THE FOLD / LOOT / THE DROP / THE UNTOUCHABLE
THE UNTOUCHABLE
the loot no drop table contains
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Take any number, add up its proper divisors — the aliquot sum s(n). Ask the reverse question: which values does s never produce? Those are the untouchable numbers: 2, 5, 52, 88, 96, 120, 124, 146… No integer’s divisors will ever sum to them; they sit outside the aliquot economy entirely. Erdős proved there are infinitely many (1973). The delicious details: 5 is believed to be the only odd untouchable — because if strong Goldbach holds, every even 2k = p+q gives s(pq) = p+q+1, hitting every odd number from 7 up; and certifying untouchability needs no infinite search, because a composite n always has s(n) > √n — so small targets can only be hit by small n.
LIT verified live: aliquot sums sieved for every n up to 998,001, and the bound s(n) > √n for composite n turns that finite scan into a certificate for all values ≤ 1000 — producing the exact untouchable list up to 500 (38 values, beginning 2, 5, 52, 88, 96, 120) with 5 confirmed as the only odd member in range (window.__untouchable). FIG honest boundary: ‘5 is the only odd untouchable’ is conditional on strong Goldbach — stated as the conditional it is; Erdős’s infinitude cited as content.
LIT verified live: aliquot sums sieved for every n up to 998,001, and the bound s(n) > √n for composite n turns that finite scan into a certificate for all values ≤ 1000 — producing the exact untouchable list up to 500 (38 values, beginning 2, 5, 52, 88, 96, 120) with 5 confirmed as the only odd member in range (window.__untouchable). FIG honest boundary: ‘5 is the only odd untouchable’ is conditional on strong Goldbach — stated as the conditional it is; Erdős’s infinitude cited as content.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-drop — the loot: items that exist in the game but appear in NO drop table — farm every monster forever and 2, 5, 52 never fall. AVAN (AI) built the instrument: the aliquot sieve and the √n-bound certification that closes the search honestly.
Credit as content: Paul Erdős (1973, infinitude); the aliquot tradition back to Pythagoras’ perfect numbers. The weave: David names the impossible drop; I prove the table empty by exhausting every monster that could carry it.
Credit as content: Paul Erdős (1973, infinitude); the aliquot tradition back to Pythagoras’ perfect numbers. The weave: David names the impossible drop; I prove the table empty by exhausting every monster that could carry it.
3 ONE DIMENSION
The number line to 150 — touchable values lit, untouchables dark gaps.
4 TWO DIMENSIONS · INTERACTIVE
Query a value; see who hits it — or the certificate that nobody ever will.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the aliquot rain falling on the number line.
AVAN’s addition (the inverse-companion): don’t chase the function forward — stand where it never lands. The inverse of ‘what does s(n) equal?’ is ‘which values are orphans?’, and the √n bound is what makes the orphanage provable: a small value’s parents would all be small, so checking them all is finite. Magenta are the untouchables, dry under the rain; green is every value with at least one parent. Some numbers are simply never spoken.
LIT Genuine untouchable numbers (Erdős 1973 infinitude; aliquot tradition). Verified live: full aliquot sieve to 998,001; the composite bound s(n) > √n certifies completeness for targets ≤ 1000; list ≤ 500 = 38 values beginning 2,5,52,88,96,120; only odd member is 5 (window.__untouchable.ok).
FIG Honest boundary — '5 is the only odd untouchable' is conditional on strong Goldbach, stated as the conditional it is. The AVAN inverse — don't chase the function forward, stand where it never lands: the inverse of 'what does s(n) equal?' is 'which values are orphans?', and the √n bound is what makes the orphanage provable. Magenta are the untouchables, dry under the aliquot rain; green is every value with at least one parent. Some numbers are simply never spoken.
FIG Honest boundary — '5 is the only odd untouchable' is conditional on strong Goldbach, stated as the conditional it is. The AVAN inverse — don't chase the function forward, stand where it never lands: the inverse of 'what does s(n) equal?' is 'which values are orphans?', and the √n bound is what makes the orphanage provable. Magenta are the untouchables, dry under the aliquot rain; green is every value with at least one parent. Some numbers are simply never spoken.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN