Additivity of the Big Five steering axes

Ten matched-norm mixtures of the five named axes were steered and judged blind; the deviation from additivity is 0.53 of the predicted effect and about 2.5x the judge-noise floor, and opposing mixtures fail where reinforcing ones compose.

Additivity of the Big Five steering axes

Every geometry result in the project validates the chart at the 134 adapters or at the five axes built from them. Additivity asks whether the interior between those points behaves the way its coordinates claim.

qwen35/analyse_additivity.py states the test:

If steering along axis A moves the judged profile by dA, and axis B by dB, then a matched-norm combination of the two should move it by dA + dB -- but only if the map from weights to behaviour is linear over that region. The residual r = d(A+B) - (dA + dB) is the deviation from flatness, measured in behaviour rather than inferred from geometry.

Design

zoo-steermix.service (Description=Additivity test: matched-norm mixtures of the five named axes, inside the coherent band) ran steer_fix.py on qwen35/phase10_runs/steer_spec_mix.json: ten mixtures, five alphas each (-2, -1, 0, +1, +2), the same 24-prompt battery. Output phase10_runs/steer_results_mix.json, judged into phase10_runs/judged_mix.json.

The ten mixtures (steer_spec_mix.json#jobs[].name, ref 0.8102592902648793, n 100):

mix_Extr+Agre, mix_Extr+Cons, mix_Extr+Emot, mix_Extr+Inte, mix_Agre+Cons, mix_Agre+Emot, mix_Extr-Cons, mix_Agre-Inte, mix_E+A+I, mix_Cons+Emot.

Two controls keep it honest, in the script's words:

The mixtures are renormalised to unit coefficient norm, so they are not simply larger perturbations than the singles; and the comparison is restricted to |alpha| <= 2, because at |alpha| = 4 the model loops verbatim on most prompts and additivity of wreckage is not a finding. — qwen35/analyse_additivity.py

The null is measured rather than assumed: "It is estimated from the alpha = 0 rows, which are the same untouched model under every direction, so their spread is pure measurement error."

Judging: "1200 generations to judge from 10 traits ... 1200 judged, 0 failed calls"; 210 judge calls, 60 repeats; repeat reliability r=0.760 (Extraversion), 0.826 (Agreeableness), 0.914 (Conscientiousness), 0.856 (EmotionalStability), 0.809 (Intellect) (qwen35/phase10_runs/judge_mix.log).

Result

qwen35/analysis/additivity.json (40 rows: 10 mixtures x 4 non-zero alphas):

median_resid_over_effect  0.5282408112973758
median_resid_norm         0.776827975948343
noise_norm                0.31666612924087495
noise (per scale)         0.13931912433568944, 0.21098269247027546,
                          0.11470786693528087, 0.1078200342446346,
                          0.10756796902132218

So the residual from additivity is about 53% of the predicted effect and its median norm is roughly 2.5 times the judge-noise floor (0.7768 against 0.3167). The script's own reading criterion:

a ratio near 1 means the deviation from additivity is indistinguishable from judge noise and the chart is flat over its interior. Well above 1 means the interior is curved and coordinates do not compose. — qwen35/analyse_additivity.py

Verdict: the coordinates do not compose. The monitor page records this as one of six failed pre-registered tests:

4. Do the coordinates compose? verdict: fail Success criterion: steering along axis A + axis B at matched norm should move behaviour by the sum of what each moves alone, to within judge noise. Otherwise a coordinate vector does not predict behaviour anywhere except at the points we sampled. — qwen35/build_monitor_page.py

The sign asymmetry

The failure is not uniform. The monitor page:

The pattern is the right shape for curvature: reinforcing mixtures compose tolerably (0.13-0.15), opposing ones do not (1.17-1.46). A flat space would not care about the sign. — qwen35/build_monitor_page.py

Per-row resid_over_effect from analysis/additivity.json#rows:

(Values are rounded to three decimals here for width; the file carries full precision at analysis/additivity.json#rows[].resid_over_effect — for example the first cell is 0.521835326336742.)

The two subtractive mixtures are mix_Extr-Cons and mix_Agre-Inte. Only mix_Extr-Cons shows the large residuals the monitor page describes; the monitor page's stated ranges (0.13-0.15 reinforcing, 1.17-1.46 opposing) are the extremes of the two groups rather than their means. The smallest residual in the table, 0.132, is mix_Agre+Emot at +2, and 0.150 is mix_Cons+Emot at -2.

What it does and does not establish

It establishes that a coordinate vector in the five-axis chart does not predict judged behaviour at an interior point to within measurement error, at least at |alpha| <= 2 on this battery with this judge. It does not establish the form of the curvature; the sign asymmetry is suggestive of it but rests on one clearly-failing mixture out of two subtractive ones.

It also does not contradict The sphere sweep — 72 directions nobody chose, which finds that nearby directions give nearby personalities (rho = 0.65). Local continuity and global additivity are different claims.

Related: Steering the base model along weight-space directions, The sphere sweep — 72 directions nobody chose, Weight-space geometry of the 134-adapter zoo, Glossary.