The Born Rule as a Structural Consequence of a Phase‑Neutral Projection of Complex Hilbert‑Space States onto a Real Measurement Arena

AbstractThis work develops a structural derivation of the Born rule that remains fully within the established Hilbert‑space formalism of quantum mechanics. Instead of treating the Born rule as an independent postulate, the paper shows that it arises necessarily from a phase‑neutral projection of a complex, unitarily evolving ontic state space (the Bulk) onto a real, measurement‑structured appearance space (the Brane). The central claim is that any measurable quantity on the Brane must satisfy phase‑invariance, positivity and normalizability, and therefore must take the form of the squared modulus of the projected wavefunction. This framework—termed the BRane Interface Substrate Model (BRISM)—demonstrates that the Born rule is a fundamental interface condition between complex amplitude information and real measurement statistics, rather than a dynamical or agent‑dependent assumption. The model reorganizes the standard quantum formalism without introducing new physics, and clarifies why the Born probability structure is an unavoidable consequence of the conditions under which reell‑valued measurement outcomes manifest.In contrast to Everett’s decision‑theoretic approaches or Zurek’s envariance‑based derivations, which rely on agent‑based rationality principles or specific assumptions about system–environment structure, BRISM avoids both conceptual dependencies. It shows that the Born rule already follows from the structural requirements of mapping complex, phase‑bearing bulk states onto a real, phase‑neutral arena of appearance. Thus, BRISM provides a pre‑instrumental, interpretation‑independent justification of quantum probabilities that does not rely on branching worlds, observer rationality, or environment‑induced symmetries.