REVIEW
A certified classical-strip comparison for the de Bruijn-Newman constant
Number Theory · Submitted 13 Oct 2026 · 16 references
Pith's verdict: A careful, fully conditional numerical improvement that honestly wears its limitations on its sleeve — the value is in the transparent code and the finite Weyl refinement, not in a claim about the de Bruijn–Newman constant itself.
T0 · AI review
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Referee report
The paper
The paper claims
New bound on de Bruijn-Newman constant: ≤0.148611789321711, under three unproven premises
The reading
What carries the argument
The central object is the de Bruijn-Newman constant Λ, which controls the reality of zeros of the heat-deformed Riemann ξ function. The proof uses the collective contraction method: a highest-zero identity converts the derivative of the squared height of the highest nonreal zero into a sum over conjugate zero pairs involving the kernel K_h. Translated-probe estimates convert lower bounds on the logarithmic derivative S_t into a differential inequality q' < -2 - c q. The inequality is integrated via a directed comparison with upward restarts. The numerical gains come from a finite Weyl tail estimate (Section 7) that improves the source floor, and a centred cone transfer that preserves the existing phase-surplus credits. All 57 fixed kernels are independently checked by rational Bernstein certificates.
Extended reading notes
Core claim
Theorem 1.1: under premises (A1)-(A3) (source correspondence, finite localisation, and uniform contact reduction), the de Bruijn-Newman constant satisfies Λ ≤ 148611789321711 / $10^{15}$ = 0.148611789321711 < 0.148612. This is the smallest classical-strip bound certified in the paper, obtained by a directed comparison that starts from the classical zero strip |Im z| ≤ 1. The improvement over the previous two-source comparison is 0.000021760452379. The paper provides a fully certified rational arithmetic layer but notes that full analytic validation of the premises remains open.
Load-bearing premise
Premise (A1) – the complete model-to-function source correspondence – asserts that the lower bounds on the logarithmic derivative S_t from Raymond's construction hold uniformly on all required physical domains, including the unbounded frequency class. This premise is not proved in the paper; the paper's Table 2 explicitly lists 'independent end-to-end analytic validation remains open'.
Editorial extensions
If this is right
- If premises (A1)-(A3) are eventually validated, the bound Λ ≤ 0.148611789321711 becomes an unconditional record, improving on all previous classical-strip estimates.
- The numerical certificate, with its exact rational checks and independent auditor, provides a template for rigorous computation in analytic number theory.
- The directed comparison method can be reused with stronger source providers to obtain smaller ceilings.
- The finite Weyl envelope derived in Section 7 gives a self-contained method to bound oscillatory sums in heat-flow contexts without importing external exponent-pair constants.
Reading between the lines
- The most significant barrier to converting this bound into an unconditional theorem is the validation of premise (A1) – the complete model-to-function source correspondence. The paper itself states that 'independent end-to-end analytic validation remains open' (Table 2).
- The numerical gain of 0.000021760452379 from the finite Weyl refinement is small compared with the total reduction of 0.000445003232418 across the four controlled comparisons, suggesting that the main improvements came from the enlarged kernel catalogue and the second provider, not from the new tail estimate.
- The paper's extensive rational certification does not compensate for the missing analytic validation. The bound is an implication under premises that are not proven here.
Original abstract
We give a directed comparison beginning with the classical zero strip \(\lvert\Ima z\rvert\le1\) for the de Bruijn--Newman heat deformation. Under explicitly stated complete model-to-function source, localisation and global contact premises, translated-probe estimates imply \[ \Lambda\le\frac{148611789321711}{10^{15}} =0.148611789321711<0.148612. \] The construction uses Raymond's finite source data and 57 fixed translated kernels, independently checked by rational Bernstein certificates. An explicit finite Weyl estimate improves the positive first-head tail and its centred logarithmic moment. A centred cone transfer retains the original physical error allowances, exceptional debit and additive cone reserve, and charges the change in existing phase-surplus credits. Directed tail enclosures are regenerated at 192 and 256 bits; a separate exact-rational auditor accepts all 19,023 comparison steps. The controlled decrease from the complete two-source comparison is \(0.000021760452379\). We also prove a compatible matched logarithmic tail transfer and identify the finite requirements for stronger exponent pairs. A supplementary section records Raymond's ceiling below \(0.116398\) under the distinct \(7/8\) zero-free hypothesis.
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.
The abstract, sentence by sentence
We give a directed comparison beginning with the classical zero strip \(\lvert\Ima z\rvert\le1\) for the de Bruijn--Newman heat deformation. Under explicitly stated complete model-to-function source, localisation and global contact premises, translated-probe estimates imply \[ \Lambda\le\frac{148611789321711}{10^{15}} =0.148611789321711<0.148612. \] The construction uses Raymond's finite source data and 57 fixed translated kernels, independently checked by rational Bernstein certificates. An explicit finite Weyl estimate improves the positive first-head tail and its centred logarithmic moment. A centred cone transfer retains the original physical error allowances, exceptional debit and additive cone reserve, and charges the change in existing phase-surplus credits. Directed tail enclosures are regenerated at 192 and 256 bits; a separate exact-rational auditor accepts all 19,023 comparison steps. The controlled decrease from the complete two-source comparison is \(0.000021760452379\). We also prove a compatible matched logarithmic tail transfer and identify the finite requirements for stronger exponent pairs. A supplementary section records Raymond's ceiling below \(0.116398\) under the distinct \(7/8\) zero-free hypothesis.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We give a directed comparison beginning with the classical zero strip \(\lvert\Ima z\rvert\le1\) for the de Bruijn--Newman heat deformation. | contribution 0.69 | 0.79 | 0.04 | asserted 0.97 |
| s1 | Under explicitly stated complete model-to-function source, localisation and global contact premises, translated-probe estimates imply \[ \Lambda\le\frac{148611789321711}{10^{15}} =0.148611789321711<0.148612. | result 1.00 | 0.88 | 0.89 | asserted 1.00 |
| s2 | \] The construction uses Raymond's finite source data and 57 fixed translated kernels, independently checked by rational Bernstein certificates. | method 1.00 | 0.67 | 0.04 | asserted 0.99 |
| s3 | An explicit finite Weyl estimate improves the positive first-head tail and its centred logarithmic moment. | contribution 0.42 | 0.76 | 0.03 | asserted 1.00 |
| s4 | A centred cone transfer retains the original physical error allowances, exceptional debit and additive cone reserve, and charges the change in existing phase-surplus credits. | method 0.96 | 0.53 | 0.02 | asserted 0.96 |
| s5 | Directed tail enclosures are regenerated at 192 and 256 bits; a separate exact-rational auditor accepts all 19,023 comparison steps. | method 0.99 | 0.83 | 0.03 | asserted 1.00 |
| s6 | The controlled decrease from the complete two-source comparison is \(0.000021760452379\). | result 0.98 | 0.85 | 0.05 | asserted 1.00 |
| s7 | We also prove a compatible matched logarithmic tail transfer and identify the finite requirements for stronger exponent pairs. | contribution 0.76 | 0.90 | 0.04 | asserted 1.00 |
| s8 | A supplementary section records Raymond's ceiling below \(0.116398\) under the distinct \(7/8\) zero-free hypothesis. | result 0.77 | 0.58 | 0.02 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v1
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- T body sentences
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References
Discussion
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