the sum of the first n triangular numbers is the nth tetrahedral number: T(1) + T(2) + ... + T(n) = n(n+1)(n+2)/6 = C(n+2, 3) — stacking triangular layers builds a tetrahedron. Verified exhaustively for n up to 100. Decidable

Interact — recompute a content-address:→ ab3e7c0b-4a21-8c31-a6a2-23155325ebb2

Type 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 · sum_of_triangular_is_tetrahedral
• content-address (receipt) · 90ef4db9-a803-8c60-a948-d3c2de8f7e1d
• 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.