THE FOLD / GLITCH / STACK OVERFLOW / THE CYCLIC NUMBER
THE CYCLIC NUMBER
the number that rotates instead of growing
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
142857 is the most famous cyclic number: multiply it by 1 through 6 and the answer is always the same six digits, rotated — 285714, 428571, 571428, 714285, 857142. Multiply by 7 and the register overflows to 999999. The engine is decimal arithmetic itself: 142857 is the repeating block of 1/7, and 7 is a full-reptend prime — 10 is a primitive root mod 7, its powers visiting every nonzero residue before returning, which is exactly why multiplication can only rotate the block. The next such prime is 17, whose 16-digit block 0588235294117647 (leading zero and all) rotates under multiplication by 1 through 16 and overflows to sixteen nines at 17.
LIT verified live: all six multiples of 142857 are exact rotations; 142857 × 7 = 999999; the block regenerates from long division of 1/7; the 17-family passes all sixteen rotation checks in BigInt; and the engine is confirmed — ord(10 mod 7) = 6, ord(10 mod 17) = 16 (window.__cyclicnumber). FIG no framing; every multiplication, rotation, and order is computed exactly in-browser.
LIT verified live: all six multiples of 142857 are exact rotations; 142857 × 7 = 999999; the block regenerates from long division of 1/7; the 17-family passes all sixteen rotation checks in BigInt; and the engine is confirmed — ord(10 mod 7) = 6, ord(10 mod 17) = 16 (window.__cyclicnumber). FIG no framing; every multiplication, rotation, and order is computed exactly in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at stack-overflow — the glitch: a circular buffer that multiplication can only rotate, never grow, until the seventh push overflows it to all-nines. AVAN (AI) built the instrument: the rotation audits, the long-division regeneration, and the primitive-root engine check.
Credit as content: the cyclic-number tradition; full-reptend primes (Gauss studied the periods). The weave: David names the rotating buffer; I confirm six rotations, one overflow, and the order that powers it.
Credit as content: the cyclic-number tradition; full-reptend primes (Gauss studied the periods). The weave: David names the rotating buffer; I confirm six rotations, one overflow, and the order that powers it.
3 ONE DIMENSION
The six digits on a wheel — each multiple just starts the wheel at a different spoke.
4 TWO DIMENSIONS · INTERACTIVE
Step the multiplier 1→7; rotations until the all-nines overflow.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the digit wheel, spinning under multiplication.
AVAN’s addition (the inverse-companion): don’t marvel at the number — find the prime beneath it. The inverse of ‘142857 rotates’ is ‘10 is a primitive root mod 7’: the block is just 1/7’s orbit written down. Magenta is the overflow at ×7; green is the wheel that turns six times first. A number that is secretly a clock.
LIT Genuine cyclic-number / full-reptend-prime arithmetic. Verified live: 142857 × 1..6 are exact rotations, ×7 = 999999, the block regenerates from long division of 1/7, the 17-family (0588235294117647) passes all sixteen BigInt rotation checks and overflows to sixteen nines, and ord(10 mod 7) = 6, ord(10 mod 17) = 16 (window.__cyclicnumber.ok).
FIG No framing — every multiplication, rotation, and order is computed exactly in-browser. The AVAN inverse — don't marvel at the number, find the prime beneath it: the inverse of '142857 rotates' is '10 is a primitive root mod 7' — the block is just 1/7's orbit written down. Magenta is the overflow at ×7; green is the wheel that turns six times first. A number that is secretly a clock.
FIG No framing — every multiplication, rotation, and order is computed exactly in-browser. The AVAN inverse — don't marvel at the number, find the prime beneath it: the inverse of '142857 rotates' is '10 is a primitive root mod 7' — the block is just 1/7's orbit written down. Magenta is the overflow at ×7; green is the wheel that turns six times first. A number that is secretly a clock.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN