Optimal Bounds for Pairs of Commuting Integer Matrices

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

$M_n(\mathbb Z)$ is the set of $n\times n$ integer matrices, and $XY=YX$ means the two products agree. The notation $\ll_\varepsilon$ bounds the left side by a constant times the right side, with allowed dependence stated by the subscript.

Statement recorded in the supplied source.

For $X\in M_n(\mathbb{Z})$, let $|X|$ be the maximum absolute value of its entries, and define $N(T)=\#\{(X,Y)\in M_n(\mathbb{Z})^2:|X|\leq T,\ |Y|\leq T,\ XY=YX\}$. Prove that, for every $\varepsilon>0$, $N(T)\ll_\varepsilon T^{n^2+1+\varepsilon}$ for arbitrary $n$. [S2]

Scope qualification.

The source fixes or varies $n$ according to its counting convention; dependence of the implicit constant on $n$ is not explicit in the supplied statement.

Source status.

The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Prove the conjectured count of bounded-entry commuting integer-matrix pairs in every dimension.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006470 (OWR-14299578-002). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Arithmetic Statistics for Algebraic Objects $2026$, Oberwolfach Reports, DOI 10.4171/owr/2025/52, pp. 2765--2842 https://doi.org/10.4171/owr/2025/52. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.

Preparation and review.

Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30006470 and source code OWR-14299578-002 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Literature and context check

The primary workshop report, printed pp. 2780--2781, states the general counting conjecture and explicitly says it is known for $n\leq3$. Thus the following $n=2$ proof reconstructs known special-case mathematics. The report's finite-field dimension comparison does not itself give an integer height bound. No recent-status claim beyond this consulted report is made.

Research attempt: Primitive-direction counting in dimension two

Put $B=\lfloor T\rfloor\geq1$. For every $n$, taking $X=aI_n$ and arbitrary $Y$ gives exactly $(2B+1)^{n^2+1}$ pairs in the box. Hence the conjectured exponent cannot be reduced.

For $n=2$, write \[ X=\begin{pmatrix}a&b\\c&d\end{pmatrix},\qquad Y=\begin{pmatrix}e&f\\g&h\end{pmatrix},\qquad v=(a-d,b,c),\quad w=(e-h,f,g). \] Direct multiplication shows that $XY=YX$ if and only if \[ bg-cf=0,\quad (a-d)f-b(e-h)=0,\quad c(e-h)-(a-d)g=0. \] These are exactly the three two-coordinate minors of $v,w$, up to sign. If $v=0$, $X$ is scalar and this contributes $(2B+1)^5$ pairs.

If $v\ne0$, choose the unique primitive integer vector $u$ on its rational line whose first nonzero coordinate is positive. Then $v=ku$, $k\in\mathbb Z\setminus\{0\}$. Vanishing minors give $w=\ell u$ with rational $\ell$. Primitivity supplies integers $z_i$ with $\sum z_i u_i=1$, so $\ell=\sum z_iw_i$ is an integer. Let $H=\|u\|_\infty$. Every coordinate of $v,w$ has absolute value at most $2B$, so \[ 1\leq H\leq2B,\qquad 0<|k|\leq2B/H,\qquad |\ell|\leq2B/H. \] The number of possible $u$ of height $H$ is at most the number of all integer vectors on that cube shell, \[ (2H+1)^3-(2H-1)^3=24H^2+2. \] There are at most $(4B/H)(4B/H+1)\leq24B^2/H^2$ pairs $(k,\ell)$, since $H\leq2B$. Once $v,w$ are fixed, choosing $a,e$ determines all other entries, so there are at most $(2B+1)^2$ matrices realizing them. Ignoring incompatible entry bounds only increases this upper count. Consequently \[ N_2(T)\leq (2B+1)^5+(2B+1)^2 \sum_{H=1}^{2B}(24H^2+2)\frac{24B^2}{H^2}=O(B^5). \] Together with the scalar lower bound, this proves $N_2(T)=\Theta(T^5)$ for $T\geq1$.

Where the attempted extension stops.

The essential reduction above was that the nonscalar centralizer modulo scalar matrices has just one rational direction. For a diagonal matrix with $n$ distinct diagonal entries, commutation forces an arbitrary diagonal matrix, whose quotient by scalars has dimension $n-1$. For $n\geq3$ that quotient has more than one direction. A naive replacement by $n-1$ independent height-bounded parameters does not retain the shell cancellation just proved; rational lattice covolumes and degenerations must be controlled. This precise missing estimate is not supplied by dimension counting.

Verification and adversarial audit

All three commutator equations were expanded, including the case $w=0$ and the distinction between rational proportionality and integer coefficients. The scalar stratum is counted separately so the primitive vector is never applied to zero. Bounds allow overcounting and hence require no converse realization. $n=1$ is immediate: all pairs commute and the count is $(2B+1)^2$. No claim is made that the failed single-direction argument rules out a different proof for larger $n$.

Audit corrections.

Counted zero and nonzero first vectors separately; justified integrality by primitivity rather than assuming rational proportionality has integer ratio.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The arbitrary-$n$ upper bound remains unproved. The special case reconstructed here is explicitly already known in the consulted source and is not new original-target progress.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://ems.press/content/serial-article-files/52436 (consulted 2026-10-11).

Model and contributor disclosure

Codex GPT-6 completed written-proof review of this attempt at its declared scope; this is model-assisted review, not independent human proof checking.

Material model collaboration.

Root Codex GPT-6 supplied the primitive-direction counting route, scalar lower bound and higher-dimensional centralizer obstruction for the known two-dimensional special case. The gpt-6-astra author inspected and reworked these arguments; this is model-assisted mathematical work, not independent human verification. OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration and repository-format review; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.