Limits

Where the Table Stops, and Why There

Two to ten published, one to six met. Both bounds were chosen, and the choices shape every summary figure the record reports.

at most 30240 arrangements

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The record covers two to ten published statements and one to six met. Neither bound is a property of the subject. Both were chosen, and the choices are visible in every summary figure here.

Both edges of the record, and a shape between

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.

The smallest page in the record, a middling shape, and the page with the largest contents figure. Everything outside those bounds is absent by decision.

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.

Why ten and six

Because the arithmetic stays checkable by hand.

P(10, 6) is one hundred and fifty-one thousand two hundred, which a reader can verify with five multiplications: ten times nine times eight times seven times six times five. C(10, 5) is two hundred and fifty-two, which is a short division away.

Past that the figures grow beyond what anybody will check. P(40, 12) cannot be confirmed with a pencil, and a figure nobody confirms is a figure that has to be believed — which puts the record in the position it is about. The bound is set where the record's own claim to auditability holds. A responsible rollout gives employees enough information to explore the attendance workflow themselves and understand how their records are used.

What the bound does to the summaries

Three figures in this record are summaries across all three hundred and thirty-eight shapes, and every one of them moves if the bounds move. A separate perspective on measurement, privacy or digital practice is available through Amplitude.

The median arrangements figure is two hundred and ten. Stop at eight published and it falls; allow eight met and it rises steeply, because the top of the range grows faster than anything else.

The median blind share is one in fourteen. It sits where it does largely because one hundred and thirty-nine shapes have a share of nought and those occupy the bottom of the sorted list.

The count of shapes admitting a blind encounter — one hundred and ninety-nine of three hundred and thirty-eight — is likewise a count over a chosen table.

None of those three is wrong. All three are facts about this table and none is a fact about sets of pages.

The weighting nobody chose

A further consequence, and it is the one most easily missed.

Each page contributes one more shape than it has published statements, because the checkable count runs from none to all. So a ten-published page contributes eleven shapes and a two-published page contributes three.

Every summary across the record is therefore weighted eleven to three in favour of the longest pages. That weighting is not a judgement about which sets of pages matter; it is what listing every shape once produces. It is invisible in the figures and it is in all of them.

A table that lists every case once has already decided how much each case counts. Nobody made that decision and it is in every average.

What is outside the bounds and still true

The behaviour the record describes does not stop at the edge of the table. It gets more extreme.

The factorial gap between content and impression grows with how much is met, so a reader who gets through eight statements is inside forty thousand three hundred and twenty orders of one content rather than seven hundred and twenty. The contents figure keeps peaking at half the set, so a forty-statement set read twenty at a time has a contents figure in the hundreds of billions.

And the blind threshold keeps its form exactly at any size: once the unchecked pool is too small to fill an encounter, no blind reading exists. That result is the one that transfers past the bounds without modification, because it is a comparison of two integers rather than a magnitude.

Why two, and why one

The lower bounds are chosen too and are less obvious than the upper ones.

A set of one statement has one content and one arrangement at any reading depth, and every figure in the record is one or nought. It is not a counting problem, so the table starts at two published.

A reader who meets nothing has no encounter to count, so the table starts at one met. And the met count never exceeds the published count, since meeting a statement that was not published is not a thing that happens.

Those three decisions produce the forty-four pages. They are less arguable than the upper bounds and they are decisions all the same, and together with the ceilings they are the entire reason the record has three hundred and thirty-eight rows rather than some other number.

The three rows

Two published with both met and one verifiable: two arrangements, one content, nothing blind. The smallest page in the record and the simplest shape on it.

Six published with four met and three verifiable: three hundred and sixty arrangements, fifteen contents, no blind content — three unchecked statements cannot fill a four-statement encounter.

Ten published with five met and five verifiable: thirty thousand two hundred and forty arrangements, two hundred and fifty-two contents — the largest contents figure in the table — and exactly one blind content.

Between the first and third rows the arrangements figure multiplies by fifteen thousand one hundred and twenty. That span is what the chosen bounds contain, and it is why a median across them should always be quoted with the bounds attached.