arxiv: 2603.16988 · v7 · submitted 2026-03-17 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

The Algebraic Landscape of Kochen-Specker Sets in Dimension Three

Michael Kernaghan

Authors on Pith no claims yet

Pith reviewed 2026-05-15 09:37 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Kochen-Speckeruncolorabilitythree-dimensional Hilbert spacenumber fieldsalgebraic islandsmodulus-2 cancellationphase cancellationcomputational survey

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The pith

Kochen-Specker sets in three dimensions appear only when coordinate generators support modulus-2 or phase cancellation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

A computational survey maps Kochen-Specker uncolorability across two-symbol alphabets drawn from quadratic, cyclotomic, and golden-ratio fields in three-dimensional Hilbert space. In every raw alphabet examined before cross-product completion, uncolorable sets appear only when the generator x satisfies either modulus-2 cancellation, where the squared modulus equals two, or phase cancellation, where a sum of unit-modulus complex numbers vanishes. Alphabets whose generators satisfy neither rule produce orthogonal triples but no KS-uncolorable configurations. The pattern accounts for the observed clustering of constructions into a handful of discrete algebraic islands, including two that yield previously unreported graph types after completion. Verification with SAT solvers supplies exact counts or upper bounds on the number of such sets within each island.

Core claim

In every tested raw alphabet before cross-product completion, KS sets arise only when x supports one of two cancellation mechanisms: modulus-2 cancellation (the generator satisfies |x|^2 = 2) or phase cancellation (a vanishing sum of unit-modulus terms). Alphabets whose generators have |x|^2 ≥ 3 and are not roots of unity produce orthogonal triples but not KS-uncolorability. Two islands yield potentially new KS graph types, and cross-product completion in the golden-ratio case introduces effective modulus-2 cancellations even though the raw alphabet satisfies neither mechanism.

What carries the argument

The two cancellation mechanisms (modulus-2 cancellation where |x|^2 equals 2, and phase cancellation where unit-modulus terms sum to zero) that govern whether a raw alphabet supports KS-uncolorable sets.

If this is right

KS constructions are confined to a small number of discrete algebraic islands across the surveyed fields.
The Heegner-7 ring and the golden-ratio field each contain new KS graph types.
Exact input counts are confirmed for three islands and upper bounds are established for three others.
The golden-ratio island is a boundary case in which cross-product completion supplies the missing cancellation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the two-mechanism pattern holds beyond the tested fields, exhaustive searches for KS sets could be restricted to generators satisfying one of the two conditions.
The pattern supplies a concrete algebraic criterion that could be used to decide whether a given number field can host KS sets.
Whether every KS set in dimension three ultimately reduces to one of these two cancellation types remains open and directly testable by extending the survey to additional fields.

Load-bearing premise

The tested alphabets and number fields are representative of the general case and the observed pattern is not produced by the specific search procedure or the cross-product completion step.

What would settle it

A single KS-uncolorable set found in a raw alphabet whose generator satisfies |x|^2 ≥ 3 and is not a root of unity.

read the original abstract

We present a computational survey of Kochen-Specker (KS) uncolorability in three-dimensional Hilbert space across two-symbol coordinate alphabets $\mathcal{A} = \{0, \pm 1, \pm x\}$ drawn from quadratic, cyclotomic, and golden-ratio number fields. In every tested raw alphabet (before cross-product completion), KS sets arise only when $x$ supports one of two cancellation mechanisms: modulus-2 cancellation (the generator satisfies $|x|^2 = 2$, as in $|\sqrt{2}|^2=2$, $|\sqrt{-2}|^2=2$, or $|\alpha|^2=2$; the integer case $1+1=2$ is the degenerate additive instance) or phase cancellation (a vanishing sum of unit-modulus terms, as in $1+\omega+\omega^2=0$). Alphabets whose generators have $|x|^2 \geq 3$ and are not roots of unity produce orthogonal triples but not KS-uncolorability in our survey. This empirical pattern explains why constructions cluster into at least six discrete algebraic islands among the tested fields (with a seventh, cubic island confirmed at higher cost). Two yield potentially new KS graph types: the Heegner-7 ring $\mathbb{Z}[(1+\sqrt{-7})/2]$ (43 vectors) and the golden ratio field $\mathbb{Q}(\varphi)$ (52 vectors, revealed only by cross-product completion); $\mathbb{Z}[\sqrt{-2}]$ provides a new algebraic realization of a known Peres-type graph. Using SAT-based bipartite KS-uncolorability, we verify the input counts of Trandafir and Cabello for three islands (exact) and establish upper bounds for three others. The golden ratio island is a boundary case: its raw alphabet satisfies neither mechanism, but cross-product completion introduces effective modulus-2 cancellations. Whether the two-mechanism pattern extends to all number fields remains an open question.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

KS sets in 3D appear tied to modulus-2 or phase cancellation in tested alphabets, yielding two new graph examples.

read the letter

The paper's core finding is that in the raw alphabets they tested, Kochen-Specker sets only turn up when the generator x allows either modulus-2 cancellation or phase cancellation. That pattern holds across the quadratic, cyclotomic, and golden-ratio fields they checked. What stands out is the new examples: a 43-vector KS graph from the Heegner-7 ring and a 52-vector one from the golden ratio field that only appears after cross-product completion. They also recover known counts exactly in three cases via SAT and give upper bounds in others. The work is honest about the limits—it treats the two-mechanism rule as an observed regularity rather than a theorem and leaves open whether it extends further. The soft spot is that the survey is finite and tied to their enumeration procedure. The golden-ratio case already shows that completion can introduce effective cancellations even if the raw alphabet does not, so the pattern might depend on how the sets are built. Still, the computations look reproducible and the new graphs are worth having on record. No load-bearing claims are overreached; they flag the generality question explicitly. This is for people working on contextuality in three dimensions or algebraic approaches to KS sets. It is worth sending to peer review; the new constructions and the empirical regularity are concrete enough to merit checking by others. The SAT verification adds some rigor to the enumeration.

Referee Report

1 major / 2 minor

Summary. The manuscript presents a computational survey of Kochen-Specker uncolorability in three-dimensional Hilbert space across two-symbol coordinate alphabets drawn from quadratic, cyclotomic, and golden-ratio number fields. It reports that, in every tested raw alphabet before cross-product completion, KS sets arise only when the generator x supports modulus-2 cancellation (|x|^2=2) or phase cancellation (vanishing sum of unit-modulus terms). This empirical pattern is used to explain clustering into at least six algebraic islands (with a seventh confirmed at higher cost), to identify new KS graph types in the Heegner-7 ring and golden-ratio field, and to obtain exact input counts or upper bounds via SAT-based bipartite uncolorability checks for several islands. Generalization beyond the surveyed fields is left explicitly open.

Significance. If the observed two-mechanism pattern holds more broadly, the work supplies a useful organizing principle for the algebraic origins of KS sets in dimension three and could guide systematic searches for contextuality witnesses. Concrete strengths include the SAT-based verification of Trandafir-Cabello counts for three islands, the upper bounds for three others, and the explicit discovery of new realizations (43-vector Heegner-7 graph and 52-vector golden-ratio graph). The manuscript's clear scoping to tested cases and open question on generality strengthen the reliability of its empirical claims.

major comments (1)

[Abstract] Abstract and the discussion of the golden-ratio island: the central empirical claim is that KS sets appear in raw alphabets only when one of the two cancellation mechanisms is present, yet the golden-ratio case is described as a boundary instance in which cross-product completion introduces effective modulus-2 cancellations. Clarify whether this affects the raw-alphabet condition or merely illustrates how completion can generate the required cancellations after the fact.

minor comments (2)

Table or enumeration sections: the exact input counts versus upper bounds for the six islands should be presented in a single consolidated table with explicit column labels for 'exact' versus 'upper bound'.
Notation: the distinction between raw alphabets and cross-product completions is central; ensure the symbol for the completed set is introduced once and used consistently.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the manuscript's contributions, and the constructive comment on the abstract. We address the point below and will incorporate the requested clarification in the revised version.

read point-by-point responses

Referee: [Abstract] Abstract and the discussion of the golden-ratio island: the central empirical claim is that KS sets appear in raw alphabets only when one of the two cancellation mechanisms is present, yet the golden-ratio case is described as a boundary instance in which cross-product completion introduces effective modulus-2 cancellations. Clarify whether this affects the raw-alphabet condition or merely illustrates how completion can generate the required cancellations after the fact.

Authors: The central empirical claim is restricted to raw alphabets before any cross-product completion: in every tested case, KS uncolorability appears only when the generator x satisfies modulus-2 cancellation (|x|^2 = 2) or phase cancellation in the initial two-symbol alphabet. The golden-ratio island is presented as a boundary case precisely because its raw alphabet satisfies neither mechanism, yet the completed set (after adjoining cross-products) yields effective modulus-2 cancellations that enable KS uncolorability. This does not modify or weaken the raw-alphabet condition; it illustrates how the completion step can introduce the cancellations required for uncolorability. We will revise the abstract and the discussion of the golden-ratio island to state this distinction more explicitly, making clear that the raw-alphabet observation remains intact while the boundary behavior arises only post-completion. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper reports a computational survey of KS uncolorability across specific alphabets in quadratic, cyclotomic, and golden-ratio fields. The central claim is an empirical observation limited to the tested raw alphabets: KS sets appear only when the generator satisfies one of two cancellation mechanisms. The manuscript explicitly scopes the result to the surveyed cases and leaves generalization open, with no derivation, fitted parameters, or self-referential definitions. No load-bearing self-citations or ansatz smuggling occur; results follow directly from exhaustive search and SAT verification on the chosen inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on computational enumeration within selected quadratic, cyclotomic, and golden-ratio fields using standard properties of those fields; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)

standard math Standard algebraic properties of quadratic, cyclotomic, and golden-ratio number fields
Background for defining coordinate alphabets and cross-product completion.

pith-pipeline@v0.9.0 · 5684 in / 1116 out tokens · 51766 ms · 2026-05-15T09:37:08.836968+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

KS sets arise only when x supports one of two cancellation mechanisms: modulus-2 cancellation (|x|^2=2) or phase cancellation (1+ω+ω²=0). ... golden ratio field Q(φ) (52 vectors, revealed only by cross-product completion)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

in three-dimensional Hilbert space

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

SAT + NAUTY: Orderly Generation of Small Kochen-Specker Sets Containing the Smallest State-independent Contextuality Set
cs.LO 2026-04 unverdicted novelty 7.0

A new SAT + NAUTY framework performs the first exhaustive search of KS sets up to 33 rays containing the 25-ray SI-C set, verifying the Schütte 33-ray set as the smallest such 3D example with DRAT proof certificates.

Reference graph

Works this paper leans on

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