The Fitted Exponent, Priced: Salpeter's Power Law, the Constant Birthrate It Rides On, and the Divergence Below Its Range

Census pricing (row CC-115) of the founding step of Salpeter's 'The Luminosity Function and Stellar Evolution,' ApJ 121, 161-167 (1955), against a fixed bench (what is postulated, what is derived, what the postulates return numerically, what they cannot reach). The step is two inputs: a star-formation birthrate history assumed constant in time, and a power-law initial mass function whose exponent 2.35 is a fit over 0.4-10 solar masses (labelled fitted wherever it appears), with universality riding on both. Verified in the notebook (5 checks, exit 0) by exact analytic integration of the declared law: stars above 1 solar mass are 0.2809 by number holding 0.5941 of the mass; above 8 solar masses, 0.004618 by number holding 0.0390 of the mass; mean mass 1.0565 solar masses; 4.37 stars above 8 solar masses per 1000 solar masses formed. Remainders at full size: across the Scalo-scale exponent band +/-0.35 the above-8 count moves by a factor 5.35; below the fitted range the total mass diverges as M_low^(-0.35) (x4.901 in mass and x147.370 in number at a 0.01 solar-mass cutoff); and the constant birthrate is spent as correction multipliers reaching 55.90 at 5 solar masses (declared bench lifetime scaling). Scout-grade verification disclosed on the manuscript's face (CC-91 form): no verbatim quotation, no page-specific claim. EXTENSION. Vacuum catastrophe NOT SOLVED; quantum gravity NOT SOLVED.