Source
Sarkar, Mukherjee and Sato (2026), LIT-1820, §3, Table 1 and Figure 1 (pp. 3–4) of the arXiv preprint 2610.37656v1; read in NOTE-1331.
What was actually shown
The task. Fourteen entities carry hidden S4 frames. A random connected set of pairs is observed, each replaced with probability p by a different uniform element, and the task is the relation between an unobserved pair (p. 3). Relations are the faithful 4×4 permutation matrices, nothing is learned, and four rounds of the update are run.
The accuracies (Table 1, p. 4). Single path / uniform average of two-hop candidates / holonomy-weighted / Singer’s spectral method / spectral with one reweighting pass:
- sparse, degree 5, p = 0.2: 0.625 / 0.872 / 0.886 / 0.897 / 0.912;
- dense, degree 8, p = 0.2: 0.645 / 0.982 / 0.988 / 0.987 / 0.990;
- dense, p = 0.3: 0.488 / 0.912 / 0.937 / 0.933 / 0.949;
- dense, p = 0.45: 0.309 / 0.623 / 0.700 / 0.678 / 0.753.
The paired holonomy − uniform gains are +0.014, +0.006, +0.026 and +0.078. The paper locates the onset where clean two-hop paths become a minority, (1 − p)² = ½ at p ≈ 0.29 (p. 4). Scoring a candidate by a triangle that contains it costs 5–12 points. Composition is the capability. Routing is a refinement, and the spectral baseline with one reweighting pass is ahead on every row. The paper says so: “Holonomy attention is not a better synchronizer” (p. 3).
What could have failed. Routing could have added nothing over uniform averaging at high corruption, or avoiding-loop scoring could have done no better than literal scoring. Neither happened.
The AUC. Clean triangles score tr H/d = 1 exactly, and one corrupted edge always gives H ≠ I. So the only corrupted triangles that tie with clean ones are those whose corruptions cancel. The paper gives 1.4% at p = 0.3, and the AUC “sits at its ceiling 0.993” (p. 4). Recomputed here by enumerating S4: 0.01422, ceiling 0.99289. The commuting-score cap, ½(1 + (|G| − |[G, G]|)/(|G| − 1)), recomputed for S4, is 0.7609, matching the paper’s 0.76 (measured 0.759). The two-cluster AUC of 0.98 is of the same kind: within-cluster relations are built consistent and cross-cluster ones random.
What this does not say
- That holonomy attention works on learned or linguistic relations. The group and its representation are supplied. The paper names operators “generated from context” as the next step (p. 4).
- That the AUC measures a detector’s power. It measures how often corruptions cancel on data whose clean part is exactly flat.
- That loop-consistency is evidence of correctness in general. Here the truth is flat by construction, so a consistent reconstruction is a correct one. CASE-179’s arm that revises correspondences “by consistency alone” is the setting where the two come apart.