December 8, 2025Open Access

Second–Variation Geometry A Structural Foundation for Quantum Theory

Key Points

Quantum theory employs a structural foundation based on second variation geometry, implying coherence and irreversibility.
Key insights include stochasticity indicating limits of representation and causality maintained in both measurement and evolution.
The analysis shows entanglement is a property of variational geometry rather than subsystem interaction.
Results propose that quantum phenomena emerge from a unified variational structure without the need for collapse postulates.

Abstract

A structural foundation for quantum theory is developed by decomposingthe second variation into symmetric and skew components. Coherence is the symmetric sector, while measurement irreversibility arises from the skew sector; no collapse postulate is required. Entanglement is global compatibility of the variational geometry, not interaction betweensubsystems. Schr\"odinger evolution and Born rule statistics are projections of the same underlying structure rather than independent axioms. The appearance of stochasticity reflects limits of representation, not limits of structure. Causality is respected both in reversible evolution and in measurement, though for distinct structural reasons. Thus quantum phenomena are unified as complementary manifestations of a single variational geometry.

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