Thresholds

The Staircase Of A Threshold

Eight optional items and a rule saying how many are needed. The nine thresholds produce a staircase that is flat in the middle and steep at both ends.

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A list of eight optional items, with a rule saying how many must be met, has nine possible rules: none of them, one of them, and so on up to all eight.

Those nine rules produce nine completely different filters from the same eight lines.

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Threshold	Admits
0 of 8	1/1
1 of 8	255/256
2 of 8	247/256
3 of 8	219/256
4 of 8	163/256
5 of 8	93/256
6 of 8	37/256
7 of 8	9/256
8 of 8	1/256

The eight lines on the page are identical in all nine cases. Only the rule changed, and the intake moved from everything to one profile in two hundred and fifty-six.

Eight optional items, every threshold

Nine thresholds drawn. The tread is flat because the threshold is a whole number: there is nothing between one item and the next. A tread at the top admits everybody, which is a list that filters nothing at all.

The threshold runs along the bottom; the height of each tread is the share admitted. The treads are flat because a threshold is a whole number: there is nothing between one item and the next.

Computed from the threshold. Each item is counted as a separate coin: this is the unit of measurement, not an estimate of any population.

The shape of the staircase

It is not a straight slope. It is flat at the top, flat at the bottom, and steep only in the middle third.

At a threshold of one, the list still admits two hundred and fifty-five profiles in two hundred and fifty-six — it excludes only the profile that meets nothing at all. At two it admits two hundred and forty-seven. Those two rules are barely filters.

At the other end, thresholds of seven and eight admit nine and one profile respectively. Those are filters so tight that the difference between them is almost invisible against the scale of what they both exclude.

Everything interesting happens between three and six, where the intake falls from two hundred and nineteen to thirty-seven — a factor of about six across three steps.

A threshold is not a dial that turns smoothly. It is nearly inert at both ends and violently sensitive in the middle, and the middle is where every real rule sits.

Why the middle is where the rules are

Because the ends are self-evidently useless and the middle is not.

Nobody sets a threshold of none; it would mean the optional items are decoration. Nobody sets a threshold of all; that is a conjunction, and if that were the intention the items would have been written as compulsory.

So real rules land somewhere between two and six of eight — which is precisely the region where moving the threshold by one changes the intake by a large factor. The parameter that is hardest to set carefully is the one that matters most.

Why the treads are flat

Because a threshold is a whole number. There is no rule that requires three and a half of eight items, so the staircase has no slope within a tread — only the jump at the edge.

That flatness is not a drawing convention. It is the shape of the thing: two lists whose thresholds differ by nothing admit identically, and two whose thresholds differ by one can differ by a factor of two or more.

Reading a tread

The height of each tread is the share admitted, printed as an exact fraction above it. Nothing is rounded and nothing is converted to a percentage, for a reason that applies throughout this record: a fraction can be checked against the binomial coefficients in a few seconds, and a percentage cannot.

The fractions here are all over two hundred and fifty-six because eight items give two hundred and fifty-six profiles. Their numerators — 256, 255, 247, 219, 163, 93, 37, 9, 1 — are running sums of the binomial coefficients from the top down, which is the whole derivation.

Reading a tread you did not expect

The treads that surprise people are the second and the seventh.

A threshold of two on eight items admits two hundred and forty-seven profiles in two hundred and fifty-six — more than ninety-six per cent. A writer who sets it there believes they have asked for something; arithmetically they have excluded only the profiles meeting nothing or one thing.

A threshold of seven admits nine. The same single step from the other end, and the filter has gone from ninety-six per cent to three and a half.

Both rules read as moderate adjustments away from an extreme. Neither is, and the asymmetry between them is covered on its own page later in this section.

Where the numerators come from

The nine numerators — two hundred and fifty-six, two hundred and fifty-five, two hundred and forty-seven, two hundred and nineteen, one hundred and sixty-three, ninety-three, thirty-seven, nine, one — are running sums of the binomial coefficients taken from the top.

The coefficients for eight items are one, eight, twenty-eight, fifty-six, seventy, fifty-six, twenty-eight, eight, one. Add them from the right and the running totals are exactly the numerators above.

That is the whole derivation, and it means any reader with a table of coefficients can reconstruct this staircase, or build one for any other number of optional items, without needing this site at all.

What this means for writing a list

The eight lines are the visible part and the least consequential. The rule over them is the invisible part and the decisive one.

Any discussion about whether to add a ninth optional item is a discussion about the minor parameter. The question that decides what the list admits is how many are needed, and the next pages take that apart case by case — including the two rules at the ends of the staircase, which look absurd and both occur in practice more often than anybody admits.