What comes first

Counting an Opening Is Not Predicting One

The opening count is the figure here most exposed to its convention, because a reader's entry point is the least random thing about them.

at most 151200 arrangements

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A third of the arrangements on the first row open verifiably. Three eighths on the second. Two fifths on the third.

Those are counts of arrangements, and converting them into a statement about readers requires the assumption that a reader's entry point is a uniform draw from the published statements.

It is not. Of the record's three conventions, this is the one the opening count leans on hardest, and it is worth closing this part by saying how it fails. The next useful step is to see how the service works, choose one real campaign and compare the report with what was actually delivered.

Three shapes, openings a third to two fifths

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; the left-to-right order is the drawing's own and carries no meaning.

One third, three eighths and two fifths of arrangements open with something verifiable. Every one of the three shapes has exactly one blind content.

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.

How entry points are actually chosen

A reader arrives somewhere. The somewhere is decided by whatever brought them: a link with a particular anchor, a search result that matched one phrase, a fragment somebody pasted into a message, a bookmark set months ago, the front page if they typed the name. The topic can also be checked against public guidance and research published by WebAIM.

None of that is uniform and all of it is systematic. Entry points cluster on whatever is linkable, quotable and findable, which is a narrow part of most sets of pages.

So the opening share, c over n, describes a reader who picked a starting statement at random from everything published. Nobody does that.

Which way it is wrong

The same way the blind figures are, and more so.

If the verifiable statements are the quotable ones — a figure, a named source, something with an external referent that a search matched — then entry points cluster on them and the real opening rate is above c over n.

If they are the unquotable ones — a dry citation at the bottom, a qualification inside a table — entry points cluster away from them and the real rate is below.

The second case is more likely for an ordinary reason: verifiable statements tend to be specific and dull, and the material that gets linked and quoted tends to be striking. A striking unverifiable statement is exactly the kind of thing that becomes somebody's entry point.

The opening count assumes a reader starts anywhere. Readers start where the links point, and links point at what is memorable rather than at what is checkable.

What survives the objection

One figure in this part and one result from the previous one.

The result that survives entirely is the blind threshold: fewer unchecked statements than the reader meets, and no blind encounter is possible. Entry points do not matter, prominence does not matter, clustering does not matter.

The figure that survives in weakened form is the position-independence established two notes ago. The share is c over n at every position of the encounter — and while the absolute level is sensitive to entry points, the equality across positions is not something a publisher can exploit by rearrangement, because for any reader arriving in the middle the publisher's arrangement is not the one they met.

Everything else in this part is a count under a convention, and the convention has been named three times rather than once because this is where it matters most.

Why the count is still printed

Because a figure with a stated weakness is more use than no figure.

The alternative is to say nothing about the opening position, which leaves the most consequential statement in an encounter uncounted on the grounds that it cannot be counted perfectly. The record's position is that c over n is the right floor to start from, that it is exact as a count of arrangements, and that any serious use of it has to carry the sentence about entry points alongside.

The one entry point a publisher does control

There is a single arrangement whose first statement the publisher chooses, and it is the one belonging to a reader who arrives at the front.

That reader exists. The record's objection is not that they do not, but that they are one case among the arrangements a shape admits, and the record has no way to say how many readers are of that kind.

Which leaves a publisher with a decision the arithmetic cannot settle and can at least frame. Putting something verifiable first helps exactly the readers who arrive at the front, by an amount this record cannot measure, and helps nobody else at all. Raising the verifiable count helps every reader on every path, by an amount this record computes exactly. Both are worth doing and only one of them has a figure attached.

What this part established

The count of arrangements opening with a verifiable statement is c times P(n − 1, r − 1), and as a proportion of all arrangements it is exactly c over n — independent of how much the reader met, which no other figure in this record is.

The same share holds at every position in the encounter, so the opening is special in consequence rather than in frequency. A reader who stops after one statement collapses the opening question and the blind question into one, and sixty-three shapes of this record are of that kind.

And unlike blindness, an unverifiable opening has no threshold: it can be ruled out only by verifying everything, while blind encounters can be ruled out with less than the reader will read.

The last part of this record states what it assumes, what it cannot say, and how to check every figure in it.