nucleation

CLAIM-310: Sorting schemes over objects with determinate properties never conflict as descriptions, since the product of any sortings refines them all; they conflict as arrangements, since a fixed number of bins realizes one partition when each item is stored once, and which costs least depends on what is searched for and on what stands out within a bin

6 min read

Properties11
StatusActive
Version2
Rolethesis
Warrantdemonstrated
Tagsinformation-retrieval, contextuality, cognition
Lineformal-apparatus, distributed-agency
GroundsCASE-095 — Two builders sorting one collection: the brick that one bins as red and two-wide and the other as a curved connector has a bin that respects both, and what the builders cannot share is an arrangement, not a description, CASE-094 — The LEGO bin: a brick owned in one copy and buried among hundreds of similar ones may as well not be owned, a black brick among white ones needs no bin of its own, and every brick can be found by going through the whole collection, THEORY-358 — In visual search among up to about 36 well-spaced items, a target that alone carries one colour or one distinctive shape part is found in time almost independent of the number of items, while a target identified only by a combination of features its neighbours carry separately takes time that grows linearly with the number of items, about twice as fast when the target is absent, the pattern of a serial self-terminating scan, and practice did not remove it; that conjunctions rather than similarity cause it is the paper's reading, not its result, THEORY-370 — On an n × n lattice whose nodes also get long-range links drawn with probability proportional to lattice distance to the power −r, a searcher that knows only the target's position and the links of nodes already reached delivers in expected time polylogarithmic in n only at r = 2: greedy routing then takes O((log n)²) steps, while for 0 ≤ r < 2 every such searcher needs Ω(n^((2−r)/3)) steps and for r > 2 Ω(n^((r−2)/(r−1))), even at r = 0 where short paths exist, THEORY-384 — A nearest-neighbour graph built by pruning each point's links against all other points with a factor α > 1, so that from every point some link is α times closer to any other point, lets greedy search from any start reach a ((α+1)/(α−1) + ε)-approximate nearest neighbour in O(log_α(∆/((α−1)ε))) steps of cost O((4α)^d log ∆), for doubling dimension d and aspect ratio ∆; the ratio is tight and the dependence on ∆ cannot be replaced by one on n, and the build takes O(n³) time done naively, THEORY-361 — When every object but one has full grade in exactly one of two attribute-sorted lists and the one has full grade in both, finding it under min takes at least about half the objects' worth of accesses for any algorithm that only reads the lists from the top and looks up objects already seen, and, with the tied objects in random order, about as many in expectation even for algorithms that may look up any object; a query on one attribute alone costs k accesses whatever the number of objects, THEORY-372 — For c-approximate near-neighbour search over n points in Euclidean space, space n^{1+ρu+o(1)} and query time n^{ρq+o(1)} are achievable on any data set whenever c²√ρq + (c²−1)√ρu ≥ √(2c²−1) (c√ρq + (c−1)√ρu ≥ √(2c−1) in Hamming space), and no list-of-points structure, the model of known data-independent hashing, does better on random instances; without that model the curve is matched only at query time n^{o(1)} by one or two cell probes, THEORY-366 — With a fixed number of data points and a query drawn independently of them, if the variance of the p-th power of the query-to-point distance divided by its mean tends to zero, the farthest point comes within any factor 1 + ε of the nearest with probability tending to one; independent coordinates meet this condition, while data on a low-dimensional subspace, or queries that fall inside well-separated clusters, escape it
Objected byCLAIM-313 — A fixed number of bins realizes one partition only when each item is stored once: with copies, overlapping partitions coexist, as hashing-based similarity search stores each item in many partitions and pays for it in space, so sortings conflict as arrangements only under a one-copy constraint, which binds bricks but not index entries
ComplementsCLAIM-131 — A compatible family of sections of the event sheaf always glues, so where a communicative object lacks a global realization the obstruction lies in its supports or distributions, not in the event sheaf, CLAIM-037 — Formal contextuality is a failure of global extension, distinct from ordinary context dependence, and needs Contextuality-by-Default when marginals shift with context
CorroboratesCLAIM-296 — What a system has, what it knows it has, and what it can deploy within a task's deadline are different, and the last is a relation among the resource, the agent, its environment, the task, the deadline and the cue, not a property of any one of them
Illustrated byCASE-094 — The LEGO bin: a brick owned in one copy and buried among hundreds of similar ones may as well not be owned, a black brick among white ones needs no bin of its own, and every brick can be found by going through the whole collection, CASE-132 — One complaint, two taxonomies: a report of an unsafe practice is correctly classified both by equipment and by workers' obligations, the intake procedure needs one primary class, and the class chosen decides whether the report reaches a body that can change working conditions

Graph View

The claim

As descriptions, no conflict. For sortings f: B → V_f and g: B → V_g of the same bricks, h = (f, g) refines both. MOM-CLM-137: “the product map is always a joint refinement”, so no pair of sortings fails to commute, and a brick’s having all its properties at once is “exactly the non-contextual” case. That is CLAIM-037’s distinction between formal contextuality and ordinary context dependence, and CLAIM-131’s observation that compatible sections of an event sheaf glue. Two builders who bin the same brick differently are attending to different coordinates of one value.

As arrangements, a real trade. A set of k bins realizes one partition of the collection at a time when each item is stored once, as a brick is. Where items can be copied, several partitions coexist at a price in space (CLAIM-313). Three things then decide which partition serves a builder, and none of them is a fact about any brick (CASE-095):

  1. The bin budget. The product of the sortings serves every query and may need far more bins than are available. If bins are found by scanning, more bins also means more scanning. The expected cost k/2 + N/(2k) is least near k = √N.
  2. The query mix. A query that names the sorted attribute opens one bin, and a query that leaves it open opens all of them. Builders who ask for different things are best served by different partitions.
  3. What the wanted item still has to be told apart by. Treisman and Gelade (LIT-1082, read in NOTE-897) found that a target with one feature no distractor shares is found almost regardless of how many items there are: 3.8 ms per item for a colour, 2.5 for a shape part, up to 30 items. A target that differs from each distractor only in a combination of features the distractors carry separately needs a serial scan: 28.7 ms per item when present, 67.1 when absent. Practice did not change this. In an unsorted pile, “the red 2×4” is such a conjunction, among red bricks of other shapes and 2×4s of other colours. Within a bin sorted on one attribute, it differs from its binmates in one attribute only. So sorting on either attribute turns a conjunction search into a feature search. The arrangements then differ in whether that one remaining attribute is a feature the eye picks out at once. A distinct colour is (CASE-094’s black brick is the paper’s easiest case). A difference only in size or proportion, which is what separates many bricks of one colour, was slow in the paper’s ellipse experiment, though not item by item. Which attributes pop out depends on the collection and on the searcher.

What survives of the monograph’s “Sheafification as Descriptive Compromise”. Builders who must share one set of bins can take the common refinement (the meet of their partitions, which costs bins) or the finest common coarsening (the join, which loses distinctions one of them needed). The partition lattice is where the non-distributivity the monograph looked for lives (MOM-ARG-047). It is a lattice of schemes, not of events, and the compromise is priced in bins and lost distinctions, not in truth.

Why it is Active

Part 1 is elementary, and part 2 is a definition. Part 3 now rests on a read source, though in a narrower form than first filed. That version said similarity across binmates makes search serial. Treisman and Gelade argue against exactly that: their serial search comes from conjunctions, and similarity on one dimension made search slow without making it serial. The similarity account, Duncan and Humphreys (LIT-1081), could not be obtained (Unavailable), so the record holds the disagreement with one side read. What is Active is the dependence: the cheapest arrangement turns on the bin budget, the query mix and what the wanted item must still be told apart by. The magnitudes are not Active. The paper’s displays were at most 30 to 36 well-spaced items with few, distinct colours, not bins of hundreds of overlapping bricks, and no one has fitted the model to a collection.

What it does not say

  • Not that sorting is never contextual. Observables with no pre-existing joint value can be. MOM-CLM-137 names disturbing measurements and partial functions on context-dependent domains. LEGO bricks are not among them.
  • Not that arrangement changes what is true of the collection. It changes what can be found in time, which is CLAIM-296’s subject. This claim corroborates that one (ADR-044): changing the arrangement alone changes the cost of retrieval, which meets CLAIM-296’s defeat condition head-on. CLAIM-296 does not rest on it.

Note of 2026-10-19: the one-copy condition

CLAIM-313, now Active, showed that “a fixed number of bins realizes one partition” is a fact about items that cannot be copied, not about bins. Hashing-based search stores every point in many tables, each a partition. The title and the arrangement clause now carry the condition. The trade this claim describes is unchanged for bricks. For copyable items it becomes a trade of space against query time.

Note of 2026-10-31: concurrent significations, and usurpation as a one-copy constraint

Bencherki (LIT-1901, pp. 782–783; THEORY-1285) holds that an action has as many significations as the individuations it takes part in, and that “Concurrent significations … are not mere misunderstandings”. The parent paper goes further: “Coexisting meanings of the same action indicate that several individuations are ongoing” (LIT-1902, p. 1358; THEORY-1278).

This claim’s first clause says why coexistence costs nothing. Judgements of one action that are each functions of its determinate features jointly refine, so they never conflict as descriptions. The spouses’ dental plan is “political activity” to the base commander and a legitimate demand to the spouses (p. 793). Those are two coordinates of one action. So coexistence alone does not tell several individuations apart from one action seen by several receivers, and the inference at p. 1358 needs more than it states.

The second clause, with CLAIM-313, says where conflict does arise: in an arrangement that holds each item once. The paper’s politics lies there. Table 1 (p. 787) asks whether an action may count toward the person’s individuation, the organization’s, or both. Its conflicting cell, usurpation, is volunteer-run daycare credited to the army and not to the volunteers, which is an attribution held once. Its harmonious cell, co-individuation, is the checklist that is at once the community’s work and a service “we offer”, which is the attribution copied. Read across, contested authorship is a one-copy constraint on credit, not a clash of meanings. This is the same structure, not shown to be the same mechanism. Whether organizational credit is exclusive is an empirical matter, and the paper’s open problem stays open: telling usurpation from legitimate incorporation without a prior standard of whose action it was. The distributed-agency line may want this as a claim of its own. No status, title or warrant changes.

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