The case
The monograph’s Example 3.4 (Part II, chapter 3): “Builder A places brick b in “red-sloped bin”. Builder B places b in “curved-connector bin”. There is no binning system in which both can be respected without contradiction.”
As descriptions. A sorting rule is a function from bricks to values, f: B → V_f. For any two, h(b) = (f(b), g(b)) is a sorting that refines both, and f and g are its two projections. So the bin red, two-wide, curved, connector respects both builders. MOM-CLM-137 makes the point in general: the product map is always a joint refinement, so the monograph’s definition of commutation failure is satisfied by no pair of sortings. MOM-ARG-047 adds that the events such sortings generate are a subalgebra of the power set of B, and so distributive; it is the lattice of partitions, not of events, that fails distributivity. A brick has all its properties at once. That is a hidden-variable model, the non-contextual case of CLAIM-037, and a compatible family of sortings glues as CLAIM-131 says sections of an event sheaf do.
As arrangements. A worked instance, fixed here for the claim it supports. The collection is 1,000 bricks: ten part shapes in ten colours, ten copies of each kind. There are ten bins, so the product’s hundred cells cannot each have one.
- By shape. Each bin holds a hundred bricks of one shape in ten colours. To fetch a red 2×4, open the 2×4 bin and find red.
- By colour. Each bin holds a hundred bricks of one colour in ten shapes. Open the red bin and find the 2×4.
If the remaining search is serial, both cost the same. The ten copies of the wanted kind sit among ninety others, and the expected number of others examined before the first copy is 90/11 ≈ 8.2. That figure is m/(c+1) for m distractors and c copies in random order, and the record re-checked it by simulation (CLAIM-307). The arrangements come apart in two ways the count leaves out:
- What the wanted brick must still be told apart by. In the unsorted pile, a red 2×4 is a conjunction: it shares its colour with red bricks of other shapes and its shape with 2×4s of other colours. Treisman and Gelade (LIT-1082, read) found such targets need a serial scan, about twice as slow when the target is absent. In either sort, the brick differs from its binmates in one attribute only, a feature search. In the shape sort that attribute is colour, which the paper found picked out at a glance when the colours are distinct (3.8 ms per item). In the colour sort it is shape, and whether a shape difference pops out depends on which feature differs. A difference only in size or proportion, like stud count, was slow in the paper’s ellipse experiment. So for many collections the shape sort leaves the easier search. Collections with near-identical colours narrow or reverse this: LEGO’s two greys, the older and the bluish ones, are the familiar instance. The paper’s displays were at most 36 well-spaced items with few colours, so this extrapolates to bins.
- What the builder asks for. A builder who wants any red piece that fits opens one bin under the colour sort and ten under the shape sort. A builder who wants any 2×4 has the opposite costs. A hundred bins would serve both, at ten times the bins. Finding the bin then becomes a search of its own: if bins are found by scanning their labels, the expected cost is about k/2 + N/(2k) for N bricks in k bins, least near k = √N. Kirsh’s hierarchical grouping (LIT-1076, NOTE-871) brings visual search from O(N) to O(log N). Labelled bins in a fixed place, which spatial memory finds at once, take most of that cost away.
What it can show
That “no shared sorting” is false of descriptions and true of arrangements under a budget. The builders do not disagree about any brick. They disagree about which partition the bins should realize. That choice is a trade among bins, search time and query mix, not a failure of global truth assignment.
What it cannot show
- Contextuality of any kind. Real contextuality needs observables with no pre-existing joint value. MOM-CLM-137 lists measurements that disturb, non-commuting operators, and partial functions on context-dependent domains. None of these is a LEGO brick.
- That builders in fact choose as the cost model predicts. Nothing was measured.