What can be asked

One Pair, One Board

Across four hundred and sixty-one schemas no two produce the same pair of figures. One figure collides for four boards in five.

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The central fact of this record is short enough to state in a line.

Across the four hundred and sixty-one schemas, the pair of figures — cells and questions — is distinct for every one. Four hundred and sixty-one schemas, four hundred and sixty-one pairs, no collisions anywhere.

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What that is worth against the alternative

Cell counts take one hundred and ninety-five distinct values across the same four hundred and sixty-one schemas. Question counts take four hundred and five.

Described by	Distinct descriptions	Schemas sharing one
cells	195	381
questions	405	—
both	461	none

So the first figure leaves four boards in five indistinguishable from some other board. The second does far better. The two together do not merely do better still; they are exact.

Why it is a theorem rather than a coincidence

It is checked rather than noticed, which is the important part: the program that builds this record counts the distinct pairs and would report a collision if one existed.

The reason behind it is the asymmetry of the two products. Cells multiply the sizes; questions multiply the sizes each raised by one. A rearrangement that preserves the first product almost never preserves the second, because the two products respond to a facet split in opposite directions.

Splitting a facet of six into facets of two and three preserves the cell count exactly: six equals two times three. The question count goes from seven to three times four, which is twelve. The first product cannot tell the two schemas apart; the second separates them by a wide margin.

Every rearrangement that fools the resolution gets caught by the question count, and the record checks that this holds without exception rather than asserting it.

What it does not claim

It does not say two figures describe a board completely. They describe its schema completely, within this record's bounds. A schema says nothing about what the facets are called, what their values mean, whether anybody uses them, or whether the listings fill the cells.

It does not extend past the bounds without checking. The record holds facets of two to six values and schemas of one to six facets. Whether the pair stays unique with facets of seven values, or seven facets, is a question somebody would have to compute. The method is the same; the result is not guaranteed by anything here.

And it does not make the figures easier to get. Both still require reading the sidebar and multiplying. The claim is about what the figures are worth once obtained, not about obtaining them.

The practical form of it

If two boards produce the same pair, they have the same schema. If they produce different pairs, their schemas differ — and the pair says exactly how, since the schema can be recovered from it.

That makes the pair a proper identifier rather than a summary. A summary loses information; this does not. Two numbers, written on a line, carry everything the record holds about a board's structure, and either of them alone loses most of it.

Recovering the schema from the pair

Because the pair is unique, the schema can be read back out of it — and the procedure is short enough to do by hand.

Factorise the question count. Every factor of seven means a facet of six, since seven appears nowhere else. Divide those out, and the remaining factors describe the rest, with each facet contributing its size plus one.

Check against the cell count. If the candidate schema multiplies to the right resolution, it is the schema; if not, no schema has that pair, and one of the two figures was read off the page wrongly.

That last case is the useful one. The pair is self-checking: a mistake in counting a dropdown produces a pair that corresponds to no schema at all, and the mismatch announces itself.

Why one figure keeps getting quoted anyway

Because one figure fits in a sentence and feels like enough.

"This board sorts across one hundred and forty-four kinds of listing" is a satisfying thing to say. It sounds specific. It is specific — to a group of seven structures running from three facets to six, permitting between two hundred and forty-five and one thousand two hundred and ninety-six questions.

Adding the second figure costs four words and converts the sentence from a classification into an identification. That is about the best return available anywhere in this record for the effort involved, and it is the single recommendation the whole thing comes down to.