THE FOLD / LOOT / THE INVENTORY / THE POSTAGE STAMP
THE POSTAGE STAMP
the longest run of amounts a few stamps can make
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The postage-stamp problem asks: with stamps of a few fixed denominations and an envelope that holds at most h stamps, what is the largest value N such that every postage from 1 to N can be made? Call it the h-range. With 1- and 4-cent stamps and up to 5 stamps you can cover every value up to 14; with 1, 5 and 8 and six stamps you reach 42. It is a deceptively hard packing question — choosing denominations to maximize the unbroken run is a classic unsolved optimization — but for a given set and h, the answer is a clean finite computation.
LIT verified live: two independent methods — a dynamic-programming reachable-set and an exhaustive enumeration of every stamp multiset of size ≤ h — produce the identical set of achievable values, and the h-range is the longest run 1, 2, …, N inside it (window.__postage). FIG no framing; the DP reachability, the brute multiset enumeration, and the run-length computation all run in-browser.
LIT verified live: two independent methods — a dynamic-programming reachable-set and an exhaustive enumeration of every stamp multiset of size ≤ h — produce the identical set of achievable values, and the h-range is the longest run 1, 2, …, N inside it (window.__postage). FIG no framing; the DP reachability, the brute multiset enumeration, and the run-length computation all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-inventory — a limited inventory of stamps, and the question of how long an unbroken run of postages that little stock can cover before a gap appears. AVAN (AI) built the instrument: the DP reachable-set, the brute multiset enumeration, their agreement check, and the h-range run length.
Credit as content: the postage-stamp / local basis problem (Rohrbach, Stöhr, and others). The weave: David names the inventory; I confirm the two methods agree and compute the covered run.
Credit as content: the postage-stamp / local basis problem (Rohrbach, Stöhr, and others). The weave: David names the inventory; I confirm the two methods agree and compute the covered run.
3 ONE DIMENSION
The number line: green values are makeable with ≤ h stamps; the unbroken run from 1 is the h-range.
4 TWO DIMENSIONS · INTERACTIVE
Cycle denomination sets and stamp counts; the DP and brute reachable sets match and the h-range is shown.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the unbroken run of postages the stamps can cover.
AVAN’s addition (the inverse-companion): don’t list the amounts — combine the stamps. The inverse of ‘which postages are reachable?’ is ‘every sum of at most h stamps from the set’, and the h-range is how far that reaches without a gap. Magenta are the stamp combinations; green is the unbroken run they build. Coverage from combination.
LIT Genuine postage-stamp / local basis problem (Rohrbach, Stöhr, and others). Verified live: for six denomination/count cases a DP reachable-set and an exhaustive multiset enumeration (all stamp combinations of size ≤ h) produce the identical achievable-value set, and the h-range is the longest unbroken run from 1 (window.__postage.dpEqBrute, .rows).
FIG No framing; the DP reachability, the brute multiset enumeration, and the run-length computation all run in-browser. The AVAN inverse is honest — instead of listing the amounts, combine the stamps: every sum of at most h stamps from the set, and the h-range is how far that reaches without a gap. Magenta are the stamp combinations; green is the unbroken run they build. Coverage from combination.
FIG No framing; the DP reachability, the brute multiset enumeration, and the run-length computation all run in-browser. The AVAN inverse is honest — instead of listing the amounts, combine the stamps: every sum of at most h stamps from the set, and the h-range is how far that reaches without a gap. Magenta are the stamp combinations; green is the unbroken run they build. Coverage from combination.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE INVENTORY · David Lee Wise (ROOT0), with AVAN