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21 lines
1.8 KiB
Markdown
21 lines
1.8 KiB
Markdown
# Proofs
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Formal mathematical arguments for the key claims.
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> 📖 All proofs use standard mathematical methods. For background on the underlying research, see [REFERENCES.md](../REFERENCES.md).
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| File | Claim | Method |
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| [`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)) |
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| [`self-reference.md`](./self-reference.md) | The QWERTY encoding is self-referential | Direct construction |
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| [`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)) |
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| [`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)) |
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| [`chi-squared.md`](./chi-squared.md) | Chi-squared goodness-of-fit and independence tests | χ² statistic / contingency tables |
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| [`lucidia.md`](./lucidia.md) | The number-theoretic identity of Lucidia (88) | Number theory: totient, Möbius, Collatz, Goldbach |
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## From the Eight Claims
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**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.
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See [`CLAIMS.md`](../CLAIMS.md) for all eight claims.
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