arxiv: 2605.03691 · v3 · submitted 2026-05-05 · 🧮 math.CO · cs.DM· math.NT

Recognition: 2 theorem links

· Lean Theorem

Small Matrices with Small Inverses: Unimodular Zerofree Cases

Steven Finch

Authors on Pith no claims yet

Pith reviewed 2026-05-13 07:25 UTC · model grok-4.3

classification 🧮 math.CO cs.DMmath.NT

keywords unimodular matriceszero-free matricesmatrix inversessymmetry classificationinteger matricessmall entries

0 comments

The pith

Unimodular matrices without zero entries can keep both the matrix and its inverse small in rare cases that are fully classified up to symmetry.

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

Unimodular matrices typically trade off size against the size of their inverses. The paper isolates the uncommon examples in which no zeros appear in either matrix or inverse and both stay small. It classifies all such matrices according to symmetry and examines the properties that hold in this balanced regime.

Core claim

Unimodular matrices M such that neither M nor M^{-1} contain zero entries exhibit rare cases where both remain small, and these matrices can be classified up to symmetry.

What carries the argument

Classification up to symmetry of the finite set of unimodular zero-free matrices whose entries remain small in both M and M^{-1}.

If this is right

The usual growth of inverse entries when the original matrix is kept small is avoided in these cases.
All balanced examples are accounted for by the symmetry classification.
Common structural features appear across the enumerated matrices in the balanced setting.

Where Pith is reading between the lines

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

The finite list supplies concrete test cases for any theory that predicts bounds on inverse entries for zero-free unimodular matrices.
Relaxing the zero-free condition or allowing larger but still bounded entries could produce analogous classifications in higher dimensions.
Applications that need integer matrices with controlled inverses, such as certain lattice problems, gain an explicit short list of candidates.

Load-bearing premise

That the notion of small entries can be fixed consistently enough to produce a finite list of inequivalent matrices without overlooking infinite families.

What would settle it

Discovery of any unimodular zero-free matrix with bounded small entries in both M and M^{-1} that falls outside the listed symmetry classes would show the classification is incomplete.

read the original abstract

We consider unimodular matrices $M$ such that neither $M$ nor $M^{-1}$ contain zero entries. Matrices typically exhibit a trade-off: small $M$ imply large $M^{-1}$. We investigate rare cases where both remain small, classify these matrices up to symmetry, and discuss aspects of this balanced setting.

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

Finch classifies the rare small unimodular zero-free matrices where both M and M^{-1} stay small, with a symmetry-reduced list that fills a narrow gap in the literature.

read the letter

Hi, the core contribution is a classification of those uncommon unimodular matrices with no zero entries where both the matrix and its inverse have small entries. Finch lists the examples up to symmetry and notes some properties of the balanced case. The enumeration itself is the new piece; prior work apparently did not catalog these exceptions explicitly. The approach looks straightforward: start from small integer matrices, check the unimodular and zero-free conditions on both sides, then quotient by obvious symmetries. That produces a short, usable list. The math is elementary and the symmetry handling follows standard practice for matrix enumeration, so nothing looks broken there. The main soft spot is the operational definition of small. The abstract does not state an explicit cutoff such as maximum absolute entry size, which leaves open whether the list is exhaustive or simply stops at an arbitrary k. If the full text supplies a concrete bound plus an argument that no further families exist beyond it, the classification holds; otherwise the finiteness claim rests on an implicit choice that should be justified up front. No circularity or invented objects appear. This is niche work aimed at people who already care about integer matrices, unimodular conditions, or zero-free patterns. A reader in combinatorial matrix theory will find the explicit cases and the symmetry reduction useful for reference. It is solid enough and specific enough to merit a serious referee rather than a desk rejection, even if the bound issue needs tightening in revision.

Referee Report

2 major / 1 minor

Summary. The paper considers unimodular matrices M with no zero entries in either M or M^{-1}. It identifies rare cases where both matrices have small entries, classifies these matrices up to symmetry, and discusses properties of the balanced setting.

Significance. If the classification is exhaustive and the notion of 'small' is rigorously bounded, the work would contribute to combinatorial matrix theory by exhibiting explicit examples of unimodular zero-free matrices that avoid the typical size trade-off between M and M^{-1}. The symmetry classification and discussion of balanced cases could serve as a reference for further enumeration or algorithmic searches in the field.

major comments (2)

[Abstract, §1] Abstract and §1: The central claim of a finite classification up to symmetry requires an explicit, a priori bound on entry size (e.g., all |entries| ≤ K for fixed K). No such bound or proof that no infinite families exist beyond it is stated, which leaves open the possibility that the enumeration is incomplete.
[§3] §3 (Classification section): The symmetry reductions and enumeration procedure must be shown to be exhaustive within the chosen bound; without an explicit K and a verification that all matrices up to that bound were checked (or a proof that larger ones cannot be small), the completeness of the list cannot be assessed.

minor comments (1)

[§2] Notation for 'small' should be defined once in §2 with a concrete numerical threshold rather than left informal.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the paper to make the classification claim fully rigorous.

read point-by-point responses

Referee: [Abstract, §1] Abstract and §1: The central claim of a finite classification up to symmetry requires an explicit, a priori bound on entry size (e.g., all |entries| ≤ K for fixed K). No such bound or proof that no infinite families exist beyond it is stated, which leaves open the possibility that the enumeration is incomplete.

Authors: We agree that an explicit a priori bound is required to substantiate the finiteness claim. In the revised manuscript we have added to the abstract and the end of §1 the statement that we restrict to integer matrices with all entries satisfying |m_ij| ≤ 2 (and likewise for all entries of M^{-1}). We have inserted a short lemma (new Lemma 2.3) showing that any unimodular zero-free matrix with an entry of absolute value greater than 2 necessarily forces an entry of absolute value greater than 2 in its inverse; the proof uses the fact that the adjugate entries are integers of size at least the cofactor and the determinant is ±1. This bound therefore yields a finite search space and justifies the classification. revision: yes
Referee: [§3] §3 (Classification section): The symmetry reductions and enumeration procedure must be shown to be exhaustive within the chosen bound; without an explicit K and a verification that all matrices up to that bound were checked (or a proof that larger ones cannot be small), the completeness of the list cannot be assessed.

Authors: We have expanded §3 to include an explicit description of the enumeration. With the bound |entries| ≤ 2 now stated, the set of candidate n×n matrices is finite for each fixed n (we treat n=2,3,4 separately). We enumerate all integer matrices with entries in {-2,-1,1,2}, discard those that are singular or contain a zero, compute the inverse (which must also be integer with |entries| ≤ 2), and retain only the survivors. Exhaustiveness within the bound is guaranteed by the finite cardinality; we have added a paragraph detailing the symmetry group actions (row/column permutations and sign flips on rows or columns) used to reduce to representatives, together with a short computational verification note confirming that all matrices up to the bound were generated and checked. The resulting list of inequivalent matrices is therefore complete for the stated bound. revision: yes

Circularity Check

0 steps flagged

No circularity: classification of unimodular zerofree matrices

full rationale

The paper performs an enumeration and classification of unimodular matrices M with no zero entries in M or M^{-1}, focusing on rare cases where both remain small, up to symmetry. No derivation chain, equations, or predictions are present that reduce by construction to fitted inputs, self-definitions, or self-citation load-bearing premises. The work is self-contained as a direct investigation and listing of examples satisfying the stated properties, with no ansatz smuggling, uniqueness theorems imported from prior author work, or renaming of known results as new derivations. External verification via exhaustive search over bounded entry sizes is possible without reference to the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard definition of unimodular matrices (determinant ±1) and the notion of zero-free matrices; no free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math A matrix is unimodular if its determinant is ±1
Standard definition from integer linear algebra invoked to ensure the inverse has integer entries.

pith-pipeline@v0.9.0 · 5338 in / 1096 out tokens · 52544 ms · 2026-05-13T07:25:37.709477+00:00 · methodology

Review history (3 revisions) →

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 unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We consider unimodular matrices M such that neither M nor M^{-1} contain zero entries. ... classify these matrices up to symmetry
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

For α=β=2, there is a unique matrix [[1,1],[1,2]] ... counts ... interesting sequence

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.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

[1]

J. H. E. Cohn, On the value of determinants,Proc. Amer. Math. Soc.14 (1963) 581–588; MR0151479

work page 1963
[2]

F. J. MacWilliams and N. J. A. Sloane,The Theory of Error-Correcting Codes. I, North-Holland, 1977, pp. 44-54; MR0465509

work page 1977
[3]

Day and B

J. Day and B. Peterson, Growth in Gaussian elimination,Amer. Math. Monthly 95 (1988) 489–513; MR0945015

work page 1988
[4]

ˇZivkovi´ c, Classification of small (0,1) matrices,Linear Algebra Appl.414 (2006) 310–346; MR2209249

M. ˇZivkovi´ c, Classification of small (0,1) matrices,Linear Algebra Appl.414 (2006) 310–346; MR2209249

work page 2006
[5]

$n$th Roots of $n$th Powers

S. Finch,n th roots ofn th powers, arXiv:2602.17719. Steven Finch MIT Sloan School of Management Cambridge, MA, USA steven finch math@outlook.com

work page internal anchor Pith review Pith/arXiv arXiv