Circularity
From the Register of the Standard, Kelling Reach. Note appended by B. Rooke, ninth Registrar, at the Lean of the sixty-first turn, for the attention of Council.
Nobody has stolen anything. I put that at the top because it is the finding, and everything below it is only the reason.
The Register requires that at the close of each quarter I set down the schedule and carry the Standard forward. I did so this morning. I have copied two older entries beneath it, in the form the book requires, so that Council can see the three together.
Lean, turn 1. Shipped protein, 4.20 the kilo. Hired hours, 3.80 the hour. Standard: 1.0000.
Lean, turn 32. Shipped protein, 4.20 the kilo. Hired hours, 3.80 the hour. Standard: 1.1680.
Lean, turn 61. Shipped protein, 4.20 the kilo. Hired hours, 3.80 the hour. Standard: 1.3506.
Those are not rounded prices. They are the prices. The Kerrera sets down every four hundred and two days, and the Reach breathes with her: protein cheap at the Landing and dear by the Lean, hours dear at the Landing when every holding is flush and cheap by the Lean when nobody is hiring. The swing is small — five in the hundred, either way — and it is stable, because the ship is stable, and it has been stable for sixty-one turns. At the Lean, the schedule comes back to where it started. It has come back sixty times. The Standard has not come back once.
The Founding Instrument defines the mark as one mark of Standard, and defines the Standard as the cost of the schedule. It does not say how to work that cost out. Article Eleven says only that the Standard shall be carried forward from turn to turn, which is the whole of the law on the matter, and which Osric Pell, the first Registrar, took to mean what any careful person would take it to mean: you compare each quarter with the quarter before it, and you multiply.
I want to be exact about Pell, because it will be tempting for Council to be unfair to him. The original schedule had forty lines. By the fourth turn, eleven of them were dead — lamp oil, sealing cord, the whole cable trade — and four more were being bought in quantities nobody had foreseen. A Registrar who insisted on pricing the founders’ basket in the founders’ proportions would have been publishing, within a decade, the cost of a way of living that no longer existed. Pell saw that immediately and wrote so. His note is in the front of volume one, and it is better reasoned than anything I have written in nine turns. The comparison must be made between neighbours, he says, because only neighbours have enough in common to be compared. He was right. He is still right. That is the difficulty.
I spent the first half of this turn looking for a thief, on the assumption that a number which only ever goes up is being pushed. There is no thief. I have re-worked every link from turn 44 forward by hand, and audited a sample of one in eight before that, and the arithmetic is clean. Every quarter, the Registrar of the day took the previous quarter’s spending as the weights, priced this quarter’s schedule against it, and multiplied the result into the running figure. Every one of those two hundred and forty steps is correct. I would have signed any of them.
Here is what the steps do.
Over one full turn, protein goes down five, up five and a bit, up five, down not quite five, and ends where it began; the product of its four ratios is one. Hours do the same in the opposite order; the product of their four ratios is also one. But each link is not a product. Each link is an average — the two ratios, weighted by what people were spending, added and halved. And the average of two numbers whose product is one is never less than one. It is equal to one only when the two numbers are the same. They are never the same. That is the entire mechanism. Half a percent a turn, compounding, for sixty turns.
There is nowhere in that sentence to put a wrongdoer.
Councillor Vey came to the office when the summary went round. I record the exchange because he asked the questions Council will ask.
“So we’ve been overcharging the tank-holders.”
“No. Every bill was correctly calculated.”
“Bettany. They’ve paid a third more than they owed.”
“They’ve paid what they owed. The instrument says the lease follows the Standard. It has followed the Standard.”
“Then the Standard’s wrong.”
“The Standard is a rule. It’s been applied properly for sixty-one turns.”
He sat down at that point, which I took to mean he had got to the part I had got to, four weeks earlier and at greater length. Then he said: “What did the founders think they were buying?”
I told him I thought they believed they were buying a fixed quantity of living, and that this belief was not available to them, and had not been available to anyone, and that Pell had understood this and had said so in a way nobody read.
“That’s not going to do,” Vey said.
I agreed that it was not going to do.
The consequences are exactly what you would expect them to be, and they are large.
Every claim in the Reach that is indexed to the Standard has grown by thirty-five in the hundred against every claim that is not. The Foundation’s water leases are indexed. The outer-tank holdings are not: they earn what protein earns, and protein, at the Lean, costs what it cost in turn one. The Torrance holding pays 1,340 marks a turn on a lease that an undrifted Standard would put at 992. Four generations have made that payment and thought it the weather.
The annuities are indexed too, so the Foundation’s pensioners have gained by the same rule that has cost the tank-holders. So has the Registry. So, by a smaller amount, have I. There is no party to this who is outside it.
There is a repair, and it is not difficult. Instead of adding the two ratios and halving, you multiply them and take the root. That figure does not drift: when the ratios come back, the product of the roots comes back, exactly, every time. It is the same information, arranged so that it cannot ratchet. I recommend Council adopt it from the Landing of turn 62, and I will publish the working.
It refunds nothing. I want Council to understand why, because I expect the next motion after adoption will be a motion for restitution.
To refund, you must say what the payment should have been. That means saying what the schedule of turn 61 costs relative to the schedule of turn 1 — not each against its neighbour, but the two ends against each other, across sixty turns of a settlement that changed everything it consumed. And the Reach does not contain the fact you would need.
I can give Council three answers, all defensible, none preferable.
If you take the two lines that survive from the founders’ schedule and price them directly, one turn against the other, the answer is 1.0000. Protein costs what it cost. An hour costs what it cost.
If you take the chain as kept, the answer is 1.3506.
If you take the chain as it would have been kept under the repair, the answer is 1.0000 again — but only because the swing happens to close. Add a single line that rose and stayed risen, and it parts from the direct comparison too.
And if you ask for the founders’ full schedule priced today, there is no answer at all, because eleven of its lines cannot be bought at any price, and a price for a thing that does not exist is not a small difficulty but the absence of the question.
The chain and the direct comparison agree for neighbours and disagree over distance. This is not a defect in our practice. It is a property of the thing: a comparison of prices across time is not a measurement of the world. It is an agreement about how to compare, and the agreement has to be made before the number exists, and different agreements — each of them honest, each of them defensible before this Council — give different numbers. We made ours in Article Eleven, in eight words, sixty-one turns ago, without knowing that we were making it.
I have not adjusted anything. The entry for the Lean of turn 61 stands as I made it this morning: protein at 4.20, hours at 3.80, Standard at 1.3506, carried forward. Council has not instructed me otherwise, and the Register must be kept.
The Kerrera is eleven days out. At the Landing, protein will fall to 3.99 and hours will go to 4.00, as they did at the Landing of turn 60, and of turn 32, and of turn 1. I will price the schedule against this quarter’s spending, and multiply, and sign.
B. Rooke, Registrar of the Standard.