Gradient 'geometry' is one shared object — the Fisher/K-FAC matrix — across GIFT and Amari, but a formally distinct construction in GRADE: not one object across all three, nor three unrelated analogies

The vault held three notes that each read a gradient's geometry rather than its magnitude, at three layers of the ML stack — GIFT (communication / quantization), Amari (optimization direction), and GRADE (inference diagnostic) — and asked whether these are one mathematical object or three superficial analogies (question-gradient-geometry-one-object-or-three-analogies). Reading each paper's own method section resolves it to neither pole cleanly:

• GIFT and Amari are the same formal object. Both build on the Fisher information matrix in its K-FAC-factored form, on GIFT's own explicit, repeated statement (claim-gift-isotropy-transform-derived-from-fisher-kfac). They deploy it for two different operations — Amari to redirect the optimizer step, GIFT only to re-coordinate a gradient before lossy compression — a boundary GIFT itself insists on (claim-gift-restricts-fisher-kfac-object-to-communication-coordinates).
• GRADE is not shown to share that object. Its stable-rank ratio is a normalized gradient-covariance spectrum with no established formal reduction to Fisher information in its own method section (claim-grade-stable-rank-not-reduced-to-fisher); it cites Fisher work as inspiration, not derivation.

So the honest answer is: one shared object across two of the three (GIFT, Amari), and a formally distinct — so far unreduced — construction for the third (GRADE). Two residuals bound this: whether GIFT's Fisher factor is the true or empirical Fisher tightens or loosens the GIFT–Amari identity (question-gift-fisher-factor-true-or-empirical); and whether GRADE could be reduced to a Fisher/K-FAC quantity is addressed by no available source.

This is the same epistemic move the vault tracks in other domains — a shared formal object versus a shared vocabulary — as in claim-perceptual-color-space-not-riemannian-bujack-2022 (is perceptual color space really Riemannian?) and observation-low-dimensional-subspace-constrains-adaptation-brains-and-nets (is the low-dimensional adaptation subspace one object across brains and nets?).