic approximation is appropriate and an exact fallback.Source: §1.3 · p. 2
• Held-out predictions are invariant to orthogonal rebasing, so a rotated basis retains the joint claim but does not inherit per-axis significance.The introduction identifies varimax as a way to obtain a word-readable basis.Source: §1.3 · p. 2
• A fixed-sequence test on the PLS extraction order supports per-component claims; strong FWER control is proved for the single-split, train-only-deflation variant.For Warriner valence, NB rejects components 3 and 4 although the split-half CV-R2 curve is flat past K=2.Source: §1.3 · p. 2
• The exact test also applies to supervised PCA and a ridge probe, whereas the asymptotic NB correction is scoped to PLS1.The introduction also reports a Rust library with Python, R, and Julia bindings and a Python text pipeline.Source: §1.3 · p. 2

PLS inference

• Table 1 compares procedures by inferential target, cost, calibration, and rotation treatment; none supplies significance for an individual varimax axis.Its cost notation uses F for CV folds, J for repeated splits, and B for permutations.Source: Table 1 · p. 3
• Component-count selection does not establish significance for each fitted component, so the paper tests held-out correlations and distinguishes single-split guarantees from repeated-split approximation.The paper also preserves joint predictions under rebasing.Source: §2 · p. 3

Supervised projections without calibrated per-axis tests

• Prior supervised projections and concept methods address different inferential targets, and the reviewed approaches do not package cheap, rotation-invariant per-axis null tests.The passage contrasts network concept methods, classical supervised dimensionality reduction, and PLS's Krylov-method characterization.Source: §2 · p. 3

What the test certifies

• A calibrated rejection concerns predictive content of the held-out supervised subspace, not the concept-causal identity of a rotated direction.Concept-level readings, such as register consistency, remain separate hypotheses and are reported separately from calibration.Source: §2 · p. 3

3.1 Setup: documents, axes, and the baseline pipeline

• The setup distinguishes fitted component counts from the selected predictive count, and uses PLS to replace the PCA basis in an OLS-on-embeddings baseline.At K=1, the test uses a direction selected by outcome covariance and refits OLS on its score; this generally differs from unrestricted OLS and OLS on the leading principal component.Source: §3 · p. 3, §3.1 · p. 3, §3.2.1 · p. 4

3.2.2 K >1: Krylov saturation and calibration axes

• The K-component PLS span is a Krylov subspace, and later components can improve prediction without identifying separate latent causes.Convergence to OLS depends on predictor spectrum and alignment with outcome covariance, not on the dimension of the linear index.Source: §3.2.2 · p. 4

3.3.1 The corrected resampled t-statistic

• Each split fits one-component PLS on half the data, predicts the held-out half, and contributes a Pearson correlation that is Fisher-transformed and aggregated.The corrected variance term approximates dependence from overlapping training folds.Source: §3.3.1 · p. 5

3.3.2 Per-axis tests via NIPALS deflation

• For each later axis, deflate the data by the preceding full-data NIPALS components, then apply the one-component test to the residualized predictor-outcome pair.The first axis on the full data supplies the omnibus test.Source: §3.3.2 · p. 5, §3.3.1 · p. 5

3.3.3 Per-axis claims and FWER control

• Fixed-sequence testing follows NIPALS extraction order and stops at the first non-rejection, controlling strong FWER if each per-step test is valid at every true null.The repeated-split NB boundary property is assessed empirically, not formally proved; a single-split Gaussian test has a finite-sample guarantee.Source: §3.3.3 · p. 5

3.3.4 When the asymptotic test is valid

• When n≥25 and standardized-X stable rank is at least 3, the corrected resampled t-test is the proposed fast approximation; otherwise use the exact test.The pre-run rule flags every level-moving cell among 25 null cells tested at 5,000 replicates, and every cleared cell holds level or is conservative.Source: §3.3.4 · pp. 5–6

3.3.5 Exact NB-permutation

• The exact fallback permutes the mean Fisher-transformed held-out correlation and is finite-sample valid for the omnibus test under outcome-predictor independence and iid rows.With fixed splits and a direction linear in training outcomes, permutations require no refits; after full-data deflation, the permutation reference is not exact.Source: §3.3.5 · p. 6

3.3.6 Rotation invariance of the held-out predictions

• Held-out OLS-refit predictions and statistics based on them are invariant to orthogonal rebasing, but a rotated axis inherits only the joint claim, not a per-axis p-value.Varimax is applied to loadings after inference; per-axis tests remain tied to the unrotated NIPALS order.Source: §3.3 · p. 4, §3.3.6 · p. 6

4.1 Mechanism — Krylov span and the suppressor fingerprint

• Higher PLS components improved prediction without recovering new outcome directions, while rotated Dim-2 missed the planted suppressor yet retained held-out outcome signal.Krylov saturation rose from approximately 0.69 at K=1 to at least 0.98 by K=10; the suppressor fingerprint appeared on both GloVe and iid geometries.Source: §4.1 · p. 7

4.2 Inference: the validity rule and null calibration

• The rule flags n<25 or stable rank<3; the exact test held level across null cells, while Tecator’s asymptotic null drift triggered a flag.Tecator had stable rank 1.014, with 98.6% of variance in its first principal component.Source: §4.2 · pp. 7–8, Table 3 · p. 8

4.3 Inference: detection power and cost

• NB matched or exceeded permutation-Q2 power across every tested n, with a 0.01–0.25 gap, while costing 115–125× less than perm-Q2 per test.The continuous power sweep compares methods across signal regimes and planted-axis counts; the gap is widest in the noisy regime.Source: Figure 3 · p. 8, §4.3 · p. 8

4.4 Inference: per-axis rejection beyond planted signal

• After one deflation, train-only NB’s chain rejection rates were near nominal; naïve full-data permutation reached 0.059 at α=.05, and some later steps were unreachable.These experiments covered at most three deflations, and step m+1 need not be a true conditional null.Source: Table 4 · p. 9, §4.4 · p. 9

4.5 Beyond PLS: supervised PCA and a ridge probe

• The exact permutation test held level in all eight non-PLS null cells and matched or exceeded perm-Q2 power, while NB lost power off PLS1.The comparisons covered supervised PCA and a ridge probe on Warriner and Tecator.Source: §4.5 · p. 9

4.6 Applied: cross-lingual VAD

• Across eight cross-lingual outcome cells, K=3 fits were significant, and each had a rotated axis carrying the technical-register versus emotion-laden contrast.The leading rotated axis was the conventional affect direction in every cell.Source: §4.6 · p. 9

Calibration boundary

• NB-asymptotic is established only at α=.05 and .10; below n=25 use the exact test, and α=.01 can inflate on flagged designs and one cleared pool.The stable-rank threshold is empirical and errs toward flagging; multi-α sweeps and procedures using the full p-value distribution should use the exact test.Source: §5 · p. 10

Rotation and interpretation

• Varimax interpretation depends on a curated target lexicon, and the named axis split is not rotation-invariant; the paper recommends pre-registering the target and reporting an unsupervised sensitivity.Loading-side varimax is not the appropriate interpretive primitive when features are wavelengths, voxels, or gene-expression probes.Source: §5 · pp. 9–10

Per-component inference

• The single-split, train-only guarantee does not establish validity for repeated-split NB; experiments measuring rejection beyond planted signals are not direct FWER estimates.The experiments cover at most three deflations.Source: §5 · p. 10

Scope of extensions and evidence

• Extensions are limited: basis-dependent refits and non-orthogonal rotations are out of scope, while evidence for real-data breadth covers one text design and two NIR chemometric designs.The exact test transfers to supervised PCA and a ridge probe under iid rows and outcome–predictor independence; the NB correction is PLS1-scoped.Source: §5 · p. 10

• Practitioners have faced biased or expensive tests, or no test, for whether a supervised subspace predicts the outcome better than chance.Source: §6 · p. 10
• The method tests held-out OLS-refit predictions, using a corrected asymptotic test with a pre-run validity check and an exact alternative.Source: §6 · p. 10
• Both tests reject planted signal at least as often as permutation-Q2 at a fraction of its cost; orthogonal rebasing preserves joint inference, while per-component claims use the unrotated sequence.Source: §6 · p. 10

Methods and notation

• Appendices A.1–A.4 cover notation, rotation invariance of the OLS refit, the NIPALS recurrence, and loading-side varimax.Source: A Technical appendices and… · p. 14

Supplementary analyses

• Appendices A.6–A.15 supply cross-lingual analyses, mechanism experiments, calibration and power benchmarks, predictive-rank analysis, and per-axis guarantees and diagnostics.Source: A Technical appendices and… · pp. 14–15

Extensions and supporting materials

• Appendices A.16–A.21 cover tests beyond PLS, embedding pipelines, a sparse-PLS sweep, broader impacts, reproducibility, and text-pipeline pseudocode.Source: A Technical appendices and… · p. 15

A.1 Notation glossary

• The notation uses r for ranks of the supervised signal as joint properties of (X, y), and K for PLS estimator component counts.The per-axis cutoff k* is the last retained PLS direction under the per-axis rule.Source: §A.1 · p. 15
• The true signal rank, r_e, is the smallest dimension through which E[y | X] factors, while r_plant counts planted directions in a synthetic response.The true rank is an unknown oracle quantity, and the planted count need not equal it.Source: §A.1 · p. 15
• K_fit counts components extracted at fit time; K* is the count selected by predictive CV-R2 or BIC, neither of which targets r_e.With correlated X, K* is generically greater than r_e.Source: §A.1 · p. 15

A.2 Rotation invariance of the OLS refit: derivation and assumptions

• The algebraic identity requires full-column-rank training scores and a consistent orthogonal basis change of training and test scores.The implementation reuses one Q across splits; applying a split-specific Qj to both score sets also preserves the statistic.Source: §A.2 · p. 15
• Orthogonal rebasing leaves held-out predictions pointwise identical, so each split correlation, its Fisher transform, and the NB statistic are unchanged.The NB statistic depends on the split Fisher transforms and a deterministic rescale.Source: §A.2 · p. 16
• Rotation invariance is exact for OLS and isotropic ridge, but generally breaks for LASSO, elastic net, and anisotropic ridge.With basis-dependent penalties, the NB statistic is only approximately invariant in the small-penalty limit; the authors advise testing on the unrotated basis.Source: §A.2 · p. 16
• Equal rotated and unrotated statistics imply the same null distribution, but do not prove the t_J−1 approximation or its convergence rate.NB calibration for this statistic is assessed separately.Source: §A.2 · p. 16
• On Toy 4, |p_unrot − p_varimax| = 0 and |p_unrot − p_rand-orth| ∼10^-78 against a baseline p ∼10^-65 across 50 random orthogonal Q.The experiment uses a planted three-axis ground truth with R2 = 0.7 on real Warriner geometry.Source: §A.2 · p. 16
• A calibration axis is an empirical diagnostic for a deflated PLS direction whose OLS-coefficient alignment is small relative to the leading axis.NIPALS does not guarantee alignment with a leading singular vector of the residual design.Source: §A.2 · p. 16
• A one-dimensional signal can require multiple PLS components to reach OLS: with independent mean-zero noise, the example has r_e = 1 but needs two components for Krylov saturation.Signal dimension does not determine the saturation depth, which depends on the predictor covariance structure and its alignment with the outcome covariance.Source: §A.2 · p. 16

A.3 NIPALS deflation recurrence

• The deflation iteration initializes X0 = X and y0 = y.Source: §A.3 · p. 17
• The recurrence then iterates over components k = 1, . . . , K.Source: §A.3 · p. 17
• The per-axis explained-outcome-variance partition is additive on the unrotated scores, and this property is invoked by the loading-side varimax recipe.Source: §A.3 · p. 17

A.4 Loading-side varimax: objective, recipe, score orthogonality

• Varimax first L2-normalizes each loading row, using a 10^-12 floor on the norm, to define the rotation objective.Source: §A.4 · p. 17
• The returned rotation is applied to the original loadings, weights, and scores; axes are sorted by absolute outcome correlation and sign-flipped to make that correlation positive.The objective is invariant under signed column permutations, and each rotated axis loads on a small set of describable tokens.Source: §A.4 · p. 17
• Orthogonal rotation preserves the joint prediction, but generally does not preserve score orthogonality when it mixes blocks with unequal score norms.Signed permutations and rotations within equal-norm blocks preserve orthogonality.Source: §A.4 · p. 17
• Per-axis R2 is reported on unrotated NIPALS scores; the same per-axis terms on rotated scores generally overshoot through cross-terms.Rotated axes inherit only the joint R2.Source: §A.4 · p. 17

A.5 Setup: document vectors, word loadings, and the PCA+OLS baseline

• Mean pooling token vectors represents each document in embedding space, making word loadings and document scores share a metric.This gives a word-level interpretation of downstream axes.Source: §A.5 · p. 17
• The document matrix X and outcome y are mean-centered, and embedding rows are ℓ2-normalized so geometry is interpretable as cosine similarity.Source: §A.5 · p. 18
• For an axis a, word loadings are inner products with normalized word embeddings, and the same inner product yields document scores.Source: §A.5 · p. 18
• The one-dimensional PCA+OLS reference projects X onto its leading K principal components, choosing K to maximize out-of-sample R2.Source: §A.5 · p. 18
• Plisiecki et al. [2025] introduced a lexicon-free, outcome-supervised OLS projection whose basis is variance-ordered rather than covariance-ordered with y.The paper replaces its PCA basis with PLS.Source: §A.5 · p. 18

A.6 Cross-lingual VAD: configurations, aggregate fit, and top words per axis

• The three configurations pair English GloVe-300 with Warriner V/A/D, Polish GloVe-800 with ANPW V/A/D, and Spanish GloVe-300 with Stadthagen-González V/A.Spanish dominance is unavailable because its norms cover only valence and arousal.Source: Table 5 · p. 18, §A.6 · p. 18
• α* FWER <.001 in every reported cell, based on per-axis NB tests on the unrotated K = 3 NIPALS extraction.The repeated-split FWER summary remains conditional on per-step validity.Source: Table 6 · p. 18, §A.6 · p. 18
• Rotated Dim-1 recovers the conventional one-dimensional supervised affect direction in every English, Polish, and Spanish rated-dimension cell.Examples include Spanish valence words felicidad, feliz, sonrisa, alegría, hermosa, and arousal words agresión, explosión, atentado, masacre.Source: §A.6 · pp. 18–19
• Hard-cardinality sparse-PLS offers a separate readability route, with a Warriner-English valence leading axis at s = 100 ranking positive and negative affect words.The axis is supported on at most s features and is read directly from w1.Source: §A.6 · p. 19
• Table 7 presents top-10 words per pole across the three rotated axes for each rated-dimension and language cell in Configurations 1–3.Per-axis breakouts appear in Tables 8–10.Source: §A.6 · p. 19
• In each language, at least one rotated K > 1 axis shows a register contrast between technical, inanimate, or descriptive content and emotion-laden, morally charged, or declarative content.The Valence group in Table 10 shows this Dim-3 contrast across English, Polish, and Spanish.Source: §A.6 · p. 19
• For English XLM-R type-level embeddings, only the leading rotated axis reads as affect; later axes reflect stable topical or part-of-speech structure across V/A/D.Those structures survive rank-1 ABTT.Source: §A.6 · p. 19

Combined leaderboard (across all three rotated axes)

• Table 7 lists the top 10 words per side at combined loadings for each rated-dimension and language cell across Configurations 1–3.Polish and Spanish entries pair English glosses with native words; Spanish dominance is not included.Source: Table 7 · p. 20
• Table 8 gives top-10 words per side for Dim-1 loadings under the same conventions as Table 7.Source: Table 8 · p. 20
• Table 9 gives top-10 words per side for Dim-2 loadings under the same conventions as Table 7.Source: Table 9 · p. 20

A.7 Dim-3 calibration axis: robustness against four counter-hypotheses

• Dim-3’s top-word vocabularies vary by supervised outcome, resist reproduction by unsupervised PCs, and appear incoherent under random-embedding controls.EN and PL show distinct outcome-specific splits; ES valence and arousal overlap more, but even the largest unsupervised-PC comparison is far below the near-one Jaccard predicted by the corpus-structure hypothesis. The interpretation is qualitative, based on top words rather than an external register taxonomy; the axes are not geometrically orthogonal, and EN PC1 partly tracks valence.Source: §A.7 · pp. 22–24, Table 13 · p. 23

A.8 Toy 1–5 mechanism experiments

• Under univariate y, Toy 1 recovers only the leading direction as R2 rises, while Toy 2’s rotated Dim-2 carries y-aware residual signal but does not recover the planted suppressor direction.Toy 1’s mean principal cosine plateaus near 1/rplant; Toy 2’s Dim-2 fingerprint persists across geometries and at ρ=0.Source: §A.8 · pp. 25–26, Table 15 · p. 25
• Toy 3 recovers an interaction only when it is added to the design; Toy 4 confirms rotation-invariant p-values, while Toy 5 shows selected K∗ varies with predictor geometry at fixed planted rank.Toy 3’s lifted interaction reaches cosine 0.88 at K=2; Toy 4 gives equality under varimax and machine-precision agreement under random orthogonal rotation. Toy 5 selects mode K∗=6 on GloVe versus 2 on iid Gaussian X despite rank-1 truth.Source: §A.8 · pp. 26–27, Table 17 · p. 26, Table 18 · p. 27

A.9 Calibration and power benchmarks (full)

• NB tracks Uniform[0, 1] under synthetic H0 (KS-uniform p≥0.46), strictly dominates permutation-Q2 across SNR, and matches or exceeds it in all eight coverage cells.The NB advantage is widest away from the power ceiling; two cells slightly exceed the target operating band, at NB=0.96.Source: §A.9 · p. 27, Table 20 · p. 28

A.10 K∗tracks predictive rank, not er

• The 1-SE selector returns predictively warranted K∗, not the planted rank or conditional-mean dimension; real-text rank-1 fits select mode K∗=6 on GloVe and 2 on iid Gaussian X.The difference reflects PLS deflation geometry, and truncating a valid sequence at K∗ preserves FWER control, subject to NB-test validity.Source: §A.10 · p. 28

A.11 Detection power on real embedding geometries

• On real embedding geometries with planted rank-1 signal, NB strictly beats permutation-Q2 in four of six cells; the two saturated big-n/noisy cells overlap within Wilson intervals.The EN-XLM-R cell tracks EN-GloVe within sampling.Source: §A.11 · p. 28, Table 21 · p. 29

A.12.1 S2 — Power point checks across eight cells

• At chemometrics defaults, NB’s closed-form approach is approximately 82× faster at n=40 and 61× faster at n=500 than permutation-Q2.At P=10^4, the predicted fit-count ratio rises to 103×; naive-t and jack.test power comparisons are non-comparable in the cited regimes because their H0 false-positive rates are inflated.Source: §A.12.1 · p. 29, Table 22 · p. 29

A.12.3 T3 — Power vs. n on Tecator NIR

• On Tecator NIR, NB and permutation-Q2 track the same power ceiling for the deliberately weak R2=0.05 signal, remaining below 80% at n=200.The synthetic Warriner sweep compares power over n across R2∈{0.05, 0.40} and rplant∈{1, 3}; the cited passages provide no numerical result for that sweep.Source: §A.12.2 · p. 29, Figure 4 · p. 30

A.12.4 Sample-size equivalence at matched power

• At matched power, NB-asymptotic needed a median 22% fewer observations across 35 reachable cells, rising to 26% at the conventional 80% power target.The all-cell reductions ranged from about 5% to 37%; among the nine cells where both curves crossed 80%, the range was 8%–37%.Source: §A.12.4 · p. 30

A.12.5 Calibration shape: NB-asymptotic on highly-collinear designs

• On Tecator’s near-rank-one design, NB-asymptotic passed the α=.05 rejection-rate check but failed the null-shape check, whereas perm-Q2 tracked uniformity.Tecator’s standardized design had stable rank 1.014 and 98.6% of variance in its first principal component; the pre-run rule flags stable rank below 3 and directs this design to the exact permutation test, which restores uniformity.Source: §A.12.5 · p. 30, Figure 5 · p. 31

A.13 Validity rule: the full null grid, the plsdof construction, and cost

• On 25 null-grid cells, the stable-rank rule cleared designs with NB-asymptotic levels of 0.035–0.054 at α=.05 and 0.074–0.100 at α=.10, while the exact test held level throughout.The rule flags n<25 or stable rank <3; flagged cells had KS p-values from 9×10^-6 to 2×10^-77, while exact-test levels were 0.042–0.057, 0.093–0.108, and 0.008–0.014 at α=.05, .10, and .01, respectively. NB-asymptotic is useful at α=.05 and .10, not .01; the stable-rank threshold was set on a separate spectrum sweep.Source: §A.13 · p. 32
• The exact degrees-of-freedom F-test construction sometimes had no defined F or poor level, and its O(n^3) cost exceeded NB-asymptotic’s measured cost at large n.On the dominant-factor pool, its numerator degrees of freedom were non-positive on 16–23% of draws; among defined tests, rejection was 0.077–0.083. Warriner timings were 0.07 s at n=500 and 13.5 s at n=4,000, versus 0.03 s at n=400 and 0.34 s at n=3,200 for NB-asymptotic. The construction is the authors’, not one proposed or level-validated by Krämer and Sugiyama.Source: §A.13 · p. 34

A.14 Per-component FWER: proof for one split with train-only deflation

• For one independent train/test split, the fixed-sequence chain controls strong FWER at α under iid Gaussian rows and train-only deflation, but the proof excludes repeated-split NB and full-data deflation.A held-out-residual permutation version is finite-sample valid under iid rows without Gaussianity, but tests independence rather than nonpositive correlation; the reported components are fitted on all n rows, unlike the proof’s training-half components.Source: §A.14 · pp. 35–36

A.15 Per-axis FWER: empirical diagnostics

• Planted-signal grids showed no chain-rejection excess over nominal resolved by Wilson intervals, but boundary-probe reach rates were often too low to assess later steps.At one deflation, the chain reached the boundary step in at least 99.8% of replicates at α=.05 across designs; at m=3 it reached the step in 0%, 5.2%, and 36% for dominant-factor, decaying-spectrum, and gasoline designs, respectively, and almost never in iid designs. These are diagnostics, not a proof of FWER control for NB aggregation or full-data deflation.Source: §A.15 · pp. 36–37

A.16 Beyond PLS: supervised PCA and a ridge probe

• The exact NB-permutation test held level in all eight non-PLS null cells and had power at or above perm-Q2 in all twelve cells at both tested signal strengths.Off PLS1, NB-asymptotic became conservative: supervised PCA power at n=40 was 0.19 versus 0.41 for perm-Q2. The one-sample t control rejected 27–40% of nulls at α=.05 in every cell.Source: §A.16 · p. 38, Table 30 · p. 38

A.17 Embedding pipelines and vocabulary cleaning

• The embedding pipelines mean-pooled token embeddings after row-normalizing E and intersected the pre-pooling vocabulary with each rated lexicon.The named lexicons were Warriner-EN, ANPW-PL, and Stadthagen-González-ES.Source: §A.17 · p. 38
• The text pipeline intersects rated lexicons with normalized embedding vocabularies, yielding n=13,365 (EN GloVe), 4,526 (Polish GloVe), 12,639 (Spanish GloVe), and 13,540 (EN XLM-R).For varimax, the loading target is the full embedding vocabulary intersected with the rated lexicon, with no external POS filter.Source: §A.17 · pp. 38–39

A.18 Sparse-PLS — per-(K, s) sweep on Warriner-EN valence

• K=5, s=50 yields five qualitatively distinct readable axes, each defined by 50 features; the dense endpoint is included as a grid endpoint, not a baseline.Sparse-PLS is intended to make more components individually readable, rather than to recover dense PLS1 at lower cardinality.Source: §A.18 · pp. 39–40, Table 32 · p. 40

A.19 Broader impacts

• The CPU-only supervised text pipeline runs on public embeddings without an LLM inference path and returns word-readable, correlational axes that may amplify spurious associations if treated causally.The released ssdiff software is a statistical library, not a generative model or scraped dataset.Source: §A.19 · p. 40

A.20 Reproducibility

• The replication record contains scripts for reported results except wall-clock timings, spanning applied configurations, toy studies, inference benchmarks, null grids, and outside-PLS pipelines.Configs 1–4 use seed 2137; the full toys, S1/S2, and Configs 1–3 sweep takes under two hours, while Config 4 XLM-R encoding used a consumer GPU.Source: §A.20 · pp. 40–41

A.21 Algorithm: PLS-K pseudocode for the supervised text pipeline

• For each canonical NIPALS axis, the pipeline applies NB split-half inference to data deflated by preceding components, uses fixed-sequence rejection for FWER, then varimax-rotates loadings without changing held-out predictions.The algorithm returns the rotated weights and loadings alongside the per-axis p-values and an FWER summary.Source: §A.21 · p. 41