The Logic of Observation: A Unified Cohomological Theory of Quantum Contextuality

A six-paper series with epilogue exploring the cohomological foundations of quantum contextuality: from geometric phases to the necessity of complex numbers.

Author: Zhou-Li Chen (co-nlang Research)

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Overview

Why does quantum mechanics require complex numbers? Why do geometric phases and Kochen–Specker contextuality arise from the same underlying structure? This series proposes a unified answer: quantum observation is governed by a cohomological obstruction ladder, where each rung corresponds to a deeper level of algebraic failure—from consistent phase patching ($H^1$), through non-commutativity ($H^2$), to non-associativity ($H^3$).

The series builds a bridge between three mathematical domains:

• Quantum logic (Bohrification topos, MASA posets)
• Group cohomology (Lyndon–Hochschild–Serre spectral sequence)
• Higher geometry (line bundles, bundle gerbes, Dixmier–Douady class)

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AI Collaboration Disclosure

This research project integrated various Large Language Models (LLMs) across multiple stages to enhance rigor and clarity. The author(s) maintain full accountability for the final content.

• Theoretical Derivation: AI was used to assist in symbolic manipulation, cross-verifying mathematical proofs, and identifying potential edge cases in formulas.
• Development & Typesetting: Code implementation and LaTeX structural optimization were supported by AI-assisted pair programming.
• Language & Refinement: Sentences were polished for academic flow and grammatical precision.
• Simulated Peer Review: AI agents were tasked to act as independent reviewers to provide critical feedback and identify logical gaps prior to publication.

Models used: Gemini 3 Pro/3.1 Pro, GPT-5.3, Claude Sonnet 4.6, Kimi K2.5/K2.6, GLM-5.0, QWen 3.5 Plus, DeepSeek V4 Pro.

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Repository Structure

research/
├── README.md
├── LICENSE                             # CC BY 4.0
├── RESEARCH_FRONTIER.md                # Toolbox & open problems
├── insight/                            # Insight Notes | Speculative thought experiment
├── papers/
│   ├── Paper1_contextuality_phase.tex
│   ├── Paper2_riemann_bohrification.tex
│   ├── Paper3_ks_central_extension.tex
│   ├── Paper4_lhs_borromean.tex
│   ├── Paper5_homological_bridge.tex
│   ├── Paper6_necessity_complex.tex
│   ├── Epilogue_algebraic_logic.tex
│   ├── LsNote_geometric_phase.tex          # L-S contraction & H¹ (Appendix to Paper I)
│   ├── LsNote_noncommutativity.tex         # L-S contraction & H² (Appendix to Paper III)
│   └── LsNote_associativity.tex            # L-S contraction & H³ + octonionic boundary
└── supplementary/
    └── construct_16cell/
        ├── LICENSE                     # MIT
        ├── README.md
        └── construct_16cell.py

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Reading Guide

Recommended Order

Paper I → Paper II → Paper III → Paper IV → Epilogue → Paper V → Paper VI

• Papers I–III establish the concrete foundation: geometric phases, Riemann surfaces, and the Kochen–Specker obstruction.
• Paper IV introduces the $H^3$ frontier (Borromean contextuality) and the Lyndon–Hochschild–Serre transgression.
• The Epilogue provides the unifying vision (EML system, Solèr's theorem, obstruction ladder).
• Papers V–VI ascend to the categorical abstraction and prove the necessity of $\mathbb{C}$.

Supplementary L-S Appendices

Three companion notes connect Lohmiller–Slotine contraction theory to the obstruction ladder. Each provides a dynamical reinterpretation of a cohomological level, framed as a companion appendix to the relevant paper:

• LsNote_geometric_phase (companion to Paper I): Liouville's theorem proves no global contraction metric exists for the Aharonov–Bohm system; the gauge mismatch Čech 1-cocycle equals the geometric phase.
• LsNote_noncommutativity (companion to Paper III): The Peres–Mermin $K_{3,3}$ nerve carries a flat $SU(2)$ connection whose 4-cycle holonomy is $-\mathbf{I}$, proving the Kochen–Specker obstruction equals the central extension class.
• LsNote_associativity (companion to Paper IV): The 16-cell $S^3$ nerve carries a global Čech 3-cocycle (conjectured pairing $-1$); octonionic dynamics break the chain rule, proving L-S theory inapplicable for $\mathbb{O}$.

Quick Reference

#	Title	Focus	Key Result
I	From Contextuality to Phase Cohomology: A Computable Bridge Between Bohrification and Geometric Quantization	Bohrification ↔ geometric quantization	$\check{H}^1(M, U(1))$ classifies geometric phases
II	Semiclassical Reconstruction of Riemann Surfaces from Bohrification	Riemann surface state spaces	Bohr-Sommerfeld orbits as divisors on spectral curves
III	Kochen–Specker Contextuality as Central Extension: The Peres-Mermin Square and the Pauli Group	Peres–Mermin square & Pauli group	KS obstruction = non-split central extension $[f] \in H^2(\bar{\mathcal{P}}_2, \mathbb{Z}/2)$
IV	The Cohomological Obstruction Ladder: Lyndon–Hochschild–Serre Transgression and the $H^3$ Frontier	Borromean contextuality & bundle gerbes	$N \geq 5$ qubit threshold; $d_2$ transgression = KS obstruction
Epilogue	The Algebraic Logic of Geometry	EML system & Solèr's theorem	$\mathcal{Q} \dashv \mathcal{B}$ adjunction and the obstruction ladder
V	Observation as Functor: The Adjunction of Quantum and Classical	LHS spectral sequence as computational engine	Conditionally unifies all obstructions via a single algebraic machine
VI	The Ultimate Axiom: Deriving Quantum Mechanics from the Logic of Observation	Solèr-Cohomology Theorem	$\mathbb{C}$ is the unique division ring permitting non-trivial continuous observation
L-S I	L-S Contraction and the Cohomological Origin of Geometric Phases	Liouville vs contraction (A-B effect)	$\check{H}^1$ 1-cocycle = geometric phase holonomy
L-S II	L-S Contraction and the Cohomological Origin of Non-Commutativity	4-cycle holonomy in PM square	$-\mathbf{I}$ = central extension class $[f]$
L-S III	L-S Contraction and the Boundary of Applicability: $H^3$, the 16-Cell, and Non-Associative Algebra	16-cell $S^3$ nerve + $\mathbb{O}$ chain rule failure	$H^3$ pairing conjecture; $\mathbb{O}$-dynamics incompatible with L-S

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The Obstruction Ladder

Level 0 (Foundation):  H¹  —  E_∞^{1,0} survivors     —  Geometric phases (Aharonov–Bohm, Berry)
Level 1 (Obstruction): H²  —  d₂ transgression         —  Kochen–Specker contextuality
Level 2 (Obstruction): H³  —  d₃ higher differential   —  Borromean non-associativity

The distinction is crucial: geometric phases are stable features that survive the entire spectral sequence filtration ($E_\infty$ page). Quantum anomalies are obstructions measured by the differentials ($d_2, d_3$) that measure the cost of forcing classical logic onto quantum systems.

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RESEARCH_FRONTIER.md

A candid inventory of the mathematical machinery powering this series and the open problems that remain. Includes:

• Part I: The Mathematical Toolbox — 11 core tools (Topos Theory, LHS Spectral Sequence, Group Cohomology, Solèr's Theorem, EML, Sheaf/Čech Cohomology, Bundle Gerbes, 16-Cell Geometry, MASA Logic, Albert Algebra, Z3 SAT)
• Part II: The "Regrets" & Open Questions — 10 unsolved problems, including the $H^3$ numerical invariant for the 5-qubit 16-cell, the $\mathcal{Q} \dashv \mathcal{B}$ adjunction proof, non-abelian EML impossibility, the octonionic frontier, and experimental realization.

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Supplementary Materials

Directory	Description
supplementary/construct_16cell/	Z3 SAT solver construction of the 16-cell nerve for $S^3$-type Borromean contextuality (Paper IV)

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DOIs

All components are archived on Zenodo:

Component	DOI
Paper I	10.5281/zenodo.20072818
Paper II	10.5281/zenodo.20073010
Paper III	10.5281/zenodo.20073127
Paper IV	10.5281/zenodo.20073184
Epilogue	10.5281/zenodo.20073253
Paper V	10.5281/zenodo.20073318
Paper VI	10.5281/zenodo.20073424
construct_16cell.py	10.5281/zenodo.20070954
L-S Note I (geometric phase)	10.5281/zenodo.20102566
L-S Note II (non-commutativity)	10.5281/zenodo.20102587
L-S Note III (associativity / $\mathbb{O}$)	10.5281/zenodo.20102638

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Build

Each paper is a standalone LaTeX document. Compile with:

pdflatex Paper1_contextuality_phase.tex
pdflatex Paper1_contextuality_phase.tex
pdflatex Paper1_contextuality_phase.tex

Requirements: TeX Live 2023+ with amsmath, amssymb, amsthm, tikz-cd, booktabs, hyperref.

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Citation

To cite the series, please reference the individual paper(s) by DOI (see above). For the series as a whole:

@misc{chen2026cohomological,
  author = {Chen, Zhou-Li},
  title  = {The Logic of Observation: A Unified Cohomological Theory of Quantum Contextuality},
  year   = {2026},
  note   = {Six-paper series with epilogue},
  url    = {https://github.com/co-nlang/research}
}

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License

• Papers (LaTeX sources in papers/): CC BY 4.0
• Supplementary code (supplementary/): MIT