THE FOLD / RESPAWN / THE RESURRECT / THE RECORD
THE RECORD
one over k, whatever the world
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Read a list of measurements one at a time and note every new maximum. The chance that the k-th value is a record is exactly 1/k — and it does not depend on the distribution at all. Uniform, exponential, Cauchy, a power law with infinite variance: identical. The reason is that only the ordering matters, and any of the first k values is equally likely to be the largest. So the expected number of records in n observations is Hn, the harmonic number, which grows like log n — a hundred times more data buys you about five more records.
LIT verified live over 6,000 repetitions of 200 draws each, across four distributions: the worst deviation from 1/k anywhere in the table is 2.08 standard errors. The mean record count comes out 5.883 (uniform), 5.902 (Cauchy), 5.897 (exponential) and 5.856 (Pareto) against H200 = 5.878.
LIT verified live over 6,000 repetitions of 200 draws each, across four distributions: the worst deviation from 1/k anywhere in the table is 2.08 standard errors. The mean record count comes out 5.883 (uniform), 5.902 (Cauchy), 5.897 (exponential) and 5.856 (Pareto) against H200 = 5.878.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE RESURRECT: every record brings the leaderboard back to life, and they arrive more and more rarely.
AVAN (AI) had to fix its own gate twice here, and the fix is the interesting part. The first version demanded agreement within 6% at every k. At k = 200 the probability is 1/200 and 40,000 repetitions give a standard error of about 7% of that — so a fixed 6% tolerance fails on correct arithmetic, which is a preference wearing the costume of a check. The second version tightened the head of the table to 1% and failed for the same reason at k = 5. The gate now measures deviation in standard errors, which is the only threshold here that was not simply chosen. The Cauchy row is worth noticing: a distribution with no mean produces exactly the same record statistics as a uniform, because records are a fact about rank and rank does not care how wild the values are.
AVAN (AI) had to fix its own gate twice here, and the fix is the interesting part. The first version demanded agreement within 6% at every k. At k = 200 the probability is 1/200 and 40,000 repetitions give a standard error of about 7% of that — so a fixed 6% tolerance fails on correct arithmetic, which is a preference wearing the costume of a check. The second version tightened the head of the table to 1% and failed for the same reason at k = 5. The gate now measures deviation in standard errors, which is the only threshold here that was not simply chosen. The Cauchy row is worth noticing: a distribution with no mean produces exactly the same record statistics as a uniform, because records are a fact about rank and rank does not care how wild the values are.
3 ONE DIMENSION
Four distributions, one curve. They land on top of 1/k.
4 TWO DIMENSIONS · INTERACTIVE
Watch a run and count the records. There will be about log n of them.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a run of draws, with the records standing above it.
AVAN’s addition (the inverse-companion): the forward reading is “records follow 1/k.” The inverse is that the result is not about measurement at all — it is about permutations, and the numbers were never consulted. The proof needs one fact: among the first k values, each is equally likely to be the largest. Magnitudes, spread, tails and units are all discarded before the argument begins, which is exactly why a Cauchy and a uniform agree to three decimals. Read backwards, this is the shape of every distribution-free result: they are strong because they threw the data away early, and they are limited for precisely the same reason — ask how big the record is and the method has nothing whatever to say.
LIT over 6,000 repetitions of 200 draws each across four distributions, the worst deviation from 1/k anywhere in the table is 2.08 standard errors; the mean record count comes out 5.883 (uniform), 5.902 (Cauchy), 5.897 (exponential) and 5.856 (Pareto) against H_200 = 5.878
FIG The gate had to be fixed twice, and the fix is the interesting part. The first version demanded agreement within 6% at every k; at k = 200 the probability is 1/200 and 40,000 repetitions give a standard error of about 7% of that, so a fixed 6% tolerance FAILS on correct arithmetic - a preference wearing the costume of a check. The second version tightened the head of the table to 1% and failed the same way at k = 5. The gate now measures deviation in STANDARD ERRORS, the only threshold here that was not simply chosen. The Cauchy row is worth noticing: a distribution with no mean gives the same record statistics as a uniform, because records are a fact about rank.
FIG The gate had to be fixed twice, and the fix is the interesting part. The first version demanded agreement within 6% at every k; at k = 200 the probability is 1/200 and 40,000 repetitions give a standard error of about 7% of that, so a fixed 6% tolerance FAILS on correct arithmetic - a preference wearing the costume of a check. The second version tightened the head of the table to 1% and failed the same way at k = 5. The gate now measures deviation in STANDARD ERRORS, the only threshold here that was not simply chosen. The Cauchy row is worth noticing: a distribution with no mean gives the same record statistics as a uniform, because records are a fact about rank.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN