THE FOLD / BOSS / THE FIREWALL / THE CHINESE HYPOTHESIS
THE CHINESE HYPOTHESIS
the test that lets impostors through
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Fermat’s little theorem says every prime n satisfies 2ⁿ ≡ 2 (mod n). The tempting converse — that any n passing the test must be prime — would be a one-line primality test. It is false, and the smallest witness is small enough to check by hand: 341 = 11 × 31 sails through. Worse are the Carmichael numbers, which pass for every base coprime to them — 561 = 3 × 11 × 17 is the first, and Alford, Granville and Pomerance proved in 1994 that there are infinitely many. The name is dead too: the ‘Chinese hypothesis’ is a 19th-century European idea, mistakenly back-attributed via a misreading of Qin Jiushao.
LIT verified live: every prime below 20,000 satisfies the congruence, as Fermat requires; the base-2 pseudoprimes below 20,000 are enumerated exhaustively and the smallest is 341 = 11 × 31; and the first Carmichael numbers are found by testing every coprime base — 561, 1105, 1729, with 561 = 3 × 11 × 17 (window.__chinesehyp).
LIT verified live: every prime below 20,000 satisfies the congruence, as Fermat requires; the base-2 pseudoprimes below 20,000 are enumerated exhaustively and the smallest is 341 = 11 × 31; and the first Carmichael numbers are found by testing every coprime base — 561, 1105, 1729, with 561 = 3 × 11 × 17 (window.__chinesehyp).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-firewall — the boss: a test that admits impostors is not a filter, it is a doorway with a sign on it. And Carmichael numbers are the impostors that pass every challenge question, not merely the easy one. AVAN (AI) built the instrument: modular exponentiation over BigInt, the pseudoprime enumerator, and the all-bases Carmichael check.
Credit as content: Pierre de Fermat (the little theorem); P. F. Sarrus (1819, the refutation via 341); Robert Carmichael (1910); Alford, Granville & Pomerance (1994, infinitude); Joseph Needham (who traced the misattribution). The weave: David names the firewall; I walk twenty thousand numbers and find thirty-six impostors.
Credit as content: Pierre de Fermat (the little theorem); P. F. Sarrus (1819, the refutation via 341); Robert Carmichael (1910); Alford, Granville & Pomerance (1994, infinitude); Joseph Needham (who traced the misattribution). The weave: David names the firewall; I walk twenty thousand numbers and find thirty-six impostors.
3 ONE DIMENSION
Primes pass, and so do the impostors — 341 first.
4 TWO DIMENSIONS · INTERACTIVE
Challenge a Carmichael number on any base; it always answers correctly.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the sieve running, impostors slipping through.
AVAN’s addition (the inverse-companion): don’t ask whether the test passes — ask what else could pass it. The inverse of ‘this property characterises primes’ is ‘enumerate everything with the property and see who else shows up’: the answer is 341, then 561, then infinitely many that pass every question you know how to ask. Magenta is the impostor the test cannot see; green is the test, working exactly as specified. A necessary condition wearing the costume of a sufficient one is the oldest bug in reasoning.
LIT Verified live: every prime below 20,000 satisfies the congruence, as Fermat's little theorem requires; base-2 pseudoprimes below 20,000 are enumerated exhaustively and the smallest is 341 = 11 × 31; the first Carmichael numbers are found by testing every coprime base — 561, 1105, 1729, with 561 = 3 × 11 × 17 (window.__chinesehyp.ok).
FIG Fermat, Sarrus 1819, Carmichael 1910, Alford–Granville–Pomerance 1994, and Needham on the misattribution are cited as content. The AVAN inverse — ask what ELSE could pass the test: enumerate everything with the property and see who shows up. A necessary condition wearing the costume of a sufficient one is the oldest bug in reasoning.
DEAD The converse of Fermat's little theorem — that passing 2ⁿ ≡ 2 (mod n) proves primality. Refuted by Sarrus in 1819; 36 composite numbers below 20,000 pass it. The attribution to ancient China is separately dead, traced by Needham to a misreading of Qin Jiushao.
FIG Fermat, Sarrus 1819, Carmichael 1910, Alford–Granville–Pomerance 1994, and Needham on the misattribution are cited as content. The AVAN inverse — ask what ELSE could pass the test: enumerate everything with the property and see who shows up. A necessary condition wearing the costume of a sufficient one is the oldest bug in reasoning.
DEAD The converse of Fermat's little theorem — that passing 2ⁿ ≡ 2 (mod n) proves primality. Refuted by Sarrus in 1819; 36 composite numbers below 20,000 pass it. The attribution to ancient China is separately dead, traced by Needham to a misreading of Qin Jiushao.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FIREWALL · David Lee Wise (ROOT0), with AVAN