On the Failure of the Upper Bound in the Refined BMV Conjecture and a Pinching Correction

Trung Hoa Dinh

arxiv: 2605.17782 · v1 · pith:44LWDDCEnew · submitted 2026-05-18 · 🧮 math.CO · math.FA· math.OA

On the Failure of the Upper Bound in the Refined BMV Conjecture and a Pinching Correction

Trung Hoa Dinh This is my paper

Pith reviewed 2026-05-20 10:01 UTC · model grok-4.3

classification 🧮 math.CO math.FAmath.OA

keywords BMV conjecturerefined BMVpinchingtrace inequalitiespositive semidefinite matricesword averagesspectral decompositionmatrix inequalities

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The pith

The refined BMV conjecture's upper bound fails because off-diagonal parts of B create spectral bridges that let mixed words outperform the clustered trace, but a pinching correction restores a valid lower bound on the averaged trace.

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

The paper shows why the refined Bessis-Moussa-Villani conjecture fails: the proposed upper bound by the clustered word Tr(A^n B^m) does not hold for the average trace over all mixed words with n A's and m B's. Counterexamples already appear for 3 by 3 positive semidefinite matrices at n equals m equals 5. The explanation is that the clustered word is not the true common part; after pinching B to its diagonal part relative to the eigenbasis of A, the off-diagonal remainder allows mixed words to distribute powers of A along closed cycles more efficiently. This mechanism motivates the corrected conjecture that the averaged trace for (A, B) is at least the averaged trace for the pinched pair (A, E_A(B)). The author proves the corrected form for the case of two B letters, establishing the sandwich A_{n,2}(A, E_A(B)) is less than or equal to A_{n,2}(A, B) which is less than or equal to Tr(A^n B^2).

Core claim

The term Tr(A^n B^m) is not the canonical common part of the pair (A, B); it is only one clustered word. After pinching B relative to A, the natural commuting contribution is A_{n,m}(A, E_A(B)). The off-diagonal complement B minus E_A(B) creates spectral bridges, and mixed words can distribute the powers of A along closed cycles more efficiently than the clustered word. This gives a mechanism for finding counterexamples. Motivated by this, the corrected pinching refinement A_{n,m}(A, B) is greater than or equal to A_{n,m}(A, E_A(B)) holds, and for two letters B it yields the sandwich refinement A_{n,2}(A, E_A(B)) less than or equal to A_{n,2}(A, B) less than or equal to Tr(A^n B^2).

What carries the argument

The pinching operator E_A(B), which extracts the diagonal part of B in the eigenbasis of A and isolates the commuting contribution to the averaged word traces.

If this is right

For m equals 2 the averaged trace over mixed words is bounded below by the pinched average and above by the clustered trace.
The pinching refinement supplies a sharper structural decomposition than the original clustered upper bound even in cases where that bound remains valid.
The spectral-bridge mechanism supplies a systematic way to construct further counterexamples to the original refined conjecture.

Where Pith is reading between the lines

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

The same pinching approach may yield corrections for the refined conjecture when the number of B letters exceeds two.
The sandwich decomposition could tighten bounds on traces of noncommuting products in settings such as quantum information where operator averages appear.
Direct computation of the averaged trace for small n and moderate matrix size would test how tight the new lower and upper bounds are.

Load-bearing premise

The off-diagonal complement B minus E_A(B) creates spectral bridges allowing mixed words to distribute powers of A more efficiently than the clustered word.

What would settle it

Positive semidefinite matrices A and B in dimension three or higher where A_{n,2}(A, B) is strictly less than A_{n,2}(A, E_A(B)) for some n would falsify the proved sandwich refinement for m equals 2.

Figures

Figures reproduced from arXiv: 2605.17782 by Trung Hoa Dinh.

read the original abstract

We analyze why the refined Bessis--Moussa--Villani conjecture fails. The refined conjecture proposed that the normalized trace average over all words with prescribed numbers of letters \(A\) and \(B\) should be bounded above by the clustered word \(\Tr(A^nB^m)\). Recent counterexamples of Cha show that this upper bound is false already for \(3\times3\) positive semidefinite matrices when \(n=m=5\). We explain the failure from the viewpoint of commutative common parts. The term \(\Tr(A^nB^m)\) is not the canonical common part of the pair \((A,B)\); it is only one clustered word. After pinching \(B\) relative to \(A\), the natural commuting contribution is \(\A_{n,m}(A,\EA(B))\). The off-diagonal complement \(B-\EA(B)\) creates spectral bridges, and mixed words can distribute the powers of \(A\) along closed cycles more efficiently than the clustered word. This gives a mechanism for finding counterexamples. Motivated by this mechanism, we propose a corrected pinching refinement \[ \A_{n,m}(A,B)\ge \A_{n,m}(A,\EA(B)). \] We prove this corrected conjecture in the case of two letters \(B\), obtaining a sandwich refinement \[ \A_{n,2}(A,\EA(B)) \le \A_{n,2}(A,B) \le \Tr(A^nB^2). \] Thus, even where the old clustered upper bound remains true, the pinching viewpoint gives a sharper structural decomposition.

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

The paper explains the refined BMV failure via pinching and proves a sandwich for the two-B case.

read the letter

The key point from this paper is that it explains the failure of the upper bound in the refined BMV conjecture by looking at commutative common parts and pinching, and it proves a corrected lower bound that gives a sandwich inequality when there are exactly two B's. The new part is the mechanistic account of why mixed words can beat the clustered one: the off-diagonal complement of the pinched B creates spectral bridges that let powers of A cycle more efficiently. They motivate a corrected conjecture A_{n,m}(A,B) ≥ A_{n,m}(A, E_A(B)), and then prove the full sandwich for m=2. That seems like a genuine addition to the literature on this conjecture. What the paper does well is stick to standard trace properties and conditional expectations to get the result without introducing new machinery. The special-case proof looks consistent with the abstract's description, and it provides a sharper structural view even in cases where the original upper bound still holds. The main limitation is that the general m>2 case remains a conjecture. The paper doesn't claim to resolve the full refined conjecture, just to correct the lower side and handle the two-B situation. Since the counterexamples are already known from Cha, the contribution is more explanatory and corrective than revolutionary. This work is aimed at researchers in operator theory and quantum information who follow the BMV conjecture and its refinements. Someone looking for concrete inequalities or ways to understand trace averages would find the sandwich useful. Given that it has a proved result for a nontrivial case and a clear explanation, it deserves to go to peer review rather than a desk reject.

Referee Report

1 major / 2 minor

Summary. The manuscript explains the failure of the refined upper bound in the Bessis-Moussa-Villani conjecture by arguing that Tr(A^n B^m) is not the canonical common part of the pair (A,B); after pinching B relative to A the natural term is A_{n,m}(A, E_A(B)). It attributes counterexamples to spectral bridges created by the off-diagonal complement B - E_A(B), which allow mixed words to distribute powers of A more efficiently. The authors propose the corrected pinching refinement A_{n,m}(A,B) ≥ A_{n,m}(A, E_A(B)) and prove the sandwich A_{n,2}(A, E_A(B)) ≤ A_{n,2}(A,B) ≤ Tr(A^n B^2) for the two-letter case.

Significance. If the central claims hold, the work supplies a concrete mechanism for the known counterexamples to the refined BMV upper bound and introduces a structurally sharper decomposition via pinching. The explicit proof of the corrected lower bound in the m=2 regime provides direct, falsifiable support for the proposed refinement and clarifies the relationship between clustered and pinched contributions even when the original upper bound remains valid.

major comments (1)

The proof of the lower bound A_{n,2}(A, E_A(B)) ≤ A_{n,2}(A,B) is the load-bearing step for the corrected conjecture. The abstract indicates it follows from standard trace and conditional-expectation identities, but the manuscript should explicitly verify that the argument does not tacitly assume commutativity or additional spectral conditions beyond positive-semidefiniteness.

minor comments (2)

The notation A_{n,m}(A,B) and E_A(B) should be defined at first appearance rather than introduced only through the displayed inequalities.
A brief comparison table or numerical example for small n,m illustrating the gap between A_{n,2}(A,B) and both the pinched term and Tr(A^n B^2) would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the work's significance, and the recommendation for minor revision. We address the single major comment below and will incorporate a clarifying addition in the revised manuscript.

read point-by-point responses

Referee: The proof of the lower bound A_{n,2}(A, E_A(B)) ≤ A_{n,2}(A,B) is the load-bearing step for the corrected conjecture. The abstract indicates it follows from standard trace and conditional-expectation identities, but the manuscript should explicitly verify that the argument does not tacitly assume commutativity or additional spectral conditions beyond positive-semidefiniteness.

Authors: We agree that an explicit verification strengthens the exposition. The lower bound is obtained from the defining properties of the conditional expectation E_A (the pinching map onto the algebra generated by A), which is a positive, unital, trace-preserving contraction. These properties, together with the linearity of the trace and the spectral decomposition of A, suffice to establish the inequality for arbitrary positive-semidefinite A and B; the argument never invokes commutativity of A and B. In the revised manuscript we will add a short remark immediately after the proof, enumerating the operator-theoretic ingredients used and confirming that no further spectral or commutativity hypotheses are required. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper explains the failure of the refined BMV upper bound via the pinching mechanism with conditional expectation E_A and proposes the corrected lower bound conjecture A_{n,m}(A,B) ≥ A_{n,m}(A,E_A(B)). It then proves the special case m=2 directly, yielding the sandwich A_{n,2}(A,E_A(B)) ≤ A_{n,2}(A,B) ≤ Tr(A^n B^2) using standard trace identities and properties of conditional expectations. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation; the central claim is an independent mathematical proof that remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of the trace functional and the conditional expectation in finite-dimensional C*-algebras; no free parameters are fitted and no new entities are postulated.

axioms (2)

standard math The trace is cyclic and linear on the algebra of matrices.
Used implicitly in the definition of the averaged trace A_{n,m} and in the comparison with Tr(A^n B^m).
standard math The conditional expectation E_A onto the commutant of A preserves positivity and the trace.
Central to the pinching construction and the proposed lower bound.

pith-pipeline@v0.9.0 · 5832 in / 1562 out tokens · 54090 ms · 2026-05-20T10:01:31.193509+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · 3 internal anchors

[1]

Bessis, P

D. Bessis, P. Moussa, and M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics,Journal of Mathematical Physics 16(1975), 2318–2325

work page 1975
[2]

E. H. Lieb and R. Seiringer, Equivalent forms of the Bessis–Moussa–Villani conjecture,Jour- nal of Statistical Physics115(2004), 185–190

work page 2004
[3]

H. R. Stahl, Proof of the BMV conjecture,Acta Mathematica211(2013), 255–290

work page 2013
[4]

Eremenko, Herbert Stahl’s proof of the BMV conjecture,Sbornik: Mathematics206 (2015), 87–92

A. Eremenko, Herbert Stahl’s proof of the BMV conjecture,Sbornik: Mathematics206 (2015), 87–92

work page 2015
[5]

Available athttps://oqp.iqoqi.oeaw.ac.at/refinement-of-the-bessis-mou ssa-villani-conjecture

IQOQI Vienna, Open Quantum Problem 40: Refinement of the Bessis–Moussa–Villani con- jecture. Available athttps://oqp.iqoqi.oeaw.ac.at/refinement-of-the-bessis-mou ssa-villani-conjecture

work page
[6]

Cha, One-parameter counterexamples to the refined Bessis–Moussa–Villani conjecture, arXiv:2603.19927, 2026

H. Cha, One-parameter counterexamples to the refined Bessis–Moussa–Villani conjecture, arXiv:2603.19927, 2026. Available athttps://arxiv.org/abs/2603.19927

work page internal anchor Pith review arXiv 2026
[7]

Sums of Hermitian Squares as an Approach to the BMV Conjecture

S. Burgdorf, Sums of Hermitian squares as an approach to the BMV conjecture, arXiv:0802.1153

work page internal anchor Pith review Pith/arXiv arXiv
[8]

Sum--of--squares results for polynomials related to the Bessis--Moussa--Villani conjecture

B. Collins, K. J. Dykema, and F. Torres-Ayala, Sum-of-squares results for polynomials related to the Bessis–Moussa–Villani conjecture, arXiv:0905.0420

work page internal anchor Pith review Pith/arXiv arXiv
[9]

P. S. Landweber and E. R. Speer, On D. H¨ agele’s approach to the Bessis–Moussa–Villani conjecture,Linear Algebra and its Applications431(2009), 1317–1324

work page 2009
[10]

C. R. Johnson and C. J. Hillar, Eigenvalues of words in two positive definite letters,SIAM Journal on Matrix Analysis and Applications23(2002), 916–928. 10

work page 2002

[1] [1]

Bessis, P

D. Bessis, P. Moussa, and M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics,Journal of Mathematical Physics 16(1975), 2318–2325

work page 1975

[2] [2]

E. H. Lieb and R. Seiringer, Equivalent forms of the Bessis–Moussa–Villani conjecture,Jour- nal of Statistical Physics115(2004), 185–190

work page 2004

[3] [3]

H. R. Stahl, Proof of the BMV conjecture,Acta Mathematica211(2013), 255–290

work page 2013

[4] [4]

Eremenko, Herbert Stahl’s proof of the BMV conjecture,Sbornik: Mathematics206 (2015), 87–92

A. Eremenko, Herbert Stahl’s proof of the BMV conjecture,Sbornik: Mathematics206 (2015), 87–92

work page 2015

[5] [5]

Available athttps://oqp.iqoqi.oeaw.ac.at/refinement-of-the-bessis-mou ssa-villani-conjecture

IQOQI Vienna, Open Quantum Problem 40: Refinement of the Bessis–Moussa–Villani con- jecture. Available athttps://oqp.iqoqi.oeaw.ac.at/refinement-of-the-bessis-mou ssa-villani-conjecture

work page

[6] [6]

Cha, One-parameter counterexamples to the refined Bessis–Moussa–Villani conjecture, arXiv:2603.19927, 2026

H. Cha, One-parameter counterexamples to the refined Bessis–Moussa–Villani conjecture, arXiv:2603.19927, 2026. Available athttps://arxiv.org/abs/2603.19927

work page internal anchor Pith review arXiv 2026

[7] [7]

Sums of Hermitian Squares as an Approach to the BMV Conjecture

S. Burgdorf, Sums of Hermitian squares as an approach to the BMV conjecture, arXiv:0802.1153

work page internal anchor Pith review Pith/arXiv arXiv

[8] [8]

Sum--of--squares results for polynomials related to the Bessis--Moussa--Villani conjecture

B. Collins, K. J. Dykema, and F. Torres-Ayala, Sum-of-squares results for polynomials related to the Bessis–Moussa–Villani conjecture, arXiv:0905.0420

work page internal anchor Pith review Pith/arXiv arXiv

[9] [9]

P. S. Landweber and E. R. Speer, On D. H¨ agele’s approach to the Bessis–Moussa–Villani conjecture,Linear Algebra and its Applications431(2009), 1317–1324

work page 2009

[10] [10]

C. R. Johnson and C. J. Hillar, Eigenvalues of words in two positive definite letters,SIAM Journal on Matrix Analysis and Applications23(2002), 916–928. 10

work page 2002