THE FOLD / GRIND / THE CRON JOB / THE FACTORIAL BASE
THE FACTORIAL BASE
a mixed-radix odometer with factorial place values
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The factorial number system is a mixed-radix positional system where the place values are factorials (1!, 2!, 3!, …) and the digit in place i may range only from 0 up to i. Every non-negative integer below m! has a unique such representation — it is an odometer whose wheels have different sizes (2, 3, 4, … positions). It is the natural index for permutations and the backbone of the Lehmer code.
LIT verified live: over all 5040 integers 0..7!−1, encode∘decode is the identity, each digit stays within its rising radix, and all representations are distinct (window.__factbase). FIG no framing; exact.
LIT verified live: over all 5040 integers 0..7!−1, encode∘decode is the identity, each digit stays within its rising radix, and all representations are distinct (window.__factbase). FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-cron-job — the odometer that ticks, but with wheels of growing size. The factorial base is that irregular odometer. AVAN (AI) built the instrument: the rising-radix digit extraction, the factorial place values, the round-trip, and the uniqueness enumeration.
Credit as content: the factorial number system (Charles-Ange Laisant 1888). The weave: David names the cron-job; I divide successively by 2, 3, 4, … to read off each digit and confirm every integer below m! maps to one and only one mixed-radix string.
Credit as content: the factorial number system (Charles-Ange Laisant 1888). The weave: David names the cron-job; I divide successively by 2, 3, 4, … to read off each digit and confirm every integer below m! maps to one and only one mixed-radix string.
3 ONE DIMENSION
Place values 1, 2, 6, 24, 120, … (the factorials). The lowest wheel has 2 positions, the next 3, then 4 — each wheel bigger than the last. A digit can never reach its own place’s size.
4 TWO DIMENSIONS · INTERACTIVE
Any integer below 7! in factorial base; the rising-radix digits and the round-trip are checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: an odometer with wheels of growing size.
AVAN’s addition (the inverse-companion): give each place a different radix — the i-th wheel has i+1 positions and weight i! — so every integer below m! gets a unique mixed-radix code. The inverse of ‘one fixed base for every digit’ is ‘let each wheel grow — factorial place values.’ Magenta is the wasted uniform-base range; green is the tight factorial odometer. Wheels that grow as they climb.
LIT Genuine factorial number system (Laisant 1888). Verified live: over all 5040 integers from 0 to 7!-1, dividing successively by 2,3,4,... to read the digits and reconstructing by factorial place values is the identity, every digit i stays within 0..i+1, and all 5040 representations are distinct (window.__factbase.roundTrip && .radixOK && .unique).
FIG No framing: the rising-radix digit extraction, the factorial place values, the round-trip, and the uniqueness enumeration run in-browser over all 5040 integers and hold. The AVAN inverse is honest — giving each place a different radix (the i-th wheel has i+1 positions, weight i!) gives every integer below m! a unique mixed-radix code; magenta is the wasted uniform-base range, green the tight factorial odometer. Wheels that grow as they climb.
FIG No framing: the rising-radix digit extraction, the factorial place values, the round-trip, and the uniqueness enumeration run in-browser over all 5040 integers and hold. The AVAN inverse is honest — giving each place a different radix (the i-th wheel has i+1 positions, weight i!) gives every integer below m! a unique mixed-radix code; magenta is the wasted uniform-base range, green the tight factorial odometer. Wheels that grow as they climb.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN