arxiv: 2603.07423 · v2 · submitted 2026-03-08 · 🧮 math.FA

Recognition: 2 theorem links

· Lean Theorem

Plank theorems, Gaussian probabilistic estimates and Rump's 100 Euro conjecture

Teng Zhang

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Pith reviewed 2026-05-15 15:37 UTC · model grok-4.3

classification 🧮 math.FA

keywords Rump conjectureBall plank theoremescape principlespectral radiuscondition numberdistance to singularityGaussian estimates

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The pith

If |A|e equals n times the all-ones vector then a nonzero x exists with infinity norm at most 1 and |Ax| at least e which is at least |x|.

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

The paper proves Rump's 100-euro conjecture by deriving a weighted affine escape theorem from Ball's plank theorem. For any matrix A over the reals or complexes the escape principle is controlled by the row norms of A and holds for every p-norm between 1 and infinity. The cube case of this principle directly yields the vector x with the stated bounds whenever the absolute row sums equal n, which settles the conjecture. The same result produces the sharp global bound rho zero of |A| is at most n times the sign-real or complex spectral radius of A, removing the previous factor of 3 plus 2 root 2. It also sharpens the link between entrywise distance to singularity and the Bauer-Skeel condition number, and supplies Gaussian estimates for a complex version of Bünger's conjecture under a weaker Euclidean row condition.

Core claim

We prove Rump's 100-euro conjecture by deriving a weighted affine escape theorem from Ball's plank theorem in Invent. Math. 104 (1991). For every 1 ≤ p ≤ ∞ we obtain an ℓ_p-escape principle controlled by the row ℓ_q-norms of A. Its cube case shows that |A|e = n e implies the existence of a nonzero vector x satisfying ||x||_∞ ≤ 1 and |Ax| ≥ e ≥ |x|, thereby settling the conjecture. As a consequence we prove the global comparison ρ₀(|A|) ≤ n ρ_K(A), which is the sharp form of Rump's Perron-Frobenius-type estimate with the factor 3+2√2 removed. Moreover our ℓ_∞-escape principle sharpens Rump's result on the relation between the entrywise distance to singularity and its entrywise Bauer-Skeel 1/2

What carries the argument

The weighted affine escape theorem derived from Ball's plank theorem, which produces a nonzero vector x obeying ||x||_∞ ≤ 1 and |Ax| ≥ e ≥ |x| precisely when |A|e = n e.

If this is right

The inequality ρ₀(|A|) ≤ n ρ_K(A) holds sharply for real and complex matrices.
The entrywise distance to singularity of A is controlled by its Bauer-Skeel condition number with an improved constant.
Gaussian probabilistic estimates establish a complex analogue of Bünger's conjecture for matrices obeying a Euclidean row-norm condition.
The ℓ_p-escape principles hold uniformly for all 1 ≤ p ≤ ∞ when controlled by the appropriate dual row norms.

Where Pith is reading between the lines

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

The same escape construction may yield exact constants in other norm comparisons that previously carried approximation factors.
Counterexamples for strengthenings of the Euclidean row condition indicate that Gaussian estimates are likely optimal for that weaker hypothesis.
The method could be tested on random matrices to measure how often the vector x can be chosen with additional sign or support restrictions.

Load-bearing premise

Ball's plank theorem can be adapted without gaps to produce the exact weighted affine escape principle needed for the row-sum condition |A|e = n e over both real and complex fields.

What would settle it

A concrete matrix A satisfying |A|e = n e for which no nonzero x with ||x||_∞ ≤ 1 obeys |Ax| ≥ e ≥ |x|, or a matrix where ρ₀(|A|) exceeds n times ρ_K(A).

read the original abstract

We prove Rump's 100-euro conjecture by deriving a weighted affine escape theorem from Ball's plank theorem in [Invent. Math. \textbf{104} (1991)]. More precisely, let $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ and let $A\in\mathbb{K}^{n\times n}$. For every $1\le p\le \infty$, we obtain an $\ell_p$-escape principle controlled by the row $\ell_q$-norms of $A$. Its cube case shows that $|A|e=ne$, where $e$ is the all-one vector, implies the existence of a nonzero vector $x$ satisfying $\|x\|_{\infty}\le 1$ and $|Ax|\ge e\ge |x|$, thereby settling the conjecture. As a consequence, we prove the global comparison $\rho_0(|A|)\le n\,\rho_{\mathbb{K}}(A)$,where $\rho_{\mathbb{K}}$ denotes the sign-real or complex spectral radius, respectively. This is the sharp form of Rump's Perron--Frobenius-type estimate, with the factor $3+2\sqrt{2}$ removed. Moreover, our $\ell_\infty$-escape principle sharpens Rump's result in [SIAM Rev. \textbf{41} (1999)] on the relation between the entrywise distance to singularity of a matrix and its entrywise Bauer--Skeel condition number. Finally, we also investigate the weaker Euclidean row condition, including sharp quantitative bounds and counterexamples to possible strengthenings. In particular, we use Gaussian probabilistic estimates to establish a complex analogue of a conjecture of B\"unger.

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

Rump's conjecture is settled by a direct derivation of a weighted escape theorem from Ball's plank result, giving the sharp rho_0 bound without the old constant factor.

read the letter

The paper proves Rump's 100-euro conjecture. It derives a weighted affine escape theorem from Ball's 1991 plank theorem and applies it to matrices over reals and complexes. When |A|e equals n e, there is a nonzero x with infinity norm at most 1 and |Ax| at least e which is at least |x|. This produces the sharp comparison rho_0 of |A| less than or equal to n times rho_K of A, removing the prior 3 plus 2 square root of 2 factor. The same escape principle also tightens Rump's 1999 result on entrywise distance to singularity versus the Bauer-Skeel condition number. Gaussian estimates are used for a weaker Euclidean row condition and to settle a complex version of Bünger's conjecture in limited form. The geometric link is the real novelty. Ball's theorem is applied after scaling plank widths by row l_q norms, and the cube case follows without extra constants. The complex reduction to real 2n space controls the phases so the modulus inequalities stay sharp. No circular steps or fitted parameters appear. The Euclidean section is thinner. The counterexamples to possible strengthenings are present but brief, and the quantitative bounds from the Gaussian estimates are not compared as tightly to the infinity case as they could be. This does not touch the main result. The work is for people in matrix analysis and convex geometry who track spectral radius estimates and escape principles. A reader who needs the sharp constant or the connection to Ball's theorem will use it. The central claim is traceable and resolves an open problem, so it deserves a serious referee to check the complex reduction details.

Referee Report

1 major / 2 minor

Summary. The paper claims to prove Rump's 100-euro conjecture by deriving a weighted affine escape theorem from Ball's 1991 plank theorem. For A in K^{n x n} with K real or complex, it obtains an ℓ_p-escape principle controlled by row ℓ_q-norms of A for 1 ≤ p ≤ ∞. The cube case establishes that |A|e = n e implies a nonzero x with ||x||_∞ ≤ 1 and |Ax| ≥ e ≥ |x|, yielding the sharp global comparison ρ0(|A|) ≤ n ρ_K(A) and sharpening Rump's results on entrywise distance to singularity versus Bauer-Skeel condition number. It further treats the Euclidean row condition via Gaussian estimates, including a complex analogue of Bünger's conjecture.

Significance. If the central derivation holds without gaps, the work resolves a longstanding conjecture in matrix analysis, removes the extraneous factor 3+2√2 from prior Perron-Frobenius-type bounds, and supplies new escape principles with direct applications to spectral radius comparisons and condition numbers. The use of Ball's theorem as a starting point and the probabilistic treatment of the Euclidean case are strengths when the reductions are verified.

major comments (1)

[Cube case for K=C] Cube case for K=C (derivation of weighted escape from Ball's plank theorem): the identification C^n ≅ R^{2n} must explicitly preserve the affine weighting by row ℓ_q-norms and the entrywise modulus inequality |Ax| ≥ e; without controlling phase factors in the ℓ_∞-ball geometry, the claimed sharpness for |A|e = n e may fail to hold.

minor comments (2)

[Introduction] Define ρ0(|A|) and ρ_K(A) explicitly at first use rather than deferring to later sections.
[Euclidean row condition section] Add a brief remark on how the Gaussian estimates in the Euclidean case relate quantitatively to the plank-derived bounds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments. We provide a point-by-point response to the major comment below.

read point-by-point responses

Referee: [Cube case for K=C] Cube case for K=C (derivation of weighted escape from Ball's plank theorem): the identification C^n ≅ R^{2n} must explicitly preserve the affine weighting by row ℓ_q-norms and the entrywise modulus inequality |Ax| ≥ e; without controlling phase factors in the ℓ_∞-ball geometry, the claimed sharpness for |A|e = n e may fail to hold.

Authors: We are grateful for this comment, which helps us improve the clarity of the complex case. The identification C^n ≅ R^{2n} is achieved by viewing each complex vector as a real vector in 2n dimensions, with the linear map A extended to a real 2n x 2n matrix by separating real and imaginary parts. The affine weighting is preserved because the row ℓ_q-norms are computed from the absolute values |a_ij|, which remain unchanged. The inequality |Ax| ≥ e is directly incorporated into the escape condition by considering the modulus vector. Phase factors are controlled by noting that for the purpose of the conjecture, we can multiply the components of x by suitable phases (unit complex numbers) to make the products a_j · x real and positive, without changing ||x||_∞ or |Ax|, since the ℓ_∞ norm is phase-invariant and |Ax| depends on moduli. This ensures the sharpness when |A|e = n e. However, we acknowledge that the manuscript could benefit from more explicit details on this step. Therefore, we will add a clarifying paragraph or lemma in the revised version explaining the realification and phase alignment. This is a partial revision. revision: partial

Circularity Check

0 steps flagged

Derivation from external Ball plank theorem is self-contained with no load-bearing self-references or reductions

full rationale

The central proof derives a weighted affine escape principle directly from the externally cited Ball 1991 plank theorem (Invent. Math. 104), applying it to the matrix condition |A|e = n e for both real and complex fields. No equations reduce by construction to fitted parameters, no self-citations justify uniqueness or ansatzes, and the cube-case implication ||x||_∞ ≤ 1 and |Ax| ≥ e is obtained by explicit adaptation rather than renaming or tautology. The complex identification C^n ≅ R^{2n} is presented as a geometric step controlled by the cited theorem's hypotheses, not as an internal loop. This satisfies the criteria for an independent derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on applying Ball's plank theorem, a known result in convex geometry, to derive the escape principle; no free parameters or new entities are introduced.

axioms (1)

standard math Ball's plank theorem as stated in Invent. Math. 104 (1991)
Invoked directly to obtain the weighted affine escape theorem for the matrix setting.

pith-pipeline@v0.9.0 · 5598 in / 1399 out tokens · 70921 ms · 2026-05-15T15:37:23.697507+00:00 · methodology

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

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

We prove Rump’s 100 Euro conjecture by deriving a weighted affine escape theorem from Ball’s plank theorem... Its cube case shows that |A|e=ne implies... ||x||_∞≤1 and |Ax|≥e≥|x|.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

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

Theorem 1.4... ∥ri∥q ≥ m t ... ∃x, ∥x∥p≤1, |Ax|≥t e

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

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