Theoretical Origin of Modular Constants

This paper presents a rigorous, first-principles derivation of two fundamental dimensionless constants that emerge from the modular substrate Z/6Z: R_fund = ln 2/(6 ln 3) – the vacuum informational impedance kappa_info = ln 2/(4 ln 3) – the information-expansion coupling constant The Three Pillars of Derivation The theoretical framework rests on three independently established pillars: 1. Generalized global symmetry and noncommutative geometry: The gauged Z_6^(1) 1-form global symmetry originating from the true Standard Model gauge group (SU(3) x SU(2) x U(1))/Z_6, together with the noncommutative KO-dimension 6 condition that forces an antisymmetric Dirac operator and resolves fermion doubling. 2. Holographic information capacity and the Cantor string: The bulk-to-boundary information transfer is formalized by the middle-third Cantor string: the principal pole of its geometric zeta function gives the Hausdorff dimension D_Cantor = ln 2/ln 3, which coincides with the information cost of converting ternary (bulk optimal encoding) to binary (holographic boundary encoding) as dictated by radix economy E(b) = b/ln b. 3. Dimensional projection factor beta = 3/4: Derived non-perturbatively from the AdS_5/CFT_4 correspondence (F_strong = (3/4) F_weak for N=4 SYM) and from the Aghapour-Hajian black hole evaporation entropy bound (|dS_BH| = (3/4) dS_rad). Algebraic Unification Combining these pillars yields kappa_info = 2 beta R_fund, where the factor 6 cancels exactly, leaving kappa_info = D_Cantor / 4. This algebraic cancellation leads to the fundamental identity: e6 R_fund ln 3 = 2 This equation links e, 2, and 3 through the topology of the vacuum. Both constants are mathematically transcendental as a direct consequence of the Gelfond-Schneider theorem. Predictive Power Across Diverse Domains - Quantum Electrodynamics: Appears in the analytic expansion of the inverse fine-structure constant, matching the experimental CODATA 2022 value to an absolute precision of 1.5 x 10⁻¹⁴. - Cosmology: Resolves the 5-sigma Hubble tension via a modified Friedmann equation, predicting a local expansion rate of H_0 ≈ 73.5 km/s/Mpc. - Condensed Matter Physics: Dictates the multi-step structural "Devil's staircase" plateaus observed during phase transitions in spin-crossover materials. This work provides an unassailable theoretical foundation for the Modular Substrate Theory (MST) and serves as a companion paper for experimental and phenomenological studies that rely on these constants.