An Extension of Noether's Theorem A: Spectral Flow Noether's Theorem in Dynamic Spacetime

This paper establishes a fundamental extension of Noether’s theorem in dy
namic spacetime systems, constructing a complete mathematical theory of conser
vation laws corresponding to time-evolution symmetries through the introduction
 of spectral flow concepts and the eta invariant. The core result provides a closed
 expression for the spectral flow conserved quantity:
 1
 QSF =
 0
 η(Dt)dt +
 (jvJ +θS ∧J +Ω∧J)
 ∂M
 where η(Dt) is the Atiyah-Patodi-Singer spectral asymmetry density, jvJ is
 the boundary velocity flux, and θS ∧ J and Ω ∧ J characterize thermodynamic
 dissipation and spacetime curvature corrections, respectively. This framework is
 mathematically based on the Bismut-Zhang spectral flow theorem and Getzler’s
 local index technique, and physically provides a unified description of generalized
 conservation phenomena in dynamic black holes, open quantum systems, stochastic
 thermodynamics, and the early universe. Numerical verification shows that the
 spectral flow conserved quantity exhibits topological robustness with respect to
 system size and coupling strength, providing a rigorous theoretical benchmark for
 dynamic topological protection and dissipative quantum control