Source
Bodnar et al. (2022), LIT-1811, §3 (pp. 3–6) and Appendices A–B (pp. 15–21) of arXiv 2202.04579v4; read in NOTE-1332.
What was actually shown
The setting. A connected graph with stalks ℝᵈ, the normalised sheaf Laplacian Δ_F, and diffusion Ẋ = −Δ_F X. In the limit the features are the orthogonal projection of the initial features onto ker Δ_F (Lemma 21, p. 16, from Hansen and Ghrist; THEORY-1070). A class of sheaves has “linear separation power” over a family of graphs if for every labelled graph in it some sheaf of the class makes the limit linearly separable for almost every initial condition (Definition 7, p. 5).
The harmonic space of an orthogonal bundle. For O(d) restriction maps, dim H⁰ ≤ d, with equality exactly when transport along paths is path-independent (Lemma 6, p. 4). This is the O(d) counterpart of THEORY-1082’s U(1) zero-mode criterion.
The hierarchy.
- Symmetric maps, d = 1, are the weighted graph Laplacians (Prop. 20). On a connected bipartite graph with |A| = |B| they cannot separate A from B for any initial condition (Prop. 9). The harmonic vector is the square root of the weighted degree, and with F_v = F_u on every edge the two sides’ weighted degrees sum to the same total, so the A-values cannot all exceed the B-values (p. 17).
- Signed maps, d = 1: −α_e at the A end and +α_e at the B end of each edge separates any two classes on any connected graph; the transport is −1 across classes and +1 within (Prop. 10). Same-signed maps fail on two nodes (Prop. 23).
- Any d = 1 sheaf fails for three classes: dim H⁰ ≤ 1, so the limiting features are multiples of one vector and some class lies in the convex hull of two others (Prop. 11). Feature channels do not help.
- Diagonal maps with d ≥ C separate C classes, one-versus-rest per coordinate (Prop. 12).
- Orthogonal bundles: if the transport from a base node to each class is a matrix from a set {±P₀, …, ±P_{d−1}} with P₀ = I, P_kᵀ = −P_k and P_kP_l = −P_lP_k (points (1)–(5), pp. 20–21), the classes’ limits are mutually orthogonal or antipodal for every projection coefficient, and up to 2d classes separate (Prop. 13). On regular graphs, rotations by multiples of 2π/C separate any C (Prop. 28).
The correction. The paper says such matrices exist “only in dimensions two and four” because the octonions are non-associative (p. 21). The conditions are those of the Hurwitz–Radon theorem: d − 1 anticommuting orthogonal complex structures on ℝᵈ, which exist exactly for d = 1, 2, 4 and 8. At d = 8 the left multiplications by the seven imaginary octonion units satisfy all five points. I built them, checked each condition, built the path-independent bundle on a random connected 60-node graph with 16 classes, and projected a random initial signal onto ker Δ_F (dimension 8, as Lemma 6 says): every class separated from the rest. So Prop. 13 holds for d ∈ {2, 4, 8}; what happens at other d is open.
Numerical checks. Prop. 9 and Prop. 10 on a 10-node balanced bipartite graph (the symmetric harmonic vector did not separate; the signed one gave opposite signs on the two sides). Prop. 3’s bound λ₀ ≤ r²/2 held on 300 random O(2) bundles.
What could have failed. A symmetric sheaf could have had a harmonic vector ordering the sides of a bipartite graph; the counting identity rules it out. A one-dimensional sheaf could have had a two-dimensional harmonic space; Lemma 6 rules it out on a connected graph with invertible maps.
What this does not say
- That a separating sheaf can be found without the labels. Every construction (Props. 8, 10, 12, 13, 28) assigns maps by class. The results bound what a class of sheaves can express. The learning problem, recovering such a sheaf from features, is not addressed, and Prop. 18 (an MLP can express any sheaf) does not address it either.
- That the geometry found is the system’s. “The right sheaf” is the one that solves the task. In QUESTION-282’s terms the correspondence is fixed by the target, not discovered; in QUESTION-130’s, it is chosen.
- Anything at finite depth. The results are for the projection; the paper’s remark that exponential convergence makes them “relevant” to finite layers (p. 4) gives no rate.
- That linear separability is accuracy. The authors note that a model that cannot perfectly fit can still score well, and that one-layer GCNs do well on bipartite graphs (p. 10).
- Anything for d outside {1, 2, 4, 8} in the orthogonal case, either way.