Paper XCII: The G2 Haar Measure Integral for the Black Hole Microstate Partition Function: The Self-Consistent Coupling a∗= 0.5333, the Infall-Rotated Fano Tilt as its Physical De- termination, and the Path to Exact BekensteinHawking Coecient from Framework-

Paper XC established that the black hole microstate partition function is ZG2 = R G2 dµ(g) e−βHV (g) and that the entropy per Fano face sface = 1/4 is the tar- get. This paper computes the partition function explicitly for the discrete 7-state Fano distribution and identies the self-consistent coupling a∗= ktilt/kBTlocal = 0.5333 that reproduces sface = 1/4 exactly. At a∗, the eective Fano tilt angle at the horizon is θhorizon = 77.22◦ the tilt has rotated from the global value θG2 = 4.00◦toward the equatorial 90◦during infall but has not completed the full rotation. The self-consistent coupling a∗is determined by the competition between the tilt rotation rate τ −1 tilt = c/rT ∗ 2 and the infall rate τ −1 fall = c3/(πGM), both xed by the framework. The exact determi- nation of a∗from these framework-native timescales is the specic calculation that will conrm whether the framework gives sface = 1/4 exactly with no free parameters. The structure is demonstrated numerically: sface(a) passes through 1/4 at a∗with denite slope, and the physical coupling is computable from the ratio τtilt/τfall evaluated at the Planck scale. Three new predictions are made (P41P43). Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.