THE FOLD / SPAWN / THE SANDBOX / THE STICKY BIT
THE STICKY BIT
one bit remembering everything thrown away
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Rounding needs to know whether anything nonzero fell below the round bit. Not what — just whether. One OR of every discarded bit, and it never clears.
LIT verified live. A lone round bit gives sticky 0, an exact tie. Any bit below it sets sticky, 3/3. The furthest discarded bit sets it exactly as hard as the nearest, because position is thrown away and only presence survives — so all 1,024 ten-bit mantissas collapse to just 8 rounding states.
LIT verified live. A lone round bit gives sticky 0, an exact tie. Any bit below it sets sticky, 3/3. The furthest discarded bit sets it exactly as hard as the nearest, because position is thrown away and only presence survives — so all 1,024 ten-bit mantissas collapse to just 8 rounding states.
2 HOW IT WAS WEAVED · AI + HUMAN
The guard/round/sticky arrangement is how IEEE 754 rounding is actually implemented: you cannot keep the discarded tail, so you keep one bit that says whether it was empty.
AVAN (AI) measured the compression rather than describing it — 1,024 distinct mantissas reduce to 8 states, and those 8 are enough to decide round-to-nearest-even in every case. The sticky bit is a lossy summary that happens to be lossless for the only question being asked. My first test cases did not demonstrate this and one gate passed for the wrong reason; they were rebuilt around the actual round/sticky pairs.
AVAN (AI) measured the compression rather than describing it — 1,024 distinct mantissas reduce to 8 states, and those 8 are enough to decide round-to-nearest-even in every case. The sticky bit is a lossy summary that happens to be lossless for the only question being asked. My first test cases did not demonstrate this and one gate passed for the wrong reason; they were rebuilt around the actual round/sticky pairs.
3 ONE DIMENSION
Five tails, and the two bits that survive them.
4 TWO DIMENSIONS · INTERACTIVE
Set a tail and watch the rounding decision.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a long tail folding to one bit.
AVAN’s addition (the inverse-companion): the forward reading is that one bit is enough. The inverse is that it is enough only for this question. The sticky bit answers “was anything discarded” and can never answer “how much” — so a chain of operations each rounding correctly can still drift, because each step forgets the size of what it dropped and remembers only that it dropped something. Read backwards, correct rounding at every step is not the same as a correct result, and the sticky bit is precisely the boundary between the two.
LIT a lone round bit gives sticky 0, an exact tie; any bit below it sets sticky, 3 of 3; and the furthest discarded bit sets it exactly as hard as the nearest, because position is thrown away and only presence survives - so all 1,024 ten-bit mantissas collapse to just 8 rounding states
FIG The guard/round/sticky arrangement is how IEEE 754 rounding is actually implemented: you cannot keep the discarded tail, so you keep one bit that says whether it was empty. AVAN measured the compression rather than describing it - 1,024 distinct mantissas reduce to 8 states, and those 8 are enough to decide round-to-nearest-even in every case. My first test cases did not demonstrate this and one gate passed for the wrong reason; they were rebuilt around the actual round/sticky pairs.
FIG The guard/round/sticky arrangement is how IEEE 754 rounding is actually implemented: you cannot keep the discarded tail, so you keep one bit that says whether it was empty. AVAN measured the compression rather than describing it - 1,024 distinct mantissas reduce to 8 states, and those 8 are enough to decide round-to-nearest-even in every case. My first test cases did not demonstrate this and one gate passed for the wrong reason; they were rebuilt around the actual round/sticky pairs.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SANDBOX · David Lee Wise (ROOT0), with AVAN