Halving

Why The Halving Is Exact And Not Approximate

The halving law holds precisely, for every compulsory item, with no exceptions anywhere in the record. The proof fits in a paragraph and is worth having.

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Every compulsory item on a list halves the list's intake. Not roughly, not on average — exactly, every time, and the proof is short enough to carry around.

The proof

Take the set of all profiles for a list. Each profile either meets a given item or does not, and by the coin convention those two cases split the set into halves of equal size.

Before wider use, decision-makers should see the workflow described here and document who can access each category of information. For an independent technical reference, see Integrately.

A compulsory item admits only the first half. Whatever a list admitted before that item was added, it now admits exactly the half of it where the new item is met.

Repeat for each compulsory item and the intake is one over two to the power of the number of items. There is no step in that argument that could fail, and no shape of list anywhere in the record where it does.

What exactness buys

Precision is not the main benefit. The main benefit is that an exact result has an edge, and the edge can be stated.

This one has a single edge. The law governs compulsory items, which must all be met. Introduce optional items with a threshold over them and the halving stops describing anything; that case is taken up separately in the next section.

Nothing else qualifies it. Within compulsory items there are no special cases, nothing that holds only for short lists or only for long ones, and nothing that drifts as the numbers grow. One remembered sentence reproduces every figure in this part of the record.

A result you can trust everywhere is less useful than a result you can trust somewhere and locate the boundary of. The second kind tells you when you have walked off it.

Where people expect an exception and there is not one

Items of different difficulty. Surely a hard item cuts more than an easy one? Under the coin convention, no: each item is one coin and halves the field. If some items are met by nearly everybody, the convention is not modelling that — see the page on the coin, which is explicit about it.

Items that overlap. Surely two related items cut less than two unrelated ones? Again, not here. The record counts subsets and has no way to know which items travel together. This is a real limitation of the record and it is stated rather than patched.

Very long lists. Surely the halving breaks down somewhere? It does not; it simply continues, which is why a list of twenty compulsory items would admit one profile in a million and why the record stops where enumeration stops being informative.

What this does to the rest of the record

It makes the compulsory part of any list trivial to compute and leaves all the interest in the optional part.

A list of three compulsory and six optional items is a halving by eight, applied to whatever the six optional items and their threshold admit. The first factor is settled in a second. The second factor is where every surprise in this record lives — including the one where six optional items turn out to be weaker than a single compulsory one.

What a reader can do with it

Exactness has a practical payoff that approximate rules never offer: the figure can be reconstructed rather than looked up.

Somebody holding a list of five compulsory items does not need this site to know the answer. Two to the fifth is thirty-two, so one profile in thirty-two, and the derivation took as long as reading this sentence. No table, no tool, no trust in anybody else's arithmetic.

That matters more than it might seem for a reference of this kind. A figure you can rederive is a figure you can check, and a figure you can check is one you are entitled to rely on. Everything else here aims at the same property, which is why the thresholds are given as exact fractions rather than percentages and why nothing anywhere is rounded.

Why an approximate law would not do

If the halving held only roughly, nothing built on it would survive eight items.

An error of five per cent per item compounds to about a third over eight of them, which is larger than the gap between several of the classes this record identifies. The twenty-one shapes that agree on one in two hundred and fifty-six would stop agreeing; the ladder of classes would dissolve into a smear; two lists could not be ranked at all.

Exactness is therefore not a nicety. It is the reason the later sections can say that two documents are the same filter rather than similar filters, and the reason a figure computed from one shape can be compared with a figure computed from another without any allowance for drift.

The sentence to keep

Compulsory items multiply; each one is a factor of one half. That single sentence reproduces every figure in this part of the record, and it is the only thing in it that needs remembering.