Non-Collapse of Time-Averaged Kernel Measures under Irrational Phase Shifts

This work proves that time-averaged kernel measures generated by phase-shifted sinusoidal signals with irrational phase increments do not collapse to a Dirac mass in the weak topology. Under minimal assumptions on amplitude boundedness and kernel regularity, we establish uniform positive lower bounds on disjoint neighborhoods, thereby excluding weak convergence to point masses. The argument combines equidistribution theory with measure-theoretic non-concentration principles. This record contains two versions: 1. A compressed technical note presenting the core theorem and proof in minimal form. 2. A supplementary extended version including detailed intermediate arguments and conceptual motivation. Both versions establish uniform non-concentration under irrational phase shifts.