Clarify the relationship between stationary phase dephasing and unitary group integral methods

Investigate and establish the precise mathematical relationship between the position-space stationary phase analysis of dephasing used in this paper and the unitary group integral saddle-point method used previously to recover dephased components and entangled states in causal fermion systems; determine whether and how these approaches are equivalent, complementary, or can be unified.

Background

The paper introduces holographic mixing and analyzes dephasing via stationary phase methods in position space, which underpin the emergence of bosonic quantum fields and canonical commutation relations. Earlier work on entangled states in causal fermion systems employed unitary group integrals, with different mathematical tools and interpretations.

The authors explicitly state that the two techniques appear complementary but their precise interrelation is not yet understood. Clarifying this connection would help unify the analytical and algebraic perspectives on dephasing and entanglement within causal fermion systems.