Structural Realizations and Invariants of Universal Obstruction Cohomology: Goodwillie Towers, Chromatic Filtrations, and Quantum Contextuality
Abstract
Contextuality is the phenomenon where information that is locally compatible---consistent on each piece of a system---nevertheless fails to assemble into a single globally consistent picture. This problem appears in quantum foundations, operator algebras, categorical semantics, and derived geometry. In this paper, we study the realization theory and structural invariants of universal obstruction cohomology within the Categorified Spectral Duality (CSD) framework, and establish four main results. First, we prove a Goodwillie Realization Theorem. For an obstruction realization functor that is compatible with both Goodwillie calculus and descent, the Goodwillie tower---which organizes information by polynomial degree---is naturally equivalent to the descent tower of the obstruction complex. Consequently, each homogeneous layer of the Goodwillie tower corresponds to a layer in the descent filtration. Second, we prove a Chromatic Realization Theorem. When the obstruction realization functor is chromatically compatible, the chromatic tower (organized by height in the sense of chromatic homotopy theory) refines the descent filtration. Each chromatic layer adds extra structure within the corresponding descent level. We also establish interchange results and define mixed obstruction layers, all under explicit compatibility hypotheses. Third, we introduce a quantitative measure called the descent complexity, denoted dc(A). This number measures the minimal number of gluing steps needed to assemble local semantic realizations into one global realization. If dc(A)=1, then the descent spectral sequence collapses at the first page, meaning no higher differential corrections are needed. We also define the obstruction spectrum, which is simply the set of positive degrees in which obstruction cohomology is nonzero, and prove that this invariant is preserved under homotopy, Morita equivalence, and descent. For the simplest systems, namely ℂ and matrix algebras Mn(ℂ), we rigorously show that dc=1 and the obstruction spectrum is empty. Even Clifford algebras reduce to ℂ via Morita invariance, while odd Clifford algebras reduce to ℂ⊕ℂ, though the latter depends on a finite-product additivity theorem that is not yet established. Explicit computations for genuinely contextual systems---such as Pauli, Specker, and Mermin--Peres configurations---as well as infinite-dimensional operator algebras like C*(F2) and the Cuntz algebras 𝒪n, remain open and require detailed descent spectral sequence analysis. Fourth, we prove an Obstruction Reconstruction Theorem. For systems belonging to the CSD reconstruction class, the enriched obstruction invariant---which includes the obstruction spectrum, the descent filtration, the cup product, and higher algebraic structure---determines the Morita class of the system up to CSD-compatible equivalence. This provides an obstruction-theoretic analogue of classical reconstruction principles such as Gelfand duality and Tannaka--Krein reconstruction, but within the specified reconstruction class. The first three realization results are conditional: they require explicit compatibility hypotheses to hold. By contrast, the reconstruction theorem is unconditional within the reconstruction class. Systems outside this class remain an open problem for future work.
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BibTeX
@misc{chang2026structural,
title = {{Structural Realizations and Invariants of Universal Obstruction Cohomology: Goodwillie Towers, Chromatic Filtrations, and Quantum Contextuality}},
author = {Shih-Yu Chang},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05739403v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Structural Realizations and Invariants of Universal Obstruction Cohomology: Goodwillie Towers, Chromatic Filtrations, and Quantum Contextuality
AU - Shih-Yu Chang
PY - 2026
DA - 2026/09/04
PB - HAL
UR - https://hal.science/hal-05739403v1
AB - Contextuality is the phenomenon where information that is locally compatible---consistent on each piece of a system---nevertheless fails to assemble into a single globally consistent picture. This problem appears in quantum foundations, operator algebras, categorical semantics, and derived geometry. In this paper, we study the realization theory and structural invariants of universal obstruction cohomology within the Categorified Spectral Duality (CSD) framework, and establish four main results. First, we prove a Goodwillie Realization Theorem. For an obstruction realization functor that is compatible with both Goodwillie calculus and descent, the Goodwillie tower---which organizes information by polynomial degree---is naturally equivalent to the descent tower of the obstruction complex. Consequently, each homogeneous layer of the Goodwillie tower corresponds to a layer in the descent filtration. Second, we prove a Chromatic Realization Theorem. When the obstruction realization functor is chromatically compatible, the chromatic tower (organized by height in the sense of chromatic homotopy theory) refines the descent filtration. Each chromatic layer adds extra structure within the corresponding descent level. We also establish interchange results and define mixed obstruction layers, all under explicit compatibility hypotheses. Third, we introduce a quantitative measure called the descent complexity, denoted dc(A). This number measures the minimal number of gluing steps needed to assemble local semantic realizations into one global realization. If dc(A)=1, then the descent spectral sequence collapses at the first page, meaning no higher differential corrections are needed. We also define the obstruction spectrum, which is simply the set of positive degrees in which obstruction cohomology is nonzero, and prove that this invariant is preserved under homotopy, Morita equivalence, and descent. For the simplest systems, namely ℂ and matrix algebras M_n(ℂ), we rigorously show that dc=1 and the obstruction spectrum is empty. Even Clifford algebras reduce to ℂ via Morita invariance, while odd Clifford algebras reduce to ℂ⊕ℂ, though the latter depends on a finite-product additivity theorem that is not yet established. Explicit computations for genuinely contextual systems---such as Pauli, Specker, and Mermin--Peres configurations---as well as infinite-dimensional operator algebras like C^*(F_2) and the Cuntz algebras 𝒪_n, remain open and require detailed descent spectral sequence analysis. Fourth, we prove an Obstruction Reconstruction Theorem. For systems belonging to the CSD reconstruction class, the enriched obstruction invariant---which includes the obstruction spectrum, the descent filtration, the cup product, and higher algebraic structure---determines the Morita class of the system up to CSD-compatible equivalence. This provides an obstruction-theoretic analogue of classical reconstruction principles such as Gelfand duality and Tannaka--Krein reconstruction, but within the specified reconstruction class. The first three realization results are conditional: they require explicit compatibility hypotheses to hold. By contrast, the reconstruction theorem is unconditional within the reconstruction class. Systems outside this class remain an open problem for future work.
ER -