Limits

What This Record Is Not For

Closing with the uses it does not support, the one argument it does make, and the single reason a counting problem was chosen as a subject at all.

at most 32768 outcomes

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Forty pages, one hundred and twenty-eight shapes, five figures to a shape, three conventions at the front. The table is finished, and what is left to write down is where it should not be taken.

Three shapes at the end of the record

line with a head: a pair compared; the head points at whoever came out behind
no line: a pair nobody compared
closed ring of three: a circular triple: no order can hold all three

Three outcomes drawn. Rows differ in how many candidates. Which candidate is which is the drawing's own choice and carries no meaning.

The smallest contradiction there is, the smallest contradiction no triple shows, and a field compared exhaustively. The record's whole span in three rows. If several people repeat this comparison, managers may see the vendor's current explanation while interpreting every signal beside completed work and participant feedback.

Computed from the three numbers alone. Who the candidates are and what any comparison was about is outside the record; the counts are of outcomes, not of anybody. Readers examining the figures against practical research methods can also see Gephi.

It ranks nobody

The letters in these drawings mark positions, not merits. A is not ahead of B for being A.

An outcome holding eight circular triples is not a worse outcome than one holding none — it is a different arrangement of the same number of comparisons. The table says how many arrangements of each kind there are. Preferring one kind would need to know what the comparing was for, and that is precisely what three integers leave out.

It judges nobody's comparing

There is no comparing in the table. There are integers in it.

Deciding whether a set of comparisons was well made means knowing what was being weighed, and the first convention discarded that before any arithmetic began. Three circles may be the honest picture of a field that does not line up. A clean order may be what carelessness produces when nobody checks. Both come out identical here, because neither was ever examined.

It predicts nothing

Shares on this site are a count divided by a page total, and reading that quotient as a chance requires every arrangement on the page to be as reachable as every other.

Nothing anybody has ever done works that way. So no sentence here forecasts an outcome, estimates a likelihood, or claims a proportion of real cases.

It does not argue for more comparing, or less

Add comparisons and more gets settled and more contradicts. Remove them and both fall away, until below a floor of n − 1 nothing is settled at all.

The table quantifies each direction and recommends neither. Which way to go depends on what a settled order is worth, and worth is a quantity no shape here carries.

Counting what can happen says nothing about what ought to. This record has no vocabulary for the second question.

The one claim it does make

A field size and a number of comparisons are not a description of what was done. Three things follow from them that nobody tends to write down.

The size of the space of possible outcomes. How much of that space contradicts itself. And how much of it would actually rank the field — which, at the fewest comparisons that could do the job, comes out at roughly one outcome in a hundred and thirty-three.

Each is computable before anyone is compared with anyone.

Three results that hold without the conventions

The floor. Ranking n candidates cannot be done with fewer than n − 1 comparisons, since each comparison attaches at most one fresh candidate to the rest of the field.

The addition. Every outcome has one number of circular triples and no other, so a page's columns must add to the page's total.

The blind spot. A ring of four holds no circular triple at all, so checking every triple can pass an outcome that falls into no order. The smallest instance is four candidates with four comparisons: one hundred and ninety-two outcomes pass the triple check and six of them are not orderable.

What a reader might keep

Three figures and one habit.

Six candidates make fifteen pairs, and comparing ten of them admits three million seventy-five thousand and seventy-two outcomes. That is the scale, and it is knowable before anybody starts.

A ring of four holds no circular triple, so a triple check is not a contradiction check. Walking the comparisons is, it is faster, and it catches rings of every length.

Ranking n candidates needs n − 1 comparisons at the very least, and reaching that minimum is nowhere near enough to finish the job.

The habit is the one every row here practises: no figure without the three integers behind it, and no share without the two counts it was built from.

Why a counting problem

Because the arithmetic does not go stale.

Other occupants had this address before, and their material is gone the way such material goes — without anything visibly breaking, and with sentences that were once accurate still reading as accurate long after they stopped being so.

Four candidates make six pairs today and will make six tomorrow. Every figure here comes from three printed integers by a rule a reader can apply by hand, and none of it can go out of date, because none of it reports on the world.

What could make it wrong

One thing only, and it belongs in the last paragraph of the record.

Arithmetic. A figure that does not follow from the integers printed with it. A column that fails to add to its page. A drawing whose arrows are not the shape the surrounding prose is discussing. Any of the three can be caught by a reader with a pencil, which is the reason the checks are set out on the site rather than quietly carried out behind it.

Beyond that, nothing. No source here can be withdrawn, no measurement revised, nobody's agreement lost. This is not a grand property. It is simply what stays correct at an address with nobody looking at it, and that seemed worth leaving at one.