Structural Origin of the Born Rule in the Theory of Structural Articulation

This work provides a structural derivation of the Born rule within the framework of the Theory of Structural Articulation (TSA). Unlike standard quantum mechanics, where the Born rule is introduced as an independent postulate, the present paper demonstrates that the quadratic probability measure arises as a mathematical and structural necessity. Working at the pre-dynamical level of alternatives (M2), the analysis does not assume a priori Hilbert space structures, unitary dynamics, operator algebras, or measurement postulates. Probability is defined instead as a measure of realizability—the structural capacity of an alternative to be ontologically manifested. We show that the quadratic form ρ=∣ψ∣2 = ||²ρ=∣ψ∣2 is the unique probability measure compatible with two minimal and physically motivated requirements: (i) Structural Disjointness, ensuring additivity for independent (orthogonal) alternatives, and (ii) Descriptive Isotropy, requiring invariance of total realizability under continuous changes of the descriptive basis. Any non-quadratic measure (including LpLₚLp-based norms with p≠2p 2p=2) is shown to violate continuous symmetry, rendering physical existence dependent on the choice of coordinates. As a consequence, the inner-product (Hilbert space) structure emerges as a derived property rather than a foundational axiom. The paper further outlines how the Born rule defines the static geometry of the simplex of realizable states and constrains metriplectic models of measurement, where probabilities correspond to volumes of attraction basins rather than subjective uncertainty. This result establishes the Born rule as a structural inevitability rather than an empirical postulate and provides a falsifiable framework via the predicted absence of higher-order interference.