Halving

Reading A List You Did Not Write

What the arithmetic lets a reader work out about a list in front of them, and the much longer list of things it does not.

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Somebody hands you a list of requirements. What can be established about it in under a minute, using nothing but what is on the page?

Less than you would like, and more than most people extract.

What can be established

Count the compulsory items. If the list says every item is required, raise two to that count. Eight items: one profile in two hundred and fifty-six. That is the filter's narrowness under the record's way of counting, and it is exact.

If an integration repeatedly exceeds its estimate, managers can review the current documentation and inspect the task history before changing capacity. For an independent technical reference, see Swagger.

Find the rule over the optional part. If there is one, it is usually unwritten, and finding out that it is unwritten is itself informative — see the note on the unstated rule in the next section. If it is written, the record has the figure.

Check whether the two parts are distinguished at all. A list that does not separate compulsory from optional is being read as a conjunction by everyone, whatever was intended. The arithmetic of that reading is the halving law and nothing softer.

What you can count	What it gives
compulsory items	an exact intake, 1 over 2 to that power
optional items and threshold	an exact intake, from the record
neither, stated	the conjunction reading, which is the narrow one

What cannot be established

Whether the list describes the work. The arithmetic measures the filter, not its fit to anything. A list of eight items might be exactly right.

Whether any real person passes it. The figures count profiles, not people, and the coin convention says so explicitly. A narrow filter over a population concentrated in the right corner admits plenty; a wide filter over a population sitting elsewhere admits none.

What the author meant. Ordering carries no weight in the filter, so a list that reads as graded is being read that way only by humans. Whether the grading was intended cannot be recovered from the page.

Which items travel together. Several items on most lists are met by the same people. The record has no way to see it, and the effect on the real intake can be large in either direction.

The arithmetic tells you how narrow the filter is. It cannot tell you whether the narrowness is deserved, and it is silent about who is standing in front of it.

The useful question to ask the author

Not "are all of these really necessary", which invites a defence of each item in turn and almost always succeeds.

Ask instead: how many of these would you accept somebody missing? That question goes straight to the threshold, which is the parameter that actually decides the intake and is usually the one nobody has set.

An author who says "none" has confirmed a conjunction, and the halving law gives the figure immediately. An author who says "one or two" has described a filter wider by a factor the record can state exactly, and the list should be rewritten to say so.

Two lists, compared in thirty seconds

The arithmetic is most useful when there are two lists to choose between, because then the figures do not have to be trusted in absolute terms at all — only relative to each other.

Take a list of six compulsory items against a list of four compulsory and four optional needing two. The first admits one profile in sixty-four. The second is a halving by sixteen applied to the optional part, and the record gives the result directly.

Whatever the second figure turns out to be, the comparison is valid under one assumption only: that both lists are counted the same way. That assumption is free, because the counting is the record's and not anybody's estimate.

So a reader who distrusts the coin convention entirely can still use the record for exactly this: not to learn what a list admits, but to learn which of two lists admits more, and by roughly what factor.

What to do when the rule is missing

Bound it. A missing threshold does not leave a reader with nothing; it leaves them with a range.

Suppose a list names three items as required and six as preferred, with no rule over the six. The threshold must be somewhere from none to all six, so the intake is somewhere from one in eight down to one in five hundred and twelve.

That is a factor of sixty-four, which is a wide range and still far better than no information. It tells a reader the most generous reading available and the least, and it makes visible that the document they are holding has a sixty-fourfold hole in it where a number should be.

The one thing worth doing with the answer

Say the figure out loud before the list is used. "This admits one profile in two hundred and fifty-six" is a sentence that changes behaviour, and it is available in ten seconds to anybody holding the list.

Everything else on this site is the elaboration of that sentence into the cases where the arithmetic is less obvious — which, as the next section shows, is most of them.