Status: exact-rational evidence package for independent review. This file strengthens the ambient variational result in RESULTS-xiprime-global-optimality.md. It does not edit or replace the publication manuscript.
Theorem
Let
\[ I=[-1/2,1/2],\qquad \mathcal H=L^2_{\mathrm{even}}(I;\mathbb R),\qquad A=I+T_{F_1}, \]
with (F_1), (T_{F_1}), and (c^*) as in RESULTS-xiprime-global-optimality.md. Thus
\[ c^*=\langle\mathbf1,A^{-1}\mathbf1\rangle, \qquad w=A^{-1}\mathbf1. \]
Let \(\mathcal A_{\mathrm{source}}\) be the scalar profiles induced by the physical windows used in Section 7.1 of the cited source paper, in the asymptotic sense made explicit below. Then
\[ \boxed{ \sup_{v\in\mathcal A_{\mathrm{source}}} \frac{\langle\mathbf1,v\rangle^2}{\langle Av,v\rangle} =\langle\mathbf1,A^{-1}\mathbf1\rangle=c^*.} \]
Equivalently, for the reciprocal quotient,
\[ \boxed{ \inf_{v\in\mathcal A_{\mathrm{source}}} \frac{\langle Av,v\rangle}{\langle\mathbf1,v\rangle^2} =\frac1{c^*}.} \]
The quotient orientation is load-bearing. The maximizing quotient is \(\langle\mathbf1,v\rangle^2/\langle Av,v\rangle\); its reciprocal is minimized, not maximized.
Exact source class
The exact cited version is:
More than two thirds of the zeros of the Riemann zeta function lie on the critical line, <https://www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc8f608f1c37.pdf>.
The relevant locations are:
- Proposition 2.1, p. 7, equation (2.7). The explicit formula takes (C_c^2(\mathbb R)) test functions supported in \([-L/2,L/2]\).
- Section 2.2, pp. 7-8, equations (2.11)-(2.14). The initial convenient realization fixes a nondecreasing (C^3) ramp \(\varrho\), chooses (1\leq r\leq L/8\), and sets \(\phi(u)=\varrho((L/2-|u|)/r)\). This produces an even, flat-topped endpoint taper with (0\leq\phi\leq1\).
- Section 7.1, p. 20, paragraph before (7.1). This is the generalized, load-bearing class. It requires \[ \phi\in C_c^2(\mathbb R),\quad \phi\text{ even},\quad 0\leq\phi\leq1, \quad\operatorname{supp}\phi=[-L/2,L/2], \] with \(\phi\) nonincreasing in \(|u|\), and with \(\|\phi''\|_1\) and \(\|(\phi^2)''\|_1\) uniformly (O(1)). Radial monotonicity gives \(\|\phi'\|_1,\|(\phi^2)'\|_1\leq2\).
- Section 7.1, p. 20, equation (7.3). The scalar profile is defined by \(\phi^2(u)=v(u/L)\). There is no integral, peak, or (L^p) normalization beyond (0\leq\phi\leq1\).
- Theorem D, p. 21, proof after (7.4). The ideal profile is realized in the limit by multiplying its square root by the fixed-width endpoint ramp from Section 2.2. The change in the scale-free constants is (O(1/L)).
- Remark 7.1(iii), p. 22. A sharp cutoff makes Proposition 4.2 fail; a fixed-width ramp is the minimal repair and costs (O(1/L)).
- Remark 7.3, p. 23. The same window mechanism is applied to \(\xi'\), with the flat and quartic profiles displayed there.
Thus (C^\infty), a flat top, and a particular ramp are not requirements. The nontrivial source constraint is radial monotonicity.
Exact strict-concavity bound
Use the exact rational trial polynomial
\[ u(s)=\sum_{j=0}^5c_js^{2j},\qquad (c_0,\ldots,c_5)= \left( \frac{427163}{446844}, -\frac{205089}{684401}, -\frac{2976898}{824779}, -\frac{13369}{15690}, -\frac{104561}{672519}, -\frac{32375}{630751} \right). \]
Let (A_0) use the (M=20) kernel truncation, let (E=A-A_0), and put
\[ r_0=\mathbf1-A_0u,\qquad z=w-u. \]
The existing tail bound is
\[ \rho=\|E\|_{2\to2}\leq \frac{45088768}{2828846926917599723269509375} <1.6\times10^{-20}. \]
The executable evidence reconstructs (A_0u) and (r_0) as exact rational polynomials. On \(|s|\leq1/2\), the coefficient absolute-sum bound gives, with every decimal below interpreted as an outward rational bound,
\[ \|r_0\|_2<7.8749770\times10^{-10},\qquad \|r_0\|_\infty<2.1710808\times10^{-5}, \]
\[ \|r_0''\|_\infty<0.005982627,qquad \|u\|_2<0.887972. \]
Since (Az=r_0-Eu\) and \(\|A^{-1}\|_{2\to2}\leq9/5\),
\[ \|z\|_2 \leq\frac95(\|r_0\|_2+\rho\|u\|_2) <1.417496\times10^{-9}. \]
The row bound \(\|T_{F_1}\|_{\infty\to\infty}\leq4/9\) and \(z=(r_0-Eu)-T_{F_1}z\) similarly give
\[ \|z\|_\infty \leq\frac95(\|r_0\|_\infty+\rho\|u\|_2) <3.907946\times10^{-5}. \]
Distributionally,
\[ F_1''=2\delta_0+q, \qquad q(x)=-8+\sum_{k\geq1}d_k|x|^{2k-1}, \qquad d_k=a_k(2k)(2k+1). \]
The coefficient ratio is exact:
\[ \frac{d_{k+1}}{d_k}=\frac{2(2k+3)}{(2k+1)^2}. \]
It decreases on the relevant tail. Hence
\[ \sum_{k>20}d_k \leq \frac{666953056256}{23065890935073171953452059375} <2.90\times10^{-17}, \]
and exact summation gives
\[ 8+\sum_{k\geq1}d_k \leq \frac{622490816315301203923339889968} {7688630311691057317817353125} <80.963. \]
The delta mass must not be dropped. Differentiating (Az=r_0-Eu\) twice gives the pointwise identity
\[ z''=r_0''-(Eu)''-2z-q*z. \]
The bounds above imply, entirely in rational arithmetic,
\[ \boxed{\|z''\|_\infty <0.006060899845<\frac{6061}{10^6}.} \]
All five nonconstant coefficients of (u) are negative, so
\[ u''(s)\leq2c_1=-\frac{410178}{684401} <-0.5993240804. \]
Therefore
\[ \boxed{w''(s)=u''(s)+z''(s)<-0.59326318<-\frac{593}{1000}} \qquad(s\in I). \]
The kernel equation and the already established positivity give
\[ \frac15\leq w(s)\leq1. \]
Indeed, the lower bound is the row-bound argument in the ambient theorem; the upper bound follows from (w=1-T_{F_1}w\), (F_1\geq0\), and (w\geq0\). Evenness gives (w'(0)=0\). Strict concavity then gives (w'(s)<0\) for \(0<s\leq1/2\), so (w\) is strictly decreasing in \(|s|\).
Explicit source-admissible sequence
Define
\[ \eta(x)= \begin{cases} 0,&x\leq0,\\ 35x^4-84x^5+70x^6-20x^7,&0<x<1,\\ 1,&x\geq1. \end{cases} \]
On \((0,1)\),
\[ \eta'(x)=140x^3(1-x)^3\geq0. \]
The first three derivatives glue to zero at both endpoints, so \(\eta\in C^3(\mathbb R)\). Exact integration gives
\[ \|\eta''\|_1=\frac{35}{8},\qquad \int_0^1\eta'(x)^2\,dx=\frac{700}{429}, \]
and therefore
\[ \|(\eta^2)''\|_1 \leq2\int_0^1\eta'(x)^2\,dx+2\|\eta''\|_1 =\frac{20615}{1716}. \]
For (L\geq8\), set
\[ \phi_L(u)= \begin{cases} \sqrt{w(u/L)}\,\eta(L/2-|u|),&|u|\leq L/2,\\ 0,&|u|>L/2. \end{cases} \]
Every source condition is preserved:
- (w\in C^2(I)\) follows from the displayed distributional identity and the uniformly convergent ordinary part (q\). Since (w\geq1/5\), \(\sqrt w\in C^2(I)\). Fourth-order endpoint vanishing of \(\eta\) gives \(\phi_L\in C_c^2(\mathbb R)\).
- Both factors are even and nonincreasing in \(|u|\), so the same holds for \(\phi_L\).
- (0\leq\phi_L\leq1\), because (1/5\leq w\leq1\) and (0\leq\eta\leq1\).
- \(\eta(x)>0\) for (0<x<1\), and (w>0\), so \(\operatorname{supp}\phi_L=[-L/2,L/2]\) exactly.
- Radial monotonicity gives \(\|\phi_L'\|_1=2\phi_L(0)\leq2\) and \(\|(\phi_L^2)'\|_1=2\phi_L(0)^2\leq2\).
- Put (p=\sqrt w\). Product-rule estimates give \[ \|\phi_L''\|1 \leq\frac{\|p''\|\infty+4\|p'\|_\infty}{L} +\frac{35}{4}, \] and \[ \|(\phi_L^2)''\|1 \leq\frac{\|w''\|\infty+4\|w'\|_\infty}{L} +\frac{20615}{858}. \] The right sides are uniform for (L\geq8\).
The induced scalar profile is
\[ v_L(s)=\phi_L(Ls)^2 =w(s)\eta(L(1/2-|s|))^2. \]
It differs from (w\) only in two endpoint strips of total length (2/L\). Since (0\leq w\leq1\),
\[ \boxed{\|v_L-w\|_2^2\leq\frac2L,\qquad v_L\to w\text{ in }L^2(I).} \]
Closing the quotient
Boundedness of (A\) gives
\[ |\langle Av_L,v_L\rangle-\langle Aw,w\rangle| \leq \|A\|\,\|v_L-w\|_2(\|v_L\|_2+\|w\|_2)\to0. \]
Also
\[ |\langle\mathbf1,v_L-w\rangle| \leq\|v_L-w\|_2\to0. \]
Since \(\langle\mathbf1,w\rangle=\langle Aw,w\rangle>0\),
\[ \frac{\langle\mathbf1,v_L\rangle^2}{\langle Av_L,v_L\rangle} \longrightarrow \frac{\langle\mathbf1,w\rangle^2}{\langle Aw,w\rangle} =\langle\mathbf1,w\rangle=c^*. \]
The ambient variational theorem supplies the upper bound (c(v)\leq c^*\) for every source-admissible (v\). The sequence supplies the reverse inequality for the supremum. This proves the theorem.
Executable evidence and lesions
admissible_closure.pyreconstructs the rational trial, residual, kernel tails, (L^2\), (L^\infty\), and second-derivative bounds usingfractions.Fractionand exact SymPy polynomials.tests/test_pub1_admissible_closure.pypins every displayed target, the endpoint ramp identities, the uniform product-rule bounds, and the (2/L\) convergence rate.- Delta lesion: replacing the coefficient (2\) of \(\delta_0\) by zero is rejected before a verdict can be produced.
- Residual lesion: multiplying every residual-derived bound by (101\) changes the concavity margin from (+0.59326318\) to (-0.01282680\), and the verdict becomes false.
Run:
.venv/bin/python hunts/wide_search/admissible_closure.py
.venv/bin/python -m pytest -q tests/test_pub1_admissible_closure.py