What does this research mean for the field?

There exists a precise categorical correspondence between the mathematical structures of quantum mechanics and general relativity, mediated by Krein spaces and established through a Grothendieck topos framework. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.ESTABLISHES_NEW_DIRECTION.

What question did this study set out to answer?

The aim is to establish a rigorous categorical framework that connects quantum mechanics and general relativity using topos theory.

March 7, 2026Open Access

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

Key Points

The aim is to establish a rigorous categorical framework that connects quantum mechanics and general relativity using topos theory.
Defined four functors on objects and morphisms within a Grothendieck topos framework.
Verified full functoriality unconditionally for the constructed functors and their morphisms.
Proved a natural transformation for intertwiner morphisms using Schur's Lemma.
Constructed a sheaf for entropy and verified its axioms, developing sheafification for entangled states.
Proved the Kryukov intertwining identity at the specified scale relating to the Bogoliubov mixing angle.
Established the Entanglement Bound Theorem linking Einstein equations and mutual information across causal horizons.
Provided a categorical reformulation of Jacobson’s thermodynamic approach.

Abstract

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).

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

Key Points

Abstract

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