THE FOLD / CHEAT / NOCLIP / THE REFLECTION
THE REFLECTION
a path folded through a wall
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Count the random walks that ever touch a level. Doing it directly means tracking the whole history of every path. Désiré André’s trick from 1887: take any path that touches level a, and reflect everything after the first touch. What comes out is a path ending at 2a−b, and the correspondence is one-to-one both ways — so a question about histories becomes a question about endpoints, which is just a binomial coefficient.
LIT verified live by enumerating all 262,144 walks of length 18: the identity P(max ≥ a) = P(Sn ≥ a) + P(Sn ≥ a+1) holds exactly at all 18 levels; the first-passage count equals (a/t) × #{St = a} exactly at all 56 (level, time) pairs tested; and the reflection map is an exact bijection onto paths ending at 2a−b in all 57 (a, b) pairs at length 14. Not approximately — equal integers, every time.
LIT verified live by enumerating all 262,144 walks of length 18: the identity P(max ≥ a) = P(Sn ≥ a) + P(Sn ≥ a+1) holds exactly at all 18 levels; the first-passage count equals (a/t) × #{St = a} exactly at all 56 (level, time) pairs tested; and the reflection map is an exact bijection onto paths ending at 2a−b in all 57 (a, b) pairs at length 14. Not approximately — equal integers, every time.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at NOCLIP: the reflected path walks straight through the barrier, and that is the entire method.
AVAN (AI) checked the bijection itself rather than only the probability identity it implies. Those are different claims: the identity could hold by coincidence of totals while the correspondence failed, and the whole force of the argument is that the map is one-to-one. Counting both sides for every (a, b) pair and getting equal integers is what establishes it. Everything here is exhaustive rather than sampled — 218 paths is small enough to enumerate, and a combinatorial identity claimed for all paths cannot be supported by a subset. The one thing worth flagging: this is the simple walk with steps ±1, where reflection is exact. For walks with other step distributions the picture breaks, because the reflected path is no longer a legal path.
AVAN (AI) checked the bijection itself rather than only the probability identity it implies. Those are different claims: the identity could hold by coincidence of totals while the correspondence failed, and the whole force of the argument is that the map is one-to-one. Counting both sides for every (a, b) pair and getting equal integers is what establishes it. Everything here is exhaustive rather than sampled — 218 paths is small enough to enumerate, and a combinatorial identity claimed for all paths cannot be supported by a subset. The one thing worth flagging: this is the simple walk with steps ±1, where reflection is exact. For walks with other step distributions the picture breaks, because the reflected path is no longer a legal path.
3 ONE DIMENSION
A path, its first touch, and the reflection of everything after it.
4 TWO DIMENSIONS · INTERACTIVE
Two counts that must agree, for every level.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: paths above the barrier, each paired with its reflection below.
AVAN’s addition (the inverse-companion): the forward reading is “reflection counts the paths that touch a level.” The inverse is that it works by destroying exactly the information the question was about. Reflection throws away where the path went after its first touch — and it is allowed to, because the question only asked whether the touch happened. Read backwards, the trick is a lesson in what a proof is permitted to forget: the reflected path is not the original and nobody claims it is, and the argument is sound precisely because the discarded part was never being counted.
LIT by enumerating all 262,144 walks of length 18, the identity P(max >= a) = P(S_n >= a) + P(S_n >= a+1) holds exactly at all 18 levels; the first-passage count equals (a/t) x #{S_t = a} exactly at all 56 (level, time) pairs tested; and the reflection map is an exact bijection onto paths ending at 2a-b in all 57 (a, b) pairs at length 14 - equal integers, every time
FIG The BIJECTION itself was checked, not only the probability identity it implies. Those are different claims: the identity could hold by coincidence of totals while the correspondence failed, and the whole force of the argument is that the map is one-to-one. Counting both sides for every (a, b) pair and getting equal integers is what establishes it. Everything is exhaustive rather than sampled - 2^18 paths is small enough to enumerate, and an identity claimed for ALL paths cannot be supported by a subset. Worth flagging: this is the SIMPLE walk with steps +-1, where reflection is exact; for other step distributions the reflected path is no longer legal. Desire Andre, 1887.
FIG The BIJECTION itself was checked, not only the probability identity it implies. Those are different claims: the identity could hold by coincidence of totals while the correspondence failed, and the whole force of the argument is that the map is one-to-one. Counting both sides for every (a, b) pair and getting equal integers is what establishes it. Everything is exhaustive rather than sampled - 2^18 paths is small enough to enumerate, and an identity claimed for ALL paths cannot be supported by a subset. Worth flagging: this is the SIMPLE walk with steps +-1, where reflection is exact; for other step distributions the reflected path is no longer legal. Desire Andre, 1887.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NOCLIP · David Lee Wise (ROOT0), with AVAN