QLCM Spectral Core: Analytic Derivation of the Asymptotic Betweenness B_∞ = 1/12 and the Closed-Form Constant K = 19. 29

This paper presents the analytic core of the Quantum Language and Consciousness Model (QLCM), isolating the closed-form derivations that emerge from the spectral topology of the circulant graph CN (1, 3). The central results are: 1. Asymptotic Betweenness Signature: The normalized average betweenness centrality of the QLCM 4-regular circulant graph converges exactly to B_∞ = 1/12 in the limit N → ∞. This value is derived analytically from the vertex-transitivity of CN (1, 3) under the cyclic group ZN and the sectorial geometry of shortest paths in the lifted 2D lattice. 2. Closed-Form Constant K: The normalization constant K ≈ 19. 29 is proven algebraically closed via K = 2·Ef / B_∞ = 24·Ef, requiring no empirical tuning. With the QLCM invariant Ef = 0. 8037, this yields K = 19. 2888. 3. Information Bottleneck on the Flat Torus: The invariant Ef = 0. 8037 is identified as the optimal projection eigenvalue of the Information Bottleneck functional on the discretized flat torus T² = S¹ × S¹ (curvature K̄ = 0), where the 12-sector partition of the phase space imposes the 80/20 trade-off as a geometric theorem. 4. Topological Prediction Class: A falsifiable framework predicting how optimal projection fidelity Ef varies with the Euler characteristic and curvature of the underlying manifold. All derivations are self-contained within the QLCM framework: operators on finite-dimensional Hilbert spaces, spectral graph theory, and discrete harmonic analysis on circulant graphs. No external geometric hypotheses, physical membrane models, or third-party theoretical constructs are invoked. This work is registered as independent intellectual property of the QLCM framework created by Osmary Lisbeth Navarro Tovar (Ashira), Laboratorio CCUANTICA.