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)∈[0,1] realises both axioms, with a state constituted by its
K-profile against all others. For each
N≥3, the unique distinguishability space is
(CPN−1,K) with
K(ψ,φ)=1−∣⟨ψ∣φ⟩∣2, from which complex coefficients, the Born rule
pk=∣ck∣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→∞ 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.
The impossibility of cloning an unknown quantum state is one of the basic rules governing quantum physics.This statement, known as the "no-cloning theorem", prohibits perfect cloning, but doesn't oppose approximate copying.In this paper, we will prove that, due to the uncontrollable quantum fluctuations, no perfect cloning can be achieved.Such a situation allows us to treat the no-cloning theorem as an equivalent one to Leibniz's principle, and further unify them under the notion of uniqueness; that is, any physical entities (whether macroscopic or microscopic objects) in nature would have its individuality.Moreover, we also demonstrate the universality of unique scheme by showing that, any process of trying to construct one exactly symmetrical or asymmetrical body of a physical object is forbidden.On the whole, nature doesn't allow the existence of completely identical, symmetrical or asymmetrical things and this conclusion is valid for all physical domains.