J-space, open models — v2

Everything on this page comes from a single measurement. Take a model reading text, nudge its residual stream at layer ℓ, and record how the final layer — the one that decides the next token — moves in response. Averaged over many positions and prompts, that response is a matrix Jℓ: the layer's typical influence on the output. Fig. 1 From a nudge to two numbers The J-lens pipeline. Everything downstream operates on Vℓ. Multiplying by the unembedding makes the influence concrete. Each of 4,096 common tokens gets a vector: the direction at layer ℓ that pushes that token's probability up. These are steering vectors in the literal sense — inject one and the model says the token, which is what E3 exploits. Together, the 4,096 vectors are the layer's dictionary. Two numbers summarize a dictionary. The first is CKA — centered kernel alignment — which asks whether two dictionaries have the same shape when you are not allowed to compare coordinates. The recipe: center a dictionary's vectors, then build its table of relations K = VVT — a 4,096 × 4,096 grid whose cell (i, j) records how strongly entry i points along entry j. The table is coordinate-free: rotate the whole dictionary and not a single cell changes. CKA is then just the cosine between two such tables, CKA(A, B) = ⟨KA, KB⟩ / ‖KA‖ ‖KB‖ — 1.0 when the geometry is identical, near zero for unrelated random tables. For calibration: two independent fits of the same model score about 0.997, and two different trained models compared at matched depth land around 0.5–0.7. Fig. 2 How CKA is computed Six toy entries, two models. Rotation changes every coordinate and not one relation, so the two tables come out identical.

Computed between every pair of a model's own layers, the same number draws a map with visible blocks — an input-side block that reads, a long middle block (the paper's "workspace"), and a small output-side block that writes. Fig. 3 A model's layer-by-layer agreement CKA between the dictionaries of every pair of qwen3-1.7b layers. Hover for values. The blocks emerge with no supervision, and their edges — "boundaries", quoted as a fraction of depth ρ — reappear in every experiment below. The second number is PR — the participation ratio — which counts how many directions a dictionary actually uses. Take the eigenvalues λi of the dictionary's covariance (the weight each independent direction carries) and form PR = (λ1 + … + λn)² / (λ1² + … + λn²) If every direction carried equal weight, PR would equal their count; if one direction carried everything, PR would be 1. Six entries spread over two directions give PR ≈ 2. Entries are not directions. Fig. 4 Why 4,096 entries is not 4,096 directions Related words point the same way, so entries share directions. A real layer's 4,096 entries span 200–600 effective directions; that count — always taken in the middle band, because near the output the measurement collapses into the output matrix — is the "capacity" number used throughout. One caveat travels with everything here: J is an average over a text distribution. Change the text and the measurement changes — that is E5.

E1 · temporal horizon How long does a nudge keep steering the output? The Verbalizable-Workspace paper reads the middle layers as a workspace that "holds things in mind". Taken literally, a nudge there should keep steering the output for many tokens — longer than a nudge anywhere else. The standard fit can't test this: it averages over all future positions and erases the time axis.

So we split it by distance: fit a separate dictionary on source–target pairs exactly Δ tokens apart and track its overall strength, effectℓ(Δ) = ‖P·Jℓ(Δ)‖ — how hard a nudge at layer ℓ still moves the output Δ tokens later. Six models, 250 prompts each, Δ from 0 to 96. Fig. 5 Why the clock starts at Δ = 1 effect(Δ1) ÷ effect(Δ0): the fraction of a nudge's immediate effect that reaches even the next token. One line per model. At Δ = 0 the measurement is dominated by the position's own path to its own output — the nudge rides the residual stream straight down. It answers "how much does a nudge change this token", not how far it travels: at most a quarter of it crosses one token, 3–6% at the deepest layers. So every curve and statistic below starts the clock at Δ = 1 and is quoted relative to it. The step down at ρ ≈ 0.6–0.7 in every model is the cliff the conclusion returns to. Fig. 6 How much a nudge still moves the output, as the target gets further away Effect at distance Δ relative to distance 1, log–log. One line per layer of the selected model; drag the slider to walk through the layers. model layer Early layers start strong and fade slowly; the deepest layers lose everything within a token. The highlighted band layer barely fades between Δ ≈ 12 and Δ ≈ 96 — but this is measured inside a 128-token window, and the next figure shows how much of that flatness the window itself manufactured. The problem with those curves: they come from 128-token sequences. An effect at Δ ≈ 96 can then only be measured from source positions squeezed into the first thirty tokens of the window, so if early positions behave differently, the far buckets are biased.