THE FOLD / GRIND / THE CRON JOB / THE PISANO
THE PISANO
Fibonacci mod m — the infinite folded into a cycle
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Pisano period. The Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, … grow forever and never repeat. But look at them through the window of a modulus — keep only the remainder mod m — and something surprising happens: the sequence becomes periodic, cycling forever. The length of that cycle is the Pisano period π(m).
The last digit of a Fibonacci number (mod 10) repeats every 60 terms; the last two digits (mod 100) every 300; the last three (mod 1000) every 1500. Why must it repeat? Because there are only m² possible consecutive pairs of remainders, so some pair recurs — and the instant the pair (0, 1) returns, the whole sequence starts over. π(m) is always even for m > 2, and never exceeds 6m.
LIT verified live: for every m up to 200 the Fibonacci sequence mod m is exactly periodic with period π(m), and the periods match the known values (π(10) = 60, π(1000) = 1500) — window.__pisano.periodic && knownMatch. FIG no framing; the periodicity and the period values are exact.
The last digit of a Fibonacci number (mod 10) repeats every 60 terms; the last two digits (mod 100) every 300; the last three (mod 1000) every 1500. Why must it repeat? Because there are only m² possible consecutive pairs of remainders, so some pair recurs — and the instant the pair (0, 1) returns, the whole sequence starts over. π(m) is always even for m > 2, and never exceeds 6m.
LIT verified live: for every m up to 200 the Fibonacci sequence mod m is exactly periodic with period π(m), and the periods match the known values (π(10) = 60, π(1000) = 1500) — window.__pisano.periodic && knownMatch. FIG no framing; the periodicity and the period values are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE CRON JOB, beside THE PERMUTATION CLOCK and THE SCHEDULE — the grind domain of things that come round on a cycle. The Fibonacci numbers, endless and non-repeating, become a clean recurring job the moment you view them mod m. AVAN (AI) built the instrument: the residue cycle, the period detection, the π(m) plot.
The weave: David names the seat (the recurring cycle); I make the infinite fold into a loop and the period checkable — the repeating strip in 1D, the cycle and π(m) in 2D, the residue loop in 3D. The sphere is the seam. Credit: named for Leonardo of Pisa (Fibonacci); the modular periodicity studied by Lagrange (1774) and D. D. Wall (1960).
The weave: David names the seat (the recurring cycle); I make the infinite fold into a loop and the period checkable — the repeating strip in 1D, the cycle and π(m) in 2D, the residue loop in 3D. The sphere is the seam. Credit: named for Leonardo of Pisa (Fibonacci); the modular periodicity studied by Lagrange (1774) and D. D. Wall (1960).
3 ONE DIMENSION
Fibonacci mod m as a strip: the residues march, then — exactly when the pair (0, 1) reappears — the whole pattern repeats. The distance between those returns is the Pisano period.
4 TWO DIMENSIONS · INTERACTIVE
Dial the modulus m and see the Fibonacci residues cycle, the period π(m) marked where (0,1) returns. Below, π(m) plotted against m — a jagged, unpredictable-looking curve hiding deep structure (it’s multiplicative over coprime m).
5 THREE DIMENSIONS + AVAN’S INVERSE
The Fibonacci residues walking a mod-m clock, their path closing into a loop of length π(m) — green, the cycle.
AVAN’s addition (the inverse-companion): the magenta mark is the return to (0,1) that closes the loop. The Fibonacci numbers are unbounded and non-repeating — an infinite line marching off to infinity. Reduction modulo m is the inverse operation: it folds that infinite line into a finite loop. It must close, because only m² consecutive pairs exist, so the pigeonhole forces a repeat — and once a pair recurs, so does everything after it. The inverse of ‘unbounded and non-periodic’ is ‘bounded and periodic’: a modulus is a window that turns endlessness into a cycle, and the length of that cycle is a hidden invariant of m. The green is the loop the infinite sequence becomes; the magenta is the pigeonhole return that makes it close.
LIT Genuine Pisano period (named for Fibonacci; modular periodicity by Lagrange 1774, D. D. Wall 1960). Verified live: for every m up to 200 the Fibonacci sequence mod m is exactly periodic with period pi(m), and the periods match the known values pi(10)=60, pi(1000)=1500 (window.__pisano.periodic && knownMatch, both true). pi(2..12) = 3,8,6,20,24,16,12,24,60,10,24. The periodicity (forced by the pigeonhole on consecutive pairs) and the period values are exact.
FIG No framing: the exact periodicity of Fibonacci mod m and the Pisano period values are real and checked. The pigeonhole argument (only m^2 pairs, so a pair must recur, and once (0,1) recurs the whole sequence does) is the genuine reason it must cycle.
FIG No framing: the exact periodicity of Fibonacci mod m and the Pisano period values are real and checked. The pigeonhole argument (only m^2 pairs, so a pair must recur, and once (0,1) recurs the whole sequence does) is the genuine reason it must cycle.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN