[33]Connectivity Geometry Framework: Pure Graph-Theoretic Derivation of the 1/4 Coefficient in Black Hole Entropy and Microscopic Output of Gravitational Coupling

Within the Connectivity Geometry framework, this paper directly derives the horizon entropy based on the pure graph-theoretic counting of the k=3 discrete spin network. We prove that k=3 is the unique allowed step length, and the density of cut edges on the horizon is locked at N_cut = (3/2)F. Combined with the Z_3 bulk spin labels, the microscopic entropy is strictly output as S_micro = (3/2)F ln 3. The second-order variation of the network coarse-grained action independently yields the gravitational coupling coefficient 1/(16πG_net) = 3 ln 3 / (8πℓ_0^2). From this, we prove that the "1/4" in the macroscopic Bekenstein-Hawking formula is the algebraic ratio output of the graph-theoretic entropy density to the independent gravitational coupling strength. This result not only explains the microscopic origin of the area coefficient but also provides the in-framework relationship between the universal gravitational constant G and the fundamental lattice spacing: G = ℓ_0^2 / (6 ln 3). The hard conclusion given by this framework is: the entropy density per cell has a locked upper bound of s_0 = (3/2) ln 3 ≈ 1.648 nats. If any future microscopic entropy calculation based on discrete spacetime yields an entropy per cell significantly higher than 1.648 nats, this framework will be falsified.