Existence as Distinguishability: Quantum Mechanics from Finite Graded Equality
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915975032471552 |
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| author | Zilly, Julian G. |
| author_facet | Zilly, Julian G. |
| contents | We derive finite-dimensional quantum mechanics from a single ontological principle, that \emph{existence is constituted by distinguishability}, together with two structural commitments: finite capacity $N$ (parametric input) and self-referential consistency (SRC, a closure schema with two equivalent forms, operational and information-theoretic). SRC unpacks into eight derived structural conditions; structural unambiguity (S5) completes the hierarchy, uniquely selecting the Born rule as the geometric/probabilistic closure. The graded distinguishability kernel $K(x,y) \in [0,1]$ realises both axioms, with a state constituted by its $K$-profile against all others. For each $N \geq 3$, the unique distinguishability space is $(\mathbb{C} P^{N-1}, K)$ with $K(ψ,ϕ) = 1 - |\langleψ|ϕ\rangle|^2$, from which complex coefficients, the Born rule $p_k = |c_k|^2$, unitary dynamics, and tensor-product composition all follow. Indeterminism is forced by capacity overflow; alternatives (e.g. Bohmian mechanics) are classified rather than refuted. Standard QM is the $N \to \infty$ limit; finite $N$ is the only free parameter. The algebraic spine is machine-checked in Lean 4 modulo five imported classical theorems and the existence direction of Stone's theorem; the Appendix states the verification scope. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11900 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence as Distinguishability: Quantum Mechanics from Finite Graded Equality Zilly, Julian G. Quantum Physics We derive finite-dimensional quantum mechanics from a single ontological principle, that \emph{existence is constituted by distinguishability}, together with two structural commitments: finite capacity $N$ (parametric input) and self-referential consistency (SRC, a closure schema with two equivalent forms, operational and information-theoretic). SRC unpacks into eight derived structural conditions; structural unambiguity (S5) completes the hierarchy, uniquely selecting the Born rule as the geometric/probabilistic closure. The graded distinguishability kernel $K(x,y) \in [0,1]$ realises both axioms, with a state constituted by its $K$-profile against all others. For each $N \geq 3$, the unique distinguishability space is $(\mathbb{C} P^{N-1}, K)$ with $K(ψ,ϕ) = 1 - |\langleψ|ϕ\rangle|^2$, from which complex coefficients, the Born rule $p_k = |c_k|^2$, unitary dynamics, and tensor-product composition all follow. Indeterminism is forced by capacity overflow; alternatives (e.g. Bohmian mechanics) are classified rather than refuted. Standard QM is the $N \to \infty$ limit; finite $N$ is the only free parameter. The algebraic spine is machine-checked in Lean 4 modulo five imported classical theorems and the existence direction of Stone's theorem; the Appendix states the verification scope. |
| title | Existence as Distinguishability: Quantum Mechanics from Finite Graded Equality |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2603.11900 |