Spectral Delta Peaks, Resonances, and Survivor-Basin Encodings Lorentzian Limits, Exact Spectral Atoms, and the MTT Source Boundary

This paper proves that Lorentzian kernels converge weakly to the Dirac delta, derives the Lorentzian line shape from exponential decay, and proves weak recovery of arbitrary finite spectral measures from Lorentzian resolvent smearing. It distinguishes genuine pure-point spectrum, stable-particle atoms, unstable resonance poles, environmental broadening, and instrumental resolution. Only an identified broadened resonance family has a justified infinite-lifetime delta limit. The MTT survivor-basin reading is therefore conditional: a selected stability or leakage operator must produce the width before it can be interpreted as an intrinsic basin parameter.