Scale-Dependent Topos-Theoretic Construction: A Rigorous Categorical Correspondence Between Hilbert Spaces and Pseudo-Riemannian Manifolds. Kryukov Intertwining Partially Resolved in the Two-Level Truncated Model via Bogoliubov Theory, Entangled State Extension, and Holographic Conjecture

We construct a rigorous categorical framework establishing a precise correspondence between the mathematical structures of quantum mechanics (QM) and general relativity (GR) via a Grothendieck topos over the scale poset ([0,1],≤) with Krein spaces as the mediating structure. Four functors (G_QM, G_GR, R, R') are defined on both objects and morphisms, with full functoriality verified unconditionally. A natural transformation η: ι∘R ⇒ ι∘R' in the functor category [Rep_irr(P), Krein_lin] is proved unconditionally for intertwiner morphisms via Schur's Lemma, and the Kryukov intertwining identity is verified at the distinguished scale λ*(ω,L) = (2/π) arctan(e^{-πωL}), identified as the Bogoliubov mixing angle of the Unruh effect — within the two-level truncated model. An entropy sheaf S_vN is constructed, its sheaf axioms verified, and its sheafification S⁺_vN developed for entangled states. The Entanglement Bound Theorem establishes that the Einstein equations hold when mutual information across the causal horizon vanishes. The Thermodynamic Global Section Theorem provides a categorical reformulation of Jacobson (1995). A Holographic Topos Conjecture is stated as a programme for future research. Paper I of the Scale-Dependent Categorical Quantum Gravity (SDCQG) Programme. Companion paper: Paper II (10.5281/zenodo.18877050, 2026).