Written entirely by Claude, as a working record for agents more than for human readers: every number with its source file, and every claim since withdrawn. Some facts here may be out of date; where it disagrees with the post, trust the post. For the readable version, see the companion.

PCA, the scree curve and the two nulls

Eleven principal components of the adapter cloud sit above a structureless null and none sits above the permuted null; the elbow is unusable and four different variance spectra circulate for the same cloud.

Status: the secondary frame

Samuel's decision of 2026-09-08 - "we're going with the factor analysis so please redo the sphere using the biggest factors now and same for the other visualisations" - makes the five-factor oblimin solution the project's primary frame and the principal components the secondary one. See The factor chart for the convention that replaced the PC chart and Factor-first migration for what moved, element by element.

Nothing on this page is withdrawn. The components are still steered, still judged, still described on their own cards, and two arguments continue to need them:

• The dimensionality argument below is PCA by nature. How many directions the cloud has above a trained null is a question about a variance spectrum. A rotated five-factor solution cannot be asked it.
• The optimised-data arms were trained against PC4 and the named Agreeableness axis (see Optimised data and the training check and qwen35/analysis/verify.json). What was run is what is reported; those sections keep their PC names.

One number on this page's subject changed with the migration. The blog page's right-hand scree panel used to plot the uncentred parallel analysis while the five factors on the same page come from the centred (ipsatised) matrix and results/fa_qwen35.json#n_factors.chosen is read off the centred one. The panel and the prose beside it now use #n_factors.parallel_analysis_centred. Retention at the reference N of 1,528 moves from 12 reduced / 9 unreduced (uncentred) to 9 reduced / 9 unreduced (centred), and the N-sensitivity sentence from 5 / 9 / 13 at N = 150 / 1,528 / 20,000 to 5 / 9 / 14. Recorded on Superseded claims.

What the scree figure shows

The blog page's left-hand scree panel plots the PCA spectrum of the adapter cloud against two trained nulls. It is fed by qwen35/analysis/scree_null_matched.json (qwen35/build_blog_page.py, scree_svg(), which reads that file by name); qwen35/analysis/scree_null.json is the earlier version and is superseded. Both arms in the current file were trained at the zoo's exact objective - see The null control arms.

First twelve components, per cent of variance, from qwen35/analysis/scree_null_matched.json:

(Values above are the first twelve of the twenty stored in each list, quoted to three decimals as they appear; n = 100.)

Counts stored in the same file:

• #n_above_structureless = 11 - eleven real components sit above the shuffled ("preference destroyed") arm.
• #n_above_null = 0 - no real component sits above the permuted arm.
• #perm_mean_abs_diff = 0.10336514494213549 - the real and permuted curves differ by that many percentage points on average over the first twelve components. (scree_null.json, the earlier objective-mismatched file, gives 0.1798544716598768; the blog page's prose quotes "0.18 percentage points", i.e. the older number - see the contradiction note below.)
• #note = "matched-objective null arms (sigmoid,sft/1.0,0.1/kl 0.001), 2026-09-04"

What the two nulls mean

• shuffled: chosen and rejected are swapped on exactly half of each trait's pairs, so no coherent preference direction is learned. Its spectrum is flat at about 1.1-1.2% per component, i.e. essentially full rank.
• permuted: each trait name is given another trait's intact pairs, so every adapter learns a real persona under the wrong label. Its spectrum lies on top of the real one.

The blog page draws the conclusion the counts force: the scree tells you how much structure a training run produces and nothing about whether that structure lines up with the trait names. Everything connecting a direction to a word rests on steering and judging (Steering the base model along weight-space directions) or on the labelled tests in Polarity, bipolarity and the trait graph, never on the eigenvalues.

The elbow is explicitly declared unusable. The blog page: the largest single drop puts the elbow after the second component, maximum curvature says the third, and a chord from the first point to the tail says the first, because the leading two eigenvalues are nearly equal - "Three criteria, three answers". The page also records the correction: with only the elbow to go on, describing six components looked generous; against the structureless null it is conservative. That reversal is logged in Superseded geometry claims.

Four different variance spectra for the same cloud

These are all real, all cited below, and they are not interchangeable. Anyone copying a "first two components take X%" figure must say which one.

The blog page's own map caption uses the 134-trait sketch figures: "The first two components take 12.7% and 11.3% of the variance ... 40% in eight dimensions" (qwen35/blog_page/index.html; the 40% is the sum of the first eight entries of analysis/alien.json#var_explained).

A fifth number exists for the raw (uncentred) 134x134 cosine matrix: qwen35/results/decomposition.json#spectrum.eigenvalue_share = 0.12523, 0.10113, 0.08096, 0.04342, ... with #spectrum.participation_ratio = 25.720397850313308 and #spectrum.top5_share = 0.3751747844019098. The participation ratio is the figure PHASE3_VERDICT keeps: "Participation ratio 25.7: five factors are a real but thin slice of a much higher-dimensional geometry."

The known trap: where the stored real curve came from

The maintainer recorded a reproduction check that could not reproduce the stored real curve from the expected Gram. .garden/journal/2026-09-05.md:

Scree null regenerated from the matched Grams; stored real came from gram_sketch100.npz, not gram_sweep (took a minute to find).

Evidence still visible in the files, recorded rather than resolved:

• #real is identical in scree_null.json (written 2026-09-02 20:20) and scree_null_matched.json (written 2026-09-05 12:15) to every printed digit (13.36216610744329 / 11.207049613198226 / 5.78913271575242... ), so the real curve was not recomputed when the matched arms were.
• Those values match qwen35/analysis/geometry_stage1.json#explained_var_top10 (0.134, 0.1122, 0.0577, 0.0385, 0.0294, 0.0224, 0.0171, 0.0154, 0.0127, 0.0122) after rounding, and geometry_stage1.json is computed from the stage1_k32 sketches over the 100 labelled traits (qwen35/geometry.py loads a sketch directory).
• results/gram_sketch100.npz (mtime 2026-08-29) is a sketch-derived Gram built by qwen35/build_gram.py; results/gram_sweep.npz (mtime 2026-08-20) is the exact one.
• qwen35/analysis/qual_identity_notes.md records the same substitution for a different analysis: "Geometry used in the interpretations was computed from results/gram_sketch100.npz (the 100x100 Gram over trait directions)".

So the real curve on the current page is a 100-trait, sketch-derived spectrum, not the exact 134-trait one. Given the reported sketch fidelity (r = 0.9996, see Weight-space geometry of the 134-adapter zoo) the shape is not expected to change, but the provenance is what it is and nothing on disk shows the curve being regenerated from gram_sweep.npz. This is flagged in Superseded geometry claims and in the section report.

PC loadings and cosines with the named axes

qwen35/analysis/pc_loadings.json#pcs holds, for each of PC1-PC6, its cosine with each of the five named Big Five axes and the five traits loading highest at either end. The largest cosine per component:

PC1 also carries -0.7001299478434748 on Conscientiousness, which is why the blog page calls it a blend rather than a factor. Per-component detail is on PC1 - Flooding and Composure .. PC6 - Feeling and Deliberation.

pc_loadings.json additionally carries a #seed block: same_trait_cos = 0.01658715905967511, same_trait_deg = 89.04958220635345, diff_trait_cos = 0.0013512710743004668, n = 40, rank1 = 40. Those are the original, objective-mismatched seed-paired numbers; the current figure is +0.01806 (see The seed floor). The blog page's prose quotes the matched 0.018 while this data file still holds 0.0166.

Dimension sweep and intrinsic coordinates

qwen35/analysis/geometry_k_sweep.json sweeps the PCA dimension k over 5, 10, 20, 30, 40, 50, 70, 99 with columns [k, whitened_axis_abscos, whitened_null, whitened_p, unwhitened_axis_abscos, unwhitened_null, unwhitened_p, loo_acc_whitened, loo_acc_unwhitened]. Its #reading:

Keying classification: read UNWHITENED (0.98 at every k). Orthogonality: read WHITENED (p<=0.01 for k=5..70). k=99 = n-1, whitening fully degenerate.

At k=5 the row is [5, 0.1106, 0.3487, 0.0033, 0.2297, 0.394, 0.01, 0.99, 0.98]. The sweep's #description records that the underlying geometry is "bilinear sketch k=32 (validated r=0.9996 vs exact)" over 100 Big Five trait adapters at LoRA seed 0, and its #null is "shuffled-keying within factor, 300 permutations".

qwen35/analysis/geometry_stage1.json carries the matched pair of readings at k=30 over n = 100:

• #cum_var_at_k = 0.6238
• #unwhitened_axis_abscos_mean = 0.1968 against #unwhitened_axis_null_mean = 0.2052, #unwhitened_axis_p = 0.4913 - unwhitened, the five axes are not more orthogonal than chance.
• #whitened_axis_abscos_mean = 0.0387 against #whitened_axis_null_mean = 0.1062, #whitened_axis_p = 0.0025 - whitened, they are.
• Leave-one-trait-out keying accuracy per factor, unwhitened (#unwhitened_loo_keying): Agreeableness 0.95, Conscientiousness 1.0, EmotionalStability 0.95, Extraversion 1.0, Intellect 1.0, each at p = 0.0099.
• #cos_same_factor_mean = -0.0014, #cos_diff_factor_mean = -0.0121.

qwen35/analysis/intrinsic_coords.json records a separate, smaller object over n_traits = 100: #eff_dim = 3.2694443860505626, #variance_shares = [0.4620546637786367, 0.2403440646151083, 0.14015058574881312, 0.11453762965028536, 0.04291305620715653], #test_retest = [0.786..., 0.872..., 0.899..., 0.833..., 0.859...], #ceiling_24prompts = 0.9926908730732167, #spearman_weight_judged = 0.5823269539942723, with #null_mean = -0.00043594766353703294 and #null_sd = 0.025062880236739825. No producing script for this file exists in the repo; the file's own keys are all the provenance there is, and the numbers are not used by the blog page.

Related: Weight-space geometry of the 134-adapter zoo, Factor analysis of the adapter Gram, The null control arms, Polarity, bipolarity and the trait graph, Superseded geometry claims, Glossary.