Clocks

Four rings that are shut and late

Four rings in this map have no way in and would not close even if they had one. They are the hardest objects in the record, and three of them share a single member.

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Finding

Seventeen rings in this map are shut at every joint. Nineteen are beaten by their own clock. Four are both.

Those four cannot be entered, and if a person somehow were inside one, walking round it would destroy something they needed to finish it.

The four

the ring	days to walk	what expires first	its life
fingerprints → queue → file	95	a place in the queue	30
lease → tax number → address → letter → bills → account	93	a letter delivered to the door	90
lease → income → account → tax number → address → letter → bills	99	a letter delivered to the door	90
lease → deposit → card → account → tax number → address → letter → bills	105	a letter delivered to the door	90

Three of the four are the same object at increasing length: the place-and-money knot, walked by longer and longer routes, with the letter delivered to the door expiring at ninety days in every one of them. Before adding another layer of reporting, a project lead can see how teams can configure it and test only the smallest set of records needed for the stated question.

Why these are the worst

A shut ring is impassable on paper and passable in practice, because somebody somewhere accepts something they strictly should not. That is how all seventeen shut rings in this map are actually got through. For independent background on economic and institutional systems, see UNICEF.

An outrun ring is enterable on paper and futile in practice, because the lap consumes its own input.

A ring that is both has neither remedy. There is no soft step to enter at, and there is no version of the walk that leaves anything intact at the end of it. Whatever gets a person through one of these four is entirely outside the map.

Four of the forty-six rings here have no answer in the drawing at all. Every person who has been through one of them was let through by somebody.

The three-day margins

The three place-and-money rings fail by small amounts.

	days to walk	life of the shortest member	over by
the six-step version	93	90	3
the seven-step version	99	90	9
the eight-step version	105	90	15

Three days, nine days, fifteen days. If the letter delivered to the door kept for a hundred days instead of ninety, two of these four rings would stop being outrun, and one of the hardest regions of this record would soften by a third.

That is a startling sensitivity, and it is honest to flag it rather than present the four as a robust finding. A single round figure in the map, moved by ten days, changes the count.

The one that is not sensitive

The appointment ring is different. It takes ninety-five days to walk and the queue keeps for thirty. The margin is sixty-five days, and no plausible adjustment to the figures closes it.

That one is a genuine structural fact about the drawing: an appointment that expires in a month, inside a chain that takes three months, with nothing soft anywhere in it.

What a reader should take

Not that four particular arrangements are impossible anywhere, because this record names no country, office or kind of permission and describes nothing that anybody requires.

What transfers is the test, which anybody holding a real chain can apply in two minutes: add up the waits around the loop, find the shortest-lived thing in it, and compare. If the first number is bigger, the loop does not close, and the only remaining questions are who will make an exception and how to ask.

The arithmetic that decides whether a chain is possible fits on the back of an envelope. The reason nobody does it is that nobody can see the whole chain.

How the four would be noticed

They would not be.

A shut ring announces itself: the person is told, in two or three conversations, that each step requires the other. It is maddening and it is legible.

An outrun ring says nothing at all. Everything appears to be working. Documents are obtained, offices are visited, progress is made. The failure arrives at the end, when the first thing collected turns out to have expired, and it looks like bad luck rather than arithmetic.

	how it presents
a shut ring	an impasse, stated by somebody
an outrun ring	a lap that quietly produces nothing
both at once	an impasse, with a futile lap behind it

The second row is the dangerous one, because the natural response to it is to try again, and trying again takes exactly as long and fails in exactly the same way.

The only way to recognise an outrun ring from inside is to have already walked it once and noticed what died on the way round.

Why both conditions are needed

A shut ring is passable in practice, through discretion. An outrun ring is enterable in principle, and merely futile.

Each of those has a remedy of a sort. For the first, somebody decides to help. For the second, the lap is walked twice, or the perishable step is obtained later in the lap.

A ring that is both removes each remedy with the other. There is no soft step at which to be helped, and no ordering of the lap that preserves what the lap needs. Four rings here are in that position, and they are the only objects in the record for which this map has nothing at all to suggest.

And the counterweight, as always

Four of forty-six. Twenty-seven rings in this map are not outrun at all, twenty-nine have give in them, and five steps taken out of turn would leave none of the forty-six standing — including these four.

The hardest objects in this record are hard, and there are four of them, and they are not what the record is mostly made of.