The loss kernel hears a character only in the model's own voice

Character training pipelines go out of their way to use on-policy data. Open Character Training (OCT) generates its introspective SFT data from the very checkpoint being trained, citing subliminal learning — where behavioural traits transfer through same-model-generated data but not across different base models. If on-policy character data really is special, the specialness should be visible in the loss landscape: responses expressing the same character should share a groove — their losses should co-fluctuate under weight perturbation — more when the kernel model itself wrote them. That is a loss kernel / BIF question, continuing this thread's arc from MNIST digits and CIFAR categories to LLM alignment data (where the kernel pattern saw topic, not valence).

Setup: a 2×4 grid of chat datasets. Two generators — gemma-3-4b-it (the kernel model itself: on-policy) and gpt-5.4-mini (off-policy) — each answer the same 60 probe questions under four OCT character system prompts: none, funny, loving, poetic. The kernel is computed once, on gemma-3-4b-it, with no system prompt anywhere and the per-response loss masked to assistant tokens. If character grooves exist, the four gemma cells should separate; the hypothesis says they separate better than the four gpt cells. It took two runs to answer, and the first run's failure is half the value of the piece.

With variable-length responses the kernel's top axis is response length (ρ = 0.81) — a pure signal-to-noise artifact that reproduces at gate 0.91. Fix the readout to exactly 64 tokens per response and a character signal appears — but only in the on-policy data (separation +0.018, CI [+0.009, +0.022] vs −0.005 off-policy), and per unit of judged character expression, on-policy responses get ~2.5× the kernel separation.

§1Eight datasets, one kernel

The characters come verbatim from OCT's pipeline: each is a hand-written constitution of ten first-person trait assertions ("I strive to approach conversations with creativity and wit…"), wrapped in OCT's distillation system prompt. OCT trains gemma-3-4b-it itself with these constitutions, so this model demonstrably absorbs them; here they stay in the prompt (the kernel never sees them — probe documents are plain user → assistant transcripts). Questions are 60 shared prompts sampled from no_robots (Open QA / Brainstorm / Generation, 20 each; OCT's own generic prompt source, LIMA, is licence-gated) — every cell answers the same questions, so topic is controlled by construction. Gemma samples at its shipped decoding config; gpt-5.4-mini at API defaults with reasoning off. The estimator is this thread's standard: localised SGLD around the trained weights, block-centred covariance of per-response losses over the retained draws, correlation-normalised, two seeds gated on agreement, embeddings frozen (the Timaeus fix) along with the unused vision tower.

§2Run 1: a reproducible artifact — the kernel measures length

The first run scored each response on however many tokens it had (up to 300). The estimator gated beautifully (0.91 between independent seeds; split-half 0.95) and clustered nothing: character NMI 0.0 in both policies. What it found instead was length. The per-token readout of a long response is a mean over more tokens, so its idiosyncratic fluctuation is smaller (ρ(fluctuation, length) = −0.70, left panel) — and once every probe's fluctuation is dominated by the chain's shared component in proportion to its length, the correlation matrix grades by length. The top eigenvector tracks token count at ρ = 0.81 (right panel). Nothing about content — a pure length-SNR artifact, and a reproducible one: gate 0.91 means both seeds measured the same artifact with high fidelity.

Two things worth keeping from the wreck. First, the mechanism is general: any loss-kernel study with heterogeneous-length probe documents will put length into the kernel through SNR — the earlier pieces in this thread never saw it because their probe docs all filled a fixed window. Second, even inside the artifact the hypothesis left a fingerprint: the raw on-policy character separation was small but positive (+0.011, question-bootstrap CI [+0.005, +0.013]) while off-policy was negative, and projecting out the top one-to-four length/heat factors always left a faint gemma-only character signal (separation +0.021…+0.027, NMI up to 0.09) with gpt at zero. Post-hoc surgery, though — so: run 2, with the artifact removed at the data level.

§3Run 2: fix the readout, and on-policy character appears

Run 2 scores every probe on exactly its first 64 assistant tokens, keeping a question only if all eight cells' responses reach 64 tokens (47 of 60 survive → 376 probes). Uniform information budget per probe: length cannot enter, by construction (the leak statistic is undefined — token count is constant). Same sampler, same calibrated lr. The openings of responses are also where these characters are loudest — gemma-poetic greets a gift-ideas question with "The edges of a birthday blur, a gentle shifting of time, / And with it, the question blooms — a tender, hopeful rhyme" where gemma-none says "Okay, let's brainstorm some birthday gift ideas for your mum!".

Three readings of that figure, in decreasing order of confidence. (1) Off-policy character structure is absent: NMI ≈ 0 whether you cluster the kernel or the text, and separation is slightly negative. (2) On-policy character structure exists but is modest: NMI 0.068, separation +0.018 with a CI clear of zero. (3) The kernel sees roughly double what surface text similarity sees, on-policy only (0.068 vs 0.037; both nil off-policy) — the grooves carry something beyond word choice. For calibration: in both policies the same kernel clusters question topic better than character (topic NMI 0.16 on-policy, 0.09 off-policy) — consistent with the alignment piece, the loss kernel's first love is topic. Notice topic is also sharper on-policy: everything is, in the model's own voice.

§4Is that just because gpt-5.4-mini ignores the persona?

The obvious confound: maybe off-policy data has no character grooves because it has no character. gpt-5.4-mini is a heavily assistant-trained model and often shrugs the persona off — especially on factual questions, where its "loving" answers are bullet-point godwit facts. So every one of the 480 probe responses was scored blind by a gpt-5.4-mini judge on humor / loving-warmth / poeticism (0–3):

So yes — gemma expresses the prompted character roughly twice as strongly, and gpt-loving barely exists in the text (+0.15). The confound is real, which is exactly why it has to be controlled rather than argued away. Plotting each character cell's kernel separation from its own generator's none cell against its judged expression contrast puts both policies on the same axis:

§5The model's own voice is the deepest groove

The single cleanest number in the experiment is also the simplest: mean within-cell correlation. Every gemma cell coheres more than every gpt cell — no overlap — and the most coherent cell of all is gemma·none: the model's own unprompted assistant voice. The loss landscape's tightest groove around gemma-3-4b-it is being gemma-3-4b-it.

This rhymes with three external results. Models recognise and prefer their own generations; subliminal learning transfers traits through same-model data only; and OCT chose on-policy introspective data on purpose. The kernel gives all three the same geometric reading: off-policy text — however well it expresses a character — sits in shallower, less differentiated grooves of the student's loss landscape, so there is simply less structure there for training to grab. The 8×8 block-mean matrix makes the asymmetry explicit: the on-policy quadrant has visible character texture, the off-policy quadrant is flat, and the cross-quadrant blocks are organised by question overlap rather than by matching characters.

§6Every probe, placed

The embedding below places each of the 376 probes by eigenvectors 2–3 of the kernel correlation. Colour is character, fill is generator; the tabs re-embed using only that policy's sub-kernel (Procrustes-aligned to the full frame, out-of-view points ghosted). Hover any point to read the response it stands for. In the on-policy view the grey none core and a poetic wing are visible; in the off-policy view colours mix.

§7Reading it honestly

The claim this experiment supports: prompted character structure is visible in a model's loss kernel for its own generations and absent for a stylistically-matched off-policy generator's, and the gap survives controlling for how strongly the text expresses the character. The claim it does not support: that the kernel cleanly clusters characters — NMI 0.068 is faint structure, found only because the design controls topic (same questions everywhere) and length (fixed 64-token readout). Run 2's gate is 0.47 — the two seeds agree on the coarse structure (their NMIs are 0.075 and 0.063) but this is a much noisier kernel than run 1's artifact (0.91) or the MNIST grooves. Other caveats, plainly: one kernel model at one scale, characters prompted rather than trained-in (the OCT-trained gemma personas would be the natural follow-up), the judge is gpt-5.4-mini scoring its own family's outputs, the question-bootstrap CI has a composition bias (visible in Fig. 2), characters bleed in expression (gemma-funny scores 1.20 on poetic), and the off-policy generator is a single model — "off-policy" and "gpt-5.4-mini-flavoured" are confounded with n=1 generators per condition.

The methods lesson is the more durable export: loss-kernel correlations on heterogeneous-length text measure length before they measure anything else — the per-token readout's noise scales as 1/√length, the chain's shared drift then dominates long documents' fluctuations, and the result reproduces seed-to-seed like real structure (gate 0.91!). Reproducibility gates catch estimator noise, not design artifacts. Fix the information budget per probe, or the kernel will happily hand you a ruler.