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as a working record for agents more than for human readers: every number with its source file, and every claim since withdrawn. Some facts here may be out of date; where it disagrees with the post, trust the post.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

The sphere sweep at iso-KL dose

The 72-direction principal-component sphere re-steered so that every point delivers the same measured KL per token instead of the same alpha; the alpha-1.5 sphere turns out to have been dosing its points over a 2.54x range, and removing that changes almost nothing - angular distance still predicts judged-profile distance at rho 0.65, the two judged fields agree per point at r 0.63 to 0.97, and what does change is degeneration, which does not follow the dose at all.

The confound

The sphere sweep — 72 directions nobody chose steered all 72 Fibonacci-lattice directions on the unit sphere of the top three principal components at one strength, ALPHA = 1.5, and found that angular distance predicts judged Big Five profile distance at Spearman 0.6511417860626274 (qwen35/analysis/sphere_page.json#smooth.rho). The sphere sweep on the factor chart reproduced that at 0.6765886845997277 on the factor chart.

Fisher norms of the steering directions then made the frame a problem. A steering direction is a unit vector in the Frobenius metric, which the model has no opinion about; measured in the metric the model does imply, the 72 points of the factor sphere spread from 0.1716923166559369 to 0.49828560642407 in Fisher norm (analysis/fisher_norms.json#comparisons.sphere_band), and F predicts how far a point's judged profile sits from the lattice centroid, Spearman 0.3706669239179368, permutation p 0.0014499275036248187 (#correlation_with_degeneration.sphere_judged_profile_distance). One alpha over a lattice is therefore not one dose over a lattice, and some of the smoothness, the coherence pattern and the lopsided coverage could be an image of how the dose field varies over the sphere rather than of where the directions point.

This run removes the confound by construction. Same 72 directions, same 8 prompts, same decoding, same judge; each direction steered at the alpha that delivers the same measured KL(base || steered) per token. Pre-registered in qwen35/PREREG_sphere_isokl.md before the steering was launched, target filled in from the calibration in a dated addendum before the same launch.

The headline: the confound was real and it did not matter. The alpha-1.5 sphere was dosing its points over a 2.54x range. Flattening that to 1.02x leaves the smoothness result, the near/far gap and the coverage essentially where they were. What it does change is degeneration and the reading of what curvature buys.

What was held fixed, and the check that it was

qwen35/build_sphere_isokl_calib.py and solve_sphere_isokl_alphas.py change the alpha and nothing else. Before anything was launched, build_sphere_spec.py's lattice and W = V[:, :3] / sqrt(w[:3]) were recomputed from the same sketches and compared against the files the 2026-09-01 run consumed (PREREG_sphere_isokl.md):

• points[*].u of phase10_runs/sphere_sweep_spec.json: maximum absolute difference 0.0.
• the three basis coefficient dicts: maximum absolute difference 0.0.
• the 72 coefficient vectors of phase10_runs/sphere_spec.json: cosine 0.9999999999999998 to 1.0000000000000002.
• ref 0.8102592902648793, phase10_runs/steer_spec.json#ref, the PC sphere's own unit - deliberately not the 0.8078003190997738 of steer_spec2_7a.json that Suppression versus amplification at matched Fisher dose used, because "the same steering magnitude" is the comparability claim here.

sphere_sweep.py gained one backward-compatible line, a_pt = float(pt.get("alpha", alpha)). A spec with no per-point alpha behaves exactly as it did.

Step one: what the alpha-1.5 sphere was actually dosing

qwen35/sphere_dose.py is fisher_dose.py with the partial checkpoint namespaced by a spec tag, a mode-agnostic progress line and its own app name; no arithmetic differs, so the measurement is on the same fixed 4,378 scored token positions (Fisher norms of the steering directions), the same 24 prompts and the same 192-token cap. Zero-alpha control exactly 0.0 for KL, symmetric KL and argmax change (analysis/sphere_isokl_alphas.json#zero_alpha_control).

Only the bf16 weight construction was measured. sphere_sweep.py writes (W0 + D*s).to(bfloat16), which is exactly fisher_dose.py's bf16 mode, so that is the dose the generating model receives; Suppression versus amplification at matched Fisher dose found bf16 delivers a median 0.9059646365364478 of the exact-weight KL at the same alpha, so matching on the exact curve would have matched an intention rather than a delivery.

72 directions at alpha 1.0, 1.25, 1.5, 1.75, 2.0 and 2.5 (phase10_runs/sphere_isokl_dose_spec.json), 432 measurements, 50.1 minutes on one A100-40GB (phase10_runs/sphere_isokl_calib.log).

At alpha 1.5 the 72 points delivered (analysis/sphere_isokl_alphas.json#summary):

So the sphere the project has been quoting since 2026-09-01 was a sphere at doses differing by a factor of 2.54. That is a slightly smaller spread than the factor sphere's Fisher-norm range, whose maximum over its minimum is 2.902 (computed here from fisher_norms.json#comparisons.sphere_band, not a key in either file, and a curvature ratio rather than a delivered-KL ratio).

Step two: the target and the alphas

Target 0.2334374880453069 nats per token (#kl_target_nats_per_token) - the median of the column above, by the rule fixed in the pre-registration before the calibration was read. The median is the statistic that keeps the aggregate dose equal to the original sphere's: half the lattice moves up in alpha and half moves down. A target taken from an external unit, as Suppression versus amplification at matched Fisher dose used (0.5 * F_random_median * dose^2), would have shifted every point the same way and confounded "iso-KL" with "a different overall dose".

The alpha is read off the measured curve by log-log interpolation between bracketing measured points - solve() from qwen35/solve_matched_alphas.py, reused unchanged. 0 of 72 extrapolated and 0 of 72 non-monotone (#summary.n_extrapolated, #summary.n_non_monotone), so no pre-registered fallback fired.

(#summary. The median landing on 1.5002 is the design working, not a coincidence: the target is the median of the KL at 1.5.)

The interpolation itself was checked at no cost: KL at alpha 1.5 predicted from the 1.25 and 1.75 measurements by the same log-log rule, over the measured value, has median 0.987947433893645 and range 0.967415326884758 to 1.0028996091629692 (#summary.interp_check_kl15_pred_over_meas). The rule is accurate to about one per cent at this spacing.

Step three: the dose actually delivered, re-measured

zoo-sphere-isokl-verify.service re-ran the same measurement once per direction at its solved alpha (phase10_runs/sphere_isokl_verify.json, 20.0 minutes). analysis/sphere_isokl.json#delivered:

Median absolute relative error 0.0022839730699117933, worst 0.019096642143344056. The 2.54x dose spread of the alpha-1.5 sphere is now 1.02x, and the residual is the size the interpolation check predicted. This is the sphere the rest of the page is about.

One caveat on that 1.02x. The delivered KL is measured with sphere_dose.py's weight delta, built in bf16 from the 134 adapters; the text is generated with sphere_sweep.py's, built in fp32 from three fp16 basis products. The direction is the same to cosine 1.0 and both are normalised to the same Frobenius length before the same bf16 rounding of W0 + D*s, so the two should agree to well inside the one per cent the interpolation already costs - but they are different rounding paths and nothing here measures the difference at alpha > 0. The S900 byte-identity below shows the two pipelines agree at alpha 0, which is a weaker statement.

The run

zoo-sphere-isokl.service, one A100-40GB container, 73 points x 8 prompts, log header "73 points x 8 prompts at per-point alpha 0.0000..1.8037 over 73 points tag=isokl basis=['PC1', 'PC2', 'PC3']" (phase10_runs/sphere_isokl.log). 584 generations, none empty, every point's alpha equal to its spec entry.

The resume hazard was checked, not assumed. sphere_sweep.py's checkpoint is namespaced by the spec's tag, here /oct/sphere_isokl/results.json; the log contains zero [resume] lines and 0 of 72 points have generations identical to the 2026-09-01 run (analysis/sphere_isokl.json#run, verified in analyse_sphere_isokl.py's own check). That is the failure The sphere sweep on the factor chart recorded as costing nothing only because it was caught.

A base condition was generated in the same container, point S900 at alpha 0, which sets s = 0 and restores the pristine bf16 base exactly. It is judged interleaved with the steered text. Its judged profile is Extraversion 4.25, Agreeableness 4.875, Conscientiousness 5.125, Emotional Stability 4.625, Intellect 5.375 (#base_profile) - on eight prompts, not the 24 Suppression versus amplification at matched Fisher dose used, so it is not the same base measurement.

The base is the base. S900's eight responses are identical, all eight, character for character, to dose_BASE from phase10_runs/steer_results_dose.json on the same eight prompts (#base_vs_dose_base, mean shared prefix fraction 1.0). Those two texts were produced by different code paths in different containers a day apart - sphere_sweep.py editing weights in place, steer_fix.py via add_adapter - and greedy decoding at enable_thinking=False puts them on the same string. It is the cleanest determinism check this project has.

Judging was judge_personas.py unchanged, anthropic/claude-sonnet-4.5 through OpenRouter, --batch 6 --workers 8, with the repeat subsample raised from 0.05 to 0.15: "584 generations to judge from 73 traits", "112 judge calls (batch=6, 87 repeats)", "584 judged, 0 failed calls". Repeat reliability over 87 duplicated units: Extraversion r=0.856, Agreeableness r=0.936, Conscientiousness r=0.876, Emotional Stability r=0.832 (n=85), Intellect r=0.868 (phase10_runs/judge_sphere_isokl.log). The judge sees only (prompt, response) and the point names are coordinates.

The statistics are computed by the original sphere's own functions

qwen35/analyse_sphere_isokl.py imports profiles, coherence and smoothness from qwen35/build_sphere_page.py and its first act is to reproduce analysis/sphere_page.json#coherence and #smooth from the 2026-09-01 files; the run stops if they do not match. They match exactly (analysis/sphere_isokl.json#selfcheck), so the two spheres are counted the same way and not merely described the same way.

One arithmetic note. analysis/sphere_isokl.json carries the original sphere's rho twice, as 0.6511417860626274 from smoothness and 0.6511433864273928 from the permutation harness, and the iso-KL sphere's as 0.6522913409368338 and 0.6522896435584883. The two code paths enumerate the 2,556 pairs by the same order but build the distances with different float operations, and ordinal ranking breaks the resulting near-ties differently. The difference is in the seventh decimal; this page quotes smooth.rho, the published function (#verdict.rho_note).

Result 1: the smoothness is position, not dose

analysis/sphere_isokl.json#smooth, against #smooth_pc_alpha1_5:

Equalising the dose moves rho by +0.0011. Each interval contains the other point estimate comfortably.

The statistical control agrees with the experimental one. Partialling the dose out of the original alpha-1.5 field, with the pairwise covariate |KL_i - KL_j| (the pre-registered primary: the confound is that a pair whose two points differ in dose differs in judged profile for that reason), gives partial rho 0.6238447733137857, a drop of 0.02729861311360704 (#dose_covariate.abs_dose_diff). The covariate is not inert - it correlates 0.28956234877940806 with angular distance and 0.25444334647827827 with profile distance - it simply does not carry the relationship. With the secondary covariate KL_i + KL_j the partial rho is 0.6518399218472459, a drop of -0.0007 (#dose_covariate.dose_sum).

The pre-registered rule was: dose if rho_iso is below rho_orig - 0.15 and the partial is below rho_orig - 0.10; position if rho_iso is within 0.10 of rho_orig and the partial is within 0.05. Both position conditions hold and neither dose condition does (#verdict). The permutation null used for the partial permutes the judged profiles across points, which breaks the position-profile and the dose-profile links at once; it is an omnibus null, not a test of the partial term alone.

The near/far gap survives too, slightly wider in both terms.

Result 2: the two judged fields agree point for point

This comparison is exact in a way The sphere sweep on the factor chart's could not be. That page had to project one lattice into the other chart and match by signed cosine, with a median match of 31 degrees. Here it is the same lattice point in both runs, so the pairing is the identity. Pearson over the 72 points (#field_agreement):

Mean Euclidean distance between a point's two five-scale profiles 0.679615261605419; 61 of 72 points keep the same top-rated scale.

Emotional Stability is again the weakest column, as it was across the two subspaces (+0.5307 to -0.5142 there, depending on the cut). Here it is +0.63 and positive, which removes one of the two candidate explanations The sphere sweep on the factor chart left open: the negative correlation there is not a property of the scale itself but of that comparison's projection, because the same scale on the same lattice at a different dose agrees at +0.63. What remains true of the scale is that its range is the smallest of the five on both spheres, so judge noise is a larger share of it.

Result 3: coverage barely moves, and Emotional Stability gains a point

Counting which scale the judge rated highest at each point (#scales.top_counts, #scales_pc_alpha1_5.top_counts), with the factor sphere from The sphere sweep on the factor chart alongside:

Per-scale range across the 72 points (#scales): Extraversion 3.375 to 6.375, Agreeableness 3.25 to 6.375, Conscientiousness 2.75 to 6.25, Emotional Stability 3.125 to 5.375, Intellect 3.625 to 6.5. Against alpha 1.5 (#scales_pc_alpha1_5): 3.5 to 6.0, 2.875 to 6.5, 3.125 to 6.25, 3.75 to 5.625, 3.625 to 6.5.

Emotional Stability's range widens from 1.875 to 2.25 and it tops three points rather than two: S001 at 5.375, S009 at 4.875 and S022 at 4.75, against S001 at 5.142857142857143 and S022 at 4.875 on the alpha-1.5 sphere. The same two lattice points carry the scale on both runs and one more joins them, which is a change of one point out of 72 and should not be read as more than "the absence of an Emotional Stability region on these spheres is not an artefact of unequal dose". The best any point does on that scale is 5.375 against this run's base of 4.625, three quarters of a point; the single highest Emotional Stability score anywhere on the alpha-1.5 sphere, 5.625 at S012, belongs to a point the judge rated highest on Agreeableness. The lopsidedness is a fact about the sphere, not about the dosing.

Result 4: equalising the dose does not equalise degeneration

This is where iso-KL sampling changes the answer.

#coherence against #coherence_pc_alpha1_5, same detector (analyse_alien_steer.looping, a 10-word window repeated four times):

The base point loops on none of its eight prompts and averages 1755.625 characters (#coherence_base), so both spheres shorten the model's output.

Three correlations say why the count fell, all with permutation p over 10,000 permutations (#dose_covariate). Write a direction's steepness as the KL it delivered at alpha 1.5 - the quantity that decides its iso-KL alpha:

Read those carefully, because two of them are one fact. On the iso-KL sphere alpha is a monotone-decreasing function of steepness by construction, so the second and third rows have to mirror each other and cannot separate alpha from flatness. What the four rows together do license is narrower than it looks:

• Holding the KL fixed does not equalise degeneration. If looping were a function of how far the output distribution moves, the iso-KL sphere would have a flat loop field. It has the opposite - the strongest loop correlation on either sphere.
• Flat directions loop more, at fixed alpha and at fixed KL. On the alpha-1.5 sphere alpha is constant and steepness still predicts looping at -0.3735, though only at p 0.064, so flatness matters independently of how far the direction was pushed.
• Pushing a flat direction further makes it worse. Equalising the dose sent the flattest directions out to alpha 1.80, and the count of completely clean points fell by three.

That last figure is a net. Sixteen of the 72 points loop on both spheres, so eleven points began looping and eight stopped (analysis/sphere_isokl.json#loop_rates). The worst point is S028 on both, at 0.375 here against 0.25 at alpha 1.5, and its solved alpha is 1.4943 - almost exactly the alpha it already had, so its extra looping is not a dose effect at all.

This sharpens rather than contradicts Fisher norms of the steering directions's "does F predict degeneration? Not by itself, no", which measured F against loop rate on the factor sphere at Spearman -0.1015, p 0.3957. The sign is the same and the magnitude here is larger. Two things differ and both help: the covariate is a measured KL at the alpha steered rather than a curvature extrapolated from |alpha| <= 0.5, and on the iso-KL arm the dose term that was working against it is gone.

Result 5: on this sphere, dose does not predict how far the persona moves

Fisher norms of the steering directions's headline positive - F against the distance of each point's five-scale profile from the 72-point centroid, Spearman +0.3706669239179368 at p 0.0014499275036248187 over the factor sphere - does not reproduce here (#dose_covariate.dose_vs_centroid_distance_pc_alpha1_5):

On the principal-component sphere at common alpha, how steep a direction is says nothing about how far it moves the judged persona. On the iso-KL sphere, where every point delivers the same KL, steepness predicts displacement negatively: the steep directions, given the smaller alphas, end up closer to the centroid.

Two readings fit these two rows and this run cannot separate them.

Cancellation. At common alpha a steep direction carries more KL, which pushes its persona further; it also converts KL into judged movement less efficiently, which pulls it back. The two effects are of similar size on this lattice and the correlation lands on zero.

Alpha, not KL. On the iso-KL sphere alpha rises as steepness falls, so "steepness predicts displacement at -0.40" is also "alpha predicts displacement at about +0.40" - the same sign and roughly the same size as alpha against loop rate, +0.5091. Together with the alpha-1.5 sphere's null, that reads as judged displacement and breakage both scaling with how far the weights moved rather than with how far the output distribution moved. What this reading has to explain away is the factor sphere's +0.37 at constant alpha, which is exactly the disagreement recorded as S28.

The page does not pick between them. Separating them needs a sphere at two doses, which is the same gap Suppression versus amplification at matched Fisher dose left open. The factor sphere's +0.37 is the same quantity measured on a different lattice with a different (extrapolated) covariate; both numbers stand and they are recorded as a source contradiction, S28 in Source contradictions.

Either way, this is the strongest single reason not to state "Fisher norm predicts how far the judged persona moves" as a general fact about this weight space. It is a fact about the factor sphere.

What this run does and does not establish

• It establishes that the smoothness result of The sphere sweep — 72 directions nobody chose and The sphere sweep on the factor chart is not a dose artefact. Both the experimental control (rho 0.6523 at 1.02x dose spread) and the statistical control (partial rho 0.6238 on the original at 2.54x spread) say so, and they were pre-registered together with the thresholds that would have said the opposite.
• It does not establish anything about more than one dose. Like Suppression versus amplification at matched Fisher dose, this is a single matched point - the alpha-1.5 median - and there is no dose-response curve. A sphere at half this dose or twice it is unmeasured.
• The 8-prompt battery and one greedy sample per prompt are the same limits the original sphere had. Every judged mean here is over eight responses.
• The judge's repeat reliability on Emotional Stability is 0.832, the lowest of the five, and that scale is also the one whose range is smallest. Its numbers carry the least weight on this page.
• Positive sign only, as the original sphere used. Nothing here bears on the sign asymmetry Suppression versus amplification at matched Fisher dose examined.

Contradictions with existing pages

• Fisher norms of the steering directions and post-draft state that F predicts judged displacement over "72 sphere points". That holds on the factor sphere and not on the principal-component sphere; see Result 5 and S28 in Source contradictions. The sentence should be scoped to the sphere it was measured on.
• Fisher norms of the steering directions's 72 sphere_S* keys are the factor sphere, not this one. Verified before this run: cosine 1.0 against phase10_runs/sphere_spec_fa.json and median 0.0094 against sphere_spec.json. F for the 72 principal-component directions has never been measured, which is why the dose covariate on this page is sphere_isokl_calib.json's measured KL and not F. That is a better covariate for this purpose in any case: it is measured at the alpha that was steered rather than extrapolated from |alpha| <= 0.5, an extrapolation Suppression versus amplification at matched Fisher dose showed overstates KL by a median factor of 1.4751663093238792.
• Fisher norms of the steering directions concludes F does not predict degeneration. Result 4 finds a considerably stronger relationship on this sphere with a measured covariate, in the same direction. This does not contradict that page - different sphere, different covariate - but its conclusion should not be carried over to the principal-component sphere unqualified.

Cost and provenance

Three Modal apps on A100-40GBs and one API judge, all on 2026-09-10.

• zoo-sphere-isokl-calib.service, app pc-qwen35-phase10-sphere-isokl-calib, launched 11:27 UTC, output written 12:17:53 UTC. 432 (direction, alpha) measurements; 50.1 minutes inside the function. Log phase10_runs/sphere_isokl_calib.log.
• zoo-sphere-isokl.service, app pc-qwen35-phase10-sphere-isokl, 12:18:26 to 14:54:54 UTC, 2 hours 36.5 minutes, 584 generations. Log phase10_runs/sphere_isokl.log.
• zoo-sphere-isokl-verify.service, app pc-qwen35-phase10-sphere-isokl-verify, 12:18:26 to 12:38:50 UTC, 20.0 minutes, 72 measurements. Log phase10_runs/sphere_isokl_verify.log.
• zoo-judge-sphere-isokl.service, 112 calls to anthropic/claude-sonnet-4.5 through OpenRouter, about 90 seconds. Log phase10_runs/judge_sphere_isokl.log.

That is 0.867 + 2.608 + 0.340 = 3.815 GPU-hours, which the spend meter's rate of $2.10 per GPU-hour (qwen35/zoo40_meter.sh, RATE) values at $8.01. The meter's rate is GPU-only; these containers also reserved 52 to 64 GiB of memory and 2 to 8 CPU cores, which Modal bills separately and which Fisher norms of the steering directions estimates at roughly $1.2 per GPU-hour at this size, taking the Modal total to roughly $12 to $13. The box-wide meter cannot confirm it: four sycophancy-forecast containers were running throughout, so its movement from $2534.00 at 11:31 UTC to $2563.05 at 14:57 UTC (phase10_runs/zoo40_meter.log) is mostly other work.

The judging is OpenRouter, whose pot has no per-project attribution (infrastructure). 112 calls carrying about 235,000 input tokens and about 45,000 output tokens; at Sonnet-tier list rates of $3 and $15 per million that is about $1.4. Authorised cap for the whole experiment: $25.

Related: The sphere sweep — 72 directions nobody chose, The sphere sweep on the factor chart, Fisher norms of the steering directions, Suppression versus amplification at matched Fisher dose, Steering the base model along weight-space directions, Additivity of the Big Five steering axes, Source contradictions, Judged evaluations of the trait adapters, infrastructure, post-draft, Glossary.