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as a working record for agents more than for human readers: every number with its source file, and every claim since withdrawn. Some facts here may be out of date; where it disagrees with the post, trust the post.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

Per-direction seed stability, and why the check is uninformative

Every named direction's trait coordinates correlate about 0.998 between the two LoRA seeds, but so does a random merge of the same adapters at 0.979, because the cross-seed Gram is a scaled copy of the within-seed Gram; the check reproduces the matrix-level result and says nothing about the factors.

The question

Cross-seed geometry establishes that the arrangement of the adapters replicates across LoRA initialisations: cross-seed cosine tracks within-run cosine at Pearson 0.9966 over the 5,320 pairs of 134 seed-0 adapters against 40 seed-1 adapters (qwen35/analysis/crossseed_arms.json#[1]). That is a statement about the whole Gram. It was never turned into a statement about the named directions - the five factor-analytic factors, PC1-PC6, the five Big Five keying axes and the grand mean - which are the objects the steering work actually uses. qwen35/analyse_direction_seed_stability.py was written on 2026-09-08 to ask the per-direction version of the question, and the answer is that the question cannot be asked this way.

What was computed

Each named direction is a weighted merge v = sum_i c_i a_i of the 134 seed-0 stage-one adapters, with coefficients taken from the steering specs (qwen35/phase10_runs/steer_spec.json and steer_spec2_7a.json, key jobs[].coef). The cosine of any adapter b with v needs only Gram entries: <b, v> / (|b| |v|) = sum_i c_i X[i, b] / (|b| sqrt(c' G c)). Seed-0 coordinates come from results/gram_sweep.npz; seed-1 coordinates come from the exact 134 x 40 cross-Gram results/cross_gram_full_root_x_data_null_seedpaired_s40_matched.npz, the matched-objective seed-1 arm (The seed floor, The null control arms).

Reported per direction over the 40 traits that exist at both seeds (analysis/direction_seed_stability.json#n_seed1_traits = 40): the Pearson correlation between the seed-0 and seed-1 coordinate vectors, a Spearman version, a permutation p, the regression slope of seed-1 on seed-0, and the two standard deviations.

The result

analysis/direction_seed_stability.json#directions.<name>:

(Values rounded from the JSON; the full precision is in the file, e.g. #directions.FA_Warmth.pearson = 0.9979502995224957. Every direction's permutation p is at the floor, #directions.*.perm_p = 0.0001999600079984003, 5,000 permutations.)

Read alone this says every named direction replicates almost perfectly. It does not say that.

Why it is uninformative

The same script draws 200 random directions of matched construction - Gaussian coefficients over the same 134 seed-0 adapters - and scores them the same way. #random_direction_pearson:

mean 0.9791758621274371, sd 0.017628602815322764, p95 0.9946732339575204, n 200, note "random Gaussian coefficients over the 134 seed-0 adapters; the shared LoRA-A makes even random merges partly reproducible"

A random merge of the zoo scores 0.979. Two of the seventeen named directions - PC4 at 0.9934 and mean_assistant_axis at 0.9908 - sit below the random directions' 95th percentile of 0.9947. Nothing here separates a factor from noise.

The reason is visible in the slope column. Every direction's seed-1 coordinates are the seed-0 coordinates multiplied by about 0.026 (the per-direction slopes run from 0.0225 to 0.0271), and 0.026 is the measured attenuation slope of the whole cross-seed Gram (analysis/crossseed_arms.json#[1] slope 0.026542111376493285), itself close to the r/d = 0.025 overlap of two random rank-64 subspaces on The seed floor. The cross-seed Gram block is, to a very good approximation, a scalar multiple of the within-seed block, so any coefficient vector c gives seed-1 coordinates proportional to its seed-0 coordinates and therefore a correlation near 1. The per-direction test is the matrix-level Pearson 0.9966 re-expressed once per direction; it adds no information beyond it.

The 0.998 must not be quoted as factor stability. It is a property of the Gram, not of the factor solution, and it would look the same if the factor analysis had returned five arbitrary directions.

What a real test would look like

The question worth asking is whether factor analysis run independently at the second seed recovers the same factors. That means factoring the seed-1 adapters among themselves - their own 40 x 40 Gram - and computing Tucker congruence between the resulting loading matrix and the seed-0 loadings for the same 40 traits, exactly as Structure of the stage-two adapter space compares stage one with stage two (qwen35/analysis/stage2_structure.json#factor_congruence_stage1_vs_stage2.best_matching, written by analyse_stage2_structure.py). That comparison is invariant to the overall scale, which is what kills the test above.

Is it feasible with 40 variables? The 40 seed-1 traits are balanced - 8 Goldberg markers for each of the five factors (from traits_primary.json joined to the npz names) - so five factors over 40 variables is a legitimate if small design: about 8 variables per factor, which is at the low end of what is usually recommended, and each Tucker congruence would be computed over 40 entries rather than 134, so the congruences would be noisy and a difference of a few hundredths would mean nothing. The 40-trait arm's own decomposition already replicates the labelled tests at ratio 1.01 and 0.96 (Cross-seed geometry), so the design is not obviously too small to see the effect.

What stops it today is which Gram exists. results/gram_data_null_seedpaired_s40.npz is a 40 x 40 within-arm Gram, but it belongs to the original-objective seed-paired arm (plain sigmoid DPO, the mismatch described on The null control arms); it is the file behind results/decomposition_seed1.json. The matched-objective seed-1 adapters, which is what the cross-Gram above uses, have no within-arm Gram on disk - only cross-Grams (results/cross_gram_full_root_x_data_null_seedpaired_s40_matched.npz and cross_gram_full_data_null_seedpaired_s40_x_data_null_seedpaired_s40_matched.npz). So the test could be run today on the original-objective arm on this box with no GPU, carrying the objective caveat that the matched retrain elsewhere changed nothing to three decimal places, or it could be run cleanly after computing one more 40 x 40 Gram on Modal. Neither has been done; it is recorded on Open questions and on What has been done to the PCs but not to the factors as open.

A second caveat for whoever runs it: analyse_fa_qwen35.py currently asserts that the input Gram has 100 or 134 traits and that exactly 100 of them are Goldberg markers, so a 40-trait input needs that assertion widened first.

Related: Cross-seed geometry, The seed floor, Factor analysis of the adapter Gram, What has been done to the PCs but not to the factors, Factor analysis of the null arms, Steering the base model along weight-space directions.