The Paper and the Pen Framework: Dimensionally Consistent Vacuum Fluctuation Scale, Variational Axion Energy Exchange, and Cosmological Diagnostics

Abstract. The Paper and the Pen framework treats the multiplier of the unimodular (four-volume-constraint) formulation of gravity as a signed record variable conjugate to the accumulated four-volume.This paper establishes three results (B1–B3 below): a Poisson model for the width of the averagedrecord density, with a canonical bound on its coefficient; the conservation of the total record in theunimodular–axion action; and the equivalence of the vacuum–axion energy exchange with a free scalarfield shifted in a quadratic potential. The width coefficient is a calibration constant, so the observeddark-energy scale is recovered up to that calibration, not predicted.