Effective Consistency Reconstruction of the Universal Admissibility Condition κc in the TAGC–LQG–RG Programme

This is Version 2.3 (June 2026) of the manuscript previously titled "Formal Derivation of the Critical Threshold κc" (Version 1, January 2026). Following the consolidation of the TAGC–LQG–RG architecture, κc is reaffirmed as a fundamental structural constant (HC2). Accordingly, the present work is reformulated as an effective consistency reconstruction rather than a first-principles derivation. Within the TAGC framework, κc = 2.04 ± 0.05 is treated as a fundamental admissibility condition. The manuscript examines how values within this interval satisfy fixed-point, spectral-stability, and holographic consistency requirements while remaining compatible with the transitional liminal regime (LT). The analysis incorporates the L1/LT/L2/L3 architecture, where LT mediates the emergence of classicality through a smooth transition governed by an accessibility functional A[Ψ] associated with informational anchoring (HC5). Support for the universality of κc is drawn from its independence from microscopic details, numerical convergence in black-hole simulations, consistency with the programme’s spectral hierarchy, and preliminary evidence for a positive Λ–ψ correlation. The principal result is formulated as a Structural Consistency Statement: within the TAGC framework, κc constitutes a necessary condition for temporal emergence, classical stability, and cosmological consistency.