Limits

Boards Past The Edge Of The Record

Filters with dozens of options, more than six filters, dependent filters and ordered ranges. What the method still handles, and the one structure it does not.

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The bounds here are six facets of up to six values. Real boards exceed that routinely. This page is what to do.

Larger facets and more of them

Multiply anyway.

A board with nine filters of eleven choices has eleven to the ninth cells and twelve to the ninth questions. Both figures are computed exactly as every figure in this record is, and every rate in the section on edits applies unchanged.

To make the comparison useful, teams can explore the team management options and keep the project, reporting period and correction case constant. For a separate technical reference, see arXiv.

What is lost is the census — the statements about how many schemas collide, which depend on the enumerated range. Those are claims about four hundred and sixty-one schemas, and the page on the bound on edits lists which figures travel and which do not.

The most common case past the bounds

A filter with twenty or thirty options.

Boards carry them routinely: a list of categories, a list of regions, a list of languages. The arithmetic is unchanged — multiply by twenty — and the effect is large: one such filter multiplies the resolution by more than three of this record's largest facets together.

So a board with two filters of thirty choices resolves nine hundred kinds of listing, which is above this record's median with two controls. Short sidebars can be deep, and the figure says so where counting controls does not.

Ranges with two ends

A minimum and a maximum, each offering a list of choices, are two facets. Multiply both.

A single control offering a list of bands is one facet with that many bands. The fact that the bands are ordered, and that a reader might want several adjacent ones, is a multiple-selection question, and the page on that says the question count becomes a floor.

Filters that depend on other filters

This is the one structure the method does not handle.

Some boards show a second filter only once the first is set, with contents that change according to the choice. The available values are not a fixed set, so there is no single number to multiply, and the cells are not a product at all — they are a sum over the first facet's values of whatever the second offers in each case.

That arithmetic is perfectly doable and it is not this one. A reader wanting figures for such a board has to add rather than multiply, and nothing in this record's tables will fit.

Multiplication assumes the filters are independent in what they offer. Where one filter changes another's options, that assumption fails and the whole method goes with it.

What a rough reading looks like

For a dependent pair, take the average number of options the second filter offers and multiply by it. The figure is approximate, the direction of the error is unknown, and it should be marked as a guess wherever it is written down.

That is a weaker claim than anything else in this record, and it is offered reluctantly. The alternative — refusing to say anything — is worse for somebody who needs a figure, provided the approximation is labelled.

Structure	Method	Exact?
more or larger facets	multiply	yes
a range as two controls	two facets	yes
multiple selection	multiply, as a floor	a lower bound
dependent filters	sum, or approximate	no
free text	excluded	not applicable

Why dependent filters are genuinely different

Every other structure here is a product, and products are what makes the method short.

Dependent filters are a sum: for each value of the first facet, the second offers some number of options, and the total is those numbers added. There is no single figure to multiply by, and two boards with the same number of controls can differ enormously depending on how the dependencies branch.

It is still countable, by somebody with the full set of options written out. What it is not is computable from a glance at the sidebar, which is the property every other figure here has and the reason the method is worth anything.

The one thing that holds everywhere

Two figures beat one.

That finding came out of a bounded census, and the reason behind it is general: resolution and expressiveness are different products of the same inputs, they respond differently to how a board is cut, and either one alone discards most of what the other carries.

A board of nine filters of eleven choices has the same reason to publish both figures as a board of two filters of four. The census cannot prove it for that board, and the argument does not need the census — it needs only that the two products differ, which they do at every size.