Certified Collision Exclusion and an Unconditional Upper Bound $\Lambda<0.156821$ for the de Bruijn--Newman Constant

Dale James Raymond; Michel Planat

REVIEW

Certified Collision Exclusion and an Unconditional Upper Bound $\Lambda<0.156821$ for the de Bruijn--Newman Constant

Michel Planat, Dale James Raymond

Number Theory · Submitted 4 Nov 2026 · 21 references

Pith's verdict: Ambitious but fragile: the collision-exclusion method is new and the bound improves the record, but the proof's narrow feasibility margin and extreme complexity mean it needs careful independent validation.

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arXiv ID2611.06245 (version 1)

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CitationDale James Raymond, Michel Planat. "Certified Collision Exclusion and an Unconditional Upper Bound $\Lambda<0.156821$ for the de Bruijn--Newman Constant." arXiv:2611.06245 (2026). https://pith.science/paper/YREDTE7G

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Claim paper

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Editor's note

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The paper's main contribution is a genuinely new method—collision exclusion—for bounding the de Bruijn–Newman constant, and it delivers an unconditional upper bound of 0.156821 that improves the previous published record of 0.2. That is a real step forward within the subfield. The architecture of the proof is logically coherent, the authors supply extensive certificates and code, and they are transparent about the narrow margin in the Gram source family. Credit where it is due.

The soft spot is right where the authors place it: the Cauchy–Schwarz condition for source 1 is nearly saturated (margin around 2% of δ₁). The entire finite-exterior argument hinges on that inequality holding. The paper acknowledges this in Remark 4.2 and calls it a narrow feasibility margin, not a flaw—but it does make the proof fragile. A rounding error or a missed term in the coefficient generation could break the torus inequality and collapse the exterior part. The stress-test note is correct to flag this; the authors already flag it themselves. The remaining steps—separator, interior propagation, infinite tail—look solid, but they depend on the source certificates being correct.

Complexity is another soft spot. 664k Fourier coefficients, 44k torus boxes, 5 source polynomials, 56 recovery leaves, 17k separator rectangles. That is a lot of moving parts for an unrefereed preprint. The authors acknowledge that their diagnostics are not an independent implementation. Independent verification will be essential before the result can be taken as settled.

Who is this paper for? Specialists in rigorous numerics for the Riemann xi function and the de Bruijn–Newman constant. It is not a paradigm shift, but it does provide a new technique that may be applicable elsewhere.

Recommendation: I would accept peer review. The paper deserves a serious referee, but the referee needs to audit the source certificates and the margin. I would not cite it in my own work until the computational parts have been independently checked.

Referee report

Major objections (3)

[Section 4.1 (page 16–17)] The premise ∥EF₀ − 1∥₁ < 1 of Lemma 4.3 is stated to be satisfied with a computed defect less than 3/1000, but the paper gives no justification for this bound. It is not explained how the product of the 53-term inverse with the 1160-term smooth fibre is computed, what the truncation error is, or how the l¹ deviation from 1 is bounded. Because this inequality is load-bearing for the recovery inequality (OR1) and hence for the entire finite-exterior argument, the authors must provide either an explicit analytic estimate or a detailed reference to the specific certificate steps that prove it.
[Remark 4.2 (page 15–16)] The Cauchy–Schwarz feasibility margin for source 1 is extremely narrow (diagnostic 2P₁₁ − S₁ ≈ 0.0013055 vs. δ₁ = 0.00128, a margin of about 0.0000255, roughly 2% of δ₁). The authors correctly note that this narrowness is an obstruction of their specific Gram family and that the certified enclosures are outward-rounded. However, the paper does not discuss whether a small error in the coefficient generation for the exponential a_n(t,x) at the exact source parameters could flip the sign of (G). Given the extreme complexity of the computation, a sensitivity analysis or a more explicit explanation of how the outward-rounding guarantees a safe lower bound across this narrow gap would strengthen confidence.
[Theorem 4.5 and (MA13) (pages 19–20)] The 56 recovery inequalities (RL) depend on the computed values b̂R_j(s_L) and I_{L,U}, whose derivation from the source certificates and the analytic gauges is only sketched. The rational floors listed in (MA13) are stated without an explanation of how they follow from the 44,182 torus-box certificates and the outward evaluations of the entrant integrals. To allow independent verification, the authors should clarify the chain from the raw box data through the recovery formula (OR4) to the final leaf margins.

Minor objections (7)

[§1.3, Eq. (3)] The fraction 156821/1000000 should be reduced to lowest terms? Not strictly necessary, but it appears several times as a literal fraction; consistency is fine.
[§3.8, after Eq. (21)] The phrase 'The coefficient drift used in every source, recovery and Euler calculation is cpol(x)=c_α/2' would be clearer if the factor-of-two conversion were explicitly restated at the start of Section 4, where the source polynomial coefficients are defined.
[§4.1, Lemma 4.3] The sentence 'Thus no assertion that a truncated formal inverse is itself zero-free is needed' is ambiguous. Please rephrase to clarify that the inverse is not truncated in the usual series sense; instead a finite product of explicitly retained terms is used and its deviation from 1 is bounded.
[§6 (page 21–23)] The sentence 'The fresh Acb source bank contains enclosures for the exact sums S_k' is unclear; 'Acb' is not defined in the text (it appears to refer to the Arb complex-ball type). Please define or reference the library.
[§8.4 (page 29)] The independent 256-bit reconstruction is said to share 'Arb/FLINT directed arithmetic'. The label 'independent' here refers to a separate code path, not a different arithmetic library; clarify to avoid confusion about independence of the rounding guarantees.
[§11 (page 34)] The declaration of AI assistance is unusual but acceptable. Please ensure that the roles of the AI systems are described specifically enough for a reader to understand how they were used in the mathematical development, not just in writing.
[General] There are occasional missing spaces after punctuation and inconsistent use of en-dashes vs. hyphens. A careful copy-editing pass is recommended.

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The paper

The paper claims

An unconditional upper bound Λ < 0.156821 for the de Bruijn–Newman constant is proved via a certified collision-exclusion method that replaces the canopy criterion with a real double-zero exclusion.

The reading

The paper proves that the de Bruijn–Newman constant Λ, the critical heat time beyond which all zeros of the deformed Riemann ξ-function become real, satisfies Λ < 0.156821. This improves the best previously published unconditional bound of 0.2 (Platt–Trudgian 2021) by a substantial margin, using only the same finite Riemann hypothesis verification up to height 3 trillion. The central innovation is a global collision-exclusion theorem that prevents multiple real zeros from appearing on a closed interval of heat times, rather than relying on a complex zero-free canopy as in earlier work. The proof combines five certified prime-polynomial source polynomials, a balanced real-jet transfer, a uniform Euler-product barrier at infinity, and a compact-interior propagation theorem, all assembled into a closed time-band exclusion that forces strictness at the proposed endpoint. The entire finite certificate is publicly archived and auditable through 664,505 structural Fourier coefficients, 44,182 torus boxes, and 17,385 separator rectangles.

What carries the argument

The central mechanism is the collision-exclusion framework: replacing the canopy criterion (which prohibits nonreal zeros crossing a complex boundary) with the global exclusion of real double zeros (H_t, H'_t) = (0,0) on a closed time interval. This is reduced, after a balanced real gauge, to a positivity statement for a Hermitian form in a pair of analytic prime-polynomial coordinates. The five stable Gram source polynomials (MA1) – each defined on a specific index interval with a synthetic fibre δ_j and weight ϑ_j – satisfy a strict torus inequality (G) that ensures, via Lemma 4.3 (ordinary weighted recovery), that the full source polynomial never vanishes on the closed unit polydisc and has modulus > δ_j. The balanced real-jet transfer (Theorem 3.1 and Corollary 3.2) converts this polynomial lower bound into a nonvanishing lower bound for the physical pair (H_t, H'_t) over the finite exterior. For the infinite exterior, a uniform Euler-product barrier (Section 8) yields a positive lower bound for u² + u_x² through a differential inequality on the completed resultant matrix. The stationary separator (Section 6) and compact interior theorem (Theorem 7.2) handle the remaining regions.

Extended reading notes

Core claim

The paper establishes that Λ < 156821/1000000 = 0.156821, and does so by excluding the possibility of a multiple real zero at any point in the time band [0.156821, 0.2] and spatial region |x| ≥ 6000000185827 ∪ [X,W] ∪ [W,∞) (with explicit constants). The argument separates three regions: a stationary separator at x = X, a compact interior where all zeros are real and simple by propagation, a finite exterior where five Gram sources guarantee a stable polynomial modulus > 1/790, and an infinite exterior where a differential inequality gives u² + u_x² > 3/10000. A critical-time compactness lemma shows that if Λ lies in the band, H_Λ would have a finite multiple real zero; the regional exclusions contradict that, forcing Λ below the band. The proof does not assume any new zero-free half-plane or unverified analytic hypotheses beyond the published Polymath approximations and Platt–Trudgian finite-RH result.

Load-bearing premise

The five prime-polynomial Gram sources (MA1) must satisfy the strict positivity conditions of the torus inequality (G) and the ordinary weighted recovery inequality (OR1) with exact, outward-rounded rational coefficients; if any single source certificate is invalid due to rounding, a missed term, or a narrow feasibility margin, the finite-exterior exclusion could fail and the bound might not hold.

Editorial extensions

If this is right

The unconditional upper bound for Λ is lowered from 0.2 to 0.156821, providing the sharpest published unconditional bound in the Polymath normalisation.
The collision-exclusion method offers a new global framework for bounding Λ that, unlike the canopy approach, does not require a nonzero imaginary part of the zero-free region, and thus may be more amenable to further refinement with stronger finite-RH data.
The proof demonstrates that the Platt–Trudgian verification up to 3 trillion is sufficient to support a substantially better bound than the previously published 0.2, showing the power of the method over a simple barrier estimate.
The public certificate archive (with 44,182 torus boxes, 17,385 separator rectangles, etc.) sets a high standard for reproducibility; an independent re-execution would either confirm the bound or reveal a flaw, making the result auditable.
If the bound is accepted, the gap between the best conditional bounds (≈0.107965 under H₇/₈) and the best unconditional bound is narrowed, highlighting the remaining distance to Λ=0 needed for the Riemann hypothesis.

Reading between the lines

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

The narrow feasibility margin (≈5.8×10⁻⁷ below the endpoint) observed for the four-prime Gram family suggests that any further improvement with the same polynomial architecture may require either a different weight distribution or the inclusion of a fifth prime in the smooth fibre, as the authors acknowledge.
The collision-exclusion method separates the global problem into independent regional certificates; therefore, future improvements in the finite-RH height or in the source polynomial construction can be plugged in directly without re-proving the entire assembly, making incremental progress on Λ a modular task.
The explicit certification of the four-prime Cauchy–Schwarz obstruction (Proposition 4.1) gives a precise feasibility ceiling for that family, which could guide the search for alternative polynomial bases that avoid the obstruction and potentially push the bound toward 0.15 or lower.
The paper's careful distinction between unconditional and conditional/unrefereed bounds (Table 1) clarifies the logical status of several recent claims, which may influence how the community evaluates progress on the de Bruijn–Newman constant.

Original abstract

The de Bruijn--Newman constant $\Lambda$ is the transition time beyond which all zeros of the heat-deformed Riemann $\xi$-function are real. We develop a computer-assisted collision-exclusion method and obtain the bound $\Lambda<156821/10^6=0.156821$, in the normalisation of Polymath. The proof combines the published finite-height verification and the starting inequality $\Lambda\leq1/5$ with a compact-interior continuation theorem, five stable prime-polynomial Gram sources, certified moving-index recovery, a balanced real-jet transfer and a uniform Euler-product barrier at infinity. The resulting closed time-band exclusion, together with uniform remote reality, forces strictness at the proposed endpoint. The finite certificate uses 664,505 structural Fourier coefficients, 44,182 closed torus boxes, 184 separator moments and 17,385 separator rectangles. We isolate a Cauchy--Schwarz obstruction intrinsic to the four-prime source family, explaining the narrow feasibility margin of the certified endpoint. We state the proof-to-certificate interface and distinguish the present unconditional logical claim from other recent, as yet unrefereed, computational bounds.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-05: 1 request, 5,781 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.81 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.79 · mentions code or data availability 0.05.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The de Bruijn--Newman constant $\Lambda$ is the transition time beyond which all zeros of the heat-deformed Riemann $\xi$-function are real. We develop a computer-assisted collision-exclusion method and obtain the bound $\Lambda<156821/10^6=0.156821$, in the normalisation of Polymath. The proof combines the published finite-height verification and the starting inequality $\Lambda\leq1/5$ with a compact-interior continuation theorem, five stable prime-polynomial Gram sources, certified moving-index recovery, a balanced real-jet transfer and a uniform Euler-product barrier at infinity. The resulting closed time-band exclusion, together with uniform remote reality, forces strictness at the proposed endpoint. The finite certificate uses 664,505 structural Fourier coefficients, 44,182 closed torus boxes, 184 separator moments and 17,385 separator rectangles. We isolate a Cauchy--Schwarz obstruction intrinsic to the four-prime source family, explaining the narrow feasibility margin of the certified endpoint. We state the proof-to-certificate interface and distinguish the present unconditional logical claim from other recent, as yet unrefereed, computational bounds.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The de Bruijn--Newman constant $\Lambda$ is the transition time beyond which all zeros of the heat-deformed Riemann $\xi$-function are real.	background 1.00	0.04	0.03	asserted 1.00
s1	We develop a computer-assisted collision-exclusion method and obtain the bound $\Lambda<156821/10^6=0.156821$, in the normalisation of Polymath.	result 0.90	0.97	0.93	asserted 1.00
s2	The proof combines the published finite-height verification and the starting inequality $\Lambda\leq1/5$ with a compact-interior continuation theorem, five stable prime-polynomial Gram sources, certified moving-index recovery, a balanced real-jet transfer and a uniform Euler-product barrier at infinity.	method 0.97	0.79	0.32	asserted 1.00
s3	The resulting closed time-band exclusion, together with uniform remote reality, forces strictness at the proposed endpoint.	interpretation 0.90	0.64	0.08	asserted 0.98
s4	The finite certificate uses 664,505 structural Fourier coefficients, 44,182 closed torus boxes, 184 separator moments and 17,385 separator rectangles.	method 0.93	0.84	0.02	asserted 1.00
s5	We isolate a Cauchy--Schwarz obstruction intrinsic to the four-prime source family, explaining the narrow feasibility margin of the certified endpoint.	interpretation 0.46	0.91	0.03	asserted 1.00
s6	We state the proof-to-certificate interface and distinguish the present unconditional logical claim from other recent, as yet unrefereed, computational bounds.	contribution 0.73	0.86	0.04	asserted 0.89

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.51.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (21)

[1]

N. G. de Bruijn,The roots of trigonometric integrals, Duke Math. J.17(1950), 197–226

View on Pith 1950
[2]

C. M. Newman,Fourier transforms with only real zeros, Proc. Amer. Math. Soc.61(1976), 245–251

View on Pith 1976
[3]

H. Ki, Y.-O. Kim and J. Lee,On the de Bruijn–Newman constant, Adv. Math.222(2009), 281–306; doi:10.1016/j.aim.2009.04.003

View on Pith doi:10.1016/j.aim.2009.04.003 2009
[4]

B. Rodgers and T. Tao,The de Bruijn–Newman constant is non-negative, Forum Math. Pi8 (2020), e6;https://arxiv.org/abs/1801.05914

View on Pith Pith review arXiv 2020
[5]

A. Dobner,A new proof of Newman ’s conjecture and a generalization, arXiv:2005.05142 (2020)

View on Pith arXiv 2020
[6]

D. H. J. Polymath,Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn–Newman constant, Res. Math. Sci.6(2019), Article 31; doi:10.1007/s40687-019-0193-1. Theorem numbering as in arXiv:1904.12438v2

View on Pith Pith review arXiv 2019
[7]

D. J. Platt and T. Trudgian,The Riemann hypothesis is true up to3 · 1012, Bull. Lond. Math. Soc. 53(2021), 792–797; doi:10.1112/blms.12460

View on Pith doi:10.1112/blms.12460 2021
[8]

D. Romik,Orthogonal polynomial expansions for the Riemann xi function, arXiv:1902.06330 (2019); https://arxiv.org/abs/1902.06330

View on Pith Pith review arXiv 2019
[9]

J. Gomila,A computer-assisted proof of the boundΛ ≤ 0.1787854for the de Bruijn–Newman constant, publicly released certificate manuscript and repository (20 August 2026), not peer reviewed AN UPPER BOUND Λ<0.156821 35 at the time of writing; https://github.com/judegomila/dbn-lambda-01787854-candidate-a udit

View on Pith 2026
[10]

D. M. Kornaus,A computer-assisted unconditional upper boundΛ < 0.172422for the de Bruijn– Newman constant, self-archived manuscript and certification package, version 2 (September 2026); https://doi.org/10.5281/zenodo.22658444

View on Pith doi:10.5281/zenodo.22658444 2026
[11]

S. Gordon,A computer-assisted upper bound for the de Bruijn–Newman constant, preprint (2026), SSRN:7608638;https://doi.org/10.2139/ssrn.7608638

View on Pith doi:10.2139/ssrn.7608638 2026
[12]

C. Hakimoglu-Brown,A conditional boundΛ ≤ 0.13963for the de Bruijn–Newman constant, arXiv:2610.12803 (2026). Conditional on absence of zeta zeros with real part greater than 7/8

View on Pith Pith review arXiv 2026
[13]

M. Planat,Collective contraction in the de Bruijn–Newman heat flow: Off-zero logarithmic derivatives and certified barrier refinements, arXiv:2609.37164 (2026); https://arxiv.org/abs/ 2609.37164

View on Pith Pith review arXiv 2026
[14]

M. Planat,A genus-two spectral curve of the RiemannΞ-function: Real structure and certified arithmetic separation of periods, arXiv:2610.32694v1 (2026);https://arxiv.org/abs/2610.32694. The paper reports the separate bound Λ≤433/2500 = 0.1732

View on Pith Pith review arXiv 2026
[15]

M. Planat,A certified classical-strip comparison for the de Bruijn–Newman constant, arXiv:2610.18214v1 (2026); https://arxiv.org/abs/2610.18214. Theorem 1.1 is conditional on its analytic premises (A1)–(A3), for which full validation is not supplied in that version

View on Pith Pith review arXiv 2026
[16]

D. J. Raymond,Correlated translated probes and the boundΛ < 0.149396for the de Bruijn– Newman constant, public, unrefereed certificate manuscript (2026), version 1.1 with erratum; https://doi.org/10.5281/zenodo.23781981

View on Pith doi:10.5281/zenodo.23781981 2026
[17]

D. J. Raymond,A computer-assisted boundΛ ≤ 0.107964910079for the de Bruijn–Newman constant, public, unrefereed certificate manuscript (2026); https://doi.org/10.5281/zenodo.23752103 . Its use of the newly asserted 7/8 zero-free region calls for separate scrutiny

View on Pith doi:10.5281/zenodo.23752103 2026
[18]

Mosaic Intelligence,Collision-exclusion assembly and certified four-prime source construc- tions, version-pinned mathematical source and certificate modules (2026), at commit 6baae4c15dccec1e94ac011de399ec382fcea7b6; https://github.com/463464q435q43/opt imizationproblems/tree/6baae4c15dccec1e94ac011de399ec382fcea7b6/certificates/cer tified0160/assembly_ce...

View on Pith 2026
[19]

Mosaic Intelligence,Collision-exclusion certificate for a reported boundΛ < 0.1595, publicly de- posited certificate package (2026), pinned commit cc8107854fedd5a96d7eb695178f405568712f35; https://github.com/463464q435q43/optimizationproblems/tree/cc8107854fedd5a96d7eb 695178f405568712f35/certificates/certified01595. Not a proof input to the present work

View on Pith 2026
[20]

E. W. Weisstein,de Bruijn–Newman Constant, MathWorld—A Wolfram Web Resource, consulted November 2026; https://mathworld.wolfram.com/deBruijn-NewmanConstant.html. Historical index only; individual claims are classified here using their primary sources

View on Pith 2026
[21]

M. Planat and D. J. Raymond,Reproducibility certificates for certified collision exclusion atΛ < 0.156821, GitHub release certificates-0156821-v1 (4 November 2026), commit 18dbc01ea36bf9a24d8f79822b5ce06b545e4ace; https://github.com/Dale- Dug/planat- r aymond-collision-exclusion-0156821/releases/tag/certificates-0156821-v1. Institut FEMTO-ST, CNRS UMR 617...

View on Pith 2026

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

N. G. de Bruijn,The roots of trigonometric integrals, Duke Math. J.17(1950), 197–226

View on Pith 1950

[2] [2]

C. M. Newman,Fourier transforms with only real zeros, Proc. Amer. Math. Soc.61(1976), 245–251

View on Pith 1976

[3] [3]

H. Ki, Y.-O. Kim and J. Lee,On the de Bruijn–Newman constant, Adv. Math.222(2009), 281–306; doi:10.1016/j.aim.2009.04.003

View on Pith doi:10.1016/j.aim.2009.04.003 2009

[4] [4]

B. Rodgers and T. Tao,The de Bruijn–Newman constant is non-negative, Forum Math. Pi8 (2020), e6;https://arxiv.org/abs/1801.05914

View on Pith Pith review arXiv 2020

[5] [5]

A. Dobner,A new proof of Newman ’s conjecture and a generalization, arXiv:2005.05142 (2020)

View on Pith arXiv 2020

[6] [6]

D. H. J. Polymath,Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn–Newman constant, Res. Math. Sci.6(2019), Article 31; doi:10.1007/s40687-019-0193-1. Theorem numbering as in arXiv:1904.12438v2

View on Pith Pith review arXiv 2019

[7] [7]

D. J. Platt and T. Trudgian,The Riemann hypothesis is true up to3 · 1012, Bull. Lond. Math. Soc. 53(2021), 792–797; doi:10.1112/blms.12460

View on Pith doi:10.1112/blms.12460 2021

[8] [8]

D. Romik,Orthogonal polynomial expansions for the Riemann xi function, arXiv:1902.06330 (2019); https://arxiv.org/abs/1902.06330

View on Pith Pith review arXiv 2019

[9] [9]

J. Gomila,A computer-assisted proof of the boundΛ ≤ 0.1787854for the de Bruijn–Newman constant, publicly released certificate manuscript and repository (20 August 2026), not peer reviewed AN UPPER BOUND Λ<0.156821 35 at the time of writing; https://github.com/judegomila/dbn-lambda-01787854-candidate-a udit

View on Pith 2026

[10] [10]

D. M. Kornaus,A computer-assisted unconditional upper boundΛ < 0.172422for the de Bruijn– Newman constant, self-archived manuscript and certification package, version 2 (September 2026); https://doi.org/10.5281/zenodo.22658444

View on Pith doi:10.5281/zenodo.22658444 2026

[11] [11]

S. Gordon,A computer-assisted upper bound for the de Bruijn–Newman constant, preprint (2026), SSRN:7608638;https://doi.org/10.2139/ssrn.7608638

View on Pith doi:10.2139/ssrn.7608638 2026

[12] [12]

C. Hakimoglu-Brown,A conditional boundΛ ≤ 0.13963for the de Bruijn–Newman constant, arXiv:2610.12803 (2026). Conditional on absence of zeta zeros with real part greater than 7/8

View on Pith Pith review arXiv 2026

[13] [13]

M. Planat,Collective contraction in the de Bruijn–Newman heat flow: Off-zero logarithmic derivatives and certified barrier refinements, arXiv:2609.37164 (2026); https://arxiv.org/abs/ 2609.37164

View on Pith Pith review arXiv 2026

[14] [14]

M. Planat,A genus-two spectral curve of the RiemannΞ-function: Real structure and certified arithmetic separation of periods, arXiv:2610.32694v1 (2026);https://arxiv.org/abs/2610.32694. The paper reports the separate bound Λ≤433/2500 = 0.1732

View on Pith Pith review arXiv 2026

[15] [15]

M. Planat,A certified classical-strip comparison for the de Bruijn–Newman constant, arXiv:2610.18214v1 (2026); https://arxiv.org/abs/2610.18214. Theorem 1.1 is conditional on its analytic premises (A1)–(A3), for which full validation is not supplied in that version

View on Pith Pith review arXiv 2026

[16] [16]

D. J. Raymond,Correlated translated probes and the boundΛ < 0.149396for the de Bruijn– Newman constant, public, unrefereed certificate manuscript (2026), version 1.1 with erratum; https://doi.org/10.5281/zenodo.23781981

View on Pith doi:10.5281/zenodo.23781981 2026

[17] [17]

D. J. Raymond,A computer-assisted boundΛ ≤ 0.107964910079for the de Bruijn–Newman constant, public, unrefereed certificate manuscript (2026); https://doi.org/10.5281/zenodo.23752103 . Its use of the newly asserted 7/8 zero-free region calls for separate scrutiny

View on Pith doi:10.5281/zenodo.23752103 2026

[18] [18]

Mosaic Intelligence,Collision-exclusion assembly and certified four-prime source construc- tions, version-pinned mathematical source and certificate modules (2026), at commit 6baae4c15dccec1e94ac011de399ec382fcea7b6; https://github.com/463464q435q43/opt imizationproblems/tree/6baae4c15dccec1e94ac011de399ec382fcea7b6/certificates/cer tified0160/assembly_ce...

View on Pith 2026

[19] [19]

Mosaic Intelligence,Collision-exclusion certificate for a reported boundΛ < 0.1595, publicly de- posited certificate package (2026), pinned commit cc8107854fedd5a96d7eb695178f405568712f35; https://github.com/463464q435q43/optimizationproblems/tree/cc8107854fedd5a96d7eb 695178f405568712f35/certificates/certified01595. Not a proof input to the present work

View on Pith 2026

[20] [20]

E. W. Weisstein,de Bruijn–Newman Constant, MathWorld—A Wolfram Web Resource, consulted November 2026; https://mathworld.wolfram.com/deBruijn-NewmanConstant.html. Historical index only; individual claims are classified here using their primary sources

View on Pith 2026

[21] [21]

M. Planat and D. J. Raymond,Reproducibility certificates for certified collision exclusion atΛ < 0.156821, GitHub release certificates-0156821-v1 (4 November 2026), commit 18dbc01ea36bf9a24d8f79822b5ce06b545e4ace; https://github.com/Dale- Dug/planat- r aymond-collision-exclusion-0156821/releases/tag/certificates-0156821-v1. Institut FEMTO-ST, CNRS UMR 617...

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