GRADE's gradient-subspace stable rank is a normalized covariance-spectrum construction with no established formal reduction to the Fisher information matrix

GRADE (arXiv:2604.02830), already in the vault for its knowledge-gap diagnostic (claim-grade-gradient-rank-gap-detection), invokes Fisher information only as inspiration for using gradients diagnostically, not as the mathematical basis of its own construction: "Inspired by pioneering works that utilize gradient-based Fisher Information to localize factual associations... we quantify the required knowledge updates for a given query via the gradient chain-rule." The construction that actually follows is a projected, normalized gradient-covariance matrix. GRADE projects the gradient g onto the sample's representation space, computes the projected covariance hg⊤gh⊤, and normalizes it as C_g = C_h† (hg⊤gh⊤) C_h†, multiplying on both sides by the Moore–Penrose pseudoinverse of the Gram matrix C_h = hh⊤ "to ensure the projected gradient covariance C_g is not being distorted by the anisotropic geometry of h." Its stable rank is then defined purely spectrally from the singular values of C_g — Eq. (6) gives two variants, srank_pre = Σλᵢ/λ₁ for the pre-response entropy objective and srank_pos = Σ(λᵢ)²/(λ₁)² for the post-response cross-entropy objective — a general-purpose numerical-linear-algebra device attributed to Sanyal et al. (2020) and Ipsen & Saibaba (2025).

Crucially, no sentence in GRADE's method section (§3.1–3.3) claims or derives an equivalence between this rank-ratio construction and the Fisher information matrix, K-FAC, or natural gradient. The "anisotropic geometry" GRADE normalizes away is the geometry of the hidden-state Gram matrix — a different anisotropy, in a different role, than the Fisher-information ellipsoid that GIFT whitens. So on the evidence read, GRADE is the formally-distinct leg of the three-way question: it operates on gradients and borrows the words "rank" and "geometry," but its object is not shown by its authors to be the Fisher matrix that ties GIFT and Amari together. Whether such a reduction could be derived is addressed by no source found. See observation-gradient-geometry-shared-object-across-two-of-three.