Treat steering as an intervention: we want maximum behavior change with minimum side effects, like incoherence, format collapse, or random off-target damage.

A useful analogy is a car on the road. A small nudge to the steering wheel gets corrected by the driver. A larger nudge causes a lane change. A very large nudge causes a crash the driver cannot recover from. Some crashes happen immediately. Some take a few seconds to develop, after the driver tries and fails to correct.

We measure the distribution shift caused by steering, especially the worst 5% of per-token KL that would cause a "crash", and find the largest scalar coefficient C that keeps that 5% below a safe threshold (default: 1 nat). At alpha = 1 the steering is delivering exactly that calibrated dose. At alpha = 2 it is doing twice the dose. Iso-KL means: comparing two methods at the same alpha = 1 means they are spending the same per-token KL budget, so any behavioural difference is not just one of them being louder.

Then comes the survival question. If the calibration only looked at the first w tokens of the rollout, do crashes still happen later? Most trajectories either stabilize or go off track within the first 50-100 tokens, but we want to see the long tail too. So at every fork t we ask: can the model still produce one of two valid answer tokens (true or false) when forced into a JSON-schema prefill? If yes, alive at t. If the probability mass on those tokens drops below 0.95, dead, and dead stays dead. Survival S(t) is the fraction still alive at token t, with right-censoring for rollouts that hit EOS before t.

Headline result on Qwen3.5-0.8B (mean_diff, w=512, n=8 held-out prompts):

alpha in {0, 0.25, 0.5, 0.75, 1.0, 1.5}: 0 of 8 trajectories die at any fork.
alpha = 2.0: 8 of 8 die. Median death time t = 64.
alpha = 4.0: 8 of 8 die at t = 0.

A phase transition at roughly 2x the iso-KL coefficient. The calibrated dose has long-horizon headroom; doubling the dose is a slow crash; quadrupling it is an instant one. Figures and table below.

Spaghetti: per-token KL trajectories, coloured by survival

Per-token KL(steered || base) for n=8 held-out long-form prompts on Qwen3.5-0.8B (mean_diff, w=512). One panel per alpha. Each thin line is one trajectory; green = alive at that token (pmass_eval >= 0.95), red = dead. Black line is the median. Dotted horizontal: KL = 1 nat (the calibration target).

How to read this plot, in order:

alpha = 0. KL is exactly zero everywhere. This is the base model; nothing has been added. A sanity check that the splicing pipeline does not itself perturb the logits.
alpha = 0.25 to 1.0. KL stays well below the 1-nat line for most tokens. The calibration window only required p95 = 1 nat, so most tokens are far below. All trajectories are green: the model is still producing schema-valid answers throughout. The "warm but not hot" zone.
alpha = 1.5. KL still mostly below 1 nat but with a fatter upper envelope. All trajectories still alive end-to-end on the held-out set, even though calibration only guaranteed survival up to roughly alpha = 1. Suggests headroom.
alpha = 2.0. KL median sits around 1 nat for most of the rollout, with a noisy plateau. The trajectories turn red part-way through. Steering is now strong enough to break format. The phase transition.
alpha = 4.0. Every trajectory is red from t = 0 and KL is roughly 4-6 nats throughout. The model is no longer answering the question; it is generating something else.

The diagnostic shape is the alpha = 2 panel: KL is only modestly above the calibration target (less than 2x in nats) but format collapse is total. KL is necessary but not sufficient as a steering budget. Going from "calibrated" to "calibrated x 2" is not a smooth degradation; it crosses a cliff.

Survival: when do trajectories die?

Kaplan-Meier S(t) on the pmass_eval death event (pmass_eval < 0.95). One curve per alpha, n = 8 trajectories each. Dotted vertical: t = 20, the upper end of the calibration window's relevance for the answer span; everything to the right is generalization.

alpha	n	died	S(mid)	S(end)	t at S<=0.5
0.00	8	0	1.000	1.000	--
0.25	8	0	1.000	1.000	--
0.50	8	0	1.000	1.000	--
0.75	8	0	1.000	1.000	--
1.00	8	0	1.000	1.000	--
1.50	8	0	1.000	1.000	--
2.00	8	8	0.586	0.321	64
4.00	8	8	0.000	0.000	0

How to read this table, in order:

Censored is always 0 here, so survival numbers are not biased by short rollouts. Every trajectory reaches every fork t in {0, 5, ..., w}.
alpha 0 to 1.5: died = 0, S = 1.0 everywhere. Format holds for the full rollout. The held-out set generalizes past the calibration window (t = 20 dotted line) all the way to t = 512. The 1-nat KL budget set on calibration prompts continues to be a survivable budget on a different prompt set, more than an order of magnitude past where calibration looked.
alpha = 2.0: died = 8 of 8 within the rollout; median death at t = 64. All trajectories eventually break, but they survive the first 64 tokens, about 3x the calibration window. Format collapse is gradual at this dose.
alpha = 4.0: median death at t = 0. The first fork already shows pmass_eval < 0.95. Format collapse is immediate.
Read t at S<=0.5 as a half-life. It scales nonlinearly with alpha: 64 tokens at 2x, 0 tokens at 4x. Doubling the dose past the calibration point does not double the half-life; it eliminates it.

The calibrated alpha (alpha = 1) earns a high-confidence survival claim within this experiment: 8 of 8 trajectories survived all 512 forks on a held-out prompt set. Doubling the dose still leaves a usable window. Quadrupling it does not.

Reproduce

uv sync --extra all
just smoke         # tiny-random model, ~1 min CPU
just calibrate     # one (model, method, seed, window) cell
just trajectory
just plot

The headline run for the figures here:

uv run --extra all python scripts/run_cell.py \
  --model Qwen/Qwen3.5-0.8B --method mean_diff --seed 0 --window 512 \
  --out-root outputs_qwen35_w512_dense \
  --run-id Qwen3.5-0.8B_mean_diff_s0_w512_dense \
  --compute-pmass --skip-pmass-qa --fork-log \
  --alphas 0.0 0.25 0.5 0.75 1.0 1.5 2.0 4.0 \
  --render-figs --render-threshold 0.95

Outputs land under outputs_qwen35_w512_dense/<run-id>/figs_auto/{survival,spaghetti,aggregate}/.

What this repo is and isn't

In scope:

One figure family (spaghetti, survival, aggregate) and one calibration script.
3 methods: mean_diff, directional_ablation, pca.
A branch_pmass metric: fork-and-teacher-force probability mass on a forced-choice answer token after schema prefill.

Out of scope:

A method zoo beyond those three.
A norm-matching baseline. Iso-KL says nothing about whether two methods of equal KL move the same distance in activation space.
A threshold sweep. The pmass < 0.95 cutoff is a single point; neighbours might disagree.

Honest caveats

Iso-KL is one fairness criterion, not the criterion. Matching p95 per-token KL means matching distributional disagreement under greedy decoding in the calibration window. It does not match intervention norm, behavioural effect size, or human-perceived quality.
Pmass is a proxy. It scores whether the model still produces one of two specific tokens after we splice in a schema. A model that has gone off-format but is otherwise coherent gets scored as dead.
n = 8. Held-out prompts, not held-out seeds. The phase-transition shape is consistent at this scale; the exact half-life at alpha = 2 is not.
One model so far. Gemma 4B, Gemma 12B (4-bit), OLMo-2 1B, and OLMo-3 7B at w = 4096 are queued. The phase-transition story may or may not survive scaling.

Glossary

Operational definitions used in this repo, not textbook ones. Skim and come back as needed.

steering coefficient (alpha): scalar multiplier on the steering vector added to residual-stream activations at one layer. alpha = 1.0 is the iso-KL-calibrated coefficient C. alpha = 0 is the unsteered base model.
iso-KL calibration: bisection on alpha such that the 95th-percentile per-token KL(steered || base) over a w-token calibration window equals 1 nat. Output is c_star, the coefficient at alpha = 1.
p95 KL: 95th percentile of per-token KL(steered || base) over the calibration window. The "worst 5%" measure used by calibration.
window (w): length in tokens of the calibration rollout. w = 512 for the figures here.
pmass: probability the model puts on the set {true, false} after we splice in a JSON schema prefill ('\nI should answer now.</think>{"value": ') at fork token t.
pmass_eval: pmass on a held-out prompt set, at every fork t.
fork point t: the token in the rollout where we splice the schema and read pmass.
death: irreversible. Alive at t means pmass_eval(t) >= 0.95. First time it drops below, dead, and stays dead. Below 0.95 the JSON schema stops being a stable attractor.
right-censoring: the rollout hit EOS at length L < t, so we never see fork t. Drops out of the at-risk denominator from L onward; not death.
survival S(t): Kaplan-Meier estimator of the fraction alive at fork t.
mean_diff: steering vector built as the difference of mean activations on a contrastive prompt pair.
spaghetti plot: every individual KL trajectory drawn as one thin line, alive segments green, dead red, black line is the median.