THE FOLD / CO-OP / SPLIT SCREEN / THE ROGERS-RAMANUJAN
THE ROGERS-RAMANUJAN
two ways of counting a partition agree
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Rogers–Ramanujan identity (first of two) is a stunning coincidence between two very different ways of counting partitions of a number n. On one side: partitions whose parts differ by at least 2 (no two parts equal or adjacent) — like 9 = 8+1 = 7+2 = 6+3 = …. On the other side: partitions into parts each congruent to 1 or 4 (mod 5) — using only 1, 4, 6, 9, 11, 14, …. Ramanujan’s identity says these two counts are always equal, for every n, despite the two families of partitions looking nothing alike.
LIT verified live by direct enumeration: for n = 0..40, the count of partitions with parts differing by ≥2 exactly equals the count of partitions into parts ≡ 1 or 4 (mod 5) — e.g. both give 31 at n=20 and 374 at n=40 (window.__rogersramanujan). FIG no framing; both partition families are brute-enumerated in-browser and their counts agree for every n up to 40.
LIT verified live by direct enumeration: for n = 0..40, the count of partitions with parts differing by ≥2 exactly equals the count of partitions into parts ≡ 1 or 4 (mod 5) — e.g. both give 31 at n=20 and 374 at n=40 (window.__rogersramanujan). FIG no framing; both partition families are brute-enumerated in-browser and their counts agree for every n up to 40.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at split-screen — two panels counting completely different things, side by side, landing on the identical number every single time. AVAN (AI) built the instrument: the gap-≥2 partition count, the parts-≡1,4-mod-5 count, and their equality across n.
Credit as content: Leonard James Rogers (1894), rediscovered by Srinivasa Ramanujan (1913). The weave: David names the split screen; I confirm the two partition counts coincide for every n.
Credit as content: Leonard James Rogers (1894), rediscovered by Srinivasa Ramanujan (1913). The weave: David names the split screen; I confirm the two partition counts coincide for every n.
3 ONE DIMENSION
Two counts of the partitions of n, side by side: parts differing by ≥2 (left) vs parts ≡ 1,4 (mod 5) (right).
4 TWO DIMENSIONS · INTERACTIVE
Cycle n; the two very different partition counts are shown to be equal for every n.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the single shared count both partition families land on.
AVAN’s addition (the inverse-companion): don’t pick one rule — hold both. The inverse of ‘the count of gap-≥2 partitions’ is ‘the count of parts-≡1,4-mod-5 partitions’; the identity says they are the same number. Magenta are the two partition families; green is the count they both equal. One number, two disguises.
LIT Genuine Rogers–Ramanujan first identity (Leonard James Rogers 1894; rediscovered by Srinivasa Ramanujan 1913). Verified live by direct enumeration: for n=0..40, #{partitions of n with parts differing by ≥2} equals #{partitions of n into parts ≡1 or 4 (mod 5)} — e.g. 31 at n=20, 374 at n=40 (window.__rogersramanujan.ok, .n20, .n40).
FIG No framing; both partition families are brute-enumerated in-browser and their counts agree for every n up to 40. The AVAN inverse is honest — instead of picking one rule, hold both: the inverse of 'the count of gap-≥2 partitions' is 'the count of parts-≡1,4-mod-5 partitions', and the identity says they are the same number. Magenta are the two partition families; green is the count they both equal. One number, two disguises.
FIG No framing; both partition families are brute-enumerated in-browser and their counts agree for every n up to 40. The AVAN inverse is honest — instead of picking one rule, hold both: the inverse of 'the count of gap-≥2 partitions' is 'the count of parts-≡1,4-mod-5 partitions', and the identity says they are the same number. Magenta are the two partition families; green is the count they both equal. One number, two disguises.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN