Five steps, and no rings left

Forty-six rings sounds like forty-six problems. It is not.

Five steps, if any one of them could be obtained out of turn, account between them for every ring in the map. Take those five out of the ordering problem and nothing circular is left anywhere.

The five, and what hangs on them

• struck through: what it requires
• solid: what requires it
• one circle, one ring

Five steps drawn. They ask for 20 things between them and answer for 23. Five of them stand on at least one ring.

Five steps. Take each out of the ordering problem and every one of the forty-six rings is gone. To compare the map with real work patterns, reviewers can see the current feature set and define the decision, access rules and review date in advance.

The five

The list is built greedily: take the step on the most rings, remove every ring through it, take the step on the most of what remains, and repeat until nothing is left. It takes five rounds. A wider policy and evidence perspective is available from Deloitte.

How the forty-six fall

The first step removes three quarters of them. The first two remove all but four. The last three deal with the appointment pair, the paper ring and a single remaining place-and-money loop.

One step accounts for thirty-four of the forty-six rings. Two account for forty-two. The knot is much smaller than the number of ways of describing it.

What "taken out of turn" means

It means that the step could be obtained on some footing that does not require the things it currently requires. A provisional registration. An account opened against something else. Cover arranged without status being settled first.

It does not mean the step is skipped. Everything in this map still has to be done; what changes is only that one of them stops being downstream of the others.

Who could do it

Not the person in the map. That is the uncomfortable part.

A person can enter a ring at a soft link — there are eight of those, and they open twenty-nine rings — but a soft link is a thing that exists or does not. Nobody applying for anything can make a step acceptable out of turn.

Whoever writes the rule for one of these five could. The change required is not large, does not cost anybody a document, a check or a fee, and does not weaken any verification: the step still gets done, and the thing it currently waits for still gets checked, afterwards.

The entire circular structure of this map could be removed by five decisions, none of which costs anybody anything, and no single office is in a position to notice that they are available.

Why the greedy list is the right one

Because it is the honest one. A smaller set might exist in principle, so the arithmetic was run exhaustively: no four steps are enough, five are, and there are forty-eight different fives that work.

An account at a local bank appears in every one of the forty-eight. Not one of the five-step sets that dissolves this map leaves it out, which is a stronger statement than any ring count — the account is not merely the busiest step here, it is unavoidable.

What matters is not the particular five. It is that the number is five and not twenty, and that one step is in all of them.

What the five have in common

Four of them are filing facts about a person — where they live, how they are paid, what they are enrolled as, whether they have been fingerprinted. The fifth is a translation.

None is slow. Between them they cost seventy-eight days, which is less than the single longest step in the map.

Seventy-eight days of errands, standing between a person and every circular dependency in a thirty-six-step map.

That is the figure this record was built to produce. Not the forty-six, not the seventeen, not the hundred and seven: five steps, seventy-eight days, and nothing circular left anywhere.

The alarming numbers on this map are all descriptions of the same small object, and the small object has five parts.

Why greedy is enough here

A greedy cover is not guaranteed to be smallest in general. In this map it happens to be, and that was checked rather than assumed: every set of four was tested and none covers all forty-six rings.

That check is cheap on an object this size, and it matters because the whole value of this page rests on the number being five rather than six or seven.

It also produced the stronger result on the same pass: forty-eight different five-step sets work, and one step is in every one of them. A fact of that kind cannot be got from a greedy run alone.

The caveat that belongs on this page

The figure rests on the drawing, and the drawing rests on judgements. Several of the thirty-four rings through the account exist because of a single hard marking — that a lease cannot be completed without a local account — and if that marking is wrong, the account's thirty- four falls and this whole page is arithmetic about something else.

That marking, and the two others carrying the most weight, are set out by name in the part of this record about the order that holds. They are printed so that a reader can disagree with one of them precisely, rather than with the record in general.

No country, office or kind of permission is named anywhere here, and nothing on this page is a proposal addressed to anybody.