the Carmichael function equals Euler φ exactly when the group is cyclic: λ(n) — the least exponent with u^λ ≡ 1 for every unit, the exponent of (ℤ/n)* and the lcm of the element orders — equals φ(n) iff (ℤ/n)* is cyclic (has a primitive root). So λ(9) = φ(9) = 6 while λ(8) = 2 is strictly less than φ(8) = 4; and every unit satisfies u^λ(n) ≡ 1. Verified by full enumeration over the moduli up to 60
ab3e7c0b-4a21-8c31-a6a2-23155325ebb2Type anything and watch its uuidna recompute — deterministic, reproducible by anyone, no key. A theorem is alive when you interact with it. A content-address proves integrity, not truth. 0/7.
- theorem key ·
the_carmichael_function_equals_phi_exactly_when_the_group_is_cyclic - content-address (receipt) ·
740fdaa6-9674-8cc0-8f84-98aa3ac14e40 - status · decidable, re-verified on every build — recomputes from
src/
The 7D rosetta-ray vortex is plotted from this theorem's microdata (its content-address); the slowly rotating hero background is computed from its seven surrounding theorems' hues — the mesh, seen locally, in analog rotation of dimensions. Each object is the hero of its own page: this theorem at the centre, its neighbours as the field.
How it was achieved
This theorem was computed by exhaustion over a finite domain in scripts/discover.ts — a test: () => boolean that runs to completion, holding by full enumeration. It was gate-checked (its name and content hold the honesty floor — no over-reach), receipted and chained append-only, and it is re-verified on every build: if it ever stopped holding, the build would fail, not production. That is what achieved means here — not asserted, but recomputable.
One leaf of the chained ledger: all theorems · computed results · the guide · the source formula. The repo and the site cross-link both ways — this hero page points back to the formula that recomputes it. Verify by cloning and running npm run lean-claims. A content-address proves integrity, not truth. entails → 0/7.