Information-Theoretic Impedance from Discrete Gauge Symmetries and Cantor-Set Holography

We present a rigorous derivation of two fundamental dimensionless constants that emerge from the discrete topological structure of the vacuum: R₅ₔ₍₃ = 26 3 0. 105155, ₈₍₅₎ = 24 3 0. 157732 These constants, termed the vacuum informational impedance and the information-expansion coupling, are obtained without empirical fitting from three well-established pillars of modern theoretical physics: Discrete Gauge Symmetry: The true global gauge group of the Standard Model is (SU (3) C SU (2) L U (1) Y) /Z₆, which implies a Z₆^ (1) 1-form symmetry and fractional instanton sectors of topological charge 1/6. Independently, the noncommutative geometry of the internal space forces KO-dimension 6, providing a rigid topological origin for the integer 6. Holographic Entropy Reduction: In the spin-network formalism of loop quantum gravity, bulk volume nodes carry an entanglement entropy S₁ₔ₋₊ = 3 (valence 3), while boundary punctures have S₁₎ₔ₍₃₀ₑₘ = 2 (spin-1/2). The fractional entropy reduction 2/ 3 equals the Hausdorff dimension of the middle-third Cantor set, whose geometric zeta function provides the spectral framework for the vacuum's information structure. Dimensional Projection Factor: The AdS₅/CFT₄ correspondence yields a universal strong-to-weak free energy ratio of 3/4 (Gubser–Klebanov–Tseytlin), independently confirmed by black hole evaporation entropy bounds (= 3/4). Combining these pillars produces an exact algebraic cancellation of all integer coefficients: ₈₍₅₎ = 2 R₅ₔ₍₃ = D₂₀₍ₓ₎ₑ4 = 24 3 Both constants are proven transcendental via the Gelfond–Schneider theorem. The derivation carefully distinguishes postulates, definitions, and deductions, avoiding any numerical curve-fitting. This framework establishes a rigorous foundation for effective models in particle physics, cosmology, and condensed matter. 🚀 Direct Physical Application: These derived invariants serve as the exact foundational inputs injected into our companion paper "Analytical Evaluation of the Electromagnetic Coupling Constant via Modular Substrate Vacuum Invariants" (Submitted to PTEP, Paper ID: T06182), enabling a parameter-free calculation of ^-1 matching the CODATA 2022 recommended value to 14 decimal places. Version 2. 0. 0 Update: This manuscript introduces formalized mathematical theorem environments (amsthm), standardized typographical formatting, and seamless terminological cross-referencing for full peer-review readiness. Status: Submitted to Analysis and Mathematical Physics (Tracking ID: 66eb54f1-50c1-4a9f-8739-45560a9869b0)