# Proofs

Formal mathematical arguments for the key claims.

| File | Claim | Method |
|------|-------|--------|
| [`ternary-efficiency.md`](./ternary-efficiency.md) | Ternary is more computationally efficient than binary | Calculus / radix economy |
| [`self-reference.md`](./self-reference.md) | The QWERTY encoding is self-referential | Direct construction |
| [`pure-state.md`](./pure-state.md) | The density matrix of the system is a pure state | Linear algebra / SVD |
| [`universal-computation.md`](./universal-computation.md) | The ternary bio-quantum system is Turing-complete | Reaction network theory |

## From the Eight Claims

**Claim 6** (Ramanujan congruences show incompleteness inside arithmetic) is a known result in number theory, not a new proof. The congruences p(5k+4)≡0 (mod 5), p(7k+5)≡0 (mod 7), p(11k+6)≡0 (mod 11) were proved by Ramanujan and later by Watson and Atkin using modular forms. The failure at 13 — p(13k+7)≢0 (mod 13) — is also established. The claim is that this structure models Gödelian incompleteness from within arithmetic: the system of partition congruences describes its own boundary.

See [`CLAIMS.md`](../CLAIMS.md) for all eight claims.