The claim
From MOM-CLM-144. Let a cluster of singular values be separated from the rest of the spectrum by a gap δ (TERM-046). The invariant is the subspace V they span, or equivalently the projector P onto V. Under a perturbation δT, P moves by at most about ‖δT‖/δ, which is Davis–Kahan. A single vector is its own cluster only when its own gaps are wide. Where values are degenerate, “the k-th vector” names nothing: any rotation within V is an equally good basis.
Why this record holds it
Davis–Kahan is standard. The 2026-10-11 re-run reproduced MOM’s table qualitatively (history note). The vector failed under degeneracy, with a maximum near a right angle, while the subspace held. Where the top value had a wide gap of its own, the vector was the more stable object, because the top-2 subspace’s boundary then fell in a narrow gap.
For this record it says what the unit is once a boundary test has been run. The unit is the block or cluster that a gap sets off, not any one member or coordinate. In the dancing couple that is the pair’s joint mode, not either dancer’s own. That application is this record’s.
What it does not say
- How large the gap must be. That is CLAIM-154.
- That spectral subspaces are the only candidates for units. It is a claim about what a gap makes definite in a linear description.
Note of 2026-10-12: evidence now linked
Two THEORYs now ground this claim. THEORY-084: a spectral threshold on the lazy-regime Fisher is well posed only at gaps between blocks, with guarantees stated only for whole eigenspaces. THEORY-078, weaker because its attribution is made by eye: a cut at the bulk edge keeps the span of the class means, not C discriminative directions.
Note of 2026-10-19: which part of THEORY-084 is established
Since the split of THEORY-084, only its first step is held Active, as CLAIM-377: the Gauss–Newton Fisher βJᵀJ and the NTK Gram matrix JJᵀ have the same non-zero eigenvalues with the same multiplicities, and the eigenvectors correspond as u ↦ Jᵀu, normalized. That gives this claim one exact instance. A gap in the Fisher’s spectrum is the same gap in the NTK Gram’s, and the subspaces the two gaps isolate correspond, so a gap test run on either is the same test. That the lazy-regime Fisher inherits the NTK’s harmonic-degree blocks, and that a threshold is well posed only between them, are still THEORY-084’s, Proposed. This claim does not need them: its core is Davis–Kahan.
Note of 2026-10-20: a gap-isolated subspace need not count slow variables
Nadler, Lafon, Coifman and Kevrekidis (NOTE-986, §5.2) show that a slow subspace isolated by a gap can be spanned by several functions of one slow variable, ψ and its powers, when the gap between that variable’s harmonics and the fast modes is wide. So the dimension of the subspace this claim makes definite is not the number of slow variables. The object is still the subspace. Reading off how many independent slow variables it holds needs a further step, which the gap does not supply.
Note of 2026-10-27: the edge says what the claim uses
The status review of this date (ADR-057, ADR-058) moved THEORY-078 and THEORY-084 from grounds to corroborated_by. As the note of 2026-10-19 says, “This claim does not need them: its core is Davis–Kahan.”
Note of 2026-10-29: Zhang et al. keep the subspace but do not require the gap
Zhang et al.’s SVD account of causal emergence, LIT-1129, is now read (NOTE-1399). Its coarse-graining keeps the span of the top r_ε singular vectors and only then clusters, so its object is the subspace, as this claim says it must be. But its definitions do not require a gap. Its Definition 4 gives vague emergence for any ε at or above σ_N and below σ₁, so every transition matrix whose singular values are not all equal shows it at some ε, and the degree varies with ε. The paper picks ε at “relatively clear cut-offs”. That is this claim’s criterion, used informally and not stated. The claim is unchanged.