On the P\'olya Frequency Order of the de Bruijn--Newman Kernel: Certified Failure at Order Five

Wojciech Michalowski

REVIEW 3 major objections 4 minor 1 cited by

On the P\'olya Frequency Order of the de Bruijn--Newman Kernel: Certified Failure at Order Five

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A certified 5×5 Toeplitz determinant shows the de Bruijn–Newman kernel is not Pólya frequency of order 5.

desk verdict Plausible and interesting claim, but the certificate has a gap in Lemma 4.1 and the paper contradicts itself on the C5 global claims; needs repair before it is a proof. read the letter →

arxiv 2602.20313 v2 pith:DEGPLZYS submitted 2026-02-23 math.CA

Wojciech Michalowski This is my paper · ORCID

classification math.CA MSC 15A1565G40

keywords PólyafrequencyfunctiontotalpositivitydeBruijn–NewmankernelToeplitzdeterminantintervalarithmeticcertifiedcomputationthetaseriesRiemannzeta

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the de Bruijn–Newman kernel K(u)=Φ(|u|), which arises in the study of the Riemann zeta function, is not a Pólya frequency function of order 5. The proof is a certified computation: a 5×5 Toeplitz matrix formed from K at nine points has determinant enclosed between −1.8472496×10⁻⁹ and −1.8472225×10⁻⁹, so it is strictly negative. Since Pólya frequency functions require all such minors to be nonnegative, this single negative determinant settles the question. The paper also shows the same configuration has positive minors of orders 2, 3, and 4, so the failure is localized to order 5, and identifies an algebraic mechanism: the leading small-spacing coefficient C₅(u₀) is negative for small u₀. Why it matters: total positivity of this kernel would have constrained the zero distribution of the Riemann ξ-function; the failure at order 5 blocks that route, while leaving order 4 open.

What carries the argument

The argument rests on three objects: (1) the Toeplitz determinant family D_r(u₀,h)=det[K(u₀+(i−j)h)] which reduces the 2r-dimensional PF_r condition to two parameters; (2) the small-h expansion giving the leading coefficient C_r(u₀) as a Vandermonde-weighted sum of Taylor coefficients of K at u₀, which isolates the order at which negativity appears; and (3) a rigorous numerical certificate combining a proved truncation bound for the theta series (tail <10⁻⁷⁰ for N=50 on [0,0.21]) with directed-rounding interval arithmetic at 80-digit precision, yielding airtight enclosures of the determinant. The generalized Vandermonde factors W(k₀,…,k_{r−1}) are the mechanism that makes the r=5 coefficient

What would settle it

Recompute det[K(0.01+(i−j)·0.05)] with an independent certified method—say, a different interval library, higher truncation N, or exact rational arithmetic on high-precision approximations—and check whether the resulting enclosure is still strictly negative. Any certified evaluation that contains zero would invalidate the counterexample. Alternatively, verify empirically that some entry c_n lies outside the claimed enclosure in Lemma 4.2; a single violated entry enclosure would break the certificate.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for K(u)=Φ(|u|), the 5×5 Toeplitz matrix M with entries K(0.01+(i−j)·0.05) has determinant −1.847236...×10⁻⁹, rigorously enclosed in [−1.8472496×10⁻⁹, −1.8472225×10⁻⁹]. Because a Pólya frequency function of order 5 must have all such determinants nonnegative, this is a direct counterexample. The authors further establish that the Toeplitz minors of orders 2–4 at the same configuration are positive, derive an exact formula for the leading coefficient C_r(u₀) of the small-h expansion of D_r(u₀,h), and prove that C₅(u₀) is negative on (0,u₀*) with u₀*≈0.031139763615 while C₂,C₃,C₄,C₆,C₇ are positive there. They also compute non-certified Gaussian-deform

Load-bearing premise

The certified negative determinant depends on two things being airtight: the tail bound showing the truncated theta series has error below 10⁻⁷⁰ on [0,0.21], and the directed rounding of every arithmetic operation in the 120-term Leibniz expansion. If either leaks, the enclosure could in principle contain zero, and the PF₅ counterexample would lose its proof.

Editorial extensions

If this is right

The kernel K is not PF₅, so any total-positivity argument for the Riemann ξ-function based on PF order ≥5 cannot work.
At (0.01,0.05), the Toeplitz PF threshold is exactly 5: D₂,D₃,D₄>0 and D₅<0, all certified.
The sign pattern of C_r suggests a resonance localized to order 5: only C₅ is negative, and only for small u₀; C₆ and C₇ are positive at u₀=0.01.
The Gaussian-deformation threshold λ*₅(u₀,h) is configuration-dependent, so there is no single 'healing time' that makes the kernel PF₅ globally; the deformed kernel's PF₅ status remains open.
Whether K is PF₄ in the full sense (all admissible configurations, not just Toeplitz) is left open and is Problem 1 of the paper.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The certification recipe here—proved tail bound plus directed-rounding interval arithmetic—could be reused to test PF thresholds for other kernels defined by theta-type series, not just Φ.
The location of the negative determinant at order 5 suggests that total positivity of Φ(|u|) may be obstructed by the rapid alternating growth of its even Taylor coefficients; a fully symbolic proof of C₅<0 would clarify whether this is a generic phenomenon or a special resonance.
Because λ*₅ depends on the configuration, testing whether a global PF₅ statement holds for some fixed t>0 would require a completely different, non-Toeplitz argument; one could start by checking whether the negative minor persists under the Gaussian deformation for t close to 6–12.
The non-symmetric Toeplitz matrices used here (u₀≠0) may probe the kernel's odd-order structure in a way that symmetric configurations cannot; scanning larger families of (u₀,h) might reveal whether order-5 is the only failing order.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know two things before reading: the paper's central negative claim about the de Bruijn–Newman kernel is probably true, but the proof as written is not. The explicit 5×5 Toeplitz minor is a nice idea, and the algebraic reduction to C_r is genuinely useful. But Lemma 4.1's tail bound has reversed inequalities—the stated bounds fail for every n>51—so the interval certificate for det(M) is not actually established. On top of that, the body still asserts a global sign claim for C5 that the abstract explicitly withdraws. Both need fixing before this is a clean contribution.

What is new and good: the paper gives a systematic way to reduce the PF_r condition to a two-parameter Toeplitz determinant family, and derives an exact formula for the leading coefficient C_r(u0). The explicit negative 5×5 minor at (0.01,0.05), if the certificate holds up, would be a real result: nobody has shown this kernel fails PF5 before. The use of interval arithmetic with an independent interval-LU check is good practice, and the code is linked. The reader's confidence in the algebraic formula and the Leibniz expansion seems justified.

The soft spots, in proportion. First, Lemma 4.1 is load-bearing and wrong as stated. For n≥51, the proof uses n^4 ≤ 51^4 e^{2log(n/51)} and n^2 ≤ 51^2 e^{log(n/51)}. Both are false for n>51; at n=52 the first gives 7.31e6 ≤ 7.03e6. So the geometric tail bound is not proven, and the ±10^-70 widening in Lemma 4.2 has no basis. The determinant enclosure in Proposition 4.3 therefore isn't a proof as written. The tail may well be small, but that needs a corrected bound.

Second, the paper has an internal inconsistency: the abstract says version 2 withdraws the certified global sign and unique-threshold claims for C5, but Theorem 1.4(ii) and Lemma 5.2 still claim C5(u0)<0 for all u0 in (0,u*0). The proof there uses 31 grid points plus continuity, which doesn't establish a continuum result without a derivative bound or monotonicity. Those sections need to be rewritten to match the abstract, or the claims removed.

Third, and minor: the 'certified' status of the extra configurations is a bit uneven—Table 2 marks many rows with '—', and only three are actually interval-certified. That's fine, but the text should be clearer about which claims are proven and which are scan-level numerics.

Who is this for? People working on total positivity of special kernels, and anyone interested in rigorous computational proofs in analytic number theory. It deserves a serious referee, because the question is real and the method is promising, but the current version needs a corrected Lemma 4.1 and a consistent treatment of the C5 claims before it can be accepted.

Referee Report

3 major / 4 minor

Summary. The paper claims a computational proof that the de Bruijn--Newman kernel K(u)=Phi(|u|) is not a Polya frequency function of order 5. The central certificate is a 5x5 Toeplitz minor at (u0,h)=(0.01,0.05) whose determinant is enclosed in [-1.8472496e-9,-1.8472225e-9] by 80-digit interval arithmetic, using a truncation of the theta series at N=50 and a tail bound. The paper also derives an exact algebraic formula for the leading coefficient C_r(u0) in the small-spacing expansion of D_r(u0,h), reports certified positivity of lower-order Toeplitz minors at the central configuration, presents additional counterexample configurations, and studies a Gaussian deformation threshold. The result is framed as a negative answer to PF_5 for this kernel and explicitly not a statement about the Riemann Hypothesis.

Significance. If the interval certificate is sound, the paper settles a natural total-positivity question for the de Bruijn--Newman kernel: K is not PF_5. The exact formula for C_r and the explicit Toeplitz reduction are useful structural observations, and the availability of reproducible code is a strength. The paper is also careful to distinguish the Toeplitz threshold from a global PF_4 statement and to separate certified computations from non-certified numerics. However, the proof of the central certificate depends on a tail bound whose stated proof is faulty, and the global sign claim for C_5 is internally contradicted by the abstract's withdrawal notice. These issues affect the rigor of the main theorem as written, although they appear repairable in scope.

major comments (3)

[§4.1, Lemma 4.1] The truncation bound is not proved as written. For n≥51 the proof asserts n^4 ≤ 51^4 e^{2 log(n/51)} and n^2 ≤ 51^2 e^{log(n/51)}. These are equivalent to (n/51)^4 ≤ (n/51)^2, which is false for n>51; at n=52 the first reads 7.31e6 ≤ 7.03e6. The geometric-tail estimate therefore does not follow from the stated inequalities. Since Lemma 4.2 widens the Phi_N enclosures by ±10^{-70} using exactly this bound, and Proposition 4.3 derives the negative determinant from those enclosures, the rigorous status of Theorem 1.1 is not established by the text as written. The determinant may well be negative, but the certificate needs a corrected tail bound.
[§5.2, Lemma 5.2 and Theorem 1.4(ii)] The claim that C_5(u0)<0 for all u0 in (0,u0*) is not supported. Negativity on 31 grid points together with continuity does not rule out positive excursions between grid points. The 'bisection-certified zero at u0*' locates a sign change at one point but does not certify uniqueness of that zero. Moreover, the abstract explicitly states that Version 2 withdraws the certified global sign and unique-threshold claims for C_5 because the derivative-tail enclosure was unsound, yet Theorem 1.4(ii) and Lemma 5.2 still assert those claims. This internal contradiction must be resolved: either a correct global proof is supplied, or the statement should be downgraded to non-certified numerical evidence on a grid.
[Abstract / §4.2] The abstract says 'Eight further configurations are certified by both an explicit Leibniz expansion and an independent interval determinant computation,' but §4.2 certifies only two further configurations by interval arithmetic, and the independent Leibniz-expansion check is not described for these configurations. Table 2 lists three certified configurations total. The count should be reconciled.

minor comments (4)

[Abstract] The manuscript is labeled arXiv:2602.20313v1 but the abstract contains a 'Version 2 withdraws...' passage. Please clarify which version is under consideration, since the withdrawal statement concerns claims that are still printed as Theorem 1.4(ii).
[§4.2, Table 2] Table 2 appears to list 24 configurations as found by scanning, but only 8 rows are shown and only 3 are marked as certified. A note on the selection criterion and the status of the remaining rows would help.
[§5.1, Proposition 5.1] The derivative magnitudes are said to be certified by 'direct differentiation,' but the bound on the tail for each derivative is referenced to Lemma 4.1, which has the gap noted above. Once Lemma 4.1 is repaired, this statement should be revisited.
[General] Several numerical values are presented with excessive digits (e.g. enclosures of width about 1e-12, while 80-digit precision is used). This is not an error, but the paper could state the precision needed for each computation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PF5 certificate is a direct interval-arithmetic evaluation of the defining Toeplitz minor; the flagged truncation-bound gap is a rigor issue, not a circular derivation.

full rationale

The central claim (Theorem 1.1) is a numerical computation: det(M) for M_ij=K(u0+(i-j)h) is evaluated by summing truncated Φ_N series terms with mpmath.iv interval arithmetic and adding a tail bound. There is no free parameter fitted to force a sign, no self-citation used as a load-bearing premise, and no definition that presumes the conclusion. The C_r coefficients in Proposition 3.2 are derived by Taylor expansion of the analytic kernel and evaluated from the exact algebraic formula; the agreement with Richardson extrapolation is a consistency check, not the proof. Lemma 5.2's grid-and-continuity inference for C5(u0)<0 on (0,u0*) is not circular, though it is underjustified as written; the abstract itself withdraws the certified global sign/unique-threshold claims for C5. The proof of Lemma 4.1 as printed also contains an apparently false bounding inequality (n^4 ≤ 51^4 e^{2 log(n/51)} for n≥51), which would undermine the rigor of the truncation enclosure if not corrected; this is a correctness/verification gap, not an equivalence-by-construction. No fitted input is relabeled as a prediction: the negative determinant is computed directly from the series terms against the fixed PF_r definition (4), and the Gaussian threshold λ*_5 is explicitly called configuration-dependent and non-universal. Hence no circular step is present.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to make the main result true; the only hand-chosen computational setting is the truncation order N=50, whose error is bounded rather than fitted. No new particles, forces, or metaphysical entities are introduced. The Gaussian-deformation threshold λ*₅ is a defined mathematical quantity, not an invented entity.

free parameters (1)

truncation order N = 50
Chosen so the tail bound in Lemma 4.1 is below 10⁻⁷⁰; it is a computational setting, not fitted to the sign of the determinant. Increasing N would only tighten the enclosure.

assumptions (4)

domain assumption The series representation Φ(u)=Σ(2π²n⁴e^{9u}−3πn²e^{5u})e^{−πn²e^{4u}} is valid for u∈[0,0.21] and can be differentiated term-by-term.
Invoked in §1.1 and used in Lemma 4.1 and §5.1 for the entries and Taylor coefficients of K.
standard math The PF_k definition (4) requires nonnegativity of all r×r minors for r≤k.
This is the standard definition of Pólya frequency functions, used in Definition 2.1 and Proposition 2.2.
standard math Alternating analytic determinants are divisible by the Vandermonde factors in x and y, giving the h^{r(r−1)} vanishing order.
Used in Lemma 3.1, citing Karlin [10]; it is a classical theorem in total-positivity theory.
domain assumption Directed-rounding interval arithmetic in mpmath.iv produces rigorous enclosures of all elementary operations at 80-digit precision.
Necessary for Lemma 4.2 and Proposition 4.3; the paper relies on the correctness of the mpmath.iv implementation.

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Pith. "Pith review of On the P\'olya Frequency Order of the de Bruijn--Newman Kernel: Certified Failure at Order Five." pith.science (2026). https://pith.science/paper/DEGPLZYS

@misc{pith2026260220313,
  author       = {Pith},
  title        = {Pith review of: On the P\'olya Frequency Order of the de Bruijn--Newman Kernel: Certified Failure at Order Five},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DEGPLZYS}},
  note         = {Machine review of arXiv:2602.20313}
}

abstract

We prove that the classical de Bruijn--Newman kernel $K(u)=\Phi(|u|)$ is not a P\'olya frequency function of order $5$ (PF$_5$). At $(u_0,h)=(0.01,0.05)$ we exhibit an explicit $5\times5$ Toeplitz minor whose determinant is rigorously enclosed in $[-1.8472496\times10^{-9},-1.8472225\times10^{-9}]$. The certificate uses 80-digit outward-rounded interval arithmetic and a proved truncation bound for the theta series. Eight further configurations are certified by both an explicit Leibniz expansion and an independent interval determinant computation. At the central configuration the determinants $D_2,D_3,D_4$ are positive, but this local sign pattern does not establish that the kernel is PF$_4$ globally. We also derive an exact finite formula for the first coefficient permitted by Vandermonde divisibility in the small-spacing expansion of $D_r(u_0,h)$. High-precision observations concerning the sign change of $C_5(u_0)$ and a Gaussian deformation are reported only as non-certified numerics. Version 2 withdraws the certified global sign and unique-threshold claims for $C_5$ made in version 1 because the derivative-tail enclosure was unsound; the direct PF$_5$ counterexample and its interval certificates are unaffected. The result concerns total positivity of this kernel and does not resolve the Riemann Hypothesis.

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Forward citations

Cited by 1 Pith paper

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

An explicit uniform cubic wedge for consecutive Toeplitz minors of the Riemann xi coefficients
math.NT 2026-07 accept novelty 7.0 of 10

For every r ≥ 2 and k ≥ 10^18 r^3, the consecutive Toeplitz minor D_{r,k} of the Riemann xi coefficients is strictly positive, proved without using verified zeta zeros.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

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