Geometry

The gauge-fixing whitening artifact for one (site, corpus) (DESIGN section 2.7.1).

mean is the reference distribution’s activation mean; sigma_sqrt and sigma_inv_sqrt are the symmetric square roots of the shrinkage-regularized activation covariance. The canonical form of a reward direction is Sigma^{1/2} w normalized (canonical.py), which weights directions by how much the reference distribution varies along them and kills directions it does not vary along at all. null_basis holds those data-null / gauge directions explicitly (an orthonormal d x m matrix, or None when none were estimated) so canonicalization can quotient them exactly rather than relying on the shrinkage floor. estimator_meta records the fit provenance: sample count, shrinkage intensity, eigenspectrum summary, and the fp32 assertion.

All arrays are fp32. The frame is frozen and carries a content-derived FrameID so it is a stable, shareable, per-(site, corpus) artifact.

Fit a Frame from a reference activation matrix (DESIGN section 2.7.1).

activations is an n x d fp32 matrix of reference-distribution activations read at site (fp16 is refused). margins, when supplied, are the per-item reward margins used to refine the null-subspace estimate to directions of near-zero margin gradient. shrinkage selects the covariance estimator; only Ledoit-Wolf ("lw") is implemented, and "none" is accepted for the raw sample covariance (which is only well conditioned when n comfortably exceeds d).

The fit computes the mean, the shrinkage covariance, its symmetric square roots, and the null basis, then derives the content FrameID from structural content (site, corpus, counts, rounded eigenspectrum summary) so identical fits share an id across platforms.

Canonical form of a reward direction in a frame: Sigma^{1/2} w normalized.

DESIGN section 2.7.1. Whitening by Sigma^{1/2} weights each direction by how much the reference distribution varies along it, so directions the distribution does not vary along contribute nothing to the canonical direction. When the frame carries an estimated null_basis the gauge components of w are projected out first, which makes canonicalization exactly invariant to adding data-null directions to w (rather than leaving the small residual the Ledoit-Wolf shrinkage floor would otherwise pass through). The result is a unit fp32 vector.

Raises NumericsError if w has no on-distribution component (Sigma^{1/2} w is zero), which means the reward direction lives entirely in the gauge subspace and has no canonical form.

Canonical angle between two reward directions with a CI and a STARC regret bound.

DESIGN section 2.7.1 / Appendix A13. Reports cos(w_tilde_a, w_tilde_b) in the shared frame (the gauge-fixed alignment), the raw cosine for contrast, a bootstrap CI obtained by refitting the frame on corpus resamples (when activations_for_bound is supplied), and a behavioural regret bound: the empirical worst-case preference disagreement on the frame corpus, which the canonical (STARC) distance upper-bounds.

The quantity is COVARIANT and the frame argument has no default; this calls require_frame_for_comparison so a frameless cross-signal comparison raises (gate 2, I3). faithful_to is “STARC 2309.15257 / partial identifiability 2411.15951”.

Top-k reward-Hessian spectrum at a site, materialization-free via HVPs.

DESIGN section 2.7.2. source is a RewardSignal, a Runtime, a SymmetricOperator, or a plain matvec callable (the synthetic-operator test path); the reward Hessian is the double-backprop of the readout scalar at site, so nothing larger than dim probe vectors is ever held, which is what keeps the spectrum within the 8B memory budget. Lanczos with full reorthogonalization gives the extreme eigenvalues; n_starts > 1 seed-averages the Ritz values across random starts for robustness; density=True adds an SLQ spectral-density estimate for the tail.

The spectrum is INVARIANT (a property of one signal at one site, not a cross-signal comparison), so no frame is required.

Participation ratio of a curvature spectrum: the effective dimension d_eff.

PR = (sum |lambda_i|)^2 / sum lambda_i^2 (Appendix A9 / A16). It counts how many eigenvalues carry the curvature: n equal eigenvalues give PR = n, a single dominant eigenvalue gives PR = 1. Magnitudes are used so a mixed-sign (indefinite) Hessian still reports the effective number of curved directions. Accepts a SpectrumResult, a raw array, or any 1-D sequence.

Fit a rank-k skew preference operator and test cyclic recovery (DESIGN section 2.7.3, T8).

activations is the n x d matrix of frozen penultimate features (one row per response item); pairs is a (P, 2) integer array of (winner_index, loser_index) preferences; rank_k is the skew rank to fit. Optionally project activations to proj_dim principal components first, which keeps the d^2 skew fit tractable at large d.

run splits the pairs into train and held-out, fits the transitive (scalar) baseline and the rank-k skew operator on the train pairs, and reports their held-out accuracies. The gap is the cyclic structure the scalar head cannot express (theorem T8). The result is INVARIANT: rank and predictive accuracy do not depend on the activation coordinate basis, so this is a within-model structural claim that needs no frame.