THE FOLD / SPAWN / GENESIS-BLOCK / THE REDUCED TOTIENT
THE REDUCED TOTIENT
the smallest exponent that resets every unit
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The reduced totient — the Carmichael function λ(n) — is the smallest exponent that resets every unit at once: the least m for which am ≡ 1 (mod n) holds for all a coprime to n. Euler’s theorem guarantees aφ(n) ≡ 1, but φ(n) is often bigger than necessary; λ(n) is the true exponent of the group of units, and it always divides φ(n). It is computed as the least common multiple of the group exponents of each prime-power factor (with the famous quirk that λ(2k) = 2k−2 for k ≥ 3, half of φ). It sets the real period of modular exponentiation — and the correct exponent bound behind RSA.
LIT verified live: for every n up to 300, aλ(n) ≡ 1 for all units, λ(n) divides φ(n), and λ is tight — some unit has order exactly λ(n) (window.__reduced_totient). FIG no framing; the prime-power lcm formula and the exhaustive order check run in-browser.
LIT verified live: for every n up to 300, aλ(n) ≡ 1 for all units, λ(n) divides φ(n), and λ is tight — some unit has order exactly λ(n) (window.__reduced_totient). FIG no framing; the prime-power lcm formula and the exhaustive order check run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at genesis-block — the smallest exponent from which every unit returns to 1, the true period of the modular world. AVAN (AI) built the instrument: the Carmichael lcm formula, the universal-exponent check, the divides-φ check, and the tightness check.
Credit as content: Robert Carmichael (the reduced totient / λ function, 1910). The weave: David names the genesis block; I confirm λ(n) is the exact, tight exponent that resets every unit — and always divides φ.
Credit as content: Robert Carmichael (the reduced totient / λ function, 1910). The weave: David names the genesis block; I confirm λ(n) is the exact, tight exponent that resets every unit — and always divides φ.
3 ONE DIMENSION
The units mod n and their multiplicative orders; λ(n) is the lcm of all orders — the smallest exponent resetting all.
4 TWO DIMENSIONS · INTERACTIVE
Pick n; see λ(n), φ(n), the unit orders — and that λ divides φ, with some unit hitting order exactly λ.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: λ(n), the exact exponent resetting every unit.
AVAN’s addition (the inverse-companion): don’t use φ(n) — use the true exponent. The inverse of ‘Euler’s aφ≡1’ is ‘λ(n), the smallest exponent that resets all units, always dividing φ.’ Magenta is φ(n) (the group size); green is λ(n) (the group exponent). The true period, tighter than φ.
LIT Genuine Carmichael function / reduced totient λ(n) (Robert Carmichael, 1910). Verified live: for every n≤300, a^λ(n)≡1 (mod n) for all units (via the prime-power lcm formula, incl. λ(2^k)=2^{k−2} for k≥3), λ(n) | φ(n), and λ is tight — the maximum multiplicative order over units equals λ(n) exactly (window.__reduced_totient.universal, .dividesPhi, .tight).
FIG No framing: the prime-power lcm formula and the exhaustive order check run in-browser. The AVAN inverse is honest — instead of Euler's φ(n) exponent, one uses the true group exponent λ(n): the smallest exponent resetting all units, always dividing φ. Magenta is φ(n) (the group size); green is λ(n) (the group exponent). The true period, tighter than φ.
FIG No framing: the prime-power lcm formula and the exhaustive order check run in-browser. The AVAN inverse is honest — instead of Euler's φ(n) exponent, one uses the true group exponent λ(n): the smallest exponent resetting all units, always dividing φ. Magenta is φ(n) (the group size); green is λ(n) (the group exponent). The true period, tighter than φ.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GENESIS-BLOCK · David Lee Wise (ROOT0), with AVAN