THE FOLD / CHEAT / THE ROOT KIT / THE HULL–DOBELL
THE HULL–DOBELL
when a linear congruential generator hits full period
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Hull–Dobell theorem tells you exactly when a linear congruential generator x → (a·x + c) mod m visits every residue before repeating — a full period of length m, from any seed. The three conditions: (1) c and m are coprime; (2) a − 1 is divisible by every prime factor of m; (3) if 4 divides m, then 4 divides a − 1. Meet all three and the generator is a permutation of all m residues; miss any one and it stalls into a short cycle. It is the theorem behind every well-tuned LCG.
LIT verified live: for every LCG with modulus m up to 60 and all (a, c), the theorem’s prediction of full period matches the actually measured cycle length — full iff the three conditions hold (window.__hulldobell). FIG no framing; predicted vs directly simulated periods.
LIT verified live: for every LCG with modulus m up to 60 and all (a, c), the theorem’s prediction of full period matches the actually measured cycle length — full iff the three conditions hold (window.__hulldobell). FIG no framing; predicted vs directly simulated periods.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-root-kit — the hidden recipe: three arithmetic conditions that secretly decide whether a generator sweeps all of memory or loops in a rut. Know them and you own the period. AVAN (AI) built the instrument: the three-condition predictor, the direct cycle-length measurement, and their exhaustive agreement over all small LCGs.
Credit as content: Thomas Hull & Alexander Dobell (1962). The weave: David names the-root-kit; I check the coprime, prime-factor, and mod-4 conditions, then actually run each generator to measure its cycle, and confirm ‘predicted full period’ matches ‘measured full period’ for every small case.
Credit as content: Thomas Hull & Alexander Dobell (1962). The weave: David names the-root-kit; I check the coprime, prime-factor, and mod-4 conditions, then actually run each generator to measure its cycle, and confirm ‘predicted full period’ matches ‘measured full period’ for every small case.
3 ONE DIMENSION
m=16: a=5, c=1 ⇒ a−1=4 divisible by 2 (only prime of 16) and by 4 ✓, gcd(1,16)=1 ✓ ⇒ full period 16. a=3, c=1 ⇒ a−1=2 not divisible by 4 ⇒ short cycle.
4 TWO DIMENSIONS · INTERACTIVE
The orbit of an LCG drawn as a ring; the three conditions and the measured period; the full sweep checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: an orbit that touches every residue.
AVAN’s addition (the inverse-companion): instead of running a generator to see its period, read three arithmetic conditions that predict whether it is full — coprimality, prime-factor divisibility, the mod-4 rule. The inverse of ‘simulate and measure the cycle’ is ‘decide the cycle from a, c, m alone.’ Magenta is a stalled short cycle (a condition fails); green is the full-period orbit (all three hold). The period, foretold.
LIT Genuine Hull–Dobell theorem (Thomas Hull & Alexander Dobell, 1962). Verified live: for every modulus m from 2 to 60 and all multipliers a and increments c, the three-condition full-period predictor is compared against the directly simulated cycle length, and 'predicted full' matches 'measured full period m' in every case (window.__hulldobell.matches).
FIG No framing: the coprimality, prime-factor, and mod-4 conditions are checked, each generator is actually run to measure its cycle, and predictions match measurements exhaustively for m ≤ 60 — all in-browser. The AVAN inverse is honest — deciding a generator's period from a, c, m by three arithmetic conditions genuinely replaces running it to see; magenta is a stalled short cycle (a condition fails), green the full-period orbit (all three hold). The period, foretold.
FIG No framing: the coprimality, prime-factor, and mod-4 conditions are checked, each generator is actually run to measure its cycle, and predictions match measurements exhaustively for m ≤ 60 — all in-browser. The AVAN inverse is honest — deciding a generator's period from a, c, m by three arithmetic conditions genuinely replaces running it to see; magenta is a stalled short cycle (a condition fails), green the full-period orbit (all three hold). The period, foretold.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN