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LIT-1803: Relational Composition as Parallel Transport: What Per-Entity Frames Force in Attention

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Properties4
StatusActive — skimmed 2026-10-29 (NOTE-1326) as the companion LIT-1820 cites for its premise. Worth keeping for its own statement that per-entity frame freedom is an inductive bias, and for the parsed-CLUTRR control in which forcing orthogonal transports costs 64.6 points out of distribution.
Tagscompositionality, representation-learning, mathematics, anthology-candidate
Cited as “paper” byNOTE-1326 — Sarkar, Mukherjee and Sato — Relational composition as parallel transport
Cited as “source” byTHEORY-1140 — If every entity's O(d) frame may change independently and relation transports change as Tᵢⱼ ↦ gᵢTᵢⱼgⱼ⁻¹, then functions of the transports alone carry no invariant information except around cycles, a triangle's invariants are exactly the class functions of its holonomy H = TᵢⱼTⱼₖTₖᵢ, and an equivariant update of an edge from one cycle through it is Ψ(H)Tᵢⱼ with Ψ conjugation-equivariant; the symmetry fixes a family of updates, not one mechanism, and says nothing about whether a task has the symmetry

Graph View

Sarkar, A., Mukherjee, A., & Sato, M. (2026). Relational composition as parallel transport: What per-entity frames force in attention. Preprint. ARXIV-2610.34945

Key takeaways

  • Per-entity frame freedom is adopted “as an inductive bias”, and the paper says it is not derived (pp. 2, 12–13).
  • Under it, invariant scores are ϕ(‖xᵢ‖, ‖xⱼ‖, ⟨xᵢ, Tᵢⱼxⱼ⟩) and equivariant messages lie in span{xᵢ, Tᵢⱼxⱼ} (Theorem 1). A covariant transport computed from the endpoints alone has rank at most the endpoints’ rank (Theorem 2), so one vector per entity can never give an orthogonal transport: “relations must be supplied”.
  • Which transports a task needs is “decided by the algebra of its relations” (p. 3). Orthogonality helps on StepGame and costs 64.6 points on CLUTRR, where the paper’s motivation is that a nonidentity idempotent has no invertible faithful representation (p. 9, App. B.4).
  • With matched anchors B = QA, an orthogonal transport is identified only on the anchored subspace (p. 9).

Standing in the record

It is the source of the premise that LIT-1820 calls “the task’s”, and its own CLUTRR control is the plainest evidence that the premise is task-relative. That bears on CLAIM-098 and CLAIM-035, which hold that translation is generally not a group action. Only the parts named in NOTE-1326 were read. anthology-candidate, as for its companion.

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