nucleation

TERM-093: correspondence structure, and how it is held: the maps that say which local states or descriptions of interacting systems count as agreeing, distinct from the physical coupling between them, and held as formed, discovered or adapted

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Properties7
StatusActive
Version3
Tagscomplex-systems, mathematics, metaphysics
Linedistributed-agency
ComplementsTERM-089 — apparatus, in the operational sense: a realized organization that selectively couples differences in one system to distinguishable outcomes in another; by function it may differentiate, retain, enforce and revise, and the four come apart
Cited as “complements” byTERM-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, TERM-097 — coherence, of a family of descriptions, transports or judgements: their agreement with one another under a stated correspondence structure (on overlaps, around cycles, between routes), distinct from fidelity to a source, from integration by dynamics, and from knowledge of the world
Cited as “uses” byCLAIM-471 — Physical interconnection can generate effective constraints on joint behaviour; measurement can estimate correspondences among local descriptions; and adaptive apparatuses can revise those correspondences; gluing expresses compatibility under specified relations, but neither the relations' physical generation nor their adequacy follows from gluing alone, CLAIM-474 — From interconnection to individuation, the relationships are distinct achievements that form a partial order, not a nested sequence: none implies the next, and a later one need not require every earlier one. Interconnection fixes what joint behaviour is possible, compatibility which local descriptions admit a coherent whole, dynamics which organizations persist, individuation which have sufficiently robust identity and boundaries, organizational autonomy which help maintain their own constitutive constraints, and inquiry which can measure and revise their own models and apparatuses under resource bounds, inquiry and reconstruction occurring without autonomy. The research program is to find the conditions for each, and their dependencies, apart from the order in which an inquiry meets them, CLAIM-538 — On a finite graph of contexts whose overlaps carry preferred real offsets a and tolerances δ_e ≥ 0, an assignment within every tolerance, |Dx − a| ≤ δ, exists exactly when no cycle's holonomy exceeds the sum of its tolerances; with every δ_e > 0 the least uniform enlargement of the tolerances that admits one is the largest ratio, over simple cycles, of holonomy to summed tolerance, so the obstruction is certified by a cycle; the feasible allocations of tolerance depend on the offsets only through their class in H¹, so local adjustment can move a mismatch among the overlaps of a cycle and never remove it; and an evaluation that omits one overlap of every witnessing cycle reports a compatibility that does not hold

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Definition

A correspondence structure on a family of locally described systems is the family of maps ρ_{i,e}: V_i → W_e that send each system’s local description into a space shared with a neighbour. Two local states agree on the edge e when ρ_{i,e}(x_i) = ρ_{j,e}(x_j). Collected over the network, the maps define a coboundary δ, and the jointly compatible assignments are its kernel, the global sections (Hansen and Ghrist, NOTE-1233). “Compatible”, “glued” and “frustrated” are all relative to a correspondence structure.

It is held in one of three ways (CLAIM-471):

how heldwhat it isexample the record has read
formedphysical interconnection fixes the maps: joining systems identifies some of their variables, so which local quantities must agree, and on what shared space, is set by how they are connected, while the law that holds at a junction is fixed beforehandBaez and Fong’s wiring, NOTE-1223: a map of terminals such as {1, 2} → {3} sets the joined potentials equal and adds the currents (Example 5.17), under the same Kirchhoff condition at every junction
discoveredmeasurement estimates a correspondence that already held; it is established only where the estimate is checked against evidence the measurement did not produce (CLAIM-143)Singer’s relative angles, NOTE-1224; Robinson’s geometric maps, NOTE-1221
adaptedan apparatus revises its own rules of comparisondiscourse sheaves, NOTE-1222

Forming sets which maps hold where. The joint behaviour they admit is their kernel, a constraint on states, and that is what CLAIM-471’s physical constraint formation names; the term keeps the two apart as map and kernel. Whether interaction among systems already connected ever changes the maps themselves, rather than which fixed kind of map holds on which edge, is QUESTION-282.

The distinctions it draws

  • From coupling. The coupling is what moves the states, for instance a dissipative Laplacian flow. The correspondence structure is what counts as agreement. A coupling attracts to the specified compatible space only if it enforces that structure (CASE-179). Setting the two apart is what makes the proposed study informative.
  • From a representation of it. An apparatus’s model of the correspondences is not the operative structure (CLAIM-471; A23 §IX: ĉ_t ≠ c_t in general). An adapted structure changes the apparatus. It changes the systems compared only if it is implemented.
  • Between the three ways of holding it. Only formation bears on the thesis that interaction generates constraints. Discovery is knowledge of a correspondence, and its adequacy needs evidence the correspondence does not itself produce (CLAIM-143). Adaptation can succeed at agreement by changing what is compared (THEORY-1088).

Why it was proposed

The word was doing three jobs in one claim before the owner’s review separated them, and four entries filed the same evening use it. QUESTION-282 asks when interaction changes the operative structure, and it needs the term to ask it. What would settle it: adoption if claims cite it to say which sense they mean; rejection if “gluing” and “compatibility” carry the distinction without it.

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