On the Structural Origin of Quantum Mechanics: A Projection-Based Reconstruction of the Quantum Formalism

This paper presents a structural reconstruction of the formal core of quantum mechanics from the properties of a bounded, linear, distinguishability-reducing projection acting on a general functional space. Effective quantum states are identified with equivalence classes under the projection ΠQM : H∗ → HQM and the Hilbert-space structure of HQM is induced by a positive operator K on the underlying space. Within this framework, the characteristic elements of quantum theory emerge as direct consequences of projection geometry and operator structure. Linearity yields superposition, quadratic norms on equivalence classes give rise to the Born rule, non-factorizability leads to entanglement and commuting local algebras ensure no-signalling. Schrödinger-type unitary dynamics arises from projection of symmetric generators, Dirac-type operators follow from algebraic compatibility with Clifford structures and violations of classical Bell inequalities together with the Tsirelson bound are enforced by operator-norm constraints in the projected C∗-algebra. No hidden variables are introduced and no modification of standard empirical quantum mechanics is proposed. Rather than postulating the quantum formalism as axiomatic, the analysis shows that its defining structures form a highly constrained and stable outcome of linear projection, equivalence-class geometry, and norm-preserving dynamics. Rather than introducing new physical dynamics, the purpose of this work is to clarify why the established quantum formalism admits so little structural freedom once a small set of minimal requirements is imposed.