Grain

Same Resolution, Different Ladder

Seven schemas resolving one hundred and forty-four listings reach between six and fifteen result sizes. Grain is a third figure.

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Seven schemas resolve one hundred and forty-four listings. The section on cells sorted them by how many questions they permit. Sort them by grain instead and the order changes.

Schema	Questions	Subsets	Sizes
4·6·6	245	8	6
3·3·4·4	400	16	9
2·2·6·6	441	16	9
2·3·4·6	420	16	13
2·2·2·3·6	756	32	13
2·2·3·3·4	720	32	15
2·2·2·2·3·3	1296	64	15

Resolution is one hundred and forty-four in every row. Grain runs from six to fifteen, and the ordering by grain is not the ordering by questions.

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The class of seven, by grain

one bar is one facet; the divisions are its values
cells: how many listings the schema can tell apart
queries: how many different questions it lets anybody ask
sizes: how many different result sizes are reachable

Seven schemas drawn; all of them tell the same 144 listings apart, across four different facet counts.

The same seven schemas as the first section, sorted by the last figure instead of the second. Resolution identical throughout; reachable sizes from six to fifteen.

Computed from the schema alone. How many facets and how many values in each is a stated convention; the three figures are products of whole numbers.

Reading the table as a choice

Somebody building a board that needs to resolve one hundred and forty-four kinds of listing has seven structures to pick from, and the table is the whole decision laid out.

Three filters at the top, six at the bottom. Questions from two hundred and forty-five to one thousand two hundred and ninety-six. Grain from six to fifteen. Nothing else about the seven differs, because the resolution is identical by construction.

Seven options, three figures each, and the comparison takes a minute once the figures exist. Without them it is a conversation about how busy a sidebar should look.

Where the orders disagree

Two, three, four and six permits four hundred and twenty questions and reaches thirteen sizes. Two, two, six and six permits four hundred and forty-one questions and reaches nine.

So the schema with fewer available questions offers a reader more positions on the ladder of coarseness. The two figures rank these boards in opposite directions, and both rankings are correct about what they measure.

Grain is not a by-product of the other two figures. It can be computed only from the facet sizes themselves, and schemas agreeing on both other figures can still differ on it.

Why two, three, four, six does well

Because its sizes are varied. Four distinct values across four facets, with the only collision being the two-times-six against three-times-four identity.

Two, two, six and six repeats both of its sizes. Every pair of identical facets collides immediately, and the products pile up on a few values: the reachable sizes are one, two, four, six, twelve, twenty-four, thirty-six, seventy-two and one hundred and forty-four.

Nine sizes, from sixteen subsets, and the whole loss comes from the repetition.

The six-facet row

Four toggles and two three-way choices reach fifteen sizes from sixty-four subsets. That is the best grain in the class — tied with two, two, three, three and four, which uses five facets to get there.

Fifteen out of sixty-four is still a heavy collapse: forty-nine subsets are producing a size something else already produced. It is simply less heavy than six out of eight, which is what the three-facet board at the top of the table manages.

So the six-facet board wins on grain, on questions, and ties on resolution. On this showing it is the best of the seven — and it is also the board asking a reader to make six separate decisions to reach a single listing, with under one question in nine doing so.

Why grain is not derivable from the pair

The pair identifies the schema, and the schema determines the grain. So in principle grain adds nothing once both figures are known.

In practice it adds a great deal, because recovering the schema from the pair takes a factorisation and counting the grain then takes an enumeration of subsets. Neither is hard; both are more work than the two multiplications that produce the pair.

So grain is best treated as a third measurement rather than a derived one: computed when it matters, from the facet sizes directly, and left out when it does not.

The four figures together

Schema	Cells	Questions	Sizes	Share
4·6·6	144	245	6	144/245
2·2·2·2·3·3	144	1296	15	1/9

Two boards, identical resolution. One permits a fifth as many questions and reaches fewer than half as many coarsenesses, and pins a single listing with more than half its questions against one in nine.

There is no ranking here that survives contact with all four columns, and that is the honest end of the comparison. The record supplies the figures; which of them matters is a question about the field, the listings and the readers, and none of those is in a schema.