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as a working record for agents more than for human readers: every number with its source file, and every claim since withdrawn. Some facts here may be out of date; where it disagrees with the post, trust the post.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

Under the activation-weighted metric a stage-one and a stage-two adapter for the same trait reach cosine +0.0090 where the Frobenius one reads +0.00017, but the frame-overlap ceiling for those two frames is 0.7855, so the cross-stage agreement is 0.92 per cent of what a perfectly reproduced update would score against 74.6 per cent for the same trait retrained at a second seed - eighty times lower; the two stages' grand means are functionally near-orthogonal too, at +0.0167; 56 of 134 stage-two adapters still find their own stage-one adapter and the arrangement correlates at 0.40, so the verdict by the preregistered thresholds is a different function with a matching arrangement.

The question

Open Character Training has two stages, and the zoo ran both for all 134 traits: stage one is DPO on the constitution's preference pairs, stage two is SFT on transcripts the trained model generates about itself (Stage two - introspection SFT and the 0.25 persona merge). They are two separate LoRAs with two separate random LoRA-A draws - stage one at seed 0, stage two at sft_seed 123456, one draw shared inside each stage (Stage-two geometry). So their Frobenius cross-Gram is zero by construction: same-trait cosine +0.00016589283708173512, different-trait +1.784355184601498e-05 (qwen35/analysis/stage2_structure.json#stage1_x_stage2_exact).

The activation-weighted Gram, and what the seed floor looks like in it showed why a cross-frame comparison reads as zero in the Frobenius metric and what happens when the metric is changed: the input covariance C of a module has a participation ratio of about ten, so two independent rank-64 frames overlap at 0.89 in C where they overlap at 0.021 in Frobenius, and the same trait retrained at a second seed goes from +0.0181 to +0.6643. Under C the two stages can be compared. This page asks whether the stage-two update is the same function as stage one's for the same trait, or a different function whose arrangement across the 134 traits happens to match - which is what the centred-cosine correlation of 0.81 on Structure of the stage-two adapter space already established. It was preregistered in qwen35/PREREG_actgram_stage2.md, with one amendment made after the smoke run and before the full run returned (below).

The answer is the second. The cross-stage cosine does rise under C, by a factor of 54, from +0.000166 to +0.0089804086688677. But the frame-overlap ceiling for these two particular frames is 0.78549746319657, so the agreement is 1.14 per cent of what a perfectly reproduced update would score, against 74.56 per cent for the same trait retrained at a second seed measured on the same run in the same metric. In relative terms the stage boundary is about sixty-five times harder to cross than the seed boundary, and the change of metric does not move that: under Frobenius the two ratios are 0.77 per cent and 84.12 per cent.

What was run

qwen35/act_gram_stages_on_modal.py, one A100-80GB, harvesting C exactly as The activation-weighted Gram, and what the seed floor looks like in it did (base Qwen/Qwen3.5-4B, no adapter, bf16 forward, fp32 accumulation, TF32 off, uncentred C for all 248 modules) on the same two token arms - pool445, the 445-prompt shared training pool with prompt tokens only, 20,612 tokens, primary; and resp4378, the 24 steering prompts with the base model's stored greedy responses, 5,106 tokens - plus a mean-centred twin of the primary arm. The participation ratios came out identical to the earlier run to two decimals (21.50, 43.83, 72.99), which is the first sign the harvest is deterministic.

Then one exact Gram over four adapter sets, 323 adapters, all at lora_alpha/r = 2.0 and 248 modules (analysis/act_gram_stage2.json#sets):

Everything else is a linear function of that Gram and is computed locally and exactly by analyse_act_gram_stage2.py. Removing stage two's shared direction needs no re-run: with M2 the stage-two grand mean, <u, v_k - M2> = <u, v_k> - mean_l <u, v_l> and ||v_k - M2||^2 = G22_kk - 2 mean_l G22_kl + mean_{l,m} G22_lm.

v_k - M2 stays inside the span of the stage-two frames, so the frame-overlap ceiling is unchanged by the removal to first order and the same ceiling is used for the raw and the residual blocks.

Validation

The Frobenius arm reproduces stage2_structure.json#stage1_x_stage2_exact, which was derived by a completely different route - that file never touches a stage-two adapter directly, it recovers the cross block from the two-volume stage-one-by-persona Gram as X12 = (X[s1, persona] - G1) / 0.25. This run reads both adapters off their volumes and contracts them. The two agree (analysis/act_gram_stage2.json#validation):

The preregistered bar was 1e-5. The stage-two within-stage Gram also reproduces results/cross_gram_full_loras_introspection_x_loras_introspection.npz to a max absolute cosine difference of 5.182098067324503e-08.

One labelling defect was found by that check and is filed, not fixed here. The reference's key is named top1_stage1_finds_own_stage2 and Structure of the stage-two adapter space renders it as "a stage-one adapter's nearest stage-two adapter is its own trait for 42 of 134", but analyse_stage2_structure.py:116 computes ranks = [1 + sum(C12[:, j] > C12[j, j])], which ranks each stage-two adapter among the 134 stage-one ones. Compared like with like this run gives exactly 42 and mean rank 9.537313432835822. The other direction - the one the name and the sentence describe - reads 59 of 134, mean rank 7.514925373134329 on the identical matrix. See S29.

Frame spread. Mean over modules of each adapter's relative deviation of A from its own set's mean A (#frame_spread): stage one 0.010178045947067138 seed 0 and 0.01010937949890391 seed 1; stage two 0.07506962422461759 seed 0 and 0.07315878054848121 seed 1. Stage two's frame is about seven times less shared than stage one's, which is the same direction as PHASE3_VERDICT.md's "stage-2 A drifts 10% from init (stage 1: 1.5%)" - though it is not that statistic, because this one measures deviation from the set mean and not from the initialisation. The Gram itself uses each adapter's own A and is exact regardless.

The numbers

analysis/act_gram_stage2.json#arms.<arm>. The 134 traits are those present in both stages.

After stage two's shared direction is removed, which holds 0.1514038882692439 of the average squared norm in the Frobenius metric and 0.2708387694659428 in C (#arms.pool445.grand_mean_share_of_mean_norm2):

Removing the shared direction makes identification much better in both metrics - 76 of 134 against 42 under Frobenius, 56 against 21 under C - and raises the arrangement correlation from 0.2175 to 0.3621 and from 0.2896 to 0.4029. The shared register is noise for the cross-stage question, exactly as the preregistration assumed. Removing both stages' shared directions gives R 0.009754181133485019, 54 of 134 and correlation 0.426311842398635 (#arms.pool445.cross_stage_both_shared_removed).

The benchmarks, measured on the same run in the same metric

This is the comparison the verdict rests on (#arms.pool445.second_seed_checks, #arms.frob.second_seed_checks):

The R column at full precision, from #arms.pool445.second_seed_checks.*.ratio_to_ceiling and #arms.frob.second_seed_checks.*.ratio_to_ceiling: cross-stage seed 0 0.01143276597268954 (Frobenius 0.007743045342250773), cross-stage seed 1 0.010126181709316171 (Frobenius 0.007644768084004354), stage one cross-seed 0.7455668811085819 (Frobenius 0.8411666943129359), stage two cross-seed 0.6869136234992818 (Frobenius 3.0196348702424674).

Three things to read off it.

1. The second seed reproduces the stage result. The 15 traits that have both a seed-1 stage-one adapter and a seed-1 stage-two adapter give R = 0.0101 against the seed-0 set's 0.0114. The stage boundary is not an artefact of one pair of frames.
2. Stage two re-learns itself almost as well as stage one does. At its own second seed, stage two reaches 0.6869 of its ceiling where stage one reaches 0.7456. So a stage-two adapter is perfectly reproducible under a change of initialisation; it simply is not stage one.
3. The Frobenius stage-two cross-seed ratio is 3.02, which exceeds 1 and is therefore not a ratio to a ceiling at all. That is the same anomaly Stage two at a second seed records as "the attenuation slope is 0.116 - 4.6x the r/d prediction, unexplained": the large shared register lifts the stage-two cross-seed cosine above what frame overlap alone allows. Under C the anomaly disappears (0.6869, comfortably below 1), because in C the shared register is no longer cheap - the ceiling itself is 0.905. Recorded here as an observation; nothing on this page depends on it.

The two stages' grand means

Frobenius +0.0001676146544225693; under C +0.01667321809542123 against a ceiling of 0.7855, which is 2.1 per cent of it. The two shared directions are functionally near-orthogonal as well as coordinate-orthogonal. Stage two's own shared component is much larger in the functional metric than in the Frobenius one: every stage-two adapter sits at cosine 0.5218264341353712 to the stage-two grand mean under C against 0.38929787448084396 under Frobenius, and the mean within-stage-two off-diagonal cosine is 0.26694991920809574 under C against 0.14519684032691033 (#arms.*.cos_to_own_grand_mean, #arms.*.within_stage_offdiag_mean_cosine). The introspection register is a bigger share of what stage two does to the model than of what it does to the weights.

The arrangement

The centred-cosine correlation between the two stages' within-stage blocks - the statistic behind the r = 0.81 on Structure of the stage-two adapter space - is 0.8137665247986187 under C against 0.8117877815269501 under Frobenius (#arms.*.centred_cosines.corr_stage1_stage2). The metric changes it by 0.002. The arrangement result was never a metric artefact, and the functional metric does not improve it either.

The verdict

PREREG_actgram_stage2.md set two bands on R_res, the ratio of the shared-direction-removed cross-stage cosine to the frame-overlap ceiling, with the benchmarks being the stage-one cross-seed ratio (0.7455668811085819, "the same function re-learned in a different frame") and the cross-seed different-trait ratio (0.0800, "unrelated"):

• "the same function as stage one" if R_res >= 0.50 and the discriminability condition held;
• "a different function with a matching arrangement" if R_res <= 0.15, with more than 13 of 134 identified and the arrangement correlation above 0.20.

Measured: R_res = 0.009222883147729255, 56 of 134 identified (mean rank 7.86 against a chance 67.5), arrangement correlation 0.40294962029699516. All three conditions of the second band hold, and R_res is below the "unrelated" benchmark of 0.0800 by a factor of nine while identification is fifty-six times chance. The verdict is "a different function with a matching arrangement" (#verdict), and it is not a close call: R_res would have to rise by a factor of 54 to reach the lower band's edge.

One preregistered condition was amended, after the smoke run and before the full run returned. The prereg's "same trait exceeds different trait by a factor of 3" is ill-defined in the residual block, because removing stage two's grand mean centres every row of the cross block exactly, forcing the different-trait mean to about -same/(n-1); the smoke returned same_over_diff = -14.39, which is arithmetic, not a finding. It was replaced by same-trait mean / sd(different trait) >= 3, reported as same_over_diff_sd and reading 2.19 under C. That condition belongs to the branch the verdict did not take, so it changes nothing here; the amendment is on the record in PREREG_actgram_stage2.md with the time it was written.

What this establishes, and what it does not

Established. (1) The stage-one and stage-two updates for one trait are not the same function, in the only metric in which the question can be asked at all: 1.14 per cent of the frame-overlap ceiling raw and 0.92 per cent with the shared register removed, against 74.6 per cent for the same trait re-learned at a second seed. (2) That is not a property of the metric - the Frobenius ratios are 0.77 and 0.74 per cent, and the ordering of the two boundaries is the same in both. (3) It is not a property of one pair of frames - the seed-1 arm reproduces it at R = 0.0101 over 15 traits. (4) What does survive the stage boundary is the arrangement and a weak trait-specific residual: 56 of 134 identified against a chance of about one, arrangement correlation 0.40 to the stage-one within-block, and the centred-cosine correlation of 0.81 unchanged by the metric. (5) The two stages' shared directions are near-orthogonal functionally as well as in coordinates.

Together with The activation-weighted Gram, and what the seed floor looks like in it this is the sharper statement: the metric was carrying the seed result and is not carrying the stage result. Changing to the functional metric multiplies the cross-seed cosine by 36.8 and leaves its ratio to the ceiling where it was; it multiplies the cross-stage cosine by 54 and leaves that ratio where it was too - two orders of magnitude lower.

Not established. (1) The dichotomy the thresholds encode is not exhaustive. Stage two's gradient was taken at merged(base + stage1 <trait>), so stage one's direction is already installed when stage two starts. A stage two that was "continuing the same function from where stage one stopped" would also score near zero against stage one's delta. R_res near zero rules out re-learning stage one's update; it does not separate "a different function" from "the remainder of the same one". (2) C is the base model's, harvested with no adapter loaded on the 445-prompt pool, while stage two was trained on the merged model and on self-generated introspective transcripts. It is a first-order approximation on the stage-two side, and every cross-stage number here inherits that; a per-trait C would be 134 different covariances and would make the Gram trait-dependent, which the design cannot allow. The resp4378 arm changes the text but not the model, and moves R from 0.0114 to 0.0081. The seed-1 check is an independent replication of the boundary - those stage-two adapters were trained on the seed-1 merged bases - but it is not a check on this approximation, which it inherits unchanged, because it uses the same base-model C. (3) Nothing here is behavioural. (4) The resp4378 arm's C is estimated from 5,106 positions, thin for the 9216-wide down_proj inputs.

Cost and provenance

zoo-actgram-stages.service, app pc-qwen35-phase11-actgram-stages, one A100-80GB, launched 2026-09-11 12:12:14 UTC, 704.2 seconds inside the Modal function (analysis/act_gram_stage2.json#wall_seconds), log phase10_runs/actgram_stages.log. One smoke run preceded it (zoo-actgram-stages-smoke.service, 54.7 s, 36 modules covering all three input widths, 12 adapters per set), which is what caught the degenerate threshold before the full run was paid for. The per-run guard was the Modal function timeout, 90 minutes, because zoo40_meter.sh is a cumulative workspace budget and not a per-run cap (costs). The meter read $2563.93 before this run and $2564.45 after, a draw of $0.52 over 0.25 counted GPU-hours at its A100-40GB rate of $2.10 per GPU-hour; the true GPU time was about 0.22 hours across two containers, which at Modal's A100-80GB rate is about $0.55. The authorised cap for the task was $10. The analysis is local CPU and free.

Related: The activation-weighted Gram, and what the seed floor looks like in it, Structure of the stage-two adapter space, Stage two, explored, Stage-two geometry, Stage two at a second seed, Stage two - introspection SFT and the 0.25 persona merge, The seed floor, Column-space structure, Full OCT persona replication, Source contradictions, costs, Glossary.