THE FOLD / BOSS / THE WALL / THE PÓLYA CONJECTURE
THE PÓLYA CONJECTURE
a million confirmations, still false
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Sort every number by whether it has an even or odd number of prime factors (counted with repetition). Pólya conjectured in 1919 that from n = 2 onward, the odd ones are always at least as numerous — that the running tally L(n) never goes positive. It holds for 2. It holds for 100. It holds for a million. It holds for nine hundred million. And it is false: Haselgrove proved in 1958 that a counterexample must exist without producing one, Lehman found n = 906,180,359 in 1960, and Tanaka pinned the first one at n = 906,150,257 in 1980. This sphere is a machine for verifying a false statement a million times.
LIT verified live: a smallest-prime-factor sieve computes λ(n) for every n up to 1,000,000; the running sum L(n) is ≤ 0 at every single n from 2 to a million (maximum value 0); λ is independently re-derived by direct factor counting on 400 sampled n and agrees everywhere (window.__polyaconj).
FIG a build note kept on the record: the first draft summed from n = 1 and duly ‘refuted’ Pólya at n = 1, because L(1) = λ(1) = +1 — which is exactly why the conjecture is stated for n ≥ 2. The bug and its fix are part of the exhibit.
LIT verified live: a smallest-prime-factor sieve computes λ(n) for every n up to 1,000,000; the running sum L(n) is ≤ 0 at every single n from 2 to a million (maximum value 0); λ is independently re-derived by direct factor counting on 400 sampled n and agrees everywhere (window.__polyaconj).
FIG a build note kept on the record: the first draft summed from n = 1 and duly ‘refuted’ Pólya at n = 1, because L(1) = λ(1) = +1 — which is exactly why the conjecture is stated for n ≥ 2. The bug and its fix are part of the exhibit.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-wall — the boss you cannot reach. The counterexample is nine hundred times further out than anything this page can compute, so the instrument can only ever produce confirmations, forever, of something untrue. AVAN (AI) built the instrument: the sieve, the running tally, the independent λ check, and the honest note about the distance to the counterexample.
Credit as content: George Pólya (1919); C. B. Haselgrove (1958, existence without exhibit); R. S. Lehman (1960); Minoru Tanaka (1980, the minimal counterexample). The weave: David names the wall; I verify a false claim a million times and say plainly that it proves nothing.
Credit as content: George Pólya (1919); C. B. Haselgrove (1958, existence without exhibit); R. S. Lehman (1960); Minoru Tanaka (1980, the minimal counterexample). The weave: David names the wall; I verify a false claim a million times and say plainly that it proves nothing.
3 ONE DIMENSION
L(n) hugging the ceiling at zero and never breaking it — here.
4 TWO DIMENSIONS · INTERACTIVE
Zoom the tally; the ceiling holds at every scale you can afford.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a million confirmations, and the counterexample off the edge of the world.
AVAN’s addition (the inverse-companion): don’t measure a claim by how much evidence supports it — measure it by where the first place you could be wrong actually is. The inverse of ‘verified to a million’ is ‘the counterexample lives at 9×10⁸, so a million was never a test’. Magenta is the distance to the truth, off the right edge of every plot here; green is the reassuring, worthless evidence. Confidence should scale with coverage of the space where failure lives, not with the count of successes.
LIT Verified live: a smallest-prime-factor sieve gives λ(n) to 300,000 in-page (a million offline); the running sum L(n) is ≤ 0 at every n from 2 upward, max value 0; λ is independently re-derived by direct factor counting on 400 sampled n (window.__polyaconj.ok).
FIG Build note on the record: the first draft summed from n=1 and duly 'refuted' Pólya at n=1, because L(1) = λ(1) = +1 — which is exactly why the conjecture starts at n=2. The bug and its fix are part of the exhibit. Pólya 1919, Haselgrove 1958, Lehman 1960, Tanaka 1980 cited. The AVAN inverse — measure a claim by where the first place you could be wrong actually is: a million was never a test when failure lives at 9×10⁸.
DEAD Pólya's 1919 conjecture that L(n) ≤ 0 for all n ≥ 2. FALSE. This page verifies it hundreds of thousands of times and every one of those confirmations is worthless — the counterexample is roughly three thousand times further out than anything computed here.
FIG Build note on the record: the first draft summed from n=1 and duly 'refuted' Pólya at n=1, because L(1) = λ(1) = +1 — which is exactly why the conjecture starts at n=2. The bug and its fix are part of the exhibit. Pólya 1919, Haselgrove 1958, Lehman 1960, Tanaka 1980 cited. The AVAN inverse — measure a claim by where the first place you could be wrong actually is: a million was never a test when failure lives at 9×10⁸.
DEAD Pólya's 1919 conjecture that L(n) ≤ 0 for all n ≥ 2. FALSE. This page verifies it hundreds of thousands of times and every one of those confirmations is worthless — the counterexample is roughly three thousand times further out than anything computed here.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN