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arxiv: 2604.22344 · v1 · submitted 2026-04-24 · 🧮 math.NT

Matrices with cyclically monotone rows and Cantor numeration systems

Pavel \v{S}\v{t}ov\'i\v{c}ek , Edita Pelantov\'a This is my paper

Pith reviewed 2026-05-08 10:06 UTC · model grok-4.3

classification 🧮 math.NT
keywords cyclically monotone rowsCantor numeration systemsParry conditiondiagonal dominanceregular matricesalternate basesnon-standard number systems
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The pith

Square matrices with cyclically monotone rows are regular when each diagonal entry strictly exceeds all other entries in its row.

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

The paper proves that square matrices with non-negative entries are regular provided each row forms a cyclically non-increasing sequence that begins with its maximum value on the diagonal and the diagonal value is strictly larger than every other entry in that row. This regularity is then applied to establish a one-to-one correspondence between alternate Cantor real bases of period p and lists of p sequences of non-negative integers that satisfy the Parry condition. The correspondence resolves an open problem in the theory of non-standard numeration systems. A reader would care because the result links a concrete linear-algebra condition to the structure of real bases in generalized number representations.

Core claim

We study a class of square matrices with non-negative elements which have cyclically monotone rows in the sense that each row consists of a cyclically non-increasing sequence of numbers starting from a maximal element on the diagonal. We prove that if every diagonal element is strictly larger than all other elements in the respective row, then the matrix is regular. This property enables us to solve an open problem that comes from the theory of non-standard numeration systems, also called Cantor numeration systems, concerning a one-to-one relationship between Cantor real bases that are alternate (periodic with period p) and lists of p sequences of non-negative integers satisfying the Parry 3

What carries the argument

Cyclically monotone rows of non-negative reals equipped with strict diagonal dominance, which together force the matrix to be regular.

If this is right

  • The matrix has full rank and is invertible over the reals.
  • Alternate Cantor real bases stand in bijection with Parry sequences of the required form.
  • The open problem on the one-to-one relationship in Cantor numeration systems is settled.
  • The regularity condition supplies a criterion for valid bases in these generalized number systems.

Where Pith is reading between the lines

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

  • The same dominance condition might serve as a test for regularity in other families of matrices whose entries obey cyclic orderings.
  • The bijection could be used to generate new families of Cantor bases by starting from known Parry sequences.
  • Extensions of the argument to block matrices or to matrices over other ordered rings would be natural to check.

Load-bearing premise

The rows must be cyclically monotone sequences of non-negative reals and the diagonal entries must be strictly larger than the remaining entries in each row.

What would settle it

A concrete matrix whose rows are cyclically non-increasing and non-negative, whose diagonal entries strictly dominate their rows, yet whose determinant is zero.

read the original abstract

We study a class of square matrices with non-negative elements which have cyclically monotone rows in the sense that each row of a matrix from the class consists of a cyclically non-increasing sequence of numbers starting from a maximal element on the diagonal. We prove that if every diagonal element is strictly larger than all other elements in the respective row, then the matrix is regular. This property enables us to solve an open problem that comes from the theory of non-standard numeration systems, also called Cantor numeration systems. The problem concerns a one-to-one relationship between Cantor real bases, which are supposed to be alternate, that is, periodic with a period p, and lists of p sequences of non-negative integers satisfying the so-called Parry condition.

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.

Referee Report

0 major / 3 minor

Summary. The paper studies square matrices with non-negative entries whose rows are cyclically monotone (each row becomes non-increasing after a cyclic shift placing the diagonal entry first). It proves that strict diagonal dominance—each diagonal entry strictly exceeds all other entries in its row—implies the matrix is regular (nonsingular). This matrix result is applied to establish a bijective correspondence between periodic alternate Cantor real bases (with period p) and lists of p sequences of non-negative integers satisfying the Parry condition, thereby solving an open problem in Cantor numeration systems.

Significance. If the central theorem holds, the work supplies a direct, parameter-free criterion for regularity of cyclically monotone matrices and resolves a concrete open question linking linear algebra to non-standard numeration systems. The explicit construction of the correspondence via the matrix property, without fitted parameters or self-referential reductions, strengthens the contribution for both matrix theory and the theory of Parry sequences.

minor comments (3)
  1. §1 (Introduction): the term 'regular' is used in the abstract and early paragraphs before its definition as nonsingular; insert an explicit definition immediately after the class of matrices is introduced.
  2. §4 (Application to Cantor systems): the exact statement of the Parry condition is referenced but not restated; include a one-sentence recall of the condition for self-contained reading.
  3. Throughout: an illustrative 3×3 example matrix satisfying the cyclic monotonicity and strict dominance hypotheses would clarify the definitions and the eigenvalue argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation for minor revision. The referee's evaluation correctly identifies the central result on strict diagonal dominance implying regularity for cyclically monotone matrices and its application to resolving the open bijection problem for periodic alternate Cantor bases and Parry sequences.

Circularity Check

0 steps flagged

No significant circularity; direct proof from definitions to regularity

full rationale

The paper defines matrices with cyclically monotone rows (non-increasing after cyclic shift to place the diagonal entry first, non-negative entries) and proves regularity under the strict diagonal dominance condition via direct arguments showing no zero eigenvalues. This is self-contained against the matrix properties without reducing to fitted parameters, self-referential definitions, or load-bearing self-citations. The numeration-systems application (bijective correspondence for alternate Cantor bases and Parry sequences) follows as a consequence of the matrix result rather than presupposing it. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard facts from linear algebra and ordered sequences; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Basic properties of real numbers, non-negative sequences, and matrix rank
    Invoked implicitly to conclude regularity from the cyclic monotonicity and strict diagonal inequality.

pith-pipeline@v0.9.0 · 5429 in / 1191 out tokens · 28673 ms · 2026-05-08T10:06:10.946045+00:00 · methodology

discussion (0)

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Reference graph

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