The Phase Offset Epsilon as an Evolutionary Classifier: A Separation Budget for Red-Giant-Branch and Clump Stars

Two red giants can share a temperature, a luminosity and even a large frequency separation while burning entirely different fuels, one fusing hydrogen in a shell around an inert helium core, the other fusing helium in the core itself. This paper computes, from stated assumptions and published stellar values, a quantitative budget for the phase offset epsilon using Kepler red-giant acoustic patterns. The headline results are: sep at 005 = 5.0; sep at 010 = 2.5; offset = 0.25. Every number in the abstract and the results section is produced by the deposited computation script from the inputs stated in the Methods, with no fitted parameters beyond those declared. The study is simulated and computed from stated assumptions; it is not a measurement, and its value lies in making the size, sensitivity and failure conditions of the effect explicit enough for observation to confirm or refute.