the only idempotents of ℤ/9 are 0 and 1: an element with e² ≡ e mod 9 must be 0 or 1, because ℤ/9 = ℤ/3² is a local ring (a prime power) with no nontrivial idempotents; by contrast ℤ/6 = ℤ/2 × ℤ/3 has four idempotents {0,1,3,4} from the CRT split. In general the number of idempotents of ℤ/n is 2 raised to the count of distinct prime factors: 9 has one prime so 2, 6 has two so 4, 30 has three so 8. Verified by full enumeration
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_only_idempotents_of_z9_are_zero_and_one - content-address (receipt) ·
7348ac6a-60a4-8d2d-b952-f2cd4f065e49 - 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.