Snapshot: 8b10ede734258e9e966f6b889176d81a4d59964d Verdict: Outcome A. The full Hilbert-space window optimization is a coercive quadratic minimization, and its unique optimizing ray has a strictly positive representative.
This note closes the functional-analytic obligation left implicit in RESULTS-xiprime.md. It does not change the source derivation of the functional or supply an interval enclosure for the reported decimal.
Theorem
Let
\[ I=[-1/2,1/2],\qquad \mathcal H=L^2_{\mathrm{even}}(I;\mathbb R) \]
with inner product \(\langle f,g\rangle=\int_I f(s)g(s)\,ds\). For \(|x|\leq1\), put
\[ F_1(x)=|x|-4x^2+ \sum_{k=1}^{\infty}\frac{(k-1)!}{(2k)!}(2|x|)^{2k+1}, \]
and define
\[ (T_{F_1}v)(s)=\int_I F_1(s-t)v(t)\,dt,\qquad A=I+T_{F_1}. \]
Then:
- \(T_{F_1}\) is a bounded compact self-adjoint operator on \(\mathcal H\), and \(\|T_{F_1}\|_{2\to2}\leq4/9\).
- \(A\) is bounded and self-adjoint, and \[ \langle Av,v\rangle\geq\frac59\|v\|2^2 \quad(v\in\mathcal H). \] In particular, \(A^{-1}\) exists on all of \(\mathcal H\) and \(\|A^{-1}\|{2\to2}\leq9/5\).
- If \(w=A^{-1}\mathbf1\), then \(w\) has a continuous even representative satisfying \[ w(s)\geq\frac15\qquad(s\in I). \]
- The exact global value of the scale-one window quotient is \[ c_1^*=\sup_{\substack{0\ne v\in\mathcal H\\v\geq0\ \mathrm{a.e.}}} \frac{\left(\int_Iv\right)^2} {\int_Iv^2+\iint_{I^2}F_1(s-t)v(s)v(t)\,ds\,dt} =\langle\mathbf1,A^{-1}\mathbf1\rangle. \] Equality holds exactly on the positive rays \(v=cw\), \(c>0\).
- For the source functional \[ c_\lambda(v)= \frac{\lambda(\int_Iv)^2} {\int_Iv^2+\lambda\iint_{I^2} F_1(\lambda(s-t))v(s)v(t)\,ds\,dt}, \qquad0<\lambda\leq1, \] the optimized value \(c_\lambda^*\) is nondecreasing in \(\lambda\). Hence the global bandwidth-one optimum occurs at \(\lambda=1\), and its optimizing profile is proportional to \(A^{-1}\mathbf1\).
Equality with the source paper's physical-window class is not a consequence of density in the whole nonnegative cone: Section 7.1 also requires radial monotonicity. RESULTS-xiprime-admissible-closure.md discharges that missing condition. It gives an exact-rational strict-concavity bound for \(w=A^{-1}\mathbf1\), proves that \(w\) decreases strictly in \(|s|\), and constructs source-admissible endpoint tapers converging to \(w\) in \(L^2\).
Farmer, Gonek, and Lee state the form-factor asymptotic for \(|x|<1\), and Chirre, Goncalves, and de Laat give the uniformly convergent infinite-series form on compact subintervals. The value at \(|x|=1\) above is its continuous extension. Changing a kernel at that measure-zero boundary would not change the quadratic form, and Remark 6.1 of the source paper separately admits the bandwidth endpoint \(\lambda=1\).
Proof
The series for \(F_1\) and its termwise integral converge uniformly on \([-1,1]\). For \(0\leq x\leq1\), isolate the \(k=1\) term:
\[ F_1(x)=x(1-2x)^2+ \sum_{k=2}^{\infty}a_kx^{2k+1},\qquad a_k=2^{2k+1}\frac{(k-1)!}{(2k)!}>0. \]
In particular, \(F_1\geq0\) on \([-1,1]\). Let
\[ M=\int_0^1F_1(x)\,dx =\frac16+\sum_{k=2}^{\infty}b_k, \qquad b_k=\frac{a_k}{2k+2}. \]
Here \(b_2=2/9\), and for \(k\geq2\),
\[ \frac{b_{k+1}}{b_k} =\frac{2k}{(2k+1)(k+2)}\leq\frac15, \]
because \((2k+1)(k+2)-10k=(2k-1)(k-2)\geq0\). Therefore
\[ M\leq\frac16+\frac{2/9}{1-1/5}=\frac49. \]
This integral also controls every row of the kernel. Write \(a=s+1/2\in[0,1]\), and let
\[ J(x)=\int_0^x u(1-2u)^2\,du. \]
Then
\[ J(1)-J(a)-J(1-a) =a(1-a)(2a^2-2a+1)\geq0. \]
For every \(q\geq1\), \(a^q+(1-a)^q\leq1\). Applying this to every positive monomial in the remaining series gives
\[ \int_I F_1(s-t)\,dt =\int_0^aF_1(u)\,du+\int_0^{1-a}F_1(u)\,du \leq M\leq\frac49. \tag{1} \]
The kernel is continuous and symmetric on \(I^2\). Hence \(T_{F_1}\) is Hilbert-Schmidt, compact, and self-adjoint. The Schur test and (1) give \(\|T_{F_1}\|_{2\to2}\leq M\leq4/9\). Notice that this says nothing about compactness of \(A\), and no such claim is needed. It follows that
\[ \langle Av,v\rangle \geq(1-\|T_{F_1}\|)\|v\|_2^2 \geq\frac59\|v\|_2^2. \]
The bounded coercive self-adjoint operator \(A\) is bijective, by the standard coercive-operator theorem, and its inverse has norm at most \(9/5\).
Set \(w=A^{-1}\mathbf1\). The integral operator maps \(L^2(I)\) into continuous functions, so \(w=\mathbf1-T_{F_1}w\) has a continuous representative. Reflection invariance and uniqueness make it even. Let \(m=\sup_s\int_I F_1(s-t)\,dt\leq4/9\). The same row bound on \(L^\infty\) gives
\[ \|w\|\infty\leq1+m\|w\|\infty, \qquad \|w\|_\infty\leq\frac1{1-m}. \]
Since the kernel is pointwise nonnegative,
\[ w(s)=1-\int_I F_1(s-t)w(t)\,dt \geq1-m\|w\|_\infty \geq\frac{1-2m}{1-m}\geq\frac15. \]
This is optimizer positivity, separate from operator coercivity.
Define \(\langle f,g\rangle_A=\langle Af,g\rangle\). For every \(v\in\mathcal H\),
\[ \langle\mathbf1,v\rangle =\langle Aw,v\rangle =\langle w,v\rangle_A. \]
Cauchy-Schwarz in the \(A\)-inner product yields
\[ \langle\mathbf1,v\rangle^2 \leq\langle Aw,w\rangle\langle Av,v\rangle =\langle\mathbf1,w\rangle\langle Av,v\rangle. \]
Equality holds exactly when \(v\) is proportional to \(w\). Since \(w\geq1/5\), the nonnegative cone contains the optimizing ray. This proves the scale-one assertion.
Finally, put \(I_\lambda=[-\lambda/2,\lambda/2]\) and \(f(u)=v(u/\lambda)\). A change of variables gives
\[ c_\lambda(v)= \frac{(\int_{I_\lambda}f)^2} {\int_{I_\lambda}f^2+ \iint_{I_\lambda^2}F_1(u-u')f(u)f(u')\,du\,du'}. \]
If \(0<\lambda_1<\lambda_2\leq1\), extension by zero embeds every admissible profile on \(I_{\lambda_1}\) into the class on \(I_{\lambda_2}\) without changing this quotient. Thus \(c_\lambda^*\) is nondecreasing and the endpoint \(\lambda=1\) is globally optimal.
Numerical status
The theorem identifies the exact constant as
\[ c^=\langle\mathbf1,A^{-1}\mathbf1\rangle, \qquad H^=2-\frac1{c^*}. \]
The existing computations in xiprime.py give
\[ c^=0.8838931253605797508\ldots, \qquad H^=0.8686415005297670641\ldots. \]
After the argument above, the latter is legitimately the numerical evaluation of the global variational optimum over \(0<\lambda\leq1\) and the full nonnegative even window class. The displayed decimal remains a converged high-precision numerical evaluation, not an interval enclosure of its last digit.
Sources and in-repository evidence
- The source paper, More than two thirds of the zeros of the Riemann zeta function lie on the critical line, equation (7.3), Theorem D, and Remark 7.3: <https://www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc8f608f1c37.pdf>.
- D. W. Farmer, S. M. Gonek, and Y. Lee, Pair correlation of the zeros of the derivative of the Riemann xi-function, Theorem 1.1: <https://arxiv.org/abs/0803.0425>.
- A. Chirre, F. Goncalves, and D. de Laat, Pair correlation estimates for the zeros of the zeta function via semidefinite programming, the infinite-series form used above before its
xi'lemma: <https://arxiv.org/abs/1810.08843>. hunts/wide_search/RESULTS-xiprime.mdfixes the functional, normalization, comparison constants, and independent numerical evaluations.hunts/wide_search/xiprime.pyevaluates the exact kernel and the Galerkin quotient.