Halving

What The First Compulsory Item Costs

The first item on a list throws away half the possibility space. It is the largest single cut any item will ever make, and it is the one nobody thinks about.

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A list with one compulsory item on it admits half of all profiles. A list with none admits all of them.

So the first item removes one half of everything. In absolute terms that is the biggest cut any item on the list will ever make — the second removes a quarter, the third an eighth, and the eighth removes less than a four-hundredth — and it is made by the line nobody deliberates over.

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Why the first line is never deliberated

Because at the moment it is written the list does not feel like a filter yet. It feels like a description of the work. Somebody writes down the thing that is obviously needed, and there is no sense of a decision being taken, because what else would the list say?

The decision is taken all the same. Half the possibility space is gone before the second line exists.

Two ways of measuring the same item

This is where the halving law can mislead, and it is worth being exact.

In proportion, every compulsory item is identical: each one halves whatever reached it. The eighth item halves the list's intake just as thoroughly as the first.

In absolute terms, they are wildly unequal. The first removes half of everything; the second removes a quarter of everything; the eighth removes less than one part in two hundred and fifty-six.

Item	Removes this much of the whole	Leaves
first	1/2	1/2
second	1/4	1/4
third	1/8	1/8
fourth	1/16	1/16

Both columns describe the same list. Which one matters depends entirely on the question. If the question is "how much does this item narrow the list", the proportional answer is right and every item is the same. If the question is "how many profiles does this item exclude that would otherwise have passed", the absolute answer is right and the early items dominate completely.

The first item and the eighth are equally strict and nothing like equally consequential. Confusing the two is how lists get written.

The practical shape of the error

The error runs in a specific direction. People scrutinise the last item added — it is the one in front of them, it is the one that feels marginal, it is the one somebody questioned — and treat the first few as settled.

Yet the first few are where almost all of the exclusion happened. By the time the list reaches its fourth item it has already discarded fifteen profiles in every sixteen. Everything after that is argument about the last sixteenth.

Anybody auditing a list would do better to start at the top and ask of the first item what they would normally ask of the last.

What makes an item worth its half

Nothing in this record can answer that. The arithmetic says what an item costs; whether the cost is justified is a judgement about the work, and the record has no access to it.

What the arithmetic can do is make the cost visible, which is more than usually happens. A list that nobody computed has an unknown selectivity. A list where somebody has noticed that line one removed half of everything is being written with its eyes open, whatever is then decided.

An objection, and what survives it

The obvious objection is that the first item is not really a coin at all — it is the thing everybody in the field has, so it excludes almost nobody in practice.

That objection is correct about the world and beside the point about the record. Under the coin convention the first item removes half the possibility space, and that is what the figure measures. If in reality nearly everybody meets it, then the real intake is wider than the figure, and the page on the coin says so in as many words.

What survives the objection is the comparison. Two lists counted the same way can be compared honestly, and a list that halves four times is narrower than one that halves twice regardless of how the real population happens to sit. The absolute figure is a ruler reading; the comparison is what the ruler is for.

What also survives is the warning. An item that "everybody has" is an item that was not worth writing down, and the arithmetic charges full price for it anyway — which is a reason to strike it rather than a reason to doubt the arithmetic.

And the mirror of it

Read backwards, the same figure is the strongest argument for removing an item. Taking the first compulsory item off a list doubles its intake — not by a few per cent, by a factor of two.

There is no other single change in this record that does as much, and the page on edits measures the rest of them against it.