What question did this study set out to answer?

The aim is to create a mathematical structure that incorporates fundamental physical equations through the Hessian in a Helmholtz field.

June 21, 2026Open Access

Physical Structures Contained in the Hessian of a Harmonic Phase: a Spectral Functional Framework

Key Points

The aim is to create a mathematical structure that incorporates fundamental physical equations through the Hessian in a Helmholtz field.
Developed a tensor decomposition of the Hessian in Euclidean ℝ⁴.
Derived emergent metrics and key equations (Maxwell, Einstein, Dirac) from a unified geometric framework.
Presented quantitative predictions including lepton mass ratios and (g−2)/2 deviation.
Predicted lepton mass ratios with a deviation of 2–5%.
Calculated (g−2)/2 = 0.001157 with a deviation of −0.20%.
Established a geometric framework that incorporates multiple fundamental equations.

Abstract

This work constructs a mathematical framework based on the Hessian of the phase of a Helmholtz field in Euclidean ℝ⁴. The tensor decomposition simultaneously contains the Maxwell, Einstein, and Dirac equations as sectors of the same geometric object. From three postulates the framework derives: emergent Lorentzian metric, closed 2-form (no monopoles), Dirac equation from Helmholtz factorisation, conserved current, and falsification criteria. Quantitative predictions with zero free parameters include lepton mass ratios (2–5% deviation) and (g−2)/2 = 0.001157 (−0.20% deviation).

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