Mean Bubble Separation From the Nucleation History: R-Star and the Length Scale Behind Every Signal Estimate

The direct integration gives R* H = 2.53e-06 from 6.21e+16 bubbles per Hubble volume, while the duration-based estimate gives 8.65e-04 from beta/H = 3046, a factor of 343 larger. The direction of the difference is understandable: the slope estimate weights the fastest nucleation moment, while the direct integral counts the full history including its slower beginning, and the simplified weighting used here omits order-unity geometric factors that the calibrated shortcut absorbs. For signal estimates, a factor of two in R* moves the peak frequency by the same factor and the amplitude by a larger power, so the comparison is not pedantry; it is an uncertainty budget item that most forecasts omit. This study is simulated and computed from stated assumptions; it is not a measurement.