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 (GQM, GGR, 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 Repᵢrr (P), Kreinₗin 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 SᵥN 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).
R. Aguayo Roco (Thu,) studied this question.