THE FOLD / LOOT / THE MINT / THE PRIMITIVE ROOT
THE PRIMITIVE ROOT
one root that generates every residue
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A primitive root modulo a prime p is a single number g whose powers g1, g2, …, gp−1 run through every nonzero residue 1, 2, …, p−1 exactly once before returning to 1. It is a generator of the multiplicative group mod p: one seed from which every residue is reached by repeated multiplication. Primitive roots underlie discrete logarithms, Diffie–Hellman key exchange, and random-number generators.
LIT verified live: for primes below 300, a primitive root’s powers form a permutation of {1,…,p−1}, and the count of primitive roots equals φ(p−1) exactly (window.__primroot). FIG no framing; exact modular arithmetic and Euler’s totient count.
LIT verified live: for primes below 300, a primitive root’s powers form a permutation of {1,…,p−1}, and the count of primitive roots equals φ(p−1) exactly (window.__primroot). FIG no framing; exact modular arithmetic and Euler’s totient count.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — one generator stamps out every coin in the group; multiply g by itself and eventually every residue has been minted, each exactly once. The primitive root is that mint. AVAN (AI) built the instrument: the order test, the permutation check on g’s powers, and the count-equals-φ(p−1) verification.
Credit as content: primitive roots (Euler conjectured, Gauss proved their existence for primes, Disquisitiones 1801). The weave: David names the-mint; I cycle a generator’s powers through the residues and confirm they hit each nonzero value once, and that exactly φ(p−1) generators exist — every group has its mints.
Credit as content: primitive roots (Euler conjectured, Gauss proved their existence for primes, Disquisitiones 1801). The weave: David names the-mint; I cycle a generator’s powers through the residues and confirm they hit each nonzero value once, and that exactly φ(p−1) generators exist — every group has its mints.
3 ONE DIMENSION
mod 7: powers of 3 are 3, 2, 6, 4, 5, 1 — all of 1..6, so 3 is a primitive root. Powers of 2 are 2, 4, 1, 2, 4, 1 — only three values, so 2 is not.
4 TWO DIMENSIONS · INTERACTIVE
The cycle of a generator’s powers around the residues; whether it hits all of them, and the φ(p−1) count, checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: every residue from one generator.
AVAN’s addition (the inverse-companion): reach every nonzero residue from a single seed by repeated multiplication — a primitive root generates the whole group, so its power sequence is a permutation of 1..p−1. The inverse of ‘list the residues 1,2,…,p−1’ is ‘pick one generator g — its powers ARE the residues, in a scrambled order.’ Magenta is the plain residue list; green is the generator’s orbit. All residues from one seed.
LIT Genuine primitive roots (Euler conjectured; Gauss proved existence for primes, Disquisitiones 1801). Verified live: for every prime p<300, a primitive root's powers form a permutation of {1,…,p−1} (window.__primroot.permutation), and the number of primitive roots mod p equals Euler's totient φ(p−1) (window.__primroot.countEqualsPhi) — exact modular arithmetic.
FIG No framing: the order test, the permutation check on g's powers, and the count-equals-φ(p−1) verification run in-browser and agree. The AVAN inverse is honest — reaching every nonzero residue from a single seed by repeated multiplication (a generator's power sequence IS a permutation of 1..p−1) genuinely replaces listing the residues; magenta is the plain list, green the generator's orbit. All residues from one seed.
FIG No framing: the order test, the permutation check on g's powers, and the count-equals-φ(p−1) verification run in-browser and agree. The AVAN inverse is honest — reaching every nonzero residue from a single seed by repeated multiplication (a generator's power sequence IS a permutation of 1..p−1) genuinely replaces listing the residues; magenta is the plain list, green the generator's orbit. All residues from one seed.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN