Kernel positivity, zero-flow entropy, Li–Keiper moment theory, and extremal Paley–Wiener mollifiers

Abstract

We present a consolidated research note articulating four complementary proof programs aimed at the Riemann Hypothesis (RH). Each program is carried to a point where a precise, sharply isolated decisive gap remains. All preliminaries and leverage points are included; intermediate lemmas, constant choices, and operator normalizations are made explicit to the extent possible without sacrificing rigor. The four programs are:

Program A (Kernel positivity via de Branges / Pontryagin dilation): Construct a matrix-valued Fourier representation of a de Branges-type kernel for a regularized Ξ and reduce RH to a finite-index positivity upgrade—a matrix Fejér–Riesz–type factorization for the theta kernel.

Program B (Lyapunov/entropy for the de Bruijn–Newman zero flow): Derive a global differential inequality for an explicit log-gas entropy along the heat flow ∂tΞt=∂x2Ξt. Prove that a bounded entropy forbids zero collisions as t↓0, which would force Λ≤0 and hence RH (given Λ≥0).

Program C (Li–Keiper coefficients as a trigonometric moment problem): Strengthen Li’s criterion from scalar inequalities λn≥0 to an infinite family of shifted Hankel positivity constraints. Prove RH ⇒ all such Hankel matrices are psd. Isolate the converse as a specific link from shifted Hankel positivity to Toeplitz/Herglotz positivity on T.

Program D (Extremal Paley–Wiener mollifiers): Pose and solve a convex variational problem for the optimal (bandlimited) mollifier in the smoothed second moment of ζ on the critical line. Show that the Euler–Lagrange solution is a projected Green’s function of an explicit positive integral operator. Completing the unconditional off-diagonal error analysis would yield improved zero-density bounds and a larger on-line zero proportion.

Each program is accompanied by computationally checkable milestones and clearly identified decisive gaps (DG-A, DG-B, DG-C, DG-D). The manuscript is self-contained: we fix notation and collect explicit preliminaries (Hadamard product, cosine representation, Riemann–von Mangoldt, explicit formula skeletons) to support the derivations.

0. Preliminaries, notation, and standard tools

0.1. Completed ξ and Ξ, critical line normalization

Let

ξ(s):=21s(s−1)π−s/2Γ(2s)ζ(s),Ξ(t):=ξ(21+it).

Then ξ is entire of order 1, satisfies ξ(s)=ξ(1−s), and Ξ is real, even, entire. Nontrivial zeros ρ of ζ (counted with multiplicity) are the zeros of ξ in 0<ℜs<1; ρ=21+iγ on RH. We frequently write γ=ℑρ.

Hadamard product. There exist real constants A,B such that

ξ(s)=eA+Bsρ∏(1−ρs)exp(ρs).

Riemann–von Mangoldt. With N(T):=#{ρ:0<ℑρ≤T},

N(T)=2πTlog2πT−2πT+O(logT).

0.2. Cosine transform and theta-kernel

There exists a smooth, rapidly decaying Φ:(0,∞)→R (explicitly expressible in terms of Jacobi theta-data) such that

Ξ(t)=∫0∞Φ(u)cos(tu)du.

The function Φ changes sign but is Schwartz.

0.3. Explicit formula (Weil–Guinand skeleton)

For test functions g with suitable decay and Fourier transform g^, an explicit formula equates sums over zeros and primes:

ρ∑g(iρ−21)=(polar terms) − n=1∑∞nΛ(n)(g^(logn)+g^(−logn)) + (archimedean).

We use it in two modes: (i) control of linear statistics of zeros with compactly supported g^, and (ii) archimedean tail analysis via Stirling’s formula for Γ(s/2).

0.4. de Bruijn–Newman deformation

Define

Ξt(x):=∫0∞etu2Φ(u)cos(xu)du(t∈R).

There exists Λ∈R such that Ξt has only real zeros iff t≥Λ. It is known that Λ≥0. RH ⟺Λ=0.

0.5. Li–Keiper coefficients

Define

λn:=(n−1)!1dsndn[sn−1logξ(s)]s=1.

Li’s criterion: RH ⟺λn≥0 for all n≥1. Moreover,

λn=ρ∑(1−(1−ρ1)n),

with symmetric pairing ensuring absolute convergence for fixed n.

1. Leverage points and equivalent formulations

We record two leverage statements frequently invoked below.

(L1) de Bruijn–Newman equivalence. RH ⟺Λ≤0. Together with Λ≥0, RH is equivalent to Λ=0.

(L2) Li positivity and pushforward. Writing T(z):=1−z1 and ν:=∑ρδT(ρ), one can rewrite

λn=∫(1−ℜ(wn))dν(w).

On RH, ν is supported on the unit circle T, so {λn} are cosine moments of a positive measure on T (up to an affine normalization).

2. Program A — Kernel positivity via de Branges and Pontryagin dilation

2.1. Regularized entire function and de Branges kernel

Fix an integer m≥2 and define the even entire function

F(z):=(41+z2)mΞ(z).

The low-frequency regularization by (41+z2)−m (with m≥2) is used to offset a finite negative index in an indefinite kernel below.

Define, for real x,y,

K(x,y):=x−yF(x)F(y)−F(−x)F(−y).

If K is positive semidefinite on R, the de Branges theory forces F (hence Ξ) to be hyperbolic (all real zeros).

Goal A-main. Prove K⪰0 on R for some fixed m∈{2,3}. This implies RH.

2.2. Fourier lifting and a matrix-valued measure

Using Ξ(x)=∫0∞Φ(u)cos(ux)du, one has

F(x)=∫0∞(41+u2)mΦ(u)cos(ux)du(x∈R).

Let vu(x):=[cos(ux),sin(ux)]⊤. A direct computation—using cos(−ux)=cos(ux), sin(−ux)=−sin(ux), and polarization—yields

K(x,y)=∫0∞vu(x)⊤dμ(u)vu(y),dμ(u):=(41+u2)mΦ(u)(0uu0)du.

Thus K is the Fourier–Bochner transform of a signed 2×2 matrix measure dμ on (0,∞).

Observation A.1 (finite negative index). Let V be the linear span of functions u↦vu(⋅) with the quadratic form induced by dμ. There exists κ=κ(m)<∞ such that this form has at most κ negative directions (equivalently, the corresponding reproducing kernel space is a Pontryagin space Πκ).

Reason. Φ is Schwartz and changes sign on a set of finite total variation; the weight (41+u2)−m with m≥2 is integrable at 0 and ∞. Negative mass is confined to finitely many small subintervals with rapidly decaying contribution; a compact-perturbation argument bounds the negative index.

2.3. Dilating to a positive kernel

Lemma A.2 (Kreĭn–Langer dilation). Let K be a reproducing kernel of a Pontryagin space Πκ. There exist polynomials P1,…,Pκ such that the augmented kernel

Kaug(x,y):=K(x,y)+j=1∑κPj(x)Pj(y)

is positive semidefinite on R×R.

Sketch. Standard: any finite-index indefinite kernel is the compression of a Hilbert-space positive kernel after adjoining κ negative directions and resolving them by a finite-rank correction. The augmentation by rank-one psd kernels achieves this at the level of scalar kernels. □

Proposition A.3 (hyperbolicity from psd kernel). Suppose Kaug⪰0 and the augmentation is finite rank with polynomial generators. Then the associated entire function F is hyperbolic (all zeros real).

Reason. The psd kernel identifies a de Branges space whose structure function’s real-zero property is equivalent to kernel positivity; the finite augmentation preserves zero locations while removing the negative index at the kernel level.

2.4. Decisive positivity step

The remaining issue is whether augmentation is actually needed and, if so, whether it affects the zero set. We isolate two equivalent forms of the decisive gap.

DG-A1 (matrix Fejér–Riesz factorization). Prove that for some fixed m∈{2,3}

(41+u2)mΦ(u)(0uu0)=a(u)a(u)⊤ du,a(u)=[a(u)b(u)],

with real a,b and a(u)2≥b(u)2 for almost every u. This would give a positive matrix measure, hence K⪰0 with no augmentation, yielding RH.

DG-A2 (augmentation neutrality). Alternatively, show that the finite psd augmentation in Lemma A.2 does not change the zero set of the canonical entire function associated to K. Establishing augmentation neutrality allows one to conclude hyperbolicity of F (hence RH) even if dμ is only almost positive (finite negative index).

Either route resolves Program A.

2.5. Milestones (computational/theoretical)

A.0 (rigorous numerics): Compute K(xi,xj) on large symmetric grids ∣xi∣≤X, certify psd up to interval arithmetic errors stemming from truncating the theta expansion for Φ. This stress-tests the feasibility of DG-A1.

A.1 (smoothed case): Convolve Φ with a small even, real-analytic, compactly supported positive mollifier. Prove positivity for the smoothed kernel (de Bruijn’s theorem guarantees hyperbolicity for sufficiently large heat time). Then attempt a limiting argument as the mollifier shrinks, keeping control of the index.

A.2 (Pontryagin index bound): Prove that, for m=2, the negative index κ is at most 2. A tiny κ narrows the scope of augmentation neutrality needed in DG-A2.

3. Program B — A Lyapunov/entropy inequality for the zero flow

3.1. Zero dynamics under ∂tΞt=∂x2Ξt

Let H(x,t):=Ξt(x). For t>Λ, all zeros {γj(t)}j∈Z are real and simple. The standard identity for heat-deformed real entire functions gives

dtdγj(t)=−∂xH∂tH(γj,t)=−∂xH∂x2H(γj,t).

Writing a (regularized) Hadamard product and differentiating logH twice in x yields

dtdγj(t)=2k=j∑γj(t)−γk(t)1−B′(γj(t),t). (B.1)

Here B′(⋅,t) is a smooth “background” drift originating from the outer factors (notably the Γ-factor) under the deformation.

Lemma B.1 (archimedean drift bound). There exist absolute constants c1,c2>0 such that for all t≥Λ and ∣x∣≥2,

∣B′(x,t)∣ ≤ c1log(2+∣x∣)+c2.(B.2)

Sketch. B′ is the x-derivative of the smooth part of logH, controlled via Stirling’s approximation for Γ(41+2ix) together with stability under the heat deformation. The derivative introduces at most logarithmic growth. □

3.2. Entropy functional and dissipation

Choose a confining potential

ψ(x):=2αx2+βlog(1+x2),α>0, β>1,

and define the log-gas energy

E({γj}):=j∑ψ(γj) − j<k∑log(γk−γj).(B.3)

With F(t):=E({γj(t)}), formal differentiation using (B.1) yields

Proposition B.2 (energy dissipation identity).

dtdF(t)=−2j=k∑(γk−γj)21 + R(t),(B.4)

where

R(t):=−j∑ψ′(γj)B′(γj,t) + j<k∑γk−γjB′(γk,t)−B′(γj,t).(B.5)

Derivation. Expand γ˙ into pairwise and background parts; the symmetric pairwise contribution reduces to the negative quadratic form in the gaps (γk−γj)−2; the remainder aggregates as R. Truncation to ∣γj∣≤R and passage R→∞ is justified in §3.4. □

Lemma B.3 (uniform background control). There exist α,β and C0>0 such that for all t≥Λ,

R(t) ≤ C0.(B.6)

Heuristic. The second term is a discrete Hilbert transform of B′, which is nearly antisymmetric across symmetric configurations; the first term is dominated by ψ′(γ)B′(γ,t) with ψ′′≥α and ∣B′∣≲log(2+∣x∣). The choice β>1 handles the logarithmic tails using the Riemann–von Mangoldt zero density. A full proof requires the tail estimates of §3.4. □

Combining,

Theorem B.4 (Lyapunov inequality — formal). For suitable α,β and some C0>0,

dtdF(t) ≤ −j=k∑(γj(t)−γk(t))22 + C0. (B.7)

Lemma B.5 (no collision under bounded entropy). If supt∈[0,T]F(t)<∞, then no collisions (coalescing zeros) occur for t∈[0,T].

Proof. Any collision forces ∑j=k(γj−γk)−2→∞, which by (B.7) would drive F→−∞, contradicting boundedness. □

3.3. Backward continuation to t=0

For t0>Λ, F(t0)<∞. Integrating (B.7) backward for t∈[0,t0]:

F(t) ≤ F(t0) − ∫tt0D(s)ds + C0(t0−t),D(s):=j=k∑(γj(s)−γk(s))22.

If F remains bounded down to t=0, Lemma B.5 forbids collisions, so the zeros stay real for t∈[0,t0]. Hence Λ≤0. Since Λ≥0 is known, this yields Λ=0, i.e., RH.

3.4. Decisive gaps for full rigor

DG-B1 (precise bound for B′). Establish (B.2) with explicit constants uniform in t≥Λ for the de Bruijn–Newman deformation of Ξ.

DG-B2 (truncation/limit). Justify the passage from truncated, finite sums to the full configuration: control tails of F and R using Riemann–von Mangoldt and ψ′′≥α. Show F is locally Lipschitz in t.

DG-B3 (uniform bound on R). Prove the discrete Hilbert transform term in (B.5) is uniformly bounded after tail renormalization by βlog(1+x2), producing a finite C0 in (B.6).

Meeting DG-B1–B3 closes Program B and forces Λ=0.

3.5. Milestones

B.0 (rigorous numerics): For band-limited truncations Ξt(N), track zeros and certify that dtdF(t)≤0 across t∈[t1,t2]⊂(0,∞) with interval ODE error bounds.

B.1 (local bound): Prove (B.7) for configurations truncated to ∣γ∣≤R with constants independent of R.

B.2 (globalization): Pass R→∞ using explicit zero density and Stirling bounds to conclude DG-B2–B3.

4. Program C — Li–Keiper as a moment/Hankel positivity theory

4.1. Pushforward representation

Define the Möbius transform T(z)=1−z1 and the pushforward measure ν:=∑ρδT(ρ). Then

λn=∫(1−ℜ(wn))dν(w).(C.1)

On RH, T(21+iγ)∈T, hence {λn} are (up to a trivial shift) cosine moments of a positive measure σ on T.

4.2. Shifted Hankel matrices and forward implication

Let mk:=λk for k≥1. For r≥0, define HN(r):=[mi+j+r]0≤i,j≤N.

Proposition C.1 (RH ⇒ Hankel psd). If RH holds, then HN(r)⪰0 for all r,N.

Proof. Under RH, there exists a finite positive measure σ on [0,2π) with

mk=∫(1−coskθ)dσ(θ).

Then for any vector α=(α0,…,αN),

i,j=0∑Nαiαjmi+j+r=∫i=0∑Nαiei(i+r)θ/22dσ(θ) ≥0.

□

Finite-rank consequences. If HN(r)⪰0 holds for all 0≤r≤R up to some large N, then the set of permissible off-critical zeros is constrained by a finite system of trigonometric moment inequalities. This yields explicit, checkable near-RH domains in the half-strip 1/2<σ<1.

4.3. Toward the converse

Let L(z):=∑n≥1λnzn, with radius 1. Formally (on RH),

L(z)=∫1−ζzz(2−ζ−ζ−1z)dσ(ζ)=∫ℜ(1−ζz1+ζz)dσ(ζ) − const.(C.2)

Thus (up to a shift) L is a Herglotz function on the unit disk: ℜL(z)≥0.

Lemma C.2 (Hamburger/Carathéodory on T). If both the shifted Hankel matrices HN(r) and the Toeplitz matrices TN=[m∣i−j∣]0≤i,j≤N are psd for all N, then there exists a positive measure σ on T with mk=∫(1−coskθ)dσ(θ).

Sketch. Hankel positivity guarantees a positive measure on [−1,1] (cosine moment problem). Toeplitz positivity upgrades support to T (trigonometric moment problem) and produces a Herglotz representation for L. □

We do not currently know that HN(r)⪰0 for all r,N alone implies Toeplitz positivity.

4.4. Decisive gap for Program C

DG-C1 (Herglotz from shifted Hankels). Prove that the family {HN(r)⪰0 ∀r,N} implies ℜL(z)≥0 on D after a fixed affine normalization. This would directly deliver a representing positive measure on T and hence RH.

DG-C2 (Hankel-to-Toeplitz positivity transform). Construct an explicit linear operator that maps the infinite block-Hankel moment array {mi+j+r} to the Toeplitz array {m∣i−j∣} while preserving positivity. Such a transform would close the circle: positivity of all shifted Hankels ⇒ Toeplitz positivity ⇒ Herglotz on T ⇒ RH.

4.5. Milestones

C.0 (rigorous verification): Compute λn with certified enclosures (using verified zero tables up to height T and explicit tail bounds) and check HN(r)⪰0 for large N,r (e.g., N=2000, r≤10). This strictly strengthens known Li-positivity checks.

C.1 (finite-rank converse): Prove that positivity of HN(r) for 0≤r≤R implies Toeplitz positivity for TN up to some N′=N′(R). This yields concrete, finite-dimensional necessary conditions squeezing off-line zeros.

5. Program D — Extremal Paley–Wiener mollifiers

5.1. Smoothed second moment and kernelization

Let W∈Cc∞([1,2]), W≥0. For a mollifier of length Tϑ, 0<ϑ<1,

M(s)=n≤Tϑ∑nsμ(n)a(n),a(1)=1,

consider

M[T;M]=∫T2T∣M(21+it)ζ(21+it)∣2W(t/T)dt.

Using the approximate functional equation for ζ and standard mean-value manipulations,

M[T;M]=T(⟨a,Kϑa⟩+E[T;ϑ]),(D.1)

where a=(a(n))n≤Tϑ, Kϑ is an explicit positive kernel

⟨a,Kϑa⟩=n,m≤Tϑ∑nma(n)a(m)(W(0)1n=m+logT1K(logTlogn,logTlogm)+⋯),(D.2)

and E[T;ϑ]=o(1) as T→∞ for fixed ϑ. The smooth kernel K encodes the archimedean Γ-weights of ξ and the support of W.

Variational problem (discrete). Minimize ⟨a,Kϑa⟩ over a supported on n≤Tϑ with a(1)=1.

5.2. Mellin reduction and Paley–Wiener constraint

Pass to x=logTlogn∈[0,ϑ] and define fT(x):=a(Tx). Then (modulo discretization error)

⟨a,Kϑa⟩ ≈ ∬[0,ϑ]2fT(x)K(x,y)fT(y)dxdy,

where K is a continuous psd kernel derived from K. In the limit T→∞ we get a continuous convex problem:

Problem D-cont. Minimize Q[f]:=∬f(x)K(x,y)f(y)dxdy on L2[0,ϑ] subject to

(i) f(0)=1 (normalization), and

(ii) Paley–Wiener bandlimit: f is the restriction of a function whose Fourier transform in x is supported in the admissible window dictated by the approximate functional equation.

Lemma D.1 (existence/uniqueness). Q is strictly convex on the Paley–Wiener subspace; there is a unique minimizer f⋆.

Euler–Lagrange equation. There exists a scalar λ such that

∫0ϑK(x,y)f⋆(y)dy = λδ0(x)in D′([0,ϑ]),(D.3)

together with the bandlimit constraint.

Proposition D.2 (extremizer as projected Green’s function). Let K denote the integral operator with kernel K, and Π the orthogonal projector onto the Paley–Wiener subspace PW. Then

f⋆ = c ΠK−1δ0,(D.4)

with c determined by f⋆(0)=1. This is canonical and computable (analytically in model weights, numerically with certification).

5.3. Consequences for zeros

Classical Levinson–Conrey arguments turn an upper bound on M[T;M] into a lower bound on the proportion κ of zeros on ℜs=21. The extremal f⋆ achieves the optimal value within the admissible Paley–Wiener class. Similarly, Montgomery-type methods with f⋆ yield improved exponents A⋆(σ) in zero-density bounds

N(σ,T) ≪ T2(1−σ)(logT)A⋆(σ).

5.4. Decisive gaps for unconditional gains

DG-D1 (error term control). Match the discrete second-moment analysis to the continuous K with unconditional control of off-diagonal errors uniform in the bandlimit. This ensures that the continuous extremum translates to a true gain.

DG-D2 (certified inversion). Invert K and apply Π with rigorous error bounds; propagate through the Levinson–Conrey machinery (which introduces derivatives of Mζ).

DG-D3 (uniformity across shifted convolutions). Show that spectral large-sieve/shifted-convolution error terms do not erase the advantage—i.e., the gain survives to the final inequality for κ or A(σ).

5.5. Milestones

D.0 (computation): Solve (D.3) numerically for representative W and ϑ, compare M[T;M] against classical mollifiers, and document a provable improvement at the continuous level.

D.1 (discretization): Prove a stability lemma showing that for T large, the discrete optimizer a is close (in the operator norm induced by Kϑ) to the sampled f⋆.

6. Concrete tasks that yield publishable partial results

Matrix positivity (C) at scale. Compute λn with certified error bounds and verify HN(r)⪰0 for large N and modest r. This would be the strongest verified Li-type (now matrix) positivity to date.

Kernel positivity (A) on grids. Evaluate K(xi,xj) with certified truncation error from the theta expansion; test psd on large grids and attempt a posteriori index estimates (upper bounds on κ).

Entropy monotonicity (B) for truncations. For Ξt(N), track zeros and compute F numerically with interval control; verify dtdF≤0 on macroscopic t-intervals.

Extremal mollifier (D). Solve (D.3) in Mellin variables, then plug into second-moment asymptotics with rigorous remainder control to obtain an explicit improved κnew or smaller A⋆(σ).

7. Risks, failure modes, and how the programs can still pay off

A (kernel positivity). It may be that no fixed m makes dμ positive (DG-A1). In that case, bounding the negative index κ and proving augmentation neutrality (DG-A2) still constitutes a nontrivial operator-theoretic advance linking theta-kernels to de Branges spaces.

B (entropy). If (B.6) cannot be proved globally, one can aim for a quantitative upper bound Λ≤ε, sharpening current estimates.

C (moment theory). Even if DG-C1/C2 is out of reach, large-scale verification of shifted Hankel positivity yields new finite necessary conditions: a stronger diagnostic than scalar Li positivity.

D (mollifiers). The variational formalism is robust; even modest unconditional improvements would be notable. The extremal object f⋆ is itself of independent interest.

Appendix A. Explicit forms and bounds used repeatedly

A.1. Theta-kernel Φ

One convenient representation (among several equivalent forms) is

Φ(u)=n=1∑∞(2π2n4e29u−3πn2e25u)e−πn2e2u+n=1∑∞(2π2n4e−29u−3πn2e−25u)e−πn2e−2u,

symmetrized to ensure Φ is even in u. This exhibits rapid decay ∣Φ(u)∣≪e−ce2∣u∣.

A.2. Archimedean term and Stirling

Stirling’s formula for Γ(σ+it) in vertical strips:

logΓ(σ+it)=(σ−21+it)log∣t∣−2π∣t∣+O(1)(uniform in σ).

Differentiating gives x-derivatives for B′ with at most logarithmic growth, supporting Lemma B.1.

A.3. Explicit formula test functions

For ψ with ψ′′≥α>0 and ψ′(x)=αx+O(log(1+∣x∣)), the linear statistics ∑jψ′(γj) are controlled by smoothed prime sums plus archimedean terms; the latter induce the O(log∣x∣) drift handled in (B.6).

A.4. Paley–Wiener spaces

Let PWΩ:={f∈L2(R):f(ξ)=0 for ∣ξ∣>Ω}. The admissible bandlimit Ω=Ω(ϑ,W) arises from the frequency window of the approximate functional equation and the support of W. The projector Π is convolution with the sinc kernel corresponding to Ω, restricted to [0,ϑ] (with boundary handling by reflection or zero-extension depending on the chosen model).

Appendix B. Derivations deferred from the main text

B.1. Derivation of the zero ODE (B.1)

Let H(⋅,t) be entire with Hadamard product

H(x,t)=ea(t)+b(t)xk∏(1−γk(t)x)exp(γk(t)x).

Then ∂xlogH=b(t)+∑k(γk−x−1+γk1). Differentiating,

∂x2logH=k∑(γk−x)2−1.

At a simple zero x=γj, ∂x2H/∂xH=∂x2logH+(∂xlogH)2 simplifies (by cancellation of principal parts) to −2∑k=j(γj−γk)−1+B′(γj,t), where B′ is the x-derivative of b(t)+∑kγk1 and other smooth parts. Using ∂tH=∂x2H yields (B.1).

B.2. From K to de Branges

Given an entire function E(z) of Hermite–Biehler type, the de Branges kernel is

KE(z,w)=2πi(w−z)E(z)E(w)−E#(z)E#(w).

For even F, one can adapt this with E(z)=F(z)+iF\*(z) for a suitable real entire F\* (a Hilbert transform–type conjugate) to match K up to a multiplicative constant. Positivity of KE implies the zeros of E lie on the real axis; hence zeros of F are real.

B.3. Hankel vs Toeplitz positivity

If mk=∫(1−coskθ)dσ(θ), then both shifted Hankel and Toeplitz matrices are psd. Conversely, Toeplitz positivity alone implies a Herglotz function on the disk. What is missing is a general theorem that shifted Hankel positivity for all shifts r yields Toeplitz positivity without extra assumptions; this is the content of DG-C2.

References (standard sources for tools invoked)

de Branges, L. Hilbert Spaces of Entire Functions. (1968).

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

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

Li, X. J. The positivity of a sequence of numbers and the Riemann hypothesis. J. Number Theory 65 (1997), 325–333.

Titchmarsh, E. C. The Theory of the Riemann Zeta-Function (2nd ed., Heath-Brown).

Montgomery, H. L. The pair correlation of zeros of the zeta function. Analytic Number Theory, Proc. Sympos. Pure Math., Vol. 24 (1973).

Conrey, J. B. More than two fifths of the zeros… J. Reine Angew. Math. 399 (1989), 1–26.

Levinson, N. More than one third of zeros… Proc. London Math. Soc. 3 (1974).

Guinand, A. P. A problem in the theory of the Riemann zeta-function. Messenger Math. 86 (1957), 56–63.

Decisive gap index (summary)

DG-A1 (Program A): Matrix Fejér–Riesz positivity for the regularized theta kernel after cos/sin lifting; prove dμ⪰0 for some fixed m.

DG-A2 (Program A): Prove that finite psd augmentation by polynomials leaves the canonical zero set unchanged (augmentation neutrality).

DG-B1 (Program B): Uniform Stirling-type bound for the background drift B′(x,t) under de Bruijn–Newman flow.

DG-B2 (Program B): Truncation-to-infinite configuration limit with tail control for F and R.

DG-B3 (Program B): Uniform bound R(t)≤C0 from discrete Hilbert transform cancellation plus confinement.

DG-C1 (Program C): Show shifted Hankel positivity for all r,N implies ℜL(z)≥0 on D (Herglotz).

DG-C2 (Program C): Positivity-preserving transform from the shifted Hankel family to the Toeplitz family on T.

DG-D1 (Program D): Unconditional reconciliation of the discrete second moment with the continuous kernel K at the level of error terms.

DG-D2 (Program D): Certified inversion of K and projection Π with propagation through Levinson–Conrey.

DG-D3 (Program D): Uniformity of shifted-convolution/large-sieve bounds ensuring the extremal gain survives to κ or A(σ).

Closing remark

Each program isolates a concrete, sharply posed mathematical statement whose resolution would either prove RH (A/B/C) or produce unconditional advances toward it (D). The kernels, flows, and moment arrays specified here are explicit; the constants (m,α,β,C0) can be fixed and optimized within the frameworks above without changing the form of the decisive gaps.

(http://www.autoadmit.com/thread.php?thread_id=5771467&forum_id=2Vannesa#49248634)