Measurement as Sheafification: Context, Logic, and Truth after Quantum Mechanics

TL;DR

This paper reinterprets quantum measurement through the lens of sheaf theory and topos logic, emphasizing the contextual nature of quantum truth and providing a mathematical framework that unifies quantum and classical perspectives.

Contribution

It introduces a sheaf-theoretic approach to quantum measurement, linking contextuality, logic, and truth, and proposes a continuous dynamics bridging quantum and classical regimes.

Findings

Quantum contextuality is modeled as the absence of global sections in a presheaf.

Sheafification of truth values explains measurement without physical collapse.

A new dynamics interpolates between quantum and classical behaviors.

Abstract

Quantum measurement is commonly posed as a dynamical tension between linear Schr\"odinger evolution and an ad hoc collapse rule. I argue that the deeper conflict is logical: quantum theory is inherently contextual, whereas the classical tradition presupposes a single global, Boolean valuation. Building on Bohr's complementarity, the Einstein--Podolsky--Rosen argument and Bell's theorem, I recast locality and completeness as the existence of a global section of a presheaf of value assignments over the category of measurement contexts. The absence of global sections expresses the impossibility of context-independent description, and \v{C}ech cohomology measures the resulting obstruction. The internal logic of the presheaf topos is intuitionistic, and the seven-valued contextual logic proposed by Ghose and Patra is exhibited as a finite Heyting algebra capturing patterns of truth, falsity and indeterminacy across incompatible contexts. Classical physics corresponds to the sheaf case, where compatible local data glue and Boolean logic is effectively restored. Measurement is therefore reinterpreted as sheafification of presheaf-valued truth rather than as a physical breakdown of unitarity. Finally, a $\sigma$--$\lambda$ dynamics motivated by stochastic mechanics provides a continuous interpolation between strongly contextual and approximately classical regimes, dissolving the usual measurement paradoxes and apparent nonlocality as artefacts of an illegitimate demand for global truth.