Spectral Flow, Dynamical Scale Generation, and Physical Observable in TCFQ

We construct the spectral mechanism underlying dynamical scale generation in the TCFQ framework and establish the emergence of a physical observable as a controlled spectral projection. Starting from the operator formulation, we derive the infrared growth of the dominant eigenvalue, identify the conditions for scale separation, complete the operator-level closure, and finally define a UV-finite, gauge-consistent observable. No external quantum field theoretic input is introduced.