Total Black Hole Mass Recalibration and the Inverse Halo Problem

Paper I (Bibb 2026) established that dark matter halo mass correlates with SMBH accretion history through the empirical relation Mhalo ≈ k · MBH · (t/τ) ² with k = 4, 120, calibrated on the Milky Way. Here we show that k is an effective constant (keff) that absorbed the Milky Way’s ratio of total black hole mass to SMBH mass. We define ktrue as the coupling to total collapsed-object mass — SMBH plus the stellar black hole graveyard — and validate it across four systems spanning dwarf galaxies to massive spirals, plus one qualitative control case (47 Tucanae). Two methodological advances are introduced. First, age-weighted stellar BH contributions, where each BH cohort contributes in proportion to (ti/τ) ² according to its formation epoch, effectively remove spin as a dominant free parameter. Second, the merger-sum generalization — in which progenitor BH contributions sum independently — explains the systematic overprediction for merger-assembled systems and establishes the single-BH formula as an upper bound. We further develop the inverse halo problem: given an observed halo mass, constrain the family of merger histories consistent with the observation. Applied to the Milky Way, the inversion recovers the Gaia–Enceladus merger parameters from halo mass alone. Applied to M31 and NGC 4258, the inversion yields no solutions under keff, correctly diagnosing these systems as requiring the ktrue recalibration. Under age-weighted ktrue, the framework achieves mean |Δ| = 0. 38 dex across three non-anchor systems at fBH = 1. 75×10⁻³. fBH is not fitted per system but constrained within population-synthesis bounds; variation reflects physical retention differences rather than free optimization. Companion to Paper I: doi: 10. 5281/zenodo. 15193025