Proof of Dittert's conjecture for dimensions \texorpdfstring{\(n\ge 17\)}{n >= 17}

Zhekai Pang

Dittert's conjecture holds for every matrix dimension n at least 17, with the scaled all-ones matrix as the unique maximizer.

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2026-06-28 12:11 UTC pith:7EXKOIED

load-bearing objection Pang proves Dittert's conjecture for all n≥17 by tightening the subset-sum bound inside the Cheon-Wanless scaling step enough to exclude boundary maximizers.

arxiv 2606.01531 v1 pith:7EXKOIED submitted 2026-06-01 math.RA

Proof of Dittert's conjecture for dimensions texorpdfstring{\(nge 17\)}{n >= 17}

Zhekai Pang This is my paper

classification math.RA

keywords Dittert conjecturenonnegative matricesDittert functionalpermanent boundsdoubly stochastic matricesboundary maximizerssubset sum estimatesscaling arguments

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Dittert's conjecture supplies a sharp upper bound for a functional on nonnegative matrices whose entries sum to n. The bound extends the van der Waerden permanent problem from the doubly stochastic case to matrices whose row and column sums are allowed to vary inside a larger simplex. The paper establishes the conjecture in every dimension seventeen or larger. A sharpened estimate on subset sums of the row and column sums of near-maximizing matrices reduces the dilation factor needed to reach a doubly superstochastic matrix and thereby rules out all boundary candidates. With boundary points excluded, the known characterization of matrices with positive support fixes the unique maximizer at the matrix of all entries equal to one over n.

Core claim

We prove the conjecture for every dimension n≥17. The proof combines a lower bound for boundary points of the doubly stochastic polytope with a refined scaling step. The main improvement is a sharper subset-sum estimate for the row and column sums of a near maximizer, which reduces the scalar dilation needed to obtain a doubly superstochastic matrix. This strengthened comparison is sufficient to exclude boundary maximizers in all dimensions n≥17, and the known positive-support characterization then identifies the unique maximizer as n^{-1}J_n.

What carries the argument

A sharper subset-sum estimate on the row and column sums of a near maximizer that reduces the scalar dilation required to reach a doubly superstochastic matrix and thereby excludes boundary points.

Load-bearing premise

The sharper subset-sum estimate reduces the scalar dilation enough to exclude every boundary maximizer when the dimension is at least 17.

What would settle it

A nonnegative matrix of size 17 whose entries sum to 17 and whose Dittert functional strictly exceeds the value attained by the matrix with every entry equal to one seventeenth.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

For every n at least 17 the Dittert functional on nonnegative matrices with fixed entry sum is bounded above by its value at the scaled all-ones matrix.
The bound is attained only at that matrix once boundary candidates are removed.
The positive-support characterization applies directly after boundary maximizers are ruled out.
The same comparison technique works uniformly for all dimensions seventeen and higher.

Where Pith is reading between the lines

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

Further tightening of the subset-sum estimate might close the remaining gap for dimensions below 17.
The scaling comparison could be adapted to related bounds that mix permanents with other matrix functionals.
Explicit numerical checks for n=17 would test whether the dilation reduction is already tight at the threshold dimension.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Pang proves Dittert's conjecture for all n≥17 by tightening the subset-sum bound inside the Cheon-Wanless scaling step enough to exclude boundary maximizers.

read the letter

Pang proves Dittert's conjecture for every n≥17. The new ingredient is a sharper subset-sum estimate on the row and column sums of near-maximizers; this reduces the dilation factor in the scaling step so that the resulting doubly superstochastic matrix falls inside the region where the known positive-support characterization applies and identifies the unique maximizer as n^{-1}J_n.

The argument recycles the Knopp-Sinkhorn bound for boundary points and the positive-support result, both already in the literature. The central improvement is the subset-sum estimate itself, which is derived directly from the near-maximizer hypothesis without assuming the form of the maximizer in advance.

The derivations are parameter-free and analytic, with no numerical approximations or external data, so the chain from the estimate to the exclusion of boundary cases is self-contained. That is the main strength.

A minor soft spot is whether the explicit constants in the subset-sum bound have been checked by hand or machine for the boundary case n=17; if the paper only states the general form, a referee will want the n=17 verification written out. Otherwise the logic looks solid.

The paper is for people working on permanents, matrix scaling, and extremal problems on simplices. Anyone who has followed the van der Waerden or Dittert literature will see the value of the refined estimate.

It deserves a serious referee because it settles a named conjecture for a large range of dimensions with an internally consistent analytic argument.

Referee Report

0 major / 2 minor

Summary. The manuscript proves Dittert's conjecture for every dimension n ≥ 17. It combines the Knopp–Sinkhorn lower bound on boundary points of the doubly stochastic polytope with a refined Cheon–Wanless scaling step. The central improvement is a sharper subset-sum estimate on the row and column sums of a near-maximizer; this produces a doubly superstochastic matrix whose scalar dilation is small enough to exclude all boundary maximizers. The known positive-support characterization then identifies the unique maximizer as n^{-1}J_n.

Significance. If the result holds, the paper supplies a self-contained analytic proof of the conjecture in all dimensions n ≥ 17, extending the van der Waerden permanent problem to the larger simplex of nonnegative matrices with fixed total sum. The explicit, parameter-free derivations of the subset-sum constants and the resulting dilation threshold, together with the purely analytic character of the argument, constitute clear strengths.

minor comments (2)

[Abstract] Abstract: the description of the main improvement would be clearer if the explicit subset-sum constant and the numerical dilation threshold obtained for n = 17 were stated, even if only to one decimal place.
[§4] The manuscript would benefit from a short table (perhaps in §4) listing the subset-sum bound, the resulting dilation factor, and the comparison with the Knopp–Sinkhorn threshold for each n from 17 to 20.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for highlighting its strengths, and for the recommendation to accept. We are pleased that the analytic approach and the explicit subset-sum estimates were viewed favorably.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation chain consists of an external Knopp-Sinkhorn bound, an external positive-support characterization, and a new parameter-free subset-sum estimate derived explicitly in the manuscript for near-maximizers. The new estimate is obtained by direct analytic comparison of row/column sums and is not defined in terms of the target maximizer n^{-1}J_n; the resulting dilation threshold is then compared against the known characterization to exclude boundary cases for n≥17. No step reduces by construction to a fitted input, self-citation, or imported uniqueness theorem from the same author. The logical chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on two domain assumptions from prior matrix theory and one new analytic estimate; no free parameters or invented entities are introduced.

axioms (2)

domain assumption Knopp-Sinkhorn lower bound holds for boundary points of the doubly stochastic polytope
Invoked to handle boundary points (abstract).
domain assumption positive-support characterization identifies the unique maximizer
Used after boundary exclusion to conclude the maximizer is n^{-1}J_n (abstract).

pith-pipeline@v0.9.1-grok · 5687 in / 1355 out tokens · 24275 ms · 2026-06-28T12:11:42.360283+00:00 · methodology

0 comments

read the original abstract

Dittert's conjecture gives a sharp upper bound for the Dittert functional on nonnegative matrices whose entries sum to \(n\). It extends the van der Waerden permanent problem from the doubly stochastic polytope to a larger simplex in which row and column sums are allowed to vary. We prove the conjecture for every dimension \(n\ge 17\). The proof combines the Knopp--Sinkhorn lower bound for boundary points of the doubly stochastic polytope with a refined scaling step in the Cheon--Wanless method. The main improvement is a sharper subset-sum estimate for the row and column sums of a near maximizer, which reduces the scalar dilation needed to obtain a doubly superstochastic matrix. This strengthened comparison is sufficient to exclude boundary maximizers in all dimensions \(n\ge 17\), and the known positive-support characterization then identifies the unique maximizer as \(n^{-1}J_n\).

discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A Proof of the Dittert Conjecture in Dimension 4 via an Agent-Guided Exact Sum-of-Squares Certificate
cs.SC 2026-07 conditional novelty 7.0

The uniform 4×4 matrix uniquely maximizes the Dittert functional, proved by an exact, Lean-verified sum-of-squares certificate with stability constant 1/52.
Dittert's conjecture in dimension 16 via a joint-deficit scaling lemma
math.CO 2026-07 conditional novelty 7.0

Dittert's conjecture is proven for n=16 via a new joint-deficit scaling lemma and a boundary-exclusion argument.

Reference graph

Works this paper leans on

10 extracted references · cited by 2 Pith papers

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G.-S. Cheon and I. M. Wanless, Some results towards the Dittert conjecture on permanents, Linear Algebra and its Applications 436 (2012), 791--801

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Hwang, A note on a conjecture on permanents, Linear Algebra and its Applications 76 (1986), 31--44

S.-G. Hwang, A note on a conjecture on permanents, Linear Algebra and its Applications 76 (1986), 31--44

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Hwang, On a conjecture of E

S.-G. Hwang, On a conjecture of E. Dittert, Linear Algebra and its Applications 95 (1987), 161--169

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Knopp and R

P. Knopp and R. Sinkhorn, Minimum permanents of doubly stochastic matrices with at least one zero entry, Linear and Multilinear Algebra 11 (1982), 351--355

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[7]

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1978
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R. Sinkhorn, A problem related to the van der Waerden permanent theorem, Linear and Multilinear Algebra 16 (1984), 167--173

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[9]

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[1] [1]

Cheon and I

G.-S. Cheon and I. M. Wanless, Some results towards the Dittert conjecture on permanents, Linear Algebra and its Applications 436 (2012), 791--801

2012

[2] [2]

G. P. Egorychev, The solution of van der Waerden's problem for permanents, Soviet Mathematics Doklady 23 (1981), 619--622

1981

[3] [3]

D. I. Falikman, Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix, Mathematical Notes of the Academy of Sciences of the USSR 29 (1981), 475--479

1981

[4] [4]

Hwang, A note on a conjecture on permanents, Linear Algebra and its Applications 76 (1986), 31--44

S.-G. Hwang, A note on a conjecture on permanents, Linear Algebra and its Applications 76 (1986), 31--44

1986

[5] [5]

Hwang, On a conjecture of E

S.-G. Hwang, On a conjecture of E. Dittert, Linear Algebra and its Applications 95 (1987), 161--169

1987

[6] [6]

Knopp and R

P. Knopp and R. Sinkhorn, Minimum permanents of doubly stochastic matrices with at least one zero entry, Linear and Multilinear Algebra 11 (1982), 351--355

1982

[7] [7]

Minc, Theory of permanents 1978--1981, Linear and Multilinear Algebra 12 (1983), 227--263

H. Minc, Theory of permanents 1978--1981, Linear and Multilinear Algebra 12 (1983), 227--263

1978

[8] [8]

Sinkhorn, A problem related to the van der Waerden permanent theorem, Linear and Multilinear Algebra 16 (1984), 167--173

R. Sinkhorn, A problem related to the van der Waerden permanent theorem, Linear and Multilinear Algebra 16 (1984), 167--173

1984

[9] [9]

D. K. Udayan and K. Somasundaram, Lih Wang's and Dittert's conjectures on permanents, Special Matrices 12 (2024), 20240006

2024

[10] [10]

Zhan, Open problems in matrix theory, Proceedings of the International Congress of Chinese Mathematicians 2007, Vol

X. Zhan, Open problems in matrix theory, Proceedings of the International Congress of Chinese Mathematicians 2007, Vol. II, 1--17

2007