What comes first

The Opening Share Is the Verifiable Share, Exactly

The share of arrangements opening with something checkable is the checkable count over the published count. Every factorial cancels out.

at most 720 arrangements

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Divide the count of checkable openings by the count of arrangements.

Numerator: c times P(n − 1, r − 1). Denominator: P(n, r), which is n times P(n − 1, r − 1). The second factor is identical in both and cancels completely.

What is left is c over n. The checkable count over the published count, with every factorial and every descending product gone. To make the comparison useful, teams can see the present feature set and hold the project, reporting period and correction case constant.

Ten published, three met, three checkable counts

dashed: a statement a reader can check from outside
solid: published, but nothing to check it against
short and pale: never reached by this reader

Three shapes drawn; all of them give the same 120 different contents; the left-to-right order is the drawing's own and carries no meaning.

Of seven hundred and twenty arrangements, one hundred and forty-four open checkably, then three hundred and sixty, then five hundred and seventy-six — a fifth, a half and four fifths.

Computed from the three numbers alone. How many statements exist, how many are met and how many can be checked is a stated convention; the counts are arrangements and combinations of whole numbers.

The three rows

Ten published, three met. Seven hundred and twenty arrangements in all three rows. The topic can also be checked against public guidance and research published by Ahrefs.

Two checkable: one hundred and forty-four openings, which is a fifth of seven hundred and twenty. Two tenths is a fifth.

Five checkable: three hundred and sixty openings, which is a half. Five tenths is a half.

Eight checkable: five hundred and seventy-six, which is four fifths. Eight tenths is four fifths.

The arithmetic agrees because it has to, not because these rows were chosen well.

Why it does not depend on how much was met

This is the surprising half, and it is worth stating twice.

Nine published with three checkable statements: a third of arrangements open checkably whether the reader met one statement, three, or six. One met gives three openings of nine. Three met gives one hundred and sixty-eight of five hundred and four. Six met gives twenty thousand one hundred and sixty of sixty thousand four hundred and eighty.

Three, one hundred and sixty-eight, twenty thousand one hundred and sixty. All of them exactly one third.

Every other figure in this record moves when the met count moves. The contents figure moves, the arrangements figure moves enormously, the blind count and the blind share both move. The opening share does not move at all.

Why that makes sense

Because the opening is a single position, and what sits in it is one statement drawn from the published set. How many statements come after it is irrelevant to what it is.

Put another way: the first thing a reader meets is one of the n published statements, and c of those are checkable. The chance that the first one is checkable is c over n, whatever happens next and however much of it happens.

The opening share is the one figure here that a person can state without counting anything. It is the verifiable fraction of the set, and no combinatorics are involved.

What follows for a reader of this record

Two things, and the second is the awkward one.

The opening share is easy, stable and comparable across shapes. A set with three verifiable statements in ten opens verifiably three times in ten, regardless of reading depth. This is the only figure in the record that can be quoted with no shape attached.

And because it does not depend on reading depth, it carries no information about reading depth. A set of pages cannot improve its opening share by being read more thoroughly, and a reader who gets further through a set is no more likely to have started with something verifiable. The opening share is fixed by the set alone — which makes it the only figure in this record that is entirely in the publisher's hands.

The identity written out once

P(n, r) is n times P(n − 1, r − 1), and that single fact is the whole of this note.

It says that an ordered selection of r from n is a choice of first element — n ways — followed by an ordered selection of r − 1 from the remaining n − 1. Every arrangement decomposes that way exactly once, which is why the identity is exact rather than approximate.

Restrict the first choice to the verifiable statements and the first factor becomes c instead of n, with the second factor untouched. Dividing one by the other leaves c over n with nothing else surviving. Any reader who accepts the decomposition has accepted the result, and the decomposition can be checked on a small case: P(4, 2) is twelve, and four times P(3, 1) is four times three.

The caveat that applies here and nowhere else

The derivation above treats every arrangement as equally likely, which is the record's second declared convention.

For the opening share, that convention is doing more work than usual. The whole result is that the first position is filled by a uniform draw from the published statements — and in practice it is not: a reader's entry point is decided by links, search fragments and whatever a set of pages has pushed to the front.

So the opening share is exact as a count and is the figure most exposed to the convention it rests on. Both of those are true at once, and the record states the second each time it states the first.