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LIT-382: A Topos Perspective on the Kochen-Specker Theorem: IV. Interval Valuations

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Properties6
StatusActive — read in full 2026-10-01 (NOTE-328); worth reading as the finished statement of the series' von Neumann-algebra framework. It proves the G ≅ CloΣ isomorphism (coarse-graining = direct image of clopen regions) that Döring–Isham's daseinisation relies on, and it shows that a state's sieve-valued truth values are fixed by its support, a global element of G. Its two theorems are correct but close to definitional, so its value lies in the framework rather than in depth.
Version2
Tagsquantum-foundations, mathematics, contextuality, logic
SupersedesLIT-380 — A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category
Cited as “paper” byNOTE-328 — A Topos Perspective on the Kochen-Specker Theorem: IV. Interval Valuations
Cited as “source” byTHEORY-619 — Over the poset of commutative von Neumann subalgebras, coarse-graining a projector to a smaller subalgebra is exactly the direct image of its clopen region of the Gelfand spectrum under restriction, so the coarse-graining presheaf is the presheaf of clopen spectral regions; and a quantum state's sieve-valued truth values are fixed by one coherent family of regions, its supports: a proposition is true at a subcontext exactly when the support there lies under the coarse-grained proposition

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Butterfield & Isham (2001), International Journal of Theoretical Physics 41(4):613–639 (April 2002), per Crossref; the journal pages were not seen — arXiv:quant-ph/0107123

Standing in the record

Filed on 2026-10-01 at the owner’s request. published: is the first appearance (ADR-002).

It was filed Deferred, unread. NOTE-328 is the close reading of 2026-10-01, and it placed the work: Active — worth reading as the finished statement of the series’ von Neumann-algebra framework. It proves the G ≅ CloΣ isomorphism (coarse-graining = direct image of clopen regions) that Döring–Isham’s daseinisation relies on, and it shows that a state’s sieve-valued truth values are fixed by its support, a global element of G. Its two theorems are correct but close to definitional, so its value lies in the framework rather than in depth.

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