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Dittert's conjecture holds in dimension 16: over K_16, Φ is uniquely maximized by the uniform matrix U_16.

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2026-08-01 14:18 UTC pith:CUO3HSGD

load-bearing objection A real proof of the n=16 Dittert case with a clever joint-deficit lemma; the inequality part is solid, but the equality case leans entirely on an unverified quote of Hwang's theorem.

arxiv 2607.19439 v1 pith:CUO3HSGD submitted 2026-07-21 math.CO math.RA

Dittert's conjecture in dimension 16 via a joint-deficit scaling lemma

Boris Kafidov This is my paper
classification math.CO math.RA MSC 15A1515B51
keywords Dittert conjecturepermanentdoubly stochastic matrixdoubly superstochastic matrixjoint-deficit scaling lemmaAM-GM inequalityboundary maximizersuniform matrix
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.

This paper proves Dittert's conjecture for 16×16 nonnegative matrices whose entries sum to 16: the functional Φ(A) = (product of row sums) + (product of column sums) − permanent(A) is at most 2 − 16!/16^16, with equality only at the uniform matrix U_16 whose entries are all 1/16. The proof's core is a 'joint-deficit scaling lemma': for a near-maximizer, the shortfall of the row-product below 1 and the shortfall of the column-product below 1 add to at most the permanent deficit, rather than being two independent quantities. That single shared budget yields a sharper scalar dilation, mapping the matrix to a doubly superstochastic matrix. A known lower bound on permanents of doubly stochastic matrices with a zero entry then rules out boundary maximizers, and a known positive-support theorem identifies the unique maximizer. A reader should care because it closes the n = 16 endpoint, leaving only 4 ≤ n ≤ 15 open.

Core claim

The central claim is Theorem 1.1: for every A in K_16, Φ(A) ≤ 2 − 16!/16^16, with equality if and only if A = U_16. The proof shows that the row-product deficit ρ and column-product deficit σ of any near-maximizer satisfy ρ + σ ≤ δ, where δ = γ − per(A) is the permanent deficit. Setting t = sqrt(16δ/(1−δ)), the lemma proves that (1−t)^{-1}A is doubly superstochastic. If a maximizer had a zero entry, this scaled matrix would also have a zero entry and, by the known boundary lower bound for permanents, its permanent could not fall below a positive constant; the resulting inequality contradicts an exact rational comparison. Hence any maximizer is positive, and the quoted positive-support theore

What carries the argument

The joint-deficit scaling lemma (Lemma 3.1). For a near-maximizer with permanent deficit δ, it sets t = sqrt(nδ/(1−δ)) and proves that (1−t)^{-1}A is doubly superstochastic. The decisive step is inequality (3.1): the row-product deficit ρ and column-product deficit σ sum to at most δ, because Φ(A) ≥ 2−γ and per(A) = γ−δ. This joint constraint, combined with a two-point entropy bound and a subset-sum estimate, yields the dilation; earlier arguments treated ρ and σ separately and obtained a weaker dilation insufficient at n = 16.

Load-bearing premise

The uniqueness conclusion depends on the quoted theorem that every entrywise-positive global maximizer of Φ on K_n is U_n; the paper does not prove or reproduce that theorem, so if that theorem is wrong or inapplicable to n = 16, the final identification fails.

What would settle it

Find a 16×16 nonnegative matrix with a zero entry and entries summing to 16 for which Φ(A) > 2 − 16!/16^16, which the theorem asserts does not exist; more locally, verify the proof's key numeric inequality m(1−t)^16 > γ−δ for every δ in (0,γ] with t = sqrt(16δ/(1−δ)), using exact bounds on γ and m — a single δ where the opposite holds would break the boundary-exclusion step.

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

If this is right

  • Dittert's conjecture holds for every n ≥ 16: this paper supplies n = 16, and a separately cited argument covers all n ≥ 17.
  • The unique maximizer in K_16 is U_16, so the inequality is tight and every other matrix in K_16 has a strictly smaller value of Φ.
  • A global maximizer of Φ on K_16 cannot have a zero entry; the proof's contradiction via the scaled permanent bound excludes all such boundary candidates.
  • The joint-deficit relation ρ + σ ≤ δ is the specific source of the improvement over earlier dilation estimates, which failed at n = 16.
  • The numerical thresholds in the proof are exact rational comparisons, making the boundary-exclusion step checkable by direct integer arithmetic.

Where Pith is reading between the lines

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

  • The joint-deficit scaling lemma is not obviously tied to n = 16; if similar dimension-specific constants were used, the same approach might close some of the remaining 4 ≤ n ≤ 15 cases.
  • A targeted computational search for zero-entry matrices in K_16 maximizing Φ would, if the theorem is correct, return values strictly below 2 − 16!/16^16, providing an independent sanity check of the boundary-exclusion step.
  • The 'one shared deficit budget' idea may transfer to other extremal problems where a sum of AM-GM product terms is coupled with a third quantity; treating the product deficits as sharing a single budget could yield sharper dilations elsewhere.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper proves Dittert's conjecture in dimension 16: for every A in K_16, Φ(A) ≤ 2 − 16!/16^16, with equality only at U_16. The main new ingredient is Lemma 3.1, a joint-deficit scaling lemma. For a near-maximizer with permanent deficit δ, the deficits ρ and σ of the row- and column-product terms satisfy ρ+σ ≤ δ, rather than two independent bounds. This yields the sharper dilation t = √(nδ/(1−δ)) and shows that (1−t)^{-1}A is doubly superstochastic. The proof then combines the Knopp–Sinkhorn boundary estimate with explicit rational bounds to exclude zero-entry maximizers; the positive case is dispatched via a cited theorem of Hwang. Together with Pang's result for n ≥ 17, the paper concludes Dittert's conjecture for every n ≥ 16.

Significance. If correct, the paper closes the previously open endpoint n = 16 and, with Pang's preprint, completes the conjecture for all n ≥ 16. The joint-deficit scaling lemma is an elegant new idea that may be useful beyond this case. The proof is largely self-contained: the Pinsker-type inequality is proved, the inclusion–exclusion step is explicit, and the boundary contradiction is numerically transparent. The main external dependency is the cited theorem of Hwang for positive maximizers; assuming that reference supports the stated proposition, the chain of reasoning is sound. The paper is concise and clearly structured.

minor comments (3)
  1. [§2, Proposition 2.1(ii)] The equality case of Theorem 1.1 rests on this proposition. Please state precisely which theorem in [4] is being invoked, ideally with a theorem number or a quotation. As written, the reader cannot tell whether [4] proves the assertion for all n or only under additional hypotheses; since the uniqueness claim A = U_16 in §4 depends entirely on this cited result, the reference should be unambiguous.
  2. [§4, Eq. (4.4)] The rational bounds for γ and m are asserted 'by exact rational comparison' but no derivation is given. These bounds are the decisive numerical step in the contradiction. Please include the exact fractions, a short derivation, or a machine-checkable computation so that the inequality m − γ − 1024m^2/(1−γ) > 0 can be verified directly.
  3. [§4, Eq. (4.3)] The displayed coefficient in the second inequality appears to be mis-set: completing the square to obtain 1024m^2/(1−γ) requires the term 64m x/√(1−γ), not 64m√(1−γ)x. The numerical conclusion is unaffected, but the formula should be corrected.

Circularity Check

0 steps flagged

No circular derivation: the joint-deficit bound is proved from the near-maximizer hypothesis, and all cited supports are external published theorems, not self-citations or fitted inputs.

full rationale

The proof is not circular. Lemma 3.1 derives the joint deficit budget ρ+σ≤δ directly from the near-maximizer inequality Φ(A)≥2−γ_n and the definition of δ, rather than assuming the conclusion. The dilation parameter t is chosen to make the doubly-superstochastic subset inequalities close, and the later numerical contradiction is an independent rational comparison. The equality case relies on Proposition 2.1(ii), attributed to Hwang [4], which is external published work and is not authored by or defined in terms of the present paper; any concern about the accuracy or applicability of that cited theorem is a correctness risk, not circularity. Similarly, Proposition 2.1(i), (iii), and (iv) cite standard external results (Egorychev–Falikman, Cheon–Wanless, Knopp–Sinkhorn). No fitted parameter is renamed as a prediction, no claim is defined in terms of its own target, and no load-bearing self-citation appears. The manuscript's own disclosure about AI assistance is unrelated to the mathematical derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 0 invented entities

The proof uses standard inequalities (AM-GM, Pinsker, Bernoulli) and four cited permanent theorems. The only assertion introduced without derivation is the exact rational bounds (4.4). There are no fitted parameters or invented entities.

axioms (8)
  • standard math AM-GM inequality
    Used in (3.2) and in AM-GM product bounds for row/column sums.
  • standard math Binary Pinsker inequality D(p∥q) ≥ 2(p−q)^2
    Used in Lemma 3.1 to bound row/column deviations; proof sketched in text.
  • standard math Bernoulli's inequality (1−t)^r ≥ 1−rt
    Used in (4.3) to lower-bound m(1−t)^16.
  • domain assumption van der Waerden theorem (per(B) ≥ n!/n^n for B doubly stochastic)
    Proposition 2.1(i), cited [2,3]; used to handle δ=0 case.
  • domain assumption Hwang's positive-support theorem (every positive global maximizer of Φ is U_n)
    Proposition 2.1(ii), cited [4]; final identification of the maximizer; not proved in paper.
  • domain assumption Cheon–Wanless characterization of doubly superstochastic matrices
    Proposition 2.1(iii), cited [1, Lemma 2.2]; used to conclude S is doubly superstochastic.
  • domain assumption Knopp–Sinkhorn boundary lower bound for permanents
    Proposition 2.1(iv), cited [5]; gives per(B) ≥ m_n for doubly stochastic B with a zero.
  • ad hoc to paper Exact rational bounds (4.4) for γ and m
    Asserted without derivation; the numerical contradiction depends on them.

pith-pipeline@v1.3.0-alltime-deepseek · 3688 in / 30909 out tokens · 228840 ms · 2026-08-01T14:18:32.325540+00:00 · methodology

0 comments
read the original abstract

Dittert's conjecture asserts that, among nonnegative $n\times n$ matrices whose entries sum to $n$, the functional $\phi(A)=\prod_{i=1}^n r_i+\prod_{j=1}^n c_j-\operatorname{per}(A)$ is uniquely maximized by the uniform matrix $J_n/n$. This paper proves the conjecture for $n=16$. The key observation is that, for a near-maximizer, the deficits of the row-sum and column-sum products satisfy a single joint constraint rather than two independent bounds. Combining this joint-deficit estimate with a Pinsker-type subset-sum bound yields a sharper scalar dilation to a doubly superstochastic matrix. The Knopp-Sinkhorn boundary lower bound for permanents then excludes maximizers with a zero entry, and Hwang's positive-support theorem identifies the unique maximizer. Together with Pang's result for $n\ge 17$ (arXiv:2606.01531), this establishes Dittert's conjecture for every $n\ge 16$.

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Reference graph

Works this paper leans on

6 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    Cheon and I

    G.-S. Cheon and I. M. Wanless,Some results towards the Dittert conjecture on permanents, Linear Algebra Appl.436(2012), 791–801. doi:10.1016/j.laa.2010.08.041

    work page doi:10.1016/j.laa.2010.08.041 2012

  2. [2]

    G. P. Egorychev,The solution of van der Waerden’s problem for permanents, Soviet Math. Dokl.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, Math. Notes Acad. Sci. USSR29(1981), 475–479

    1981

  4. [4]

    Hwang,A note on a conjecture on permanents, Linear Algebra Appl.76(1986), 31–44

    S.-G. Hwang,A note on a conjecture on permanents, Linear Algebra Appl.76(1986), 31–44. doi:10.1016/0024-3795(86)90212-0

    work page doi:10.1016/0024-3795(86)90212-0 1986

  5. [5]

    Knopp and R

    P. Knopp and R. Sinkhorn,Minimum permanents of doubly stochastic matri- ces with at least one zero entry, Linear Multilinear Algebra11(1982), 351–355. doi:10.1080/03081088208817459

    work page doi:10.1080/03081088208817459 1982

  6. [6]

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

    Z. Pang,Proof of Dittert’s conjecture for dimensionsn≥ 17, arXiv:2606.01531v1 [math.RA], 1 June 2026. doi:10.48550/arXiv.2606.01531

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.01531 2026