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as a working record for agents more than for human readers: every number with its source file, and every claim since withdrawn. Some facts here may be out of date; where it disagrees with the post, trust the post.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

Where no word goes - the hole on the factor chart

Redone on the five-factor chart, the widest gap in the 134 words' coverage is 54.8 degrees against a 40.3-degree null, but the direction is 23.1 degrees from the principal-component one, the k=2 result that the words tile better than chance disappears, and the three externally suggested candidate names now miss the hole outright.

Samuel's decision of 2026-09-08 makes the factor analysis the primary frame and the principal-component chart secondary (The factor chart, Factor-first migration). This page redoes "where no word goes" in that frame. The principal-component version, which stands as the record of what was done first, is Where no word goes - the hole and the alien direction; the steering of the new direction is Steering the unnamed direction on the factor chart.

What changed, in one paragraph

The question, the estimator and the null are unchanged. deepest_hole is imported from qwen35/analyse_alien.py so one solver produces both charts. What changes is the chart the adapters are placed in. analyse_alien.py uses the top five eigenvectors of the double-centred sketch Gram; qwen35/analyse_alien_fa.py uses fa_chart.FAChart - the orthonormal Gram-Schmidt basis of the span of the five oblimin factor directions, in the fixed order Warmth, Competence, Fearful withdrawal, Arousal, Imagination, with inner products taken from the exact Gram results/gram_sweep.npz rather than from sketches.

Two consequences are definitional and are stated rather than hidden:

• The factor chart does not centre. FAChart.trait_coords places the raw adapters. The five basis rows are contrasts (they sum to zero over the 134), so every coordinate is a contrast, but the 134 points are not centred in the chart. analysis/alien_fa.json#offset: the mean chart coordinate has length 0.2543682278747567 against a mean trait chart length of 0.9369798382181508 (ratio 0.27147673567713826), and the mean cosine of a trait with that mean direction is 0.2619522101666195. The 134 lines lean one way, and the uniform-random null does not, which is a reason the observed gap can sit above the null that has nothing to do with the lexicon.
• The chart has exactly five dimensions, so the sweep stops at k=5 instead of running to k=16.

Coverage: the k sweep

analysis/alien_fa.json#k_sweep, observed gap against the mean of 134 uniformly random unit directions in the same k, 24 draws, the same restart budget as the principal-component run:

The null columns are bit-identical to the principal-component run's at every k from 2 to 5 (compare analysis/alien.json#k_sweep): the null does not look at the data, only at k, and both scripts draw it from the same seeded generator in the same order. So every difference in the z column below is a difference in the observed gap alone.

The k=2 finding does not survive the change of chart. On the principal-component chart the observed gap at k=2 was smaller than random, z = -2.280849861251385 (analysis/alien.json#k_sweep.2), and the blog page's caption read "A 134-word sample of the lexicon covers two dimensions better than chance and every dimension after that worse." On the factor chart the same statistic is z = +0.19788810289715242: in the plane of Warmth and Competence the words tile no better and no worse than 134 arbitrary directions. At k=3 the z-score falls from +3.0570268913767125 to +0.6738544262288416. Only from k=4 does the effect reappear, and at k=5 it is larger than on the PC chart (+13.717953930316737 against +11.581198742161746) against the same null mean.

The alien direction

analysis/alien_fa.json#alien_fa. Gap 54.76378041380744 degrees inside the chart, against 52.51479642189155 for the principal-component direction inside its own chart (analysis/alien.json#alien_k5.gap_deg). Its coordinates on the Gram-Schmidt basis, in factor order, are

[-0.169506532387767, 0.05396084672322733, -0.4669660186218202, 0.5621744687966154, -0.6589828272303582]

so it is mostly Imagination against Arousal against Timidity, with almost no Competence. Its coefficients over the 134 adapters sum to -2.7755575615628914e-17 and it has unit norm in the exact Gram (#alien_fa.coef_sum, #alien_fa.g_norm); they are steerable with exactly the machinery the named axes use. The nearest trait lines in the chart are immodest at 54.76378041380744, high_strung at 54.77724333415994 and artful at 54.8888341942092 - a flat floor, not one word standing out.

In the full adapter space, measured exactly from the Gram with each adapter a line, the direction is 69.65795787499941 degrees from its nearest word, anxious (#alien_fa.full_gap_deg, #alien_fa.full_nearest).

analysis/alien_v_fa.npy is the direction's image in the 253,952-dimensional sketch space, written only for parity with analysis/alien_v_k5.npy. Nothing downstream reads it: every angle on this page and on Steering the unnamed direction on the factor chart is computed from the exact Gram instead.

It is not one factor. The sharpest thing Where no word goes - the hole and the alien direction could say about the principal-component hole was that it is 98% PC5 - a component the project had already steered, generated along and judged (PC5 - Contemplation and Scheduling) - so the hole was not unexplored terrain and the test it performed was of the chart rather than of the space. That deflation does not carry over. The largest single loading here is -0.6589828272303582 on Imagination, with 0.5621744687966154 on Arousal and -0.4669660186218202 on Timidity: three comparable terms, not one. The factor-chart hole is a blend no single steered direction stands in for.

Is it the same hole?

analysis/alien_fa.json#vs_pc. No, and the comparison is the point of the page:

• the two unnamed directions are 23.124232986519818 degrees apart as lines (cosine -0.9196554778769139);
• but the principal-component direction lies 0.998062868506644 inside the five-factor span, so the two charts are very nearly the same five-dimensional subspace - what moved is where the deepest hole in it is, not the subspace;
• measured in the factor chart, the principal-component direction is a 49.5426484650169-degree hole with mothering nearest, which is shallower than the 54.76378041380744 the factor-chart search finds.

So the subspace survived the change of frame and the winning direction inside it did not. That is a statement about how flat the minimax surface is, and the flat floor in the nearest-trait table above says the same thing.

Angles to the vocabulary

analysis/direction_gaps_fa.json, the factor-chart twin of analysis/direction_gaps.json. The angles here are computed in the full adapter space from the exact Gram, with adapters uncentred, each treated as a line; the principal-component table measures the same thing in the centred 253,952-dimensional sketch space, so the two are not interchangeable.

The row for alien_k5 is a useful check on the whole exercise: measured this way it is 68.68571215660741 degrees, against 68.93287342956332 for the same direction in the sketch space (analysis/direction_gaps.json#alien_k5.deg). Two quite different measurements of one direction agree to a quarter of a degree, which is how much of the difference elsewhere on this page can be blamed on the sketch. Almost none of it.

The chart_len column in the same file records how much of each direction the five-factor chart sees: 1.0 for the five factors by construction, 0.9468 to 0.9602 for the five named Big Five axes, and 0.5579675797893804 for the personality axis - the mean adapter is only half inside the chart.

The hole words on the factor chart

qwen35/analyse_hole_fa.py, analysis/hole_geometry_fa.json. The same three adapters the externally suggested names produced - cavalier, blase, insouciant, trained on the zoo's shared prompt pool at the matched objective - are placed in the factor chart. The gate is the one analyse_hole.py runs and is imported unchanged: LoRA-A drift 0.0130, 0.0154, 0.0153 against the zoo's own 0.0146.

Placing an external adapter in this chart needs the exact cross-Gram column <a_i, b> over the 134, which did not exist. It was computed with the project's own cross_gram_full_on_modal.py against the zoo root on pc-qwen35-sweep, into results/cross_gram_full_root_x_pc-qwen35-adapters_data_hole_common.npz. Every angle below is exact; no sketch is used. As a check, the cosines from that file agree with the sketch cosines to a mean of 0.005 and a maximum of 0.020.

Values in this table are rounded as printed by analyse_hole_fa.py; the file carries full precision under #<word>.deg_to_u_chart, #<word>.frac_in_chart, #<word>.chart_len_frac_of_trait_mean, #<word>.deg_to_v_full, #<word>.nearest and #<word>.deg_nearest.

Benchmarks in the same file: #_hole_nearest_existing_deg = 54.76378041380744 (what the nearest of the 134 manages inside the chart) and #_hole_nearest_existing_full_deg = 69.65795787499941 (the full-space figure).

The verdict flips. #_verdict is "the names miss". On the principal-component chart insouciant sat 39.56272020815669 degrees from the hole against 52.514796421891425 for anything that already existed, and PHASE3_VERDICT.md recorded "Suggestive in-plane, unconfirmed" (Where no word goes - the hole and the alien direction). On the factor chart the best of the three is insouciant at 66.4505 degrees against 54.76378041380744 for the nearest existing adapter: the zoo already had something closer to its own hole than any of the three new words. The chance level moves with it - #_p_one = 0.43257832813236446 and #_p_any_of_three = 0.817308745873875, against 0.07270091967910823 and 0.20263074303625694 on the PC chart - which is another way of saying 66 degrees is not a hit.

The negative finding on the PC chart survives and hardens. The three words are still further from each other than from the hole, now measured exactly (#_pairwise, from results/cross_gram_full_data_hole_common_x_data_hole_common.npz): cavalier/blase 86.86256389717796, cavalier/insouciant 74.0348470906374, blase/insouciant 83.0585738681165. The scale to read those against has to be computed the same way - exact Gram, uncentred, adapters as lines - because analysis/trait_angles.json is centred and sketched and does not belong beside these numbers. analysis/alien_fa.json#benchmarks_exact_uncentred: two adapters at random 82.92066894680495, fifth percentile 69.45162642724753, a trait to its own nearest neighbour 63.51233996451681, and the closest pair in the whole zoo composed/imperturbable at 51.52127547983898. Two of the three hole words are further apart than two zoo adapters drawn at random.

The alignment adapters on the factor chart

qwen35/analyse_alignment_fa.py, analysis/alignment_geometry_fa.json, on the aligncommon arm - the four adapters retrained on the zoo's exact shared prompt pool. The gate: 0.0146, 0.0143, 0.0150, 0.0151. Cross-Gram from results/cross_gram_full_root_x_pc-qwen35-adapters_data_alignment_common.npz.

Basis coordinates of the unit-normalised adapter, in factor order, with how much of the adapter the chart sees and where it sits relative to the hole:

(Every number in that table is rounded as printed by analyse_alignment_fa.py. The file carries full precision at #<trait>.chart for the coordinates, #<trait>.frac_in_chart, #<trait>.chart_len_frac_of_trait_mean, #<trait>.deg_to_u_chart, #<trait>.deg and #<trait>.nearest; #<trait>.chart_oblique gives the same point in coordinates on the five oblique factor directions themselves. The same applies to the "Why the raw angles moved" table below, whose #_centred_exact entries are stored in full.)

None of the four is near the hole - the closest, sycophantic, is 71.2 degrees from it, where a random 5-d direction lands within that angle 53.3% of the time (#_p_one).

The four preregistered verdicts are unchanged: 1 held, 2 failed, 3 held, 4 failed, the same pattern as Alignment-relevant traits in the geometry.

1. Sycophancy loads on warmth/agreeableness. HELD: FA_Warmth is the largest of the five factor cosines at +0.423, and axis_Agreeableness is +0.433 (#sycophantic.cos).
2. The synonym pair under 54 degrees. FAILED: #_pair_deg = 67.1, exact, against 61.867099876005376 for the sketch figure on the PC page.
3. Power-seeking over 60 degrees from all 134. HELD: 65.5.
4. Corrigible under 60. FAILED: 66.6.

Why the raw angles moved, and what it is not

The nearest-of-134 angles above are several degrees off the ones on Alignment-relevant traits in the geometry and Where no word goes - the hole and the alien direction, and the cause is centring, not the change from sketches to the exact Gram. analyse_alignment_fa.py and analyse_hole_fa.py both recompute the same angle on exactly-centred vectors as a check (#_centred_exact):

The centred-exact column reproduces the sketch column to within a degree everywhere, so the sketch was accurate and the difference is entirely the factor chart's decision to work on raw adapters.

Related: Where no word goes - the hole and the alien direction, Steering the unnamed direction on the factor chart, The factor chart, Alignment-relevant traits in the geometry, Steering the unnamed direction, Factor-first migration, What has been done to the PCs but not to the factors, Weight-space geometry of the 134-adapter zoo.

Trained directly in activation space

Persona sliders - LoRAs trained to an activation-space target trained a rank-64 LoRA to produce the activation image of this direction -- alien_fa.coeffs applied to the 134 trait-centred persona vectors -- rather than merging the adapters. The slider reaches its target at cosine +0.8259 on held-out prompts and sits 55.1032 degrees from the nearest of the 134 persona vectors against a permuted-coefficient null median of 47.8617, but its judged behaviour is not distinguishable from two shuffled-coefficient sliders built the same way (qwen35/analysis/persona_sliders.json#behaviour.hole). Its cosine with the weight-space alien_fa merge is +0.0541 against the controls' +0.0146 and +0.0095 -- a lean, not a location.