EPS Dashboard

2026-07-28. Scope: the fitted maps from the #779 (up to 1M rows) and #1092 (21,193-row crossed real-conversation corpus) lines. Grounding: the methods survey (docs/ideas/2026-07-06-context-answer-map-analyses.md), the consolidated results (docs/results_summaries/2026-07-22-prefix-query-context-answer-map-consolidated.md), and the empirical paper's M_{C,A} object. Analysis plan over banked artifacts plus a small set of new fits; any execution routes through the standard task pipeline.

1. The object: one conditional expectation, four conditionings

Every row has generative structure a = F(p, q, ε): answer summary a from prefix p, user query q, decode noise ε. The four maps estimate conditional expectations of this one object:

Arm	Input	Estimand	Plain reading
context	c(p,q), pooled prefix+query	E[a \| p, q]	the full-information map
prefix-end	e(p), last prefix token (pre-query)	E[a \| p]	query-marginalized: what the persona alone fixes
bare query	v(q), query with no prefix	E[a \| q]	prefix-marginalized: what the task alone fixes
query-averaged	v̄_P = mean_q c(p,q)	E[a \| p]	same estimand as prefix-end, richer input

The deviation from the snapshot literature. DMD/Koopman fits assume an autonomous system: the state at t determines the distribution at t+1. Here an exogenous input — the user turn — enters between input and output. The correct template is DMD with control: a = A·p + B·q + interaction + noise (Proctor–Brunton–Kutz DMDc; verify arXiv id before paper use). The estimator-level toolkit (ResDMD certification, bagged CIs, constrained fits) imports unchanged because it is about regression operators, not physics; the dynamical readings (iterating the map, attractors, mixing times) do not. The conditional-expectation framing (Grünewälder 1205.4656; Mollenhauer & Koltai 2012.12917; Kostic 2302.02004) is what absorbs the user turn cleanly: for the prefix arms the query is marginalized, so it is structured noise with three knowable consequences — an R² ceiling equal to the between-prefix variance share, attenuation of the fitted operator, and anisotropic residuals that break vanilla Marchenko–Pastur nulls. Each consequence is a testable prediction, not a nuisance.

Measured facts the theory must explain (consolidated writeup, all held-out, novel-prefix folds):

Context map R² 0.74–0.81 per-context; prefix-end R² 0.07–0.10 per-context, 0.35–0.58 of its own ceiling even on averaged targets.
Additivity to 91%: disjoint prefix + bare-query forwards stitch to R² 0.833 vs 0.910 full.
Variance shares: query 79% / prefix 11% / interaction 10%.
The prefix and context operators share output subspaces (angles 22–34° vs ~78° null) but read near-orthogonal inputs; the prefix arm's predictions are a global ~0.6× shrinkage of the context arm's.
The independently-fit averaged map is a noisier estimate of the context map, not a second mechanism (operator-coincidence check).

2. Q1 — Are the four maps one object? (algebraic consistency)

Three relations the one-object view predicts, all computable from banked stores:

Jensen/commutativity gap. For a linear true map, averaging inputs then mapping equals mapping then averaging: M_ctx(v̄_P) = mean_q M_ctx(c(p,q)). Compute both sides per prefix; the gap is a direct curvature witness — how nonlinear the true map is across each prefix's query cloud — obtained with zero new fits. Interprets as: where (which prefixes, which answer directions) the linear abstraction bends.
Factorization of the prefix-end deficit. Fact 1 says e(p) reaches only 0.35–0.58 of ceiling. Two rival explanations: the pre-query state does not contain the information (state deficit), or it contains it in a form the linear readout misses (readout deficit). Fit the intermediate map e(p) → v̄_P and chain it: if R²(e→v̄_P→a) ≈ R²(e→a), the deficit is in the state; if chaining through v̄_P recovers skill, the information is present but linearly inaccessible from e. Interprets as: locating WHERE persona information is lost before the query arrives — the single most monitoring-relevant unknown in the line.
Marginal vs joint prefix operator. In the crossed design p ⊥ q, so the prefix block A of the joint (stitch) fit and the marginal prefix-end map should coincide (omitted-variable term B·Cov(q,p) vanishes). Compare them. Interprets as: a validity check on the crossing — divergence measures confounding from sparse-crossing imbalance, which would contaminate every cross-arm comparison.

3. Q2 — What does the user turn do? (the DMDc layer)

Low-rank bilinear interaction. Extend the additive stitch with a rank-r interaction term: a = A·p + B·q + Σᵢ (uᵢᵀp)(vᵢᵀq)·wᵢ, sweeping r. Fact 2 caps the interaction at ~0.08 R²; the question is its shape. Interprets as: how many degrees of freedom of prefix-modulation the query has — r=1 means the query is a single gain knob on the persona; small r means a few named modulation channels (inspect uᵢ, vᵢ, wᵢ against the trait dictionary); diffuse means genuine mixing only attention explains.
Where the interaction lives. Project the interaction component onto the persona-vector dictionary and answer PCs. Interprets as: whether prefix×query interaction is persona-bearing (bad news for linear monitors: persona expression is query-gated) or stylistic residue.

4. Q3 — Channel structure per arm

For each of the four arms, on the same folds and targets:

Held-out predictable-variance spectrum (whitened; number of canonical/PLS components with out-of-sample R² > 0). Interprets as: the honest channel count — how many independent things each input determines about the answer. Robust to λ and to the estimator's mechanical rank cap.
Linear maximal-correlation ceiling ρ₁² per layer. Interprets as: the hard cap on any linear readout of the answer from that arm's input, comparable across arms and layers; the arm-to-arm ρ₁² gap is "how much the query unlocks," as a bound rather than a fit.
Per-trait-direction held-out R² table (project true and predicted answers onto each persona/trait direction; directions fixed on train folds). Interprets as: which behaviors ride which arm — a trait predictable from prefix-end is pre-query monitorable; a trait predictable only from context requires seeing the query; a trait predictable from neither is (linearly) spontaneous. This is the leakage-relevant deliverable.
Cross-arm subspace geometry. Principal angles between arms' predictive input and output subspaces (extends Fact 4 to all four arms, with spectrum-matched nulls). Interprets as: whether the query adds new channels or reweights existing ones — Fact 4 says reweighting for prefix-vs-context; test whether bare-query behaves the same.

5. Q4 — Ceilings and null spaces

Import of the Round-3 null-space program, now legitimate on the n ≫ d arms (context, bare-query; the prefix arm stays rank-limited by 1,145 distinct prefixes — widen the pool before any prefix-side kernel claim):

Co-kernel ceiling per behavior: fraction of each trait read-out r_B outside range(M), per arm. Interprets as: a per-behavior upper bound on any linear context-side monitor; the cross-arm differences say how much of each behavior's expression is query-gated.
Kernel causal tests: LEACE-erase or inject along candidate kernel directions (predicted: nothing happens — the null-intervention control) vs top singular directions (predicted: behavior moves). Interprets as: upgrades "small singular value" to "causally inert direction" — the cheap-to-falsify negative prediction that distinguishes a mechanism claim from a fit description.

6. Q5 — Eigen-structure, singular directions, and the fixed point (endomorphism reads)

All four arms map into the same residual-stream space at the same layer, so each fitted operator is an endomorphism and eigen-analysis is licensed — the one affordance the layer-lens literature lacks (methods survey §"What W is"). Pooling grains differ across arms (last-token input for prefix-end vs span-mean elsewhere); state this beside any cross-arm eigen comparison. Every eigen read runs AFTER the non-normality gate (normality gap, pseudospectra, Σ-metric sym/antisym split) and is ResDMD residual-certified with bagged CIs; gate failure ⇒ report singular values/subspaces only.

Singular directions, decoded. Top right singular vectors = the context directions the map reads; their paired left singular vectors = the answer directions it writes. Decode both through the tuned lens and cosine them against the trait dictionary. Interprets as: the map's input–output vocabulary in human terms — what it attends to and what it says in response; the concrete content behind Q3's channel counts.
Eigen reads (the copying detector). Fraction of eigenvalue mass with positive real part and trace(W)/d (mean copying score); |λ|≈1 modes = directions carried into the answer near-unchanged, |λ|≈0 = discarded, complex pairs = rotational re-expression that SVD alone cannot see. Near-eigenvector trait test: is W·v_trait ∝ v_trait (trait copied), rotated, or killed? Interprets as: per-direction transport verdicts — the operator-level answer to "does the persona pass through verbatim," the OV-eigenvalue copying read applied to a fitted map.
The affine fixed point. With intercept b and 1 ∉ spec(W), x* = (I−W)⁻¹b is the unique state the map sends to itself, and the map rewrites as a − x* = W(x − x*). Interprets as: the operator's neutral point — the answer state predicted for a context carrying no signal, and the natural origin all displacement reads are measured from; decoding x* says what the map treats as the "default assistant." Caveat carried from the survey: W^k / convergence-toward-x* language describes the operator's geometry, never multi-turn dynamics — the map is one-shot.
Maintained directions, graded, and invariant subspaces. "The map keeps it" comes in three strengths, per arm: (i) pointwise maintained — Wv ≈ v (eigenvalue ≈ +1): the answer re-expresses the direction at the same strength; (ii) direction-maintained — cos(Wv, v) high, gain ‖Wv‖/‖v‖ free: kept but re-weighted; report the (cos, gain) pair for every trait direction and every top singular direction — the per-direction preservation map; (iii) subspace-invariant — W·S ≈⊆ S: the map may mix within S but does not leave it, found from eigenvalue clusters / Schur structure (certified, gate-conditional). For the trait dictionary specifically, the trait gain matrix G = U_Bᵀ·W·V_B decomposes transport in one artifact: diagonal = each trait maintained as itself, off-diagonal = rotation into other traits, energy outside span(B) = trait content re-encoded outside the dictionary. Interprets as: the graded answer to "does the answer stay in character along this direction" — and, jointly with Q4's kernel, a complete per-direction taxonomy: every context direction is classified maintained / re-weighted / rotated into other traits / re-encoded elsewhere / discarded. Cross-arm comparison of G says whether the user turn changes which traits survive or only their gains.

7. Q6 — Where linearity ends (kernel and MLP)

Detection before estimation. HSIC / distance correlation between each arm's input and its linear-map residuals, group-respecting permutation p-values. Interprets as: is there ANY structure the linear map missed — answered exactly, before spending on nonlinear estimators.
Estimation ladder per arm, identical folds and nested-CV tuning on every rung: ridge → random-features/Nyström kernel ridge (exact kernel methods do not scale past n ~ 10⁴; RFF/Nyström do) → MLP. Note the regime reversal: at n = 21K–1M, MLPs are legitimate estimators (unlike the n≈2.5K per-example grain, where optimizer variance dominates); report seed spread. The nonlinearity gain per arm = held-out R²(nonlinear) − R²(linear), both against the identity+bias baseline and kNN-retrieval reads per the standing mapping rules. Interprets as: (a) which of the four relations are genuinely linear vs linear-only-as-approximation; (b) the prefix-arm gain is the monitoring number — persona→answer transport that NO linear probe can see; (c) if the MLP's gain over linear-on-context is closed by the rank-r bilinear model of Q2, the "nonlinearity" is named — it is the prefix×query interaction, not a black box.
Decode-noise ceiling. K ≥ 5 answer draws per context on a subset; per-direction noise floor. Interprets as: separates "the map fails here" from "nothing could predict this" — required before reading any per-direction R² table (Q3) or nonlinearity gain as a map property.

8. Validity gates (binding on every read above)

Group-level folds by prefix id; λ discipline (df(λ) reported, headline reads at ~3 λ values, same λ refit inside every permutation draw); permutation + matched-n nulls for every spectrum and subspace overlap; non-normality gate before any eigen-read; identity+bias baseline + kNN retrieval for every fitted map; directions selected on train folds only; n_train vs d stated per fit. Full text: methods survey §"Validity gates".

9. Compute and reuse policy

Per-family reuse (every reuse conditional on the artifact-reuse fitness check at execution):

Ridge — reuse the banked fitted operators (#779 up-to-1M-row; #1092 21K crossed) as the objects of study for every operator-geometry read (Q4, Q5). Reads that need held-out predictions (Q1, Q3) refit cheaply per fold from the banked activation stores — per-row held-out predictions were not persisted (#1092 banked-checkpoint note) — CPU, batched per the vectorize-first rule.
Kernel — banked at both grains on the context arm. Exact-RBF KRR at n_train=50k (held-out R² 0.807 vs ridge 0.760; eval_results/issue_779/fitter-fair-comparison-n50k/) and streaming Nyström RBF KRR at n_train=963,444 (R² 0.807 vs ridge 0.754; fitter-fair-comparison-n1m/n1m_fits.json, val/test pinned byte-identical to the original round), plus multilayer n=963k fits with persisted weights (n1m-nonlinear-map-behavior-readout/). Reuse as the Q6 kernel rungs — after verifying the split respects novel-prefix grouping. New kernel fits: the other three arms only.
MLP — banked at 1M in two widths. w=8192 (protocol arm, R² 0.810) and w=32768 (capacity arm, R² 0.813) at n_train=963,444, plus the n50k MLP (0.779) and #1092's pca48 companion ceiling. Reuse as the Q6 MLP rungs and ceilings; re-fits only where the matched-folds + nested-CV protocol demands it (the nonlinearity gain is defined only under identical folds and tuning on both rungs).
New compute: multi-draw answers for the noise ceiling (small GPU); nonlinear fits for the prefix-end / bare-query / query-averaged arms (cheap GPU band); kernel causal tests (GPU, steering rig).

The banked comparisons already answer Q6 for the context arm at two grains: nonlinearity gain ≈ +0.05 R² at 50k and ≈ +0.06 at 963k — and at 1M all three nonlinear families (Nyström KRR, both MLP widths, residual-skip) converge to ≈ 0.81, consistent with ONE shared missing component rather than family-specific expressivity. Q2's rank-r bilinear interaction is the named candidate: test whether it closes the same ≈ 0.06 gap. Pending the fold-structure check before any of this is quoted as a result.

Execution shape: two tasks. Task A — linear/operator characterization (Q1, Q3, Q4, Q5 + the multi-draw noise ceiling, which gates Q3's per-trait tables): ~5–12 GPU-h, majority 0-GPU on banked stores. Task B — nonlinearity (Q2 bilinear + Q6 ladder on the three remaining arms + residual HSIC + the banked-datapoint fold check): ~7–15 GPU-h, reusing Task A's draws and folds. Q2 rides with Task B because its headline test — does the rank-r bilinear model close the same gap the MLP/KRR find — must be scored under one protocol with the ladder. Both tasks route through the standard pipeline; nothing here pre-authorizes a run.

10. Summary: computation → what it tells us

Computation	What it tells us about the mapping
Jensen gap (avg-then-map vs map-then-avg)	curvature of the true map over query clouds — where linearity bends
e(p)→v̄_P chain	whether persona information is absent from the pre-query state or just linearly unreadable
Joint-vs-marginal prefix operator	whether the crossed design cleanly identifies the prefix's own effect
Rank-r bilinear interaction	how the user turn modulates the persona: gain knob, few channels, or diffuse mixing
Predictable-variance spectra (4 arms)	honest channel count per input
ρ₁² per arm/layer	hard linear-monitor ceiling; what the query unlocks, as a bound
Per-trait R² table (4 arms)	which behaviors are pre-query monitorable, query-gated, or spontaneous
Cross-arm principal angles	query adds channels vs reweights them
Co-kernel fraction per trait	per-behavior monitor ceiling from unreachable answer directions
LEACE/injection on kernel vs top directions	which directions are causally inert vs load-bearing
Decoded top singular pairs	what the map reads and what it writes, in vocabulary/trait terms
Eigen/copying reads + near-eigenvector trait test	which directions pass to the answer verbatim, rotated, or killed
Affine fixed point x*	the map's neutral point — the default answer state displacements are measured from
(cos, gain) preservation map + trait gain matrix G	per-direction taxonomy: maintained / re-weighted / rotated into other traits / re-encoded / discarded
Almost-invariant subspaces (eigen-cluster / Schur)	subspaces the map keeps closed — candidate persona macro-axes
Residual HSIC/dCor	whether anything nonlinear remains, exactly
RFF/MLP nonlinearity gain per arm	the persona transport invisible to any linear probe; whether nonlinearity = the named interaction
Multi-draw noise ceiling	what no map could ever predict