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# Proofs Formal mathematical arguments for the key claims. > 📖 All proofs use standard mathematical methods. For background on the underlying research, see [REFERENCES.md](../REFERENCES.md). | File | Claim | Method | |------|-------|--------| | [`ternary-efficiency.md`](./ternary-efficiency.md) | Ternary is more computationally efficient than binary | Calculus / radix economy ([Knuth, 1980](https://en.wikipedia.org/wiki/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 ([von Neumann, 1932](https://en.wikipedia.org/wiki/Density_matrix)) | | [`universal-computation.md`](./universal-computation.md) | The ternary bio-quantum system is Turing-complete | Reaction network theory ([Turing, 1936](https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf)) |