Module row_metric 

RowMetric — the single provenance-carrying per-row inner product shared by the SAE-manifold likelihood (residual whitening) and the gauge (isometry pullback weight).

Why this exists

The SAE-manifold machine historically carried two independent inner products:

• the likelihood measured reconstruction residuals isotropically — a single scalar dispersion φ̂ = RSS / residual-dof, the data-fit loop summing the bare ½ rᵀr; there was no per-row metric at all; and
• the gauge carried its own per-row metric in IsometryPenalty.weight: WeightField — a low-rank W_n = U_n U_nᵀ pullback g_n = J_nᵀ W_n J_n, settable independently of anything the likelihood saw.

Nothing structurally forced “the metric the likelihood whitens by” to equal “the metric the gauge pulls back through”. That is exactly the objective↔gradient-desync bug class wearing geometry clothing: a likelihood-metric ≠ gauge-metric state was representable.

RowMetric collapses the two into one object. The likelihood whitens through it; the gauge WeightField is constructed from it. A divergent-metric state is therefore unrepresentable — there is only one per-row factor stack U_n, with one MetricProvenance tag.

Magic-by-default selector

There is no flag. The provenance is chosen by whether per-row Fisher factors exist:

• no factors supplied ⇒ MetricProvenance::Euclidean; W_n = I_p; whitening is the identity, so φ̂ and the data-fit loop are bit-for-bit the prior isotropic path; and
• per-row Fisher factors supplied ⇒ MetricProvenance::OutputFisher; the residual is whitened by U_nᵀ and the gauge pulls back through the same U_n.

Validation

Every metric block is constructed through crate::normalize_fisher_rao_blocks, which broadcasts and eigenvalue-validates PSD-ness. RowMetric does not reimplement that validation; it materializes W_n = U_n U_nᵀ (which is PSD by construction) and runs it through the shared normalizer as the single point of truth for “is this a valid precision metric”.

Any rank floor used to make a block invertible for an internal solve is solver-only (mirroring RidgePolicy::solver_only, #747): it never enters the residual the objective sums, so δ cannot bias the criterion.

Rung 1 — the behavioral metric in the reconstruction loss (nats currency)

MetricProvenance::OutputFisher installs the output-Fisher inner product as a gauge metric only: it whitens nothing (whitens_likelihood() is false), by deliberate #980 contract, so reconstruction stays the isotropic ½‖r‖². That answers “what coordinate is canonical”, not “what does a reconstruction error cost”.

MetricProvenance::BehavioralFisher is the opposite deliberate choice: the same low-rank output-Fisher factors, but installed as the reconstruction likelihood weight. Plain MSE prices a reconstruction error e = x − x̂ by its Euclidean size; the model, however, reads the activation only through the rest of the network, so the behavioral cost of e is the KL between the clean and corrupted next-token distributions, KL ≈ ½ eᵀ G(x) e with G = JᵀFJ the network-Jacobian pullback of the output Fisher F (units: nats). Minimizing (x−x̂)ᵀ G (x−x̂) instead of ‖x−x̂‖² is generalized least squares: for a fixed per-row G it is still a linear Gaussian model in the coefficients, so the entire REML/evidence/EDF/certificate stack survives verbatim — this is why the metric rides the identical whitens_likelihood() plumbing the MetricProvenance::WhitenedStructured noise model uses, and why the G=I limit reproduces the plain-MSE fit bit-for-bit (see the module tests).

This is the principled form of Braun’s end-to-end KL + MSE objective. Anchoring to the activation keeps it reconstruction (it does not collapse to “match the logits by any means” — the decoder still has to reproduce x), while pricing the residual in nats through G. The payoff is automatic selection for mattering: G’s null directions — activation structure the rest of the network cannot read — are penalized nothing, because eᵀ G e = 0 there. MSE in a behaviorally-inert direction goes free, which is the correct behavior, not a bug: nothing downstream changes, so nothing should be paid.

The d×d G is never materialized. G is sketched by s random probes, vᵢ = Jᵀ F^{1/2} uᵢ (uᵢ iid, s ≈ 4…16), computed by s backward passes per token at harvest time (the model-interaction boundary) and stored as the columns of the per-row factor U_n = [v₁ … v_s] ∈ ℝ^{p×s}. Then G ≈ Σᵢ vᵢ vᵢᵀ = U_n U_nᵀ and the criterion-facing eᵀ G e ≈ Σᵢ (vᵢᵀ e)² = ‖U_nᵀ e‖² is exactly what RowMetric::quad_form / RowMetric::whiten_residual_row already compute — zero train-time model cost, O(p·s) per row. See RowMetric::behavioral_fisher and the probe-packing helper pack_probe_factors.

Structs

RowMetric

The single per-row metric object. Holds one low-rank factor stack U_n (or none, for Euclidean) plus the validated PSD blocks, tagged with its MetricProvenance.

Enums

MetricProvenance

Where the per-row metric came from — the provenance that makes “likelihood-metric ≠ gauge-metric” diagnosable instead of silent.

WeightField

Per-observation behavioral-metric field W_n ∈ ℝ^{p × p}, stored in low-rank factored form W_n = U_n U_n^T with U_n ∈ ℝ^{p × r_n}.

Functions

pack_probe_factors

Pack a harvest-emitted probe stack into the row-major factor layout RowMetric::behavioral_fisher expects.