The Compton–Gravity Threshold as a QDL Admissibility Anchor: Reduced Compton Wavelength, Schwarzschild Radius, and the mP/√2 Boundary

This technical note develops the reduced-Compton/Schwarzschild equality as a narrow admissibility anchor within the Quantized Dimensional Ledger (QDL). The reduced Compton wavelength, λ̄C = ℏ/(mc), is treated as the localization scale associated with rest-energy recurrence, while the Schwarzschild radius, RS = 2Gm/c², is treated as the gravitational radius associated with a non-rotating, uncharged mass. Equating these two standard length scales gives m* = mP/√2, where mP = √(ℏc/G) is the Planck mass. The contribution of the paper is not the discovery of the Compton–gravity threshold, which follows by standard algebra. Rather, the paper classifies the threshold as a QDL admissibility anchor: a model that uses the reduced-Compton and Schwarzschild definitions while claiming to connect rest-energy recurrence and gravitational radius formation should recover λ̄C = RS, hence mP/√2, or state exactly which definition, role assignment, or dynamical correction changes the threshold. The paper is Stage 3 in a conservative QDL admissibility-anchor sequence. Stage 1 formulated the closure-deficit theorem, which states that dimensional homogeneity is necessary but not sufficient for QDL admissibility. Stage 2 applied that theorem to the gravitational parameter μ = GM as a closure-stable observable. The present paper applies the same discipline to the Compton–gravity threshold. It does not derive quantum gravity, black-hole microstates, the Standard Model, Newton’s constant, Planck’s constant, or the speed of light. Its purpose is to provide a source-checkable Planck-scale boundary condition that future QDL completion work must reproduce, respect, or explicitly modify.