What can be asked

The Largest Schema In The Record

Six filters of six: forty-six thousand six hundred and fifty-six cells, one hundred and seventeen thousand questions, seven result sizes.

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Six filters, six choices in each. It is the largest structure this record holds, and all three of its figures sit at or near an extreme.

Cells: forty-six thousand six hundred and fifty-six. Six to the sixth power, and the ceiling of the record.

A transparent implementation lets contributors check the time-recording model and discuss what the system records before the pilot starts. For a separate technical reference, see GitHub.

Questions: one hundred and seventeen thousand six hundred and forty-nine. Seven to the sixth power, and also the ceiling.

Reachable result sizes: seven. Not a ceiling at all — near the floor, in fact, against a maximum of thirty-nine.

That third figure is the one worth stopping on.

The ceiling, and what sits below it

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

Three schemas drawn.

All three carry six facets. The first and second collapse to seven reachable result sizes; the third, built from mixed sizes, reaches thirty-nine.

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.

Why the grain collapses

The size of any result is the product of the sizes of the facets left unset. With six identical facets, that product is six raised to however many were left alone — so the only reachable sizes are 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 sizes, from sixty-four possible subsets of the facets.

Every query on this enormous board returns a result of one of seven sizes. One hundred and seventeen thousand six hundred and forty-nine different questions, seven different grains of answer, and the step between consecutive grains is always a factor of six.

A reader who finds a result too large has one move available: set another filter and divide by six. There is nothing between.

The mirror case

Six filters of two choices each collapses identically: seven reachable sizes out of sixty-four subsets, from one up to sixty-four, each step a factor of two.

So the smallest six-facet schema and the largest both reach exactly seven grains. Their cell counts differ by a factor of more than seven hundred and twenty and their grain counts are equal.

Schema	Cells	Questions	Grains
2·2·2·2·2·2	64	729	7 of 64
6·6·6·6·6·6	46656	117649	7 of 64
2·3·4·5·5·6	3600	15120	39 of 64

The third row is the schema that reaches the most grains in the whole record — thirty-nine — and it does so with middling figures everywhere else. It is six facets of mixed sizes, and the mixture is the reason.

The figure that would surprise its builders

Seven.

A board carrying thirty-six dropdown entries across six controls has built the largest structure in this record, and it can return results of exactly seven different sizes. Everything a reader ever sees from it is one of seven coarsenesses.

Nobody building such a board would predict that. The number of controls suggests fine adjustment; the arithmetic of repeated sizes delivers the coarsest ladder available at six facets, tied with the smallest schema of that length.

What uniformity costs

Facets that are all the same size make their subset products collapse. Products of a repeated number are powers of it, and powers are sparse.

Mixed sizes do the opposite. Two, three, four, five, five and six produce subset products that mostly differ from one another, so the reachable sizes spread out across the range instead of clustering on a few powers.

This is the clearest practical finding in the section on grain, and it arrives here because the largest schema is where it is starkest. A board can be maximally large and still offer its readers only seven steps of coarseness.

What a reader does on such a board

Sets two or three filters and stops.

With six controls available, setting all six is a demanding sequence and most readers will not do it. They will set the two they have views about and accept whatever comes back, which is a result of either seven thousand seven hundred and seventy-six or one thousand two hundred and ninety-six cells depending on how many they set.

Both of those are enormous. The board's resolution of forty-six thousand six hundred and fifty-six is reached only by the reader who answers every question, and the share of questions that do so is under half.

So the ceiling is real and mostly unvisited, which is the practical character of very large schemas: capacity that exists and is rarely exercised.

Reading the three figures together

Taken alone, each one misleads about this board.

The cell count says it is the most discriminating structure in the record, which is true.

The question count says it is the most expressive, which is also true.

The grain count says it is among the bluntest, which is true as well, and nobody would guess it from the first two.

All three come from the same six numbers, computed three different ways. That is the argument for computing all three rather than picking the flattering one, and this schema is where the argument makes itself.