Formal Derivation of the Critical Threshold κc from Renormalization Group Principles and Spectral Stability

We present an effective derivation of the universal informational critical threshold κc based on renormalization group (RG) analysis, spectral stability, and universality arguments. Under minimal physical assumptions—informational primacy, objective irreversibility, and the requirement of stable classical emergence—it is argued that the value κc ≈ 2.04 ± 0.05 emerges as the unique consistent solution of a coupled system of algebraic and differential conditions, defining a well-characterised informational universality class. The threshold κc simultaneously governs multiple physical domains: the emergence of spacetime from a pregeometric substrate, horizon formation in black holes, cosmological anchoring processes (Multibang framework), the spectral hierarchy of fundamental interactions, coherence limits in ER↔EPR configurations, and the selection of effective physical laws as informational attractors. No adjustable parameters are introduced: κc is fixed by the soft-mode condition λ_min(H[κc]) = 0 of the informational Hessian, the RG flow halting condition βκ(κc) = 0, and holographic occupancy constraints. Values κ ≪ 1 lead to trivial collapse, while κ ≫ 1 generate informational runaway, leaving only a narrow critical interval around κ ∼ 2. Universality is supported by independence from microscopic details, numerical convergence in black hole simulations, preliminary observational evidence for a Λ–ψ correlation, and consistency with unification models based on spectral hierarchy. We argue that κc belongs to the hard core of the TAGC–LQG–RG research programme in the Lakatosian sense, with epistemological status analogous to universal critical exponents in phase transitions. The result is formulated as an Inevitability Theorem for temporal emergence, classicality, and cosmological stability.