The sphere sweep on the factor chart

The 72-direction sphere resampled on the span of the top three oblimin factors after the 2026-09-08 decision to make the factor analysis primary; 45 of 72 loop on no prompt, angular distance predicts judged-profile distance at rho = 0.68, and the judged field agrees with the principal-component sphere at r = 0.71 to 0.82 on three of five scales where the two subspaces actually overlap.

The sphere sweep on the factor chart

Samuel's decision of 2026-09-08 makes the factor analysis the primary frame for the whole project and the principal-component chart secondary. The sphere sweep was the largest behavioural result stated in principal-component coordinates, so it was re-run on the factor chart. This page is the record of that run. The sphere sweep — 72 directions nobody chose remains the record of the 2026-09-01 principal-component sphere; its data are unchanged and its numbers still stand.

The rendered page is served at /sphere-fa.html (qwen35/sphere_page/index_fa.html, 881,729 bytes, copied by wiki/tools/build_site.py STATIC_FILES). Every point's eight generations ship inside it, so a reader can click any of the 72 directions and read the model there.

What the chart is

qwen35/fa_chart.py fixes one convention for every analysis that places directions in factor space, and this sweep uses nothing else:

The factors are oblique, so the chart uses an orthonormal basis of their 5-dimensional span obtained by Gram-Schmidt in the G inner product, in the order above (Warmth first). ORDER MATTERS and is fixed here. — qwen35/fa_chart.py

The five factor directions are the oblimin PAF factors as they were actually steered in phase 10, phase10_runs/steer_spec2_7a.json jobs FA_Warmth, FA_Competence, FA_FearfulWithdrawal, FA_Arousal, FA_Imagination, ordered by oblimin sum of squared loadings (results/fa_qwen35.json#solutions.centred_k5.ss_loadings.oblimin = 10.76210106481124, 8.421873312354228, 7.020338408333558, 6.814511494042316, 5.798806928556024; also copied to analysis/sphere_layout_fa.json#ss_loadings_oblimin). Inner products come from the exact Gram results/gram_sweep.npz, never from coordinates.

The sphere is the unit sphere of the span of the first three basis vectors — Warmth, Competence, Fearful withdrawal (analysis/sphere_layout_fa.json#axis_names).

What was held fixed, and why

qwen35/build_sphere_spec_fa.py changes the subspace and nothing else, so the two spheres are comparable point for point:

• The same 72-point Fibonacci lattice. Identical function, identical N = 72, so the two runs differ only in which subspace the lattice is laid on.
• alpha = 1.5 (analysis/sphere_layout_fa.json#alpha).
• The same 8 prompts, PROMPT_IDX = [0, 1, 5, 9, 11, 14, 15, 22].
• The same reference norm. ref = 0.8102592902648793 (analysis/sphere_layout_fa.json#ref), taken from phase10_runs/steer_spec.json#ref — what the principal-component sphere used — and deliberately not the 0.8078003190997738 of phase10_runs/steer_spec2_7a.json#ref. The difference is 0.3%, but "the same steering magnitude" is the comparability claim, so it is the PC sphere's number that was kept.

Each sampled direction's coefficients over the 134 adapters are u @ basis[:3], which is unit norm in G by construction, written to phase10_runs/sphere_spec_fa.json. The container's own input is phase10_runs/sphere_sweep_spec_fa.json: three basis coefficient dicts plus the 72 unit vectors.

Landmarks

Fourteen named directions were projected into the same three-space and normalised (analysis/sphere_layout_fa.json#landmarks). Each is a direction the study has actually generated from — the five factors from steer_spec2_7a.json, everything else from steer_spec.json — not a construction. #landmark_cos records how much of each one this three-space actually sees (the cosine between the direction and its own projection):

Two entries in that table matter more than the rest.

The grand mean is inside this sphere. mean_assistant_axis — the direction of having a persona at all, see What the stage-two shared direction does — has cosine 0.5067 with the top-3 factor space. The principal-component sphere could not have had this: its subspace comes from the eigenvectors of the double-centred Gram and is orthogonal to the grand mean by construction. The factor chart works in the uncentred G (qwen35/fa_chart.py), so half of the grand mean lies in the sphere the lattice covers. That is a real difference in what is being sampled, and it is the most likely mechanism behind the coherence and coverage changes below. It is recorded here, not resolved.

PC3 barely appears. PC1 and PC2 are almost inside this three-space (0.9482, 0.9724) but PC3 is at 0.2979. The two three-dimensional subspaces share roughly two dimensions, not three. Every comparison below is conditioned on that.

Coverage of the zoo

analysis/sphere_layout_fa.json#var3 = 0.11590344397319126, 0.10223504799677975, 0.044322108527744895 — the share of the total centred adapter variance (trace of the double-centred Gram) projecting onto each of the three basis vectors, summing to 0.2624606004977159. The rendered page shows this as "26.2%".

The principal-component sphere's own var3 (analysis/sphere_layout.json#var3 = 0.12668915056443317, 0.11252214481119611, 0.04955491287468896, summing to 0.2887662082503183) is not directly comparable: it was computed from the eigenvalues of the k=32 sketch Gram (build_sphere_spec.py loads analysis/sketches/stage1_k32), not the exact one. Two like-for-like references were therefore computed on the exact centred Gram and stored alongside:

• #var3_pc_exact_gram = 0.1249462838655816, 0.10984693652476066, 0.049679397531247095 (sum 0.28447261792158934) — the top three eigenvalues, i.e. the best possible three-dimensional subspace.
• #var3_pc_steered_dirs = 0.12025180802811421, 0.1115120780857482, 0.0487418136518603 (sum 0.2805056997657227) — the space actually spanned by the three steered PC directions.

So the factor sphere sees 26.2% of the centred variance against 28.1% (#var3_pc_steered_dirs) for the space the principal-component sphere actually sampled; 28.4% (#var3_pc_exact_gram) is the best any three-space could do. The factor chart gives up about two points of variance for an axis set with names.

The factors are oblique, and the sphere is not orthogonal in the sense a reader might assume. #factor_pairwise_cosines gives the cosines between the five oblique factor directions as they sit in the chart; Warmth and Fearful withdrawal are at -0.6661, Competence and Arousal at -0.728.

The run

zoo-sphere-fa.service (Description=Sphere sweep on the FACTOR chart: 72 sampled directions in one container) ran modal run --detach sphere_sweep.py --spec phase10_runs/sphere_sweep_spec_fa.json --out phase10_runs/sphere_results_fa.json. The log's header line confirms what it consumed: "72 points x 8 prompts at alpha=1.5 tag=fa basis=['B1_FA_Warmth', 'B2_FA_Competence', 'B3_FA_FearfulWithdrawal']" (qwen35/phase10_runs/sphere_fa.log). It ends "wrote phase10_runs/sphere_results_fa.json: 72 points".

One code change was needed and it was a real hazard. sphere_sweep.py hardcoded its resume checkpoint at /oct/sphere/results.json. Re-run under that path, the container would have found the 2026-09-01 run's 72 finished points and returned those generations under the new point names — silently, at no cost and with no error. The checkpoint is now namespaced by a tag from the spec (/oct/sphere_fa/results.json here); untagged runs keep the original path so the PC run stays resumable. Verified empirically, not from the log line: the FA log contains zero [resume] lines, and of the 72 points, 0 have generations identical to the PC run.

Thinking is off — enable_thinking=False in the apply_chat_template call (qwen35/sphere_sweep.py), the failure mode recorded in the project's method lessons.

Judging was the same blind pipeline with the same arguments as the PC sphere (zoo-judge-sphere-fa.service, --batch 6 --workers 8 --repeat-frac 0.05). qwen35/phase10_runs/judge_sphere_fa.log: "576 generations to judge from 72 traits", "101 judge calls (batch=6, 28 repeats)", "576 judged, 0 failed calls". Repeat reliability: Extraversion r=0.852 (n=28), Agreeableness r=0.963 (n=28), Conscientiousness r=0.857 (n=27), EmotionalStability r=0.912 (n=28), Intellect r=0.937 (n=28). The judge is anthropic/claude-sonnet-4.5 (judged_sphere_fa.json#model).

Result 1: most of the space is still habitable, slightly less so

qwen35/analysis/sphere_page_fa.json#coherence, computed with the same looping detector as every other degeneration number in the study (looping in qwen35/analyse_alien_steer.py: a 10-word window repeated 4 times):

none       45
any        27
worst      0.375
mean       0.06597222222222222
len_mean   1389.6996527777778

Against the PC sphere (analysis/sphere_page.json#coherence: none 48, any 24, worst 0.25, mean 0.046875, len_mean 1261.234375). Three fewer directions are completely clean, the worst direction loses 37.5% of its responses rather than 25%, and the mean looping rate rises from 4.7% to 6.6%. Responses are longer: 1389.7 characters against 1261.2.

The conclusion the PC sphere reached is unchanged and if anything stronger for being reproduced on a different subspace: most directions through this space give intact output at alpha = 1.5, so the coherence of the directions the study chose is not, by itself, evidence of anything.

Result 2: personality varies continuously, and slightly more so

qwen35/analysis/sphere_page_fa.json#smooth:

rho        0.6765886845997277
n_pairs    2556
n_scored   72
near_mean  0.9462620103993861
far_mean   2.917385198267297

Over all 2,556 pairs of the 72 scored directions, angular distance and judged Big Five profile distance correlate at rho = +0.68, against +0.65 on the PC sphere (analysis/sphere_page.json#smooth rho = 0.6511417860626274). Directions less than 30 degrees apart differ by 0.95 on the Big Five (PC sphere: 0.97); more than 120 degrees apart, by 2.92 (PC sphere: 2.81). Dividing those two quoted values gives a near/far ratio of 3.08 against 2.91 — computed here, not a key in either file.

This is the load-bearing result of the sweep and it survives the change of chart intact. As on the PC sphere, it is a local continuity claim and does not conflict with Additivity of the Big Five steering axes.

Result 3: the coverage is lopsided, and more so

Counting which scale the judge rated highest at each of the 72 points (analysis/sphere_page_fa.json#judged.<point>.top), with the PC sphere alongside:

Emotional stability is now the top-rated scale nowhere on the sphere. Note that axis_EmotionalStability has cosine 0.604 with this three-space and axis_Intellect only 0.3968 (#landmark_cos), so part of this shift is the subspace not containing those axes rather than the model not going there.

Result 4: the two judged fields, compared directly

This is the one measurement the redo makes possible. Each of the 72 factor-sphere directions was matched to the nearest of the 72 principal-component directions, after projecting the latter into this chart from their coefficient dicts in phase10_runs/sphere_spec.json (not from their PC-space u), using the signed cosine — steering at +u and -u are different personalities, so matching on |cos| would pair a direction with its opposite. #vs_pc:

• median_match_deg = 31.23512652729555, max_match_deg = 79.55102833275029
• median_pc_cos_into_fa3 = 0.76, min_pc_cos_into_fa3 = 0.047
• n_close = 27 of 72 matched within close_deg = 20 degrees

Pearson correlation of the judged score over the matched pairs, per scale (#vs_pc.pearson, and #vs_pc.pearson_close restricted to the 27 pairs under 20 degrees):

How to read this. Because PC3 sits at cosine 0.2979 with this three-space, the projected principal-component lattice is squashed towards a great circle: the median match is 31 degrees apart and the worst is 80. Over all 72 pairs the correlations are therefore a mixture of real agreement and points that have no counterpart at all. Restricted to the 27 pairs that genuinely land near each other, three of five scales agree at +0.71 to +0.82 — two directions close in weight space give close judged personalities, whichever chart named them, which is the smoothness result of Result 2 seen across the two runs.

Emotional stability does not agree on either column, and its correlation is negative. This is unresolved. Two candidates, neither tested: it is the scale with no top-rated point anywhere on the factor sphere and the fewest on the PC sphere (2), so its range across both fields is small and the correlation is dominated by judge noise; and it is the scale whose named axis is furthest outside both subspaces after Intellect. It should not be quoted as evidence that the two spheres disagree about emotional stability until someone has separated those.

Cost

The sweep ran as one A100-40GB container. zoo-sphere-fa.service started 2026-09-08 13:32:04 UTC and phase10_runs/sphere_results_fa.json was written at 17:25:28 UTC: 3.890 hours of unit wall clock, including roughly four minutes of image build and base-model download before the GPU did any work. At the meter's rate of $2.10 per GPU-hour (qwen35/zoo40_meter.sh, RATE=2.10) that is an upper bound of $8.17. The box-wide meter cannot confirm this: as many as 16 containers were active during the window (phase10_runs/zoo40_meter.log, 2026-09-08T14:31:03Z), so its $2251.20 to $2346.57 movement over the same period is mostly other work.

The FA run was slower per point than the PC run (about 3.1 minutes against about 1.7) because its responses are longer — 1389.7 mean characters against 1261.2 (#coherence.len_mean) — and generation is capped by tokens, not time.

Judging cost is OpenRouter, not Modal: 101 calls to anthropic/claude-sonnet-4.5.

Where it is rendered

qwen35/build_sphere_page.py now takes --suffix ("" for the PC sphere, _fa for this one) and --html. It writes analysis/sphere_page_fa.json (874,865 bytes) and the standalone qwen35/sphere_page/index_fa.html, served at /sphere-fa.html. The standalone page shares no asset with the blog page on purpose: the blog page's _js.txt colours sphere landmarks by Big Five name, which is wrong on a sphere whose axes are factors.

Verified against the PC sphere before use: re-running the refactored builder with no arguments reproduces analysis/sphere_page.json byte for byte.

Related: The sphere sweep — 72 directions nobody chose, Factor analysis of the adapter Gram, Steering the base model along weight-space directions, Additivity of the Big Five steering axes, What the stage-two shared direction does, Steering the unnamed direction, Weight-space geometry of the 134-adapter zoo, Glossary.