Structural Origin of the Born Rule within the Order-Projection Principle

This work investigates the structural origin of the Born rule within the framework of the Order-Projection Principle (OPP). We consider outcome-indexed measurements associated with orthonormal bases in finite-dimensional Hilbert spaces and analyze the constraints imposed by:(i) outcome-information separation (OIS),(ii) positivity and completeness of measurement operators, and(iii) non-contextuality of probability assignments. Under these conditions, we show that measurement operators reduce to rank-1 projectors and that, within an overlap-based family of candidate probability functions, the quadratic form P(e|ψ) = |⟨e|ψ⟩|² is uniquely compatible with non-contextuality. The result is obtained without introducing additional auxiliary axioms beyond the outcome-level formulation of the OPP framework. The role of linear structure (A4) and quadratic observation (A5) is also discussed, including their structural relationship to more primitive assumptions. Numerical experiments are provided to corroborate the consistency of the derivation across multiple dimensions. This work focuses on finite-dimensional, orthonormal-basis-indexed measurements and pure states. Extensions to mixed states, generalized POVMs, and continuous observables are left for future investigation. Note: Parts of the manuscript were linguistically and structurally refined with the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own. Note: This work represents Version 1.0 of an ongoing research program on the Order-Projection Principle (OPP). Minor typographical corrections and clarifications may appear in later versions. The core conceptual claims remain unchanged.