The Quantum Realization Functor and the κ-Obstruction Theorem: Why Quantum Computation is Bounded by the Algebraic Geometry of Motives

We introduce the quantum realization procedure Hₐ₅ₓ, which packages the Betti and étale realizations of a motive into a single Hilbert space and implements a computational analogue of the comparison isomorphism at polynomial cost via the quantum Fourier transform. The central result is the κ-obstruction theorem, a purely categorical statement: any exact functor from an abelian category to a semisimple target category annihilates all higher extension groups Extⁿ for n 1. This applies in particular to quantum computation, where measurement maps quantum states to classical outputs in a semisimple category (finite-dimensional vector spaces). We show that the complementarity index of the Algorithmic Motives (AM) framework corresponds canonically to a nontrivial extension class arising from the derived inverse limit ¹, which embeds into Ext¹ of the motivic category. When = 0, the obstruction is fully captured by morphisms and can be resolved via the comparison isomorphism, yielding polynomial-time quantum algorithms (e. g. , Shor, Biasse–Song). When > 0, the residual obstruction lies in Ext¹, which is annihilated by any realization functor and therefore cannot be accessed by any computational model whose output lands in a semisimple category. This establishes a structural limitation on quantum computation: quantum speedup is possible precisely when the relevant motivic obstruction is visible at the level of realizations, and impossible when it resides in extension data invisible to all such functors. The result is independent of physical assumptions and depends only on standard homological algebra and the semisimplicity of the output category.