A Baseline $T\log^2 T$ Upper Bound for KL-Regularized Prime--Zero Optimal Transport

Zhejun Yang (University of Sydney)

REVIEW 3 major objections 3 minor 4 references

A Baseline $T\log^2 T$ Upper Bound for KL-Regularized Prime--Zero Optimal Transport

T0 review · 3 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper claims an unconditional T log^2 T upper bound for the KL-regularized prime–zero optimal transport cost, obtained by Fejér-averaged dual feasibility and an L1-controlled smoothed explicit formula.

desk verdict The Fejer-averaging idea is a nice twist, but the central bound collapses because the probe scaling is internally inconsistent and the main term in the explicit formula is never bounded. read the letter →

arxiv 2509.07329 v1 pith:7CLGPVY2 submitted 2025-09-09 math.NT math.CAmath.PR

Zhejun Yang (University of Sydney) This is my paper · ORCID

classification math.NTmath.CAmath.PR MSC 11M2611M0649Q22

keywords KL-regularizedoptimaltransportprimezeroszetaBeurling–SelbergkernelFejéraveragingsmoothedexplicitformulaunconditionalupperbound

topics The Riemann Hypothesis

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

The paper aims to prove that the KL-regularized optimal transport cost between two arithmetic measures—one built from prime log-density, one from zeros of the Riemann zeta function—grows at most like T log^2 T, with no assumption such as the Riemann Hypothesis. If correct, this is the natural baseline upper bound, matching the scale of known explicit-formula errors. The proof works by averaging the transport cost with a Fejér kernel, calibrating away the main density, and applying a smoothed explicit formula whose error is controlled by L1 norms of the test function and its Fourier transform. The result matters because it turns an optimal-transport question about zeros and primes into a quantitative counting statement that can be attacked with standard analytic number theory.

What carries the argument

The load-bearing identity is the two-way Fejér identity, 1 - hat F_Lambda(t) cos(gamma t) = integral of (1 - cos(xi t)) F_Lambda(gamma - xi) dxi. It shows that, after convolution with the Fejér kernel, the pointwise cost kernel is exactly an average of simpler cosine kernels, so a dual-feasibility inequality can be integrated instead of enforced pointwise. The companion machinery is the L1-controlled explicit formula, which asserts that the difference between the zero-side integral of the Fourier transform and the prime-side integral is a main term plus an error bounded by (||h||_1 + ||hat h||_1) log^2 T. A Beurling–Selberg kernel eta, equal to 1 on [0,T] with nonnegative Fourier transform,

What would settle it

For a concrete T, construct the probe h as in Sections 1–3 and separately compute the zero-side integral of its Fourier transform and the prime-side integral. If their difference exceeds a constant times T log^2 T, the bound fails. The same computation also tests Proposition 4.2's norm claims directly.

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Extended reading notes

Core claim

The central claim is Theorem 5.1: for the smoothed measures nu_T and mu_Omega with Omega = kappa T, the KL-regularized prime–zero transport cost satisfies OT_eta(T) << T log^2 T unconditionally. The author's route is to normalize the cost kernel to eta(t)(1 - cos gamma t), average over zero frequencies with a Fejér kernel so the cosine term becomes an average of cos(xi t), calibrate the probe's zero-frequency mass so the main density cancels, and then bound the residual by an L1-controlled smoothed explicit formula. On the paper's telling, the log^2 T factor comes from gamma/zero bookkeeping in the explicit formula, while the Paley–Wiener mass of the probe supplies the scale T. No use of RH

Load-bearing premise

The proof depends on the main contribution in the smoothed explicit formula being no larger than the L1 norms of the test function and its Fourier transform times log^2 T; the paper states this but never supplies the bound on that main contribution.

Editorial extensions

If this is right

The KL-regularized transport cost between prime-log-density and zeta-zero density is brought down to the same order as the classical counting-error scale, T log^2 T, rather than a larger power of T.
The Fejér-averaging step makes the dual inequality integrated, so the proof never needs a pointwise separable envelope for the cost kernel.
The zero-frequency calibration turns the main-density contribution into an explicit-formula difference, reducing the transport bound to a test-function estimate.
Because the bound is unconditional, it stands independently of the truth or falsity of the Riemann Hypothesis.
The L1 norms of the test function and its Fourier transform are the quantities that determine the final power of log T, suggesting a direct route to sharper bounds if those norms can be improved.

Reading between the lines

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

The same Fejér-averaging plus L1 explicit-formula scheme would likely transfer to other arithmetic measures with explicit formulas, such as primes in arithmetic progressions or Hecke eigenvalues, yielding analogous T log^2 T transport bounds.
A numerical evaluation of the explicit-formula difference for the constructed h at moderate T would show whether the log^2 T factor is saturated or an artifact of the bookkeeping estimate.
If the main-term contribution in the explicit formula is eventually shown to be always O((||h||_1 + ||hat h||_1) log^2 T), the theorem follows as stated; if not, the calibration argument would need modification. This is an editorial inference, not a result in the paper.

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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

The construction is genuinely new in spirit: Fejer-synchronizing the zero variable before taking the dual, then calibrating the zero frequency to kill the main density, is a clever way to turn a pointwise transport bound into an averaged one. The paper also correctly recognizes that an L1-controlled explicit formula would be the right tool. If the technical pieces worked, the T log^2 T bound would be a reasonable baseline result. So the paper is not empty.

The problem is that the technical pieces do not work. Proposition 4.2 is internally inconsistent with (1.5). Since \hat F_Λ is supported on [-Λ,Λ]=[-T^α,T^α], the function h(t)=η(t)f(t)\hat F_Λ(t) vanishes outside that interval. And the normalization S=∥\hat f∥_1≍T^{-1} forces |f(t)|≤S/(2π)=O(T^{-1}) on the whole real line. So ∥h∥_1≤C·Λ·T^{-1}=O(T^{α-1})=o(1), which contradicts the asserted ∥h∥_1≍T. This is not a subtle gap; the claimed scaling cannot hold for the very definitions in the paper.

Even if that were fixed, Theorem 5.1 has a second load-bearing problem. Proposition 4.1 is itself asserted without proof, and it gives the difference as M+E with E bounded by the L1 norms times log^2 T. The proof then silently drops M. No lemma bounds M in terms of the L1 norms or anything else. The zero-frequency calibration (3.1) only controls the integral against m, not the archimedean/main term M. So the final chain of inequalities is a non sequitur even granting (4.1).

The paper also does not compare against any prior prime-zero optimal transport work, which limits its novelty claim, but that is a minor issue compared to the proof failures. The citations themselves are fine; the references are standard and appropriate.

Who is this for? A reader curious about whether optimal transport methods can say anything about primes and zeros might find the setup worth a skim, but the paper does not establish the advertised result. My recommendation is to reject. The author is clearly thinking about an interesting problem, and the Fejer-averaging idea might be salvageable, but as it stands the central proof is internally unsound. A rewrite that proves the explicit formula and fixes the scaling would deserve another look.

Referee Report

3 major / 3 minor

Summary. The paper claims an unconditional upper bound OT_η(T) ≪ T log² T for a KL-regularized unbalanced optimal-transport cost between smoothed prime and zero distributions. The argument proceeds through a Fejér-averaged dual bound (Proposition 2.1), a zero-frequency calibration (Section 3), and an L¹-controlled smoothed explicit formula (Proposition 4.1) whose error is bounded by (∥h∥₁+∥ĥ∥₁)log² T. Theorem 5.1 combines these ingredients with Paley–Wiener size estimates (Proposition 4.2) to conclude the T log² T bound without assuming RH.

Significance. If the proof were correct, the result would be a notable baseline: an unconditional polynomial bound for a prime–zero transport distance, with the T log² T form obtained through explicit-formula error terms rather than RH. The Fejér synchronization identity and the averaged dual inequality are interesting and potentially reusable. However, the central analytic input is asserted rather than proved, the size estimates used to close the argument are internally inconsistent with the construction, and the main term in the explicit formula is never bounded. As it stands, the manuscript does not establish Theorem 5.1.

major comments (3)

[§4, Proposition 4.2 versus (1.5)] The asserted size ∥h∥₁ ≍ T is inconsistent with the definitions. Since ĥ₀? More precisely, h = η f RF_Λ, and RF_Λ(τ) = (1 - |τ|/Λ)_+ is supported on [-Λ, Λ] = [-T^α, T^α]. With S = ∥Rf∥₁ ≍ T^{-1}, the even probe f satisfies |f(t)| ≤ S/(2π) = O(T^{-1}) pointwise. With η bounded on the support of RF_Λ, this gives ∥h∥₁ = ∫ |η(t) f(t) RF_Λ(t)| dt ≤ C Λ S = O(T^{α-1}) = o(1). This contradicts (4.3), so the proof of Theorem 5.1 cannot invoke Proposition 4.2 to obtain a T log² T bound.
[§5, Theorem 5.1 proof, Eq. (4.1)–(4.2)] The proof drops the main term M(h;T,Ω) of Proposition 4.1 without any estimate. Proposition 4.1 gives ∫ĥ dμ_Ω - ∫h dν_T = M + E with |E| ≤ C(∥h∥₁+∥ĥ∥₁)log² T. The proof then writes the left side as O((∥h∥₁+∥ĥ∥₁)log² T). This is valid only if M is of the same order. No lemma or argument bounds M; the zero-frequency calibration (3.1) relates only S∫ηm and ∫hm, and does not control M. Thus the final inequality chain is a non sequitur.
[§4, Proposition 4.1 and §3, Corollary 3.1] Proposition 4.1 is the load-bearing analytic result: it is an L¹-controlled smoothed explicit formula with explicit error dependence on ∥h∥₁+∥ĥ∥₁ and uniformity in T. It is asserted without proof or a precise reference to an existing theorem that delivers this exact form. Moreover, Corollary 3.1 already invokes “the same L¹-controlled explicit formula” to bound the calibration error, before Proposition 4.1 has been stated or proved. This is not a presentation gap but a missing proof of the central estimate.

minor comments (3)

[§1, construction of η] The Beurling–Selberg majorant η is asserted to exist with all required properties (η=1 on [0,T], ĥη≥0, ∥ĥη∥₁ ≍ T, and L¹-controlled boundary ringing) but no construction or Vaaler reference is given with enough detail to verify these properties simultaneously. A precise lemma with constants would help.
[§3, Corollary 3.1 proof] The proof writes an explicit-formula error as O(∥h∥₁+∥ĥ∥₁) after stating it is ≪ (∥h∥₁+∥ĥ∥₁)log² T. The justification “under the final bound” is nonstandard; the log² T factor should be retained throughout or the absorption explicitly quantified.
[General notation] The primal definition of OT_η(T) is never stated. The reader must infer it from the dual formulation. Please include the Kantorovich-type definition, the role of η, and the normalizations, so that the claimed upper bound is unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected; the proof contains serious non-circular gaps (unbounded main term M, questionable L1 sizes).

full rationale

The derivation chain does not reduce the claimed bound to its own inputs by construction or by a load-bearing self-citation. The zero-frequency calibration in Section 3 is an explicit normalization (choosing the bump h0 and root c*) rather than a renamed version of the target bound; no fitted quantity is later called a prediction. Proposition 4.1 is stated without proof and leaves an explicit main term M(h;T,Omega) unestimated; the proof of Theorem 5.1 silently passes from 'difference = M + E' to 'difference = O((||h||_1+||ĥ||_1) log^2 T)' without bounding M. That is a non-sequitur, a missing proof, not a circular reduction: the conclusion does not follow from the stated inputs, but it is not logically identical to them. Similarly, Proposition 4.2's claim that ||h||_1 ≍ T appears inconsistent with the normalization S=||f̂||_1 ≍ T^{-1} and the support [-Λ,Λ] of F̂_Λ, since the bound |f(t)| ≤ S/(2π) would give ||h||_1 ≤ C Λ S = O(T^{α-1}); this is a size-estimate error, not a self-definitional equivalence. The paper cites Vaaler for the Beurling–Selberg kernel and standard texts for explicit-formula background; there is no self-citation chain that forces the result. These defects are correctness risks and would need to be fixed by proving Prop. 4.1, bounding M, and justifying the L1 sizes, but they do not constitute circularity under the stated rubric.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on an unproved L1-controlled explicit formula, an ad hoc calibration parameter, and an assumed Beurling-Selberg construction with specific L1 scalings. These are substantial inputs pulled from references or constructed within the paper without independent verification.

free parameters (4)

calibration amplitude c* = small root of L(c)=0, not computed
Section 3: chosen to enforce S * integral(eta m) = integral(h m); existence requires tuning xi_0 and a nonconstancy assumption.
frequency bump h0 and support width xi_0 = unspecified
Section 3: introduced ad hoc to make L affine and nonconstant; affects the calibration and the resulting probe.
smoothing exponent alpha = arbitrary in (0,1)
Section 1: Delta = T^{-alpha} sets the Fejer width Lambda = T^alpha; the bound is claimed uniform for fixed alpha but the construction depends on it.
zero cutoff kappa = fixed positive constant
Section 1: Omega = kappa T; bounds are claimed uniform for fixed kappa.

assumptions (5)

standard math Riemann-von Mangoldt formula N(Gamma) = (Gamma/2pi) log(Gamma/2pi) - Gamma/2pi + O(log Gamma)
Used in Lemma 1.1 to bound ||b||_{L1}.
standard math de la Vallee Poussin zero-free region
Used in Lemma 1.1 to get e^{-c sqrt(t)} decay in G(t).
domain assumption Existence of Beurling-Selberg majorant eta with eta=1 on [0,T], hat eta >= 0, and ||hat eta||_1 of order T
Invoked at (1.5) and Proposition 4.2; cited to Vaaler, but the L1-norm scaling is not proven in the text.
ad hoc to paper Proposition 4.1: L1-controlled explicit formula with |E| << (||h||_1 + ||hat h||_1) log^2 T
The central analytic engine; stated without proof and its main term M is left unbounded.
domain assumption Duality formulation of KL-regularized unbalanced optimal transport
Section 2 starts from the KFR dual without defining the primal problem or citing a theorem.

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Pith. "Pith review of A Baseline $T\log^2 T$ Upper Bound for KL-Regularized Prime--Zero Optimal Transport." pith.science (2026). https://pith.science/paper/7CLGPVY2

@misc{pith2026250907329,
  author       = {Pith},
  title        = {Pith review of: A Baseline $T\log^2 T$ Upper Bound for KL-Regularized Prime--Zero Optimal Transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CLGPVY2}},
  note         = {Machine review of arXiv:2509.07329}
}

abstract

We prove that $\mathsf{OT}_\eta(T)\ll T\log^2 T$ unconditionally via a band-limited test scheme with Fej\'er averaging. The approach normalizes $\|\widehat f\|_1=\Theta(T^{-1})$ to ensure $\|h\|_1\asymp T$ and $\|\widehat h\|_1\ll 1$, and applies an $L^1$-controlled smoothed explicit formula to bound the error terms. As a result, the prime--zero optimal transport distance admits the baseline $T\log^2 T$ upper bound without additional assumptions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

Chizat, G

L. Chizat, G. Peyr´ e, B. Schmitzer, and F.-X. Vialard, Unbalanced optimal transport: Dynamic and Kantorovich formulations,J. Funct. Anal.274(2018), no. 11, 3090–3123

work page 2018
[2]

Iwaniec and E

H. Iwaniec and E. Kowalski,Analytic Number Theory, AMS Colloquium Publications, Vol. 53, 2004

work page 2004
[3]

E. C. Titchmarsh (revised by D. R. Heath-Brown),The Theory of the Riemann Zeta- Function, 2nd ed., Oxford Univ. Press, 1986

work page 1986
[4]

J. D. Vaaler, Some extremal functions in Fourier analysis,Bull. Amer. Math. Soc. (N.S.) 12(1985), no. 2, 183–216. 5

work page 1985

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Chizat, G

L. Chizat, G. Peyr´ e, B. Schmitzer, and F.-X. Vialard, Unbalanced optimal transport: Dynamic and Kantorovich formulations,J. Funct. Anal.274(2018), no. 11, 3090–3123

work page 2018

[2] [2]

Iwaniec and E

H. Iwaniec and E. Kowalski,Analytic Number Theory, AMS Colloquium Publications, Vol. 53, 2004

work page 2004

[3] [3]

E. C. Titchmarsh (revised by D. R. Heath-Brown),The Theory of the Riemann Zeta- Function, 2nd ed., Oxford Univ. Press, 1986

work page 1986

[4] [4]

J. D. Vaaler, Some extremal functions in Fourier analysis,Bull. Amer. Math. Soc. (N.S.) 12(1985), no. 2, 183–216. 5

work page 1985