Extended PBH mass function boosts high-z halo formation and predicts few LISA IMBH events per year to z~50

The study computes Poisson fluctuations from the extended PBH mass function and propagates them through the halo mass function to the cosmic star-formation rate density and Thomson optical depth. Results indicate a mild increase in high-redshift star formation that remains within current reionization bounds while populating the stellar-mass binary regime already accessible to LIGO-Virgo-KAGRA. For LISA the signal is dominated by 10^3-10^4 solar-mass binaries whose high-redshift detections would distinguish primordial from astrophysical seeds. This connects directly to the long-standing puzzle of supermassive black hole formation by z~7, where PBH seeds could bypass standard Pop-III pathways. Earlier analytic work on monochromatic PBH populations underestimated the high-mass tail contribution to intermediate-mass mergers; the extended function naturally supplies that tail without violating microlensing or dynamical constraints. Future LISA data releases will test the prediction that cumulative rates above z=20 exceed one event per year if PBHs comprise even a few percent of dark matter.