On the Born Rule: Projection, Coherence and the Emergence of Quantum Probability

We present a first-principles derivation of the Born rule as a geometric consequence of projection-based physics. In this framework, quantum probability emerges from the coherent overlap between interference-stable boundary modes and projection kernels mapped from a timeless, scale-invariant conformal boundary. Rather than postulating probability axiomatically, we show that the squared modulus of the amplitude arises naturally from the geometry of projection itself. This approach resolves the quantum measurement problem without invoking collapse or many-worlds branching, reinterpreting quantum probabilities as coherence weights in informational projection. The Born rule, , is shown to be an inevitable result of the geometry underlying emergent quantum fields, spacetime, and measurement. This paper builds on the foundational projection framework introduced in The Mathematical Principles of Natural Philosophy: On the Origin of Spacetime and extends its implications into the core foundations of quantum mechanics. Author: Daniel Toupin 📧 [email protected] 🔗 ORCID: 0009-0003-7682-9579