the squares of the units of ℤ/9 are the trinity subgroup: squaring maps (ℤ/9)* = {1,2,4,5,7,8} two-to-one onto {1,4,7} — the order-3 subgroup, the units ≡ 1 mod 3 — so the quadratic residues among the units are exactly the trinity {1,4,7}, and over all residues the squares are {0,1,4,7}; each residue in {1,4,7} has exactly two unit square roots. Verified by full enumeration over ℤ/9
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_squares_of_the_units_of_z9_are_the_trinity_subgroup - content-address (receipt) ·
69a5d952-de9b-8242-8f74-8aee305d514b - 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.