Conrmation of sface = 1/4: The Continuous G2/SU(3) Bessel Function Integral, the 7-State Discrete Model, and Their Reconciliation Paper XCIV of the Brane-Bulk Octonionic Series

Two independent computations of the Fano face entropy are compared: (1) the 7-state discrete Shannon entropy H7(a) of the nite orbit {e1, . . . , e7} under G2, and (2) the continuous G2/SU(3) = S6 Haar measure integral with partition function ZS6(a) ∝I3(a)/a3. At a = 0 (no coupling), both are normalised to give ln 7, conrming consistency. At a = a∗, the discrete model gives H7(a∗) = π p 3/8 (Paper XCIII), while the continuous integral gives a slightly dierent value due to the spread of the continuum distribution over S6 beyond the 7 special xed points of SU(3). The reconciliation: the physical microstate count is the discrete 7-state model, because the Fano lattice is discrete at the Planck scale. The continuous integral approximates this but overestimates by 3.9% at a∗due to entropy contributions from non-xed-point states on S6. The discrete 7-state model is therefore the correct model for the microstate count, and the master relation H7(a∗) = π p 3/8 gives sface = 1/4 exactly. Two new predictions are made (P45P46). 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.