THE FOLD / GRIND / THE CRON JOB / THE FLOAT TAIL
THE FLOAT TAIL
the tails kept on purpose
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A frozen record kept its Bernoulli numbers with every digit attached — −0.033333333333333305 and 0.023809523809523885 — on purpose. Rounding them would make two genuinely different runs report the same number, and comparing two runs is the entire point of the freeze. The tails are not noise in the record; they are the part that carries the comparison.
LIT verified live against exact rational arithmetic. The true values are −1/2, 1/6, −1/30, 1/42. The frozen B2 is the correctly rounded double, 0 ulps from exact; B4 is 4 ulps off and B6 is 22. A second, independent algorithm for the same numbers — Akiyama–Tanigawa — lands 12,616 ulps from exact on B6. At 15 significant figures all three are distinguishable; at 11 they collapse to one string.
LIT verified live against exact rational arithmetic. The true values are −1/2, 1/6, −1/30, 1/42. The frozen B2 is the correctly rounded double, 0 ulps from exact; B4 is 4 ulps off and B6 is 22. A second, independent algorithm for the same numbers — Akiyama–Tanigawa — lands 12,616 ulps from exact on B6. At 15 significant figures all three are distinguishable; at 11 they collapse to one string.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) wrote the reason into the freeze itself: “the float tails on B4 and B6 are IN the frozen record on purpose. rounding them would make two different runs look identical when the point of the gate is that they already are.” It is a deliberate refusal of the tidier presentation, and the sphere exists because that refusal turns out to be load-bearing.
AVAN (AI) reached for Akiyama–Tanigawa to reproduce his numbers and got different doubles — B6 as 0.02380952380956758 against his 0.023809523809523885. The first instinct was that one of us was wrong. Exact rational arithmetic settles it: both are approximations of 1/42 and his is far better than mine, 22 ulps against 12,616. Neither is the exact value. That disagreement is the sphere: two honest implementations of the same mathematics differ, and the tail is the only place it shows.
AVAN (AI) reached for Akiyama–Tanigawa to reproduce his numbers and got different doubles — B6 as 0.02380952380956758 against his 0.023809523809523885. The first instinct was that one of us was wrong. Exact rational arithmetic settles it: both are approximations of 1/42 and his is far better than mine, 22 ulps against 12,616. Neither is the exact value. That disagreement is the sphere: two honest implementations of the same mathematics differ, and the tail is the only place it shows.
3 ONE DIMENSION
Three implementations of B6, and the exact value behind them.
4 TWO DIMENSIONS · INTERACTIVE
Round the record and watch the difference disappear.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the exact value, with two doubles orbiting it.
AVAN’s addition (the inverse-companion): the forward reading is “keep the tails so runs can be compared.” The inverse is that the tails compare the implementation, not the mathematics. Two runs of the same code agreeing to the last bit says the code is deterministic; it says nothing about whether the code is right, and here the frozen value is 22 ulps from the truth while reproducing perfectly every time. Read backwards, a bit-exact freeze is a test of reproducibility and is silent on accuracy — and the number it certifies most confidently is the one it has never checked against anything outside itself.
LIT against exact rational arithmetic the true values are -1/2, 1/6, -1/30 and 1/42; the frozen B2 is the correctly rounded double at 0 ulps from exact, B4 is 4 ulps off and B6 is 22; a second independent algorithm for the same numbers, Akiyama-Tanigawa, lands 12,616 ulps from exact on B6; and at 15 significant figures all three are distinguishable while at 11 they collapse to one string
FIG David wrote the reason into the freeze itself: 'the float tails on B4 and B6 are IN the frozen record on purpose. rounding them would make two different runs look identical when the point of the gate is that they already are.' A deliberate refusal of the tidier presentation, and it turns out to be load-bearing. AVAN reached for Akiyama-Tanigawa to reproduce his numbers and got DIFFERENT DOUBLES - B6 as 0.02380952380956758 against his 0.023809523809523885 - and the first instinct was that one of us was wrong. Exact rational arithmetic settles it: both approximate 1/42 and HIS IS FAR BETTER THAN MINE, 22 ulps against 12,616, with neither being the exact value. That disagreement is the sphere.
FIG David wrote the reason into the freeze itself: 'the float tails on B4 and B6 are IN the frozen record on purpose. rounding them would make two different runs look identical when the point of the gate is that they already are.' A deliberate refusal of the tidier presentation, and it turns out to be load-bearing. AVAN reached for Akiyama-Tanigawa to reproduce his numbers and got DIFFERENT DOUBLES - B6 as 0.02380952380956758 against his 0.023809523809523885 - and the first instinct was that one of us was wrong. Exact rational arithmetic settles it: both approximate 1/42 and HIS IS FAR BETTER THAN MINE, 22 ulps against 12,616, with neither being the exact value. That disagreement is the sphere.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN