Source
Three papers by Xin Li (2026), read together on 2026-10-01 as one programme, at the owner’s request and charitably:
- Local Urysohn Width (arXiv 2603.15412 v1), LIT-376, read in NOTE-323: the first definition of width and its cleanest proofs.
- Structural Learning Theory (arXiv 2602.07974 v2), LIT-375, read in NOTE-324: width for contractive cells with a risk bound, the error floor, the spectral estimator.
- Structural Decoupling (arXiv 2506.20699 v2), LIT-374, read in NOTE-322 with a joint re-reading: the additive bound and the decoupling prescription.
What was actually shown
Width as defined first (LIT-376). For a margin-γ problem, the safe region is the union of the γ/2-neighbourhoods of the class sets. Local Urysohn width is the least number of connected sets of diameter ≤ D₀, each carrying a continuous classifier correct on the safe region it meets, needed to cover the safe region (Defs 3.2–3.3). On a bouquet of w circles with safe balls at the antipodes, width is exactly w when 3γ/2 ≤ D₀ < L/2 − 3γ/4 (Thm 4.1); with m balls per loop it is at least wm, so Ω(w·L/D₀) (Cor 5.1). Width and VC dimension do not determine each other (Thm 6.1). Ω(w log w) samples are needed (Thm 7.1, coupon collector). The proofs are correct (NOTE-323).
Why it is a covering number. Class safe regions are closed and more than γ apart, so on any connected patch Urysohn’s lemma supplies a correct continuous classifier. Local correctness is automatic, and width is the least number of connected D₀-diameter sets covering the safe region: labels enter only through γ (NOTE-323, C6). The bouquet lower bound uses only geodesic distance through the wedge point; Remark 8.11 says the obstruction “is metric (diameter bound) rather than cohomological”. The same argument gives width w on a star-shaped tree with w arms, where β₁ = 0. The bouquet’s safe region is disjoint arcs, so it does not meet the homological condition of the Betti bound (Thm 8.5), and the general Betti bound is a conjecture (Conj 8.9).
The contractive-cell version (LIT-375, LIT-374). Width is redefined as the least open cover by cells on each of which some predictor is γ-contractive with risk ≤ δ. Below width there is an n-independent error floor η(w, K) under Assumption 2.3, which states that any K-cell cover mixing two basins has excess risk ≥ η (Thm 2.4). That is close to definitional. The spectral estimator converges to a count relative to a given predictor, not to width (Thm 3.5). The bound R̂ ≤ √(2 log Δ_str(n)/n) + LK·R̂_n(G) for routed predictors holds, with a fixable gap in its proof (LIT-374, Thm 4.5).
What could have come out otherwise. The bouquet bound could have used the loops’ homology. It uses distance only.
What this does not say
- That topology forces classifier complexity. The abstract and §10 of LIT-376 say so; the proof is metric confinement, as the paper concedes.
- That a monolithic model needs w experts. Remark 3.4 of LIT-376: a single global Urysohn function makes every margin problem width 1. “Any system that maintains fewer than uw experts cannot correctly classify” (§9) restates the definition’s locality constraint. The question whether a capacity-matched monolithic network shows the floor is open (NOTE-324).
- That routing should be trained apart from the task loss. LIT-374 reads the additivity of an upper bound as a rule that its two terms be reduced by separate mechanisms on separate signals (Prop 4.7, §4.2). The bound holds for joint training too, and an upper bound’s form does not fix the error’s.
- That width is estimated. The contractive-similarity Laplacian counts prediction discontinuities of a fixed predictor, w_G(P), and no result ties that to width (NOTE-324, C6).
- That the alignment section follows. LIT-374’s recasting of alignment as scaffold alignment is asserted, with internal contradictions.
Connections
- LIT-373 (“The two dragons of cognition”, Rejected) is the programme’s informal origin. LIT-376 concedes the global-Urysohn triviality that LIT-373’s linear-readout claim missed; none of the three papers repairs LIT-373’s other grounds (NOTE-322).
- THEORY-028. Every guarantee here is class-level or assumes correct routing. A claim about which gradients should update which parameters would need algorithm-dependent analysis of that account’s kind, and the programme has none.
- THEORY-019. The spectral estimator reads structure from eigenvalue multiplicity at zero, which counts components, not a group; consistent with that account’s point that multiplicities carry dimension counts.
- THEORY-039. LIT-374 identifies grokking with a width transition by shape only, which that account warns against.