What question did this study set out to answer?

The aim is to demonstrate how probabilities relate quadratically to amplitudes within quantum mechanics.

May 9, 2026Open Access

The Born Rule from Projection Measures: Why Probabilities are Quadratic in Amplitudes

Key Points

The aim is to demonstrate how probabilities relate quadratically to amplitudes within quantum mechanics.
Presented a theorem detailing the quadratic projection measure.
Applied the Hilbert embeddability ansatz under specific structural constraints.
Discussed implications of Gleason's theorem in the context of Hilbert representation.
Established that probabilities are derived quadratically from projection measures under structural constraints.
Confirmed the emergence of the standard Born rule through logical propositions.
Highlighted the foundational role of axioms A4, A5, A6 in deriving these results.

Abstract

Theorem (Quadratic Projection Measure) + Ansatz (Hilbert Embeddability) + Proposition (Standard Born Rule). Under three structural constraints (basis invariance, projection-integration stability, orientation independence), the projection-induced measure on coherent informational states is forced to be ||². The standard quantum-mechanical Born rule emerges as a Proposition under the Hilbert Embeddability Ansatz, with Gleason's theorem providing the converse on the Hilbert representation. Inherits from Paper 0 (axioms A4, A5, A6).

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Travis Bergen

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The Born Rule from Projection Measures: Why Probabilities are Quadratic in Amplitudes

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Abstract

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