Limits

Nothing Here Was Quoted From Anywhere

Every figure on this site was produced from three integers. That removes one kind of error, introduces another, and both are worth stating.

11520 outcomes

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Not one figure on this site was taken from anybody else. No survey, no study, no benchmark, no number read off a report.

Everything was produced from three integers — by formula where a formula exists, by generating every outcome where none does. Teams comparing routes, outcomes and corrections can read how the platform handles corrections, then check whether the resulting evidence improves the decision in practice.

Five candidates, eight pairs compared

column height: how many outcomes have exactly that many circular triples
the figure under each: how many triples came out circular

Five candidates with eight pairs compared. The columns drawn are every one the page has, and add to all 11 520 of its outcomes. 3330 of the 11 520 fall into an order. The shortest column is about 124 times shorter than the tallest, and the scale is linear. A separate technical or research perspective relevant to the comparison can be found at Graphviz.

Five columns adding to all eleven thousand five hundred and twenty outcomes the page allows. None of the five came from anywhere but enumeration.

The column is the count of outcomes with exactly that many circular triples. The columns of one page add to the page's own total, and the drawing checks it.

What comes attached to a borrowed number

A history the borrowing page has not carried across.

Somebody measured something, under conditions nobody downstream can inspect, at a time that may or may not be recorded, by a method that decides whether the figure is a measurement or a forecast in disguise. All of that travels with the number and none of it is printed beside it.

Checking even one such figure means tracking down the original, reading far enough to see what was counted, and deciding whether the borrowing sentence does it justice. The work is seldom done, and the number carries on regardless.

There is a worse property. When the source is revised, the borrowing sentence does not change: it goes on sounding exactly as sure of itself, with no mechanism for learning that it has become false and none for telling anybody.

Which figures here come from where

Two of the five on every row are a line of arithmetic.

The pair count is C(n, 2). The outcome count is C(C(n,2), r) multiplied by two to the r. Both can be done on the back of anything.

Two more come from generating outcomes. How many hold exactly a given number of circular triples, and how many admit an order, are counts no short expression produces, so every outcome of every page was produced and examined — fourteen million three hundred and forty-eight thousand nine hundred and seven of them for a field of six.

The fifth is the third divided by the second.

Which figures a reader can redo in a line and which would need a machine is itself printed. Both kinds can be checked; they cannot be checked the same way.

What is given up

Any claim about the world, and it should be said rather than implied.

Nothing borrowed means nothing described. The record has no view on how large fields usually are, how many comparisons typically get made, how often results really clash, or whether anybody minds when they do.

Exactness and evidence are not interchangeable. A count under a printed convention answers a small question completely; a measurement answers a large one roughly. Readers wanting the second will not find it here.

The four quantities that would be needed

Worth naming, since the absence is the point.

Typical field size. Typical number of comparisons. How often real results contradict each other. Whether a contradiction changes what anybody then does.

With those four the counts here could be weighted into expectations and the shares would approach frequencies. Lacking them, the shares stay counts — and a count printed with its convention beside it is a better thing to publish than an expectation resting on four quantities nobody can verify.

Safe borrowing would need four things at once

The rule here is not that borrowing is always wrong.

It is that borrowing safely requires knowing what was measured, under what conditions, when, and by what inspectable method — all four together. A number with a method and no date goes quietly stale; one with a date and no method cannot be questioned at all.

A page that borrows without supplying those is asking for belief on somebody else's behalf, which is more than it sounds like, and a counting problem has no need of it.

One figure did come from outside

A blanket claim invites a search for the exception, so here it is.

The colours of the drawings were tested against a published minimum for contrast, and the minimum was not derived here. It is a condition the drawings had to meet rather than a quantity any sentence reports, and no count, share or claim on this site rests on it.

The columns above

Five candidates with eight of their ten pairs compared: eleven thousand five hundred and twenty outcomes.

Three thousand three hundred and ninety with no circular triple, three thousand seven hundred and twenty with one, three thousand five hundred and forty with two, eight hundred and forty with three, thirty with four.

Those five add to the page exactly, and each was arrived at by producing fifty-nine thousand and forty-nine outcomes and counting them.

What this costs a reader

Two things, and neither of them is trivial.

There is no reading list. Anybody who wants to go further gets no pointer from here, and arithmetic this plain has certainly been set down elsewhere by people better at it.

And there is no corroboration. A reader who doubts a figure cannot hold it against a published table; they can only recompute it. That is why the method behind each figure sits on the page beside it rather than being referenced. Recomputing is slower than following a citation, and it does not depend on a link that may stop resolving.

What the restriction buys

One thing: nobody else can make this record wrong.

Pages built on quotations are hostages to their sources — revised, withdrawn, reinterpreted or simply aged, with the page none the wiser. Nothing here has that exposure, because nothing here was taken from anywhere.