Grain

Sixty-Four Ways To Ask, Seven Answers

Six identical filters give sixty-four subsets and seven reachable result sizes, whatever the filters are. The collapse is caused by repetition, not by size.

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Take a board with six filters of two choices each. Sixty-four cells, seven hundred and twenty-nine questions.

A query leaves some subset of the six toggles alone. There are sixty-four such subsets. The result size is two raised to however many were left — so the only sizes reachable are one, two, four, eight, sixteen, thirty-two and sixty-four.

Anyone applying a counting model to people should look at the setup details and preserve the difference between a measurable signal and a judgement. For a separate technical reference, see PyTorch.

Seven sizes, from sixty-four subsets. Fifty-seven of the subsets are producing a size some other subset already produced.

The same size, reached two ways

one cell: one combination of values
the heavy outline with corner ticks: what the query returns

Four grids drawn; one of them sets every facet and so reaches a single cell; one sets none, and so returns everything.

A result of four cells arrives from a four-value facet left alone, and also from two toggles left alone. The board cannot tell a reader which route produced the number.

Computed from the two facet sizes. A query sets a facet to one value or leaves it unset: that is the convention, not a claim about how anybody searches.

The same collapse at the other end

Six filters of six choices: forty-six thousand six hundred and fifty-six cells, one hundred and seventeen thousand six hundred and forty-nine questions.

The reachable sizes are the powers of six: one, six, thirty-six, two hundred and sixteen, one thousand two hundred and ninety-six, seven thousand seven hundred and seventy-six, and forty-six thousand six hundred and fifty-six.

Seven again.

Schema	Cells	Subsets	Sizes
2·2·2·2·2·2	64	64	7
6·6·6·6·6·6	46656	64	7

The two boards differ by a factor of more than seven hundred in resolution and agree exactly on grain. The collapse does not care how big the filters are; it cares that they are all the same.

Repetition is what collapses the grain. Six identical filters give seven sizes whether each offers two choices or six.

Why repetition does it

Because the products of a repeated number are its powers, and a schema of six identical facets has only seven of those: the number raised to zero through six.

Any two subsets of the same size produce the same product. There are fifteen subsets of size two among six filters, and all fifteen give the same result size. The information about which two filters were left alone is real and is not reflected in the size at all.

What a reader loses

Fine adjustment.

Somebody looking at thirty-two cells' worth of results on the toggle board, who wants roughly twenty, has two options: stay at thirty-two or go to sixteen. The ladder has no rung in between and cannot be given one without changing the filters.

On the six-of-six board the jumps are worse in absolute terms: from one thousand two hundred and ninety-six the next rung down is two hundred and sixteen. Each step divides by six.

This is why "narrow it a bit" is not a thing a faceted board can do. Every narrowing divides by a whole facet, and the facet sizes are the step sizes.

The contrast case

Six filters of two, three, four, five, five and six: three thousand six hundred cells, and thirty-nine reachable sizes from the same sixty-four subsets.

Mixed sizes mean subset products mostly differ. Leaving the two alone gives two; leaving the three alone gives three; leaving the two and three alone gives six, which collides with leaving the six alone — but most combinations do not collide, and the ladder ends up with thirty-nine rungs instead of seven.

More than five times as many positions, from a board that looks identical in the sidebar: six controls either way.

The figure a reader would want instead

How many rungs lie between their current result and one listing.

That is answerable from the schema: it is the number of distinct products reachable from where they are. On a board of six toggles holding thirty-two cells, there are five rungs below — sixteen, eight, four, two and one — and every step is a halving.

On a board of six mixed facets at a comparable position there are more rungs and the steps vary, so a reader can take a small step or a large one depending on which filter they set.

That choice does not exist on a uniform board. Every available step is the same size, and the only decision is whether to take one.

The design reading

Varied facet sizes buy grain for free.

They cost nothing in resolution — the product is whatever it is — and nothing in expressiveness beyond what the sizes already determine. They are not a design choice anybody usually makes deliberately, because nobody is looking at this figure.

But a board choosing between a six-value facet and two facets of two and three has a real decision in front of it, and the grain is one of the few things that distinguishes the options after the resolution has been fixed. It is the cheapest improvement available in this record, and it is invisible without doing the arithmetic.