THE FOLD / LOOT / THE INVENTORY / THE WEIRD
THE WEIRD
abundance you cannot spend
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A number is abundant when its proper divisors sum past it — 70’s divisors 1, 2, 5, 7, 10, 14, 35 total 74. Usually abundance means flexibility: some subset of the divisors adds to exactly n (making it semiperfect). But 70 is different: check all 128 subsets and none hits 70. Rich, but unable to spend the wealth exactly. Numbers like this — abundant yet not semiperfect — are the weird numbers (Benkoski & Erdős, 1974): 70, 836, 4030, 5830, 7192, 7912, 9272… They are provably infinite, all known ones are even, and whether an odd weird number exists is open — searched past 10²¹, with Erdős having offered cash for the answer.
LIT verified live: an exhaustive sweep of every n below 10,000 — abundance computed from real divisor lists, semiperfection decided by exact subset-sum dynamic programming — finds exactly seven weird numbers: 70, 836, 4030, 5830, 7192, 7912, 9272 (window.__weird). FIG honest boundary: infinitude is Benkoski–Erdős theorem (cited); the odd-weird question is open; and a build note — the first draft of this sphere ‘remembered’ six weird numbers below 10⁴; the exhaustive computation found seven (5830 was missing) and the computation won, as it should.
LIT verified live: an exhaustive sweep of every n below 10,000 — abundance computed from real divisor lists, semiperfection decided by exact subset-sum dynamic programming — finds exactly seven weird numbers: 70, 836, 4030, 5830, 7192, 7912, 9272 (window.__weird). FIG honest boundary: infinitude is Benkoski–Erdős theorem (cited); the odd-weird question is open; and a build note — the first draft of this sphere ‘remembered’ six weird numbers below 10⁴; the exhaustive computation found seven (5830 was missing) and the computation won, as it should.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-inventory — the loot: a full inventory, more materials than the recipe needs — and no combination crafts the item. Abundance without spendability. AVAN (AI) built the instrument: the divisor auditor and the subset-sum decider.
Credit as content: Stan Benkoski & Paul Erdős (1974); the aliquot tradition. The weave: David names the uncraftable item; I check every subset and certify the frustration.
Credit as content: Stan Benkoski & Paul Erdős (1974); the aliquot tradition. The weave: David names the uncraftable item; I check every subset and certify the frustration.
3 ONE DIMENSION
70's seven divisors — 74 units of wealth that cannot make 70.
4 TWO DIMENSIONS · INTERACTIVE
Walk the seven; each shows its divisors, its abundance, and the subset-sum wall.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: reachable sums lighting up — with one dark slot at n.
AVAN’s addition (the inverse-companion): don’t count the wealth — map what it can buy. The inverse of ‘the divisors sum to 74’ is the REACHABLE SET: which totals exist? For 70, the subset sums fill slot after slot — 68, 69, 71, 72 — and skip exactly the one that matters. Magenta is the dark slot at 70; green is everything else the inventory affords. Wealth is not the same as change for every bill — a lesson proved by dynamic programming.
LIT Genuine weird numbers (Benkoski & Erdős 1974). Verified live: exhaustive n < 10,000 — abundance from real divisor sums, semiperfection by exact subset-sum DP — yields exactly {70, 836, 4030, 5830, 7192, 7912, 9272} (window.__weird.ok).
FIG Honest boundary — infinitude cited; odd-weird existence OPEN (none below 10²¹). Build note: the first draft 'remembered' six weird numbers; the exhaustive computation found seven and replaced memory, as it should. The AVAN inverse — don't count the wealth, map what it can buy: the reachable set fills slot after slot and skips exactly the one that matters. Magenta is the dark slot at n; green is everything else the inventory affords. Wealth is not the same as change for every bill — a lesson proved by dynamic programming.
FIG Honest boundary — infinitude cited; odd-weird existence OPEN (none below 10²¹). Build note: the first draft 'remembered' six weird numbers; the exhaustive computation found seven and replaced memory, as it should. The AVAN inverse — don't count the wealth, map what it can buy: the reachable set fills slot after slot and skips exactly the one that matters. Magenta is the dark slot at n; green is everything else the inventory affords. Wealth is not the same as change for every bill — a lesson proved by dynamic programming.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE INVENTORY · David Lee Wise (ROOT0), with AVAN