Grain

The Twenty-Seven That Do Not Collapse

In twenty-seven schemas every subset of filters produces a distinct result size. All of them have four facets or fewer, and most of them contain a five.

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Twenty-seven of the four hundred and sixty-one schemas reach every result size their structure could possibly offer: the number of distinct sizes equals the number of subsets exactly.

Facets	Schemas without collapse
one	5
two	10
three	9
four	3
five	0
six	0

The single facets are trivial — two subsets, two sizes. The interesting end is the four-facet row, which holds exactly three schemas.

A fair trial begins when the team can see the platform in context, understand every active setting and agree how exceptions will be discussed. For a separate technical reference, see Astronomer.

Four facets, no collisions at all

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

Four schemas drawn.

The first three are the only four-facet schemas where every subset of filters gives a different result size. The fourth has the same number of facets and reaches five sizes from sixteen subsets.

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.

The three survivors

Two, three, four and five. Two, four, five and six. Three, four, five and six.

All three contain a five, and that is not a coincidence. Five is the only facet size in this record that shares no factor with any other: two, four and six are all even, three and six share a three, and five stands alone.

So any subset containing the five produces a product divisible by five, and no subset without it can match. The five partitions the sixteen subsets into eight that carry it and eight that do not, and no collision can cross that line.

Five is the only size in the record that cannot be built from the others, and the schemas that avoid collapse are the ones that lean on it.

Why five facets is already too many

There are only five sizes available. A schema of five facets either repeats a size — which collapses products immediately — or uses each size once.

Using each exactly once gives two, three, four, five and six. Its subset products collide anyway: the subset holding two and six multiplies to twelve, and so does the subset holding three and four. Two different ways of leaving filters alone, one result size.

Four facets can avoid that by leaving out one of the colliding pieces. Five facets cannot, because all five sizes are present and the collision is built into them.

What the twenty-seven have in common

They are small. None has more than four facets, and their resolutions run from two cells up to three hundred and sixty.

That is the trade in its plainest form. A board can have every result size its structure allows, or it can have substantial resolution. Not both, inside these bounds — and the arithmetic that forces it is just the shortage of sizes that do not share factors.

Schema	Cells	Sizes	Subsets
3·4·5·6	360	16	16
2·3·4·5·5·6	3600	39	64

The second resolves ten times as much and reaches more than twice as many sizes — and it is collapsing sixty-four subsets into thirty-nine rather than running clean. The clean board is the smaller one.

The ten two-facet cases

Ten schemas of two facets have no collapse, which is every pair except the five that repeat a size.

Two facets of different sizes give four subsets — nothing, each one, and both — and their products are one, the first size, the second size, and the product. Those four are distinct unless the two sizes are equal.

So at two facets the rule is simply: do not repeat. At three facets nine of the thirty-five survive, and at four only three. The condition tightens fast, because every extra facet adds subsets that must all avoid colliding with everything already there.

Why no five-facet schema survives

Five facets means either a repeated size or all five sizes used exactly once.

A repeat collides immediately: two facets of the same size give two single-element subsets with the same product.

All five used once gives two, three, four, five and six — and the subset holding the two and the six multiplies to twelve, as does the subset holding the three and the four. The collision is unavoidable because it is built into the sizes themselves.

So at five facets both routes fail, and at six there is no route at all.

What to do with this

Nothing directly. It is a structural fact rather than a recommendation.

What it is good for is calibration. Perfect grain is not a target anybody should aim at, because reaching it means staying under four facets, and four facets of mixed sizes cap out at three hundred and sixty cells.

The schema with the most reachable sizes in the whole record collapses most of its subsets and is the better board for it. Grain is worth having more of; it is not worth having all of, and the twenty-seven schemas that have all of it are the demonstration.