Nothing to check

Forty-Four Shapes Where Every Path Is Blind

With nothing checkable published every encounter is blind. One such shape sits on every page, which is why the count equals the page count.

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A set of pages publishes nothing a reader can check from outside. Every statement either has to be taken on the publisher's word or set aside.

The blind count is then the whole contents figure and the share is one. Two published with one met: two blind contents out of two. Six with three met: twenty out of twenty. Ten with six met: two hundred and ten out of two hundred and ten.

The bar is dashed from end to end, which is the only place in the record where that happens. Teams testing how order changes interpretation may compare team time tracking software with completed work, deadlines and the explanations supplied by contributors.

Nothing checkable, three page sizes

the whole bar: every content a reader might come away with
the dashed part: contents holding nothing checkable

Three shapes drawn; three be blind whatever the reader meets.

The bars are dashed end to end. Two published with one met, six with three, ten with six — in each the whole contents figure is blind, because nothing was published that a reader could verify.

Computed from the three numbers. A blind content is one holding no checkable statement: that is a property of the page, not a prediction about any reader.

Forty-four, and why it has to be

There are forty-four such shapes, one on every page, because a page has exactly one way to publish nothing checkable. Readers looking beyond this site's model can compare it with material from IEEE.

That is where the number comes from, and it needs saying because the record also has forty-four pages and the agreement looks like a coincidence. It is not. A shape is wholly blind precisely when the checkable count is nought, each page admits that count once, and so the tally of wholly blind shapes is forced to equal the tally of pages.

The record's own notes got this wrong at first and recorded the two figures as unrelated. They are the same count arrived at from two directions, and writing them as a coincidence would have been presenting an identity as a surprise.

What is left when verifiability is gone

Everything the first two parts of the record counted, untouched.

Ten published with six met still has one hundred and fifty-one thousand two hundred arrangements, two hundred and ten contents and seven hundred and twenty orders per content. A reader of such a set is in one of two hundred and ten states of information, each reachable in seven hundred and twenty ways, and not one of those states contains anything they could audit.

So the ordering problem and the subset problem are entirely independent of the verifiability problem, and a set of pages can have all three at once. It usually will.

Nothing checkable does not mean nothing said. It means everything said rests on the reader's willingness to accept it, and the record can count how many different unverifiable accounts are available.

The smallest case

Two published statements, one met, nothing checkable. Two contents, both blind, two arrangements.

This is the whole problem in miniature and it is worth holding onto for that reason. A reader arrives, meets one of two statements, and cannot confirm it. The record's entire apparatus — factorials, binomial coefficients, thresholds — reduces here to a reader who was told one thing and has no way to find out whether it is true.

Everything else in this record is that situation counted at larger sizes.

Why these shapes are drawn rather than dropped

A wholly blind shape has no internal structure. Every content is blind, the share is one on every page it appears on, and no figure distinguishes one such shape from another except the contents figure underneath it.

They are still tabulated, drawn and counted, for the same reason the shapes with a blind count of nought are. A record that listed only the cases with interesting fractions would have quietly removed both ends of its own range, and the ends are where the extremes of the subject are.

Forty-four at one, one hundred and thirty-nine at nought, and one hundred and fifty-five in between. The two ends hold more than half the record, and the fractions that get discussed most are the minority.

The figure that does distinguish them

Forty-four wholly blind shapes, and one number separates them: the contents figure, which is the whole blind count.

So the record can still say which wholly blind shapes are worse than others, in one narrow sense. Two published with one met has two blind contents; ten published with six met has two hundred and ten. A reader of the second comes away with one of two hundred and ten unverifiable accounts rather than one of two.

Whether that is worse is a judgement the record does not make. What it can say is that the number of distinct unverifiable impressions available grows exactly as fast as the contents figure, which the first two parts showed can reach two hundred and fifty-two inside this table alone.

The mirror

The opposite end — everything checkable published — is also forty-four shapes, one per page, for the identical reason.

Between the two ends the third number does its work, and the record's middle part described what that work is: a count of ways to avoid everything verifiable at once, which collapses to nought once there is too little to avoid and sits at the whole contents figure when there is nothing to avoid at all.