EPS Dashboard

A typology for cross-cell transfer of fitted context→answer maps (base↔instruct, chat↔plain-text, character↔character, story↔chat). A "cell" is one (model × framing × persona) setting; in each cell we fit an affine map

f(x) = W x + b        x = context activation, y = answer activation

(ridge, held-out evaluation). Transfer = carry the source cell's map to a target cell while allowing only a restricted transformation class to be refit. The tiers are ordered by the size of that class: the smaller the class that reaches the target's own ceiling, the stronger the claim that the two cells share one map. Every tier is reported as a fraction of the target's own held-out ceiling and judged against a matched null; the measurement is "the smallest tier that reaches ceiling."

Above Tier 1 every tier refits the offset. Scaling (Tier 3) and rotation (Tier 4) are not nested in each other — the ladder is a lattice with two branches that meet at Tier 5.

Tier 1 — Direct transfer

Apply the source map verbatim: ŷ = W_s x + b_s. Zero fitted parameters.
Reading: the target writes the same information along the same directions, at the same scale, around the same origin — literally one map serves both cells. The strongest sameness available from a readout.

Tier 2 — Constant shift

The source map is anchored to where the source's data lived: inputs centered at x̄_s, predictions anchored at ȳ_s. A constant shift can re-anchor the context side, the answer side, or both (all refit means come from target train folds only):

Context side: recenter only the inputs to the target's context mean; keep the source answer anchor: ŷ = W_s(x − x̄_t) + ȳ_s. Corrects the context-mean displacement as seen through the frozen map (W_s Δx, Δx = x̄_t − x̄_s). Recovers transfer iff the settings differ by where their CONTEXTS sit.
Answer side: keep the source input centering; re-anchor predictions at the target's answer mean: ŷ = W_s(x − x̄_s) + ȳ_t. Corrects the answer-mean displacement (Δy = ȳ_t − ȳ_s). Recovers transfer iff the settings differ by where their ANSWERS sit.
Both sides: apply both recenterings — equivalently, refit the single least-squares-optimal intercept given the frozen slope: ŷ = W_s(x − x̄_t) + ȳ_t = W_s x + b*, b* = ȳ_t − W_s x̄_t. The correction relative to naive transfer is b* − b_s = Δy − W_s Δx — one constant in answer space that bundles both sides, which is why the combined form alone cannot say which side moved; the one-sided forms split it.
Decomposition report: ‖W_s Δx‖, ‖Δy‖, and cos(W_s Δx, Δy). Large and aligned ⇒ the two terms cancel inside Δy − W_s Δx and the map itself already transports the shift — naive transfer then carries little constant error even though both clouds moved.
Reading: same map, displaced operating point(s). Held-out R² punishes a constant displacement without bound, so catastrophically negative naive transfer is still compatible with a Tier-2 relationship — this tier is what separates "different origin" from "different map."
Steering view: because the map is linear, the shift is equivalently an activation-steering intervention — add a fixed vector at the context representation (−Δx) or at the answer representation (+Δy); any split of the net correction between the two sides is observationally equivalent at this tier. The vectors involved are difference-of-means directions between two conditions, the same construction persona-vector steering uses, so a Tier-2 pass says the two settings differ by something steering-shaped. Which side carries it is the one-sided question above; whether imposing the displacement moves behavior is a causal claim that needs an intervention, not a readout. The steering view is only meaningful within one model (cross-model cells have no shared space to steer).

Tier 3 — Scaling

Global gain: one scalar, ŷ = c·W_s x + b*. Same map, different overall gain — absorbs the confound that regularized fits shrink differently across cells (different n / SNR), so two estimates of the same operator can differ by a near-scalar factor.
Per-dimension rescale: ŷ = W_s D x + b* with D diagonal. Axis-aligned gain changes only — no direction mixing. Higher-variance version of the same idea; many free scales can also mismatch where one scalar would not.
Reading: same map after gain changes; the relation's directional structure is untouched.

Tier 4 — Rotation

Orthogonal input/output alignments fit from paired activations around the frozen map: ŷ = Q_out W_s Q_in x + b*, QᵀQ = I (Procrustes); optionally plus one global scale.
Rotations are rigid: they preserve all lengths, angles, and singular values; they can re-label which directions carry which features but cannot stretch, shear, or mix beyond a rigid turn.
Reading: the same computation written in a rotated basis — functional geometry intact, coordinate frame turned. The companion similarity statistic is the data-paired rotation-aligned operator cosine, judged against a random-rotation null and a shuffle-fit null. A rotation-invariant spectrum-only similarity (comparing sorted singular values) is quasi-mechanical — maps fit on scrambled pairings score just as high — and carries no shared-structure evidence.

Tier 5 — General linear / affine

General linear input/output alignments fit from paired activations ONLY (never end-to-end through the prediction objective), wrapped around the frozen map: ŷ = B W_s A x + b*, judged against matched-capacity nulls (the identical alignment recipe around a spectrum-matched, structure-destroyed middle; and alignments refit on shuffled correspondence).
Reading: the predictive content and the input→output relation are preserved; each cell encodes them in its own linear coordinate system. This is the weakest linear sameness: as pure algebra, ANY two full-rank maps are related by some invertible A, B (rank is the only two-sided GL invariant), so this tier is informative only through its constraints — correspondence-fit alignments plus failing nulls. It cannot distinguish "same circuit with re-encoded inputs/outputs" from "different circuit computing an equivalent function."
One-sided forms: input-only ŷ = W_s(Ax) + b* vs output-only ŷ = B(W_s x) + b* localize which side of the map the re-encoding lives on, and each keeps a real algebraic invariant the both-sided form lacks: input-only composites are confined to W_s's column space, output-only to its row space — one whole side of the operator stays frozen by construction. Well-defined only for within-model pairs (both cells must share one activation space). The alignment fits themselves double as item-level similarity reads per side (alignment held-out R², the identity-plus-learned-bias baseline, kNN retrieval).
Rank-k form: constrain A, B to rank k and sweep k — the recovery-vs-capacity curve measures the dimension of the transported core, with the null curve quantifying how non-trivial the transport is at every capacity.

Tier 6 — Nonlinear

Nonlinear alignments (or a nonlinear map) in place of the linear pieces.
Reading: bounds the linear story from above. Where Tier 5 fails, Tier 6 separates "the information is absent in the target representation" from "present but not linearly accessible."

Reading the ladder

Each tier hands one more piece of the map over to refitting; the "same map" claim is only as strong as what stays frozen:

Tier	What is refit	What stays frozen
1	nothing	everything (W and b)
2	where the map is anchored (b)	the entire slope W
3	gains (one scalar, or per-dimension)	W's direction-mixing
4	the basis, rigidly	W's singular structure and all scales
5	the entire encoding, one or both sides	W's core — its action on the predictive subspace
6	even linearity	only "some function relates them"

At Tiers 4–5 the map's entries are never edited — W stays frozen and transformations wrap around it; what shrinks as you climb is the invariant residue that must still do the work in the target cell. "How much of the map did we have to modify to make it work over there" and "how much of the map is shared" are the same measurement read from opposite ends.

Standing caveats

Every tier is a property of a fitted readout, not of circuits. Tier 1 is the strongest correlational signal of mechanism reuse; no tier is causal. Upgrading a tier claim to a mechanism claim needs interventions (patch the aligned subspace, test behavioral transport).
Higher tiers need their nulls. A Tier-5 "recovery" without matched-capacity + shuffled-correspondence nulls is uninterpretable; a similarity statistic without a shuffle-fit null can be quasi-mechanical.
Recoveries can be asymmetric (source→target succeeding while target→source fails). A genuine invertible coordinate change would be symmetric; asymmetry is an errors-in-variables signature of direction-specific representation noise.