A mathematical preprint reports counterexamples to a conjecture that would place a universal inverse-square limit on a graph quantity used in Pauli shadow tomography. In its main constructed family, the fractional chromatic number—written χ_f—scales at least as Ω(ε_m^-2.07598). That growth is faster than the conjectured ε^-2 scale: multiplying by ε² does not keep the quantity below a single constant for every state and error covered by the conjecture. Theorem A presents this constructed family as a violation of the conjecture.
The bound under challenge
The conjecture is expressed as χ_f(B_ε(ϱ)) · ε² ≤ C for every state ϱ and error ε. For a general reader, the key point is the quantifier: a finite C would have to work across all the states and errors in the claim. χ_f is the fractional chromatic number, a graph-based score in the paper’s test; in this story, it is simply the quantity the conjecture tries to keep under control. The question is whether that expression can remain bounded, not whether a particular example behaves well.
How the construction works
This is a modeling result, not a trial. No empirical participant sample is reported. The authors instead construct families of theoretical quantum states, Pauli strings and anticommutation graphs, then examine the graph invariants that enter the bound. Their central mathematical operation is an m-fold lexicographic graph product, a repeated graph-building operation. Under that product, fractional chromatic numbers and expectation magnitudes multiply as χ_f(G)^m and a^m.
The anti-heptagon example
The anti-heptagon construction supplies the clearest concrete route. Its seed consists of seven Pauli strings realizing the anti-heptagon C7 as an anticommutation graph. The associated seed state is a Hamiltonian ground state, and the seed observables have equal expectation values in that state. In the amplified family, the proof reports a lower-bound factor of approximately (1.046918)^m, unbounded as m tends to infinity.
The same construction is summarized in the paper’s main stated scaling result. Theorem A says the family has χ_f(B_εm) scaling as Ω(ε_m^-2.07598), violating the conjectured bound. The exponent 2.07598 is above the conjectured power of 2. This is an asymptotic claim about the behavior of the constructed family as amplification grows.
A broader graph condition
The paper then turns to a more general graph condition. Its β-number argument uses a threshold lemma to select observables with sufficiently high squared expectation values. For a graph on N > 0 vertices, the construction supplies states σ_m and thresholds ε_m > 0. The stated lower bound has a prefactor of 0.99, a denominator of 1 + m log N, and a graph term equal to [β(G)/α(G)]^m.
Theorem B states that any graph satisfying α(G) < β(G) is a counterexample to the conjecture. The anti-heptagon C7 is given as an example satisfying that condition. The authors’ own qualification is that failure of the conjecture does not rule out a triply efficient shadow-tomography protocol for arbitrary Pauli sets.
What remains unresolved
The finding does not amount to a verdict on quantum tomography as a whole. The authors say that failure of the conjecture does not rule out a triply efficient shadow-tomography protocol for arbitrary Pauli sets. Nor does the paper show that no polynomial bound of the form O(ε^-κ) can exist. The open question is whether some positive κ gives a universal fractional-colouring bound of that form.
In other words, the preprint rejects the proposed exponent-2 guarantee for its constructed families, while leaving weaker polynomial guarantees open. The authors’ interpretation is that triply efficient Pauli shadow-tomography protocols may still exist under those weaker bounds. The evidence remains limited to the constructed mathematical families and their analytic graph and quantum-state properties.
The status of the evidence
The manuscript is an arXiv version 1 preprint dated 20 August 2026. It reports no empirical participant sample; the central objects are constructed quantum states and Pauli observables. The headline exponents therefore describe mathematical scaling in those families.
The front matter reports that GPT Sol 5.6 was used to derive Theorems A and B, with the authors saying they verified and contextualized the results. The authors also report funding from the National Science Centre, Poland, and the Polish National Agency for Academic Exchange through the named grants.
Paper data and sources
Original title: Counterexamples to the fractional coloring conjecture for triply efficient shadow tomography
Authors: Jędrzej Stempin, Santiago Llorens, Felix Huber
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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