nucleation

TERM-091: relational vocabulary, of a class of descriptions: the arities and kinds of relation it can state (pairwise or higher-order, among states or among histories, of which declared types), and so which constraints are expressible in it at all

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Properties6
StatusActive
Version2
Tagscompositionality, mathematics, design-rationale
Linenucleating-argument
ComplementsTERM-090 — effective boundary constraint, of a composite: the relation its interconnected parts impose on the variables an outside system can reach, once the interior is eliminated; approximately instantaneous or history-dependent, and, independently, possibly of a kind the parts' relations could not state
Cited as “uses” byCLAIM-480 — Composing interactions can turn an interior into a constraint, and sometimes into a constraint of a new kind: in the chain E = ½k₁(x₁ − x₂)² + ½k₂(x₂ − x₃)², with x₂ internal and equilibrated, the boundary sees E_eff = ½·k₁k₂/(k₁ + k₂)·(x₁ − x₃)², a coupling between x₁ and x₃ that neither part has; if the interior relaxes slowly, eliminating it leaves a history-dependent constraint, a memory kernel, so interconnection fixes an effective constraint on the boundary's trajectories and separation of time scales decides only whether that constraint is approximately memoryless; and for cellular sheaves the reduced constraint can relate three boundary values where every relation among the parts is pairwise

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Definition

The relational vocabulary of a class of descriptions is the set of relations it can state. It has three dimensions:

  • arity: pairwise only, as in a sheaf on a graph, or higher-order, as in a hypergraph or a sheaf on a cell complex;
  • domain: relations among states, or among histories (a memory kernel, a behaviour in Willems’s sense);
  • type: which declared kinds of relation exist. A luria record’s vocabulary is its relation types and fields (CLAIM-479).

A constraint is expressible in a vocabulary if some description in the class states exactly it. Hansen and Ghrist’s reduced constraint v₁ = v₂ + v₃ is not expressible in the pairwise vocabulary of sheaves on the boundary graph (THEORY-1090).

The distinctions it draws

  • Inexpressible is not false or unknown. A description can fail because it states a relation that does not hold, because it lacks evidence, or because no description in its class states the relation at all. The last is a fact about the vocabulary. The response to it is to enlarge the vocabulary, not to revise a value. In CEGAR’s terms (NOTE-1231) it is the case refinement within a fixed answer space cannot reach.
  • Forced by the composite or by the class. A richer vocabulary may be needed because of what the composite does, or only because the class excluded what would have stated it, here the interior vertex. Telling these apart is part of QUESTION-283.

Why it is proposed

The owner’s review called the higher-order reduction “a remarkably direct mathematical analogue of the sort of event we keep encountering in Luria: the existing relation scheme cannot express what the composed organization requires.” Stating that analogy needs a name for what the scheme can express. CLAIM-419 and CLAIM-469 describe the record’s amendments as revising the grammar, and this term says what the revision enlarges. Adopted 2026-10-29: CLAIM-480 and TERM-090 state the demonstrated part of QUESTION-283’s answer in it, and “grammar” and “scheme” name only its third dimension, for a luria record. It would be revised if QUESTION-283’s answer needed a dimension these three do not give.

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