Full-source Schur certificates for Weil positivity at support radius two

1. Result and relation to prior work

Let $\mathfrak a_a$ be the classical geometric Weil form on complex smooth functions supported in $(-a,a)$, with the normalization in (2.1), and let $A_a$ be its closed Friedrichs realization.

Theorem 1.1. Both parity restrictions of $A_2$ have strictly positive spectral infimum. Consequently

$$ \inf\sigma(A_a)>0\qquad(0<a\leq2). \tag{1.1} $$

Every prime power with $\log n<4$, every archimedean boundary response, the signed pole, and all couplings to the infinite Fourier complement are included.

This is a computer-assisted fixed-window theorem. It does not prove positivity at arbitrary support, a new zero-free region, or RH. The small finite-core bounds below are not asserted to be lower bounds for $A_2$. The full-operator conclusion follows from two complete Schur arguments.

Hongmin Li [1] establishes the corresponding complete-form theorem through $a=(\log11)/2$. We use that paper's exponential Green-response identity and its cosine and half-integer sine bases. The new endpoint is $a=2$. The principal additional mechanism is to sum the entire response family before eliminating the core: the full matrix retains displacement rank two, and the exact core elimination is described by four rational functions. This allows interval assembly of a large intermediate Schur matrix while retaining the actual core inverse.

The operator and finite-to-infinite precedents are substantial. Yoshida [2] proves small-window positivity with interval pivots, tail estimates and coupling control; Suzuki [3] supplies a modern operator-theoretic framework. Zhu [4, v2] proves positivity at half-width $0.8$ and reports geometric-side variational upper bounds down to $3.2\times10^{-283}$ at half-width $2$. Our result concerns the opposite inequality at that endpoint; it does not improve those variational upper bounds. Liu [5] proves bounds at half-widths $1$ and $17/16$ using source-exact block Schur comparisons, retained tails and positive congruences. We make no claim that finite-to-infinite reduction, Schur complementation, displacement algebra, or residual positivity tests are new. Rigorous floating-point verification has the general precedents surveyed by Rump [6]. The contribution is the complete support-two certificate and the full-response inverse calculation that makes it possible. These version-specific comparisons were refreshed on 26 October 2026.

The certificate has three scales:

$$ K=512,\qquad N=131072,\qquad M=524288. \tag{1.2} $$

The first $K$ modes in each parity are validated by a high-precision congruence. Their exact elimination leaves a structured operator on $q\geq K$. A finite Schur matrix on $K\leq q<N$ retains all entries through $M-1$, and an analytic bound pays the entire tail $q\geq M$. Its certified remaining gap exceeds $1/20$. No positive floating pivot or finite eigenvalue is used by itself as an operator certificate.

2. The form, its domain, and the complete source

Use $\widehat f(t)=\int f(x)e^{-itx}\,dx$ and $C_f(u)=\int f(x+u)\overline{f(x)}\,dx$. Then

$$ \begin{aligned} \mathfrak a_a[f]={}&\frac1{2\pi}\int_{\mathbb R} \bigl(\Re\psi(1/4+it/2)-\log\pi\bigr)|\widehat f(t)|^2\,dt\\ &+2\Re\bigl(\widehat f(i/2)\overline{\widehat f(-i/2)}\bigr) -2\sum_{n\geq2}\frac{\Lambda(n)}{\sqrt n}\Re C_f(\log n). \end{aligned} \tag{2.1} $$

Reflection splits the form orthogonally. Identify each parity space unitarily with $L^2(0,2)$; write $\epsilon=1$ for even and $-1$ for odd. The bases are

$$ e_q^+(x)=\gamma_q\cos(q\pi x/2),\quad \gamma_0=1/\sqrt2,\quad\gamma_q=1\ (q>0); \qquad e_q^-(x)=\sin((q+1/2)\pi x/2). \tag{2.2} $$

For the odd sector all $\gamma_q=1$. Put $k=q$ or $q+1/2$ respectively, $\kappa=\pi k/2$, and

$$ d_k=\Re\psi(1/4+i\pi k/4)-\log\pi,\qquad c_j=2j+1/2. \tag{2.3} $$

The positive boundary responses and pole vectors are

$$ r_j^\epsilon(x)= \frac{e^{-2c_j}(e^{c_jx}+\epsilon e^{-c_jx})} {\sqrt{1-\epsilon e^{-4c_j}}}, \qquad p_\epsilon(x)=e^{x/2}+\epsilon e^{-x/2}. \tag{2.4} $$

The Green identity in [1, Lemma 3.1] gives on the smooth compact core

$$ A=D+B_p^\epsilon+\sum_{j\geq0}r_j^\epsilon\otimes r_j^\epsilon +\epsilon p_\epsilon\otimes p_\epsilon. \tag{2.5} $$

Here $D$ has diagonal (2.3), and

$$ \begin{aligned} (B_p^\epsilon v)(x)=-\sum_{\log n<4}\frac{\Lambda(n)}{\sqrt n} \bigl[& \mathbf1_{x+\log n<2}v(x+\log n) +\mathbf1_{x>\log n}v(x-\log n)\\ &+\epsilon\mathbf1_{0<\log n-x<2}v(\log n-x)\bigr]. \end{aligned} \tag{2.6} $$

The identity follows termwise from the convergent digamma difference expansion and the difference between the exponential kernel and its Neumann, or mixed Dirichlet-Neumann, Green kernel.

We record the domain justification because the following computation uses the complete response family. Its kernel is bounded in absolute value by

$$ \frac4{1-e^{-2}}\sum_{j\geq0}e^{-c_j(4-x-y)} \leq \frac4{1-e^{-2}}\left(1+\frac1{2(4-x-y)}\right). \tag{2.7} $$

Indeed $1/(1-e^{-s})\leq1+1/s$. The constant kernel on $(0,2)$ has norm $2$, and the Carleman kernel $1/(s+t)$ has norm at most $\pi$ on that interval: Schur's test with weight $t^{-1/2}$ proves the bound directly. Thus the response sum is a bounded positive operator, with norm less than $17$. The prime and pole operators are bounded as well. Accordingly (2.5) defines a closed form on the diagonal form domain of $D$.

Smooth compact tests are a core for this form. Finite basis combinations are dense for the logarithmic diagonal norm. Cutting such a combination off near the endpoints converges in $H^s$ for any fixed $0<s<1/2$, hence in the weaker logarithmic norm; parity can be preserved. The bounded terms preserve this convergence. Thus the closure of (2.1) agrees with (2.5), and all subsequent block arguments concern the full target form.

3. Rank-two displacement of the full matrix

Conjugate by the diagonal unitary $\mathcal E_{qq}=(-1)^q$. Let

$$ \alpha_+=\sinh^2 1,\qquad \alpha_-=-\cosh^2 1, $$

and define

$$ \begin{aligned} F_\epsilon(x) &=\sum_{j\geq0}\frac{x^2(1-\epsilon e^{-4c_j})}{c_j^2+x^2} -\frac{\alpha_\epsilon}{1/4+x^2}\\ &=\frac{x}{2}\Im\psi(1/4+ix/2) -\epsilon x^2\sum_{j\geq0}\frac{e^{-4c_j}}{c_j^2+x^2} -\frac{\alpha_\epsilon}{1/4+x^2}. \end{aligned} \tag{3.1} $$

Direct integration gives

$$ \langle r_j^\epsilon,e_q^\epsilon\rangle =(-1)^q\gamma_q\frac{c_j\sqrt{1-\epsilon e^{-4c_j}}}{c_j^2+\kappa^2}, \tag{3.2} $$

and pole coefficients $(-1)^q\gamma_q\sinh1/(1/4+\kappa^2)$ in the even sector and $(-1)^q\cosh1/(1/4+\kappa^2)$ in the odd sector. The divided difference of $x^2/(c^2+x^2)$ is $c^2/((c^2+x^2)(c^2+y^2))$. Hence (3.1) includes the full response and signed pole matrix, with no discarded response tail.

For the 24 active prime powers set

$$ w_n=\frac{\Lambda(n)}{\sqrt n},\qquad \phi_n=\frac{\pi\log n}{2},\qquad s_k=\sum_nw_n\sin(k\phi_n). $$

Define

$$ \lambda_q=k^2,\quad v_q=\gamma_q,\quad u_q=\gamma_q\left(\frac{2ks_k}{\pi} +\frac4{\pi^2}F_\epsilon(\pi k/2)\right). \tag{3.3} $$

Integration of the three terms in (2.6) gives, for $q\ne r$,

$$ A_{qr}=\frac{u_qv_r-v_qu_r}{\lambda_q-\lambda_r}, \qquad \Lambda A-A\Lambda=uv^T-vu^T. \tag{3.4} $$

For $k>0$ the exact diagonal is

$$ A_{qq}=d_k-2\sum_nw_n(1-\log n/4)\cos(k\phi_n) +\frac{s_k}{\pi k}+\gamma_q^2R_\epsilon(\kappa), \tag{3.5} $$

where

$$ R_\epsilon(x)= \frac{\Im\psi(1/4+ix/2)}{4x} +\frac{\Re\psi'(1/4+ix/2)}8 -\epsilon\sum_{j\geq0}\frac{c_j^2e^{-4c_j}}{(c_j^2+x^2)^2} +\frac{\alpha_\epsilon}{(1/4+x^2)^2}. \tag{3.6} $$

At even $k=0$ omit $s_k/(\pi k)$ and use $R_+(0)=\psi'(1/4)/4-\sum_j e^{-4c_j}/c_j^2+16\alpha_+$. The constant mode has its own normalization; it is not obtained from the continuous limit of the nonconstant prime diagonal.

Only the exponential correction in (3.1), (3.6) is truncated numerically. After $j=J-1$, rigorous absolute errors are

$$ \delta_J=\frac{e^{-4c_J}}{1-e^{-8}} \quad\hbox{for }F,\qquad \delta_J/c_J^2 \quad\hbox{for }R. \tag{3.7} $$

The source generator adds these as Arb error balls. It uses $J=280$ at 3000 bits for the core and $J=24$ at 160 bits for the distant source.

4. A validated finite core, without a pivot-to-gap inference

Let $P_0$ project onto $0\leq q<K$ and let $M_0=P_0AP_0$. Point structured elimination proposes a triangular factor, and an approximate solve proposes an exactly specified dyadic matrix $T_0$. Neither proposal is trusted. Independent Arb multiplication encloses

$$ E_0=T_0M_0T_0^T-I. \tag{4.1} $$

An absolute row-sum bound gives $\|E_0\|\leq\eta<1$. Therefore $T_0M_0T_0^T$ is positive definite; since $T_0$ is square, this also proves its invertibility and $M_0>0$. Moreover

$$ M_0\succeq \frac{1-\eta}{\|T_0\|_F^2}I. \tag{4.2} $$

The complete 512-mode enclosures are:

| sector | upper bound for $\eta$ | upper bound for $\|T_0\|_F^2$ | certified core lower bound | |---|---:|---:|---:| | even | $5.30\,10^{-615}$ | $3.026\,10^{282}$ | $3.3055\,10^{-283}$ | | odd | $1.45\,10^{-619}$ | $1.744\,10^{277}$ | $5.7369\,10^{-278}$ |

The last column is obtained from the more precise retained enclosures, not by dividing the shortened bounds in the preceding columns. This calculation uses the full source (3.1)–(3.7).

We also need two certified solves:

$$ a=M_0^{-1}u_{P_0},\qquad b=M_0^{-1}v_{P_0}. \tag{4.3} $$

For each proposed solution $x_0$, direct interval multiplication encloses $r=M_0x_0-f$. The bound $\|x_0-M_0^{-1}f\|\leq \|M_0^{-1}\|\|r\|$, using (4.2), certifies every coordinate. The respective coordinate errors for $a,b$ are below $1.39\,10^{-417},1.26\,10^{-422}$ in the even sector and $2.12\,10^{-425},1.94\,10^{-430}$ in the odd sector. These solves retain the true inverse; no scalar approximation to it is substituted.

5. Exact rational transfer after eliminating the core

For $z$ outside the finite label set define

$$ \begin{array}{ll} A_0(z)=\displaystyle\sum_{i<K}\frac{v_i a_i}{z-\lambda_i}, &B_0(z)=\displaystyle\sum_{i<K}\frac{u_i a_i}{z-\lambda_i},\\ C_0(z)=\displaystyle\sum_{i<K}\frac{v_i b_i}{z-\lambda_i}, &D_0(z)=\displaystyle\sum_{i<K}\frac{u_i b_i}{z-\lambda_i}, \end{array} $$$$ \mathcal T(z)= \begin{pmatrix}1-A_0(z)&B_0(z)\\-C_0(z)&1+D_0(z)\end{pmatrix}. \tag{5.1} $$

Write $C_{\rm core}=P_0A(I-P_0)$ and

$$ S=(I-P_0)A(I-P_0)-C_{\rm core}^TM_0^{-1}C_{\rm core}. \tag{5.2} $$

Its generators are

$$ \binom{U_q}{V_q}=\mathcal T(\lambda_q)\binom{u_q}{v_q}. \tag{5.3} $$

This follows by taking the Schur complement of (3.4); equivalently $U=u-C_{\rm core}^Ta$, $V=v-C_{\rm core}^Tb$. For $R(z)=(zI-\Lambda_{P_0})^{-1}$, $[M_0,R]=-R(uv^T-vu^T)R$, whence $D_0-A_0=A_0D_0-B_0C_0$ and

$$ \det\mathcal T(z)=1. \tag{5.4} $$

The exact diagonal correction is also determined by these four sums:

$$ \mathcal Q_q= g_q^T\mathcal T(\lambda_q)^T \begin{pmatrix}0&1\\-1&0\end{pmatrix} \mathcal T'(\lambda_q)g_q,\qquad g_q=(u_q,v_q)^T, \quad S_{qq}=A_{qq}-\mathcal Q_q. \tag{5.5} $$

Write $J$ for the displayed skew-symmetric matrix. The prime denotes differentiation in $z$. To verify (5.5), take the diagonal limit in the off-diagonal Schur identity. Terms differentiating the original generator cancel by (5.4). One may instead hold $g$ constant near $z$. For one eliminated pivot $p$, the formula reduces to $(u(z)v_i-v(z)u_i)^2/[p(z-\lambda_i)^2]$, fixing the sign. Thus (5.2) retains all original mixed terms.

The coefficients in (5.1) can be as large as $4\,10^{147}$ and cancel. Their convolution is consequently performed at 1400 bits, not in ordinary floating arithmetic. For $k=q+\omega$, $\ell=i+\omega$, $\omega=0$ or $1/2$,

$$ \frac1{k^2-\ell^2}= \frac1{2k}\left(\frac1{q-i}+\frac1{q+i+2\omega}\right). \tag{5.6} $$

Polynomial convolution evaluates these Toeplitz and Hankel sums; squared kernels give the derivatives. The source, polynomial products and conversion errors are all enclosed. A constant replacement $U\mapsto U+110000V$ is used for storage only and is undone with interval arithmetic before the estimates below.

6. A lower bound for the whole distant block

6.1 Prime norm

Let $\mathcal B$ be (2.6) with all signs positive. Then $|B_p^\epsilon f|\leq\mathcal B|f|$ and $\|B_p^\epsilon\|\leq\|\mathcal B\|$. The four-step positive graph enumeration yields

$$ \|\mathcal B\|^4<6331.639210,\qquad \|B_p^\epsilon\|<\rho=8.920292. \tag{6.1} $$

This is an exhaustive bound, not a sampled row maximum. At successive steps the cell counts are $30,375,3719,30023$. Endpoints are labelled exactly as $2e+\log r$ with rational $r$; every preimage under the active translation and reflection maps is included. Arb proves interval membership and strict ordering. All floating endpoint and logarithm errors are checked below $10^{-14}$. A lookup farther than $10^{-12}$ from a breakpoint is therefore correct for the translated values in $[-4,6]$; closer lookups are checked directly in Arb. Weights and every positive addition and multiplication are rounded upward. Self-adjointness and Schur's test applied to $\mathcal B^4$ prove (6.1). The active list is exact because $54<e^4<55$.

6.2 Complete core covariance, through normalized moments

Let $\ell_*=(K-1+\omega)^2$ and $\theta_i=\lambda_i/\ell_*\leq1$. For any coefficient list $f_i$ in (5.1), use 96 moments $\mu_r=\sum_i f_i\theta_i^r$. With $t=\ell_*/z$,

$$ \sum_i\frac{f_i}{z-\lambda_i} =\frac1{\ell_*}\sum_{r=0}^{95}\mu_rt^{r+1}+\mathcal R(t). \tag{6.2} $$

Writing $A_f=\sum_i|f_i|$, for $0\leq t<1$,

$$ |\mathcal R(t)|\leq\frac{A_f}{\ell_*}\frac{t^{97}}{1-t},\qquad |\mathcal R'(t)|\leq\frac{A_f}{\ell_*} \frac{t^{96}(97-96t)}{(1-t)^2}. \tag{6.3} $$

Moments are summed before absolute values are taken. The large cancelled coefficients are never replaced by their absolute sum except in (6.3).

For the resulting polynomial approximation $\mathcal T(t)$, form $-\mathcal T(t)^TJ\mathcal T_t(t)$. Its symmetric quadratic coefficients, after multiplication by $\ell_*$, bound (5.5) as

$$ 0\leq\mathcal Q_q \leq\frac{L_{11}u_q^2+2L_{12}|u_q|+L_{22}}{k^4}. \tag{6.4} $$

Here $v_q=1$ for the distant modes. Each polynomial is bounded by the sum of absolute coefficients times powers of $t_N=\ell_* /(N+\omega)^2$; the product errors from (6.3) are added. This gives a uniform inequality for every $q\geq N$.

The source has a simple phase decomposition

$$ u_q=k h_q+r_q,\qquad h_q=\tfrac12+\tfrac2\pi s_k,\qquad |r_q|<1. \tag{6.5} $$

For the last assertion, monotone integral comparison puts $\sum_jx^2/(c_j^2+x^2)$ between $\pi x/4-1/4$ and $\pi x/4+1$. The exponential and pole corrections then give $|r_q|<0.461$ for $k\geq512$.

Finite geometric sums give constants $V_h,E_h$ such that $|\sum_{q=r}^s(h_q^2-V_h)|\leq E_h$ for every interval. Explicitly, put

$$ \begin{aligned} V&=\tfrac12\sum_nw_n^2,\qquad E_1=\sum_n\frac{w_n}{|\sin(\phi_n/2)|},\\ E&=\sum_{n\ne m}\frac{w_nw_m}{2|\sin((\phi_n-\phi_m)/2)|} +\sum_{n,m}\frac{w_nw_m}{2|\sin((\phi_n+\phi_m)/2)|},\\ V_h&=\tfrac14+\frac4{\pi^2}V=1.6833134137\ldots,\qquad E_h=\frac2\pi E_1+\frac4{\pi^2}E<232.761. \end{aligned} \tag{6.6} $$

The half-integer shift changes phases but not these bounds. Abel summation and integral comparison yield, for $k_N=N+\omega$,

$$ J_2=\frac{V_h}{k_N-1}+\frac{E_h}{k_N^2} \geq\sum_{q\geq N}\frac{h_q^2}{k^2},\qquad J_4=\frac1{3(k_N-1)^3}\geq\sum_{q\geq N}\frac1{k^4}. \tag{6.7} $$

Using (6.4), (6.5) and Cauchy-Schwarz,

$$ \sum_{q\geq N}\mathcal Q_q \leq L_{11}J_2+2(L_{11}+L_{12})\sqrt{J_2J_4} +(L_{11}+2L_{12}+L_{22})J_4. \tag{6.8} $$

At $N=131072$ the bound is below $0.226010$ in the even sector and $0.222615$ in the odd sector. This bounds the trace, hence the operator norm, of the entire positive core covariance on this tail. The negative odd pole contributes less than

$$ \frac{16\cosh^2 1}{3\pi^4(k_N-1)^3}<5.8\,10^{-17}. \tag{6.9} $$

Positive responses may now be dropped on this distant diagonal block. Equations (6.1), (6.8), (6.9) prove

$$ S_{q\geq N}\succeq T,\qquad Te_q=t_qe_q,\quad t_q=d_k-\rho-0.227. \tag{6.10} $$

The $d_k$ increase, by their digamma partial fraction expansion. In both sectors $t_N>1.2499$ and $t_M>2.6362$.

7. The intermediate Schur matrix and its entire omitted tail

Let $P$ project onto $K\leq q<N$ inside the post-core space and let $C=PS(I-P)$. Replace only the distant diagonal block by (6.10). The required finite Schur operator is

$$ \mathcal S=PSP-CT^{-1}C^* =\mathcal S_M-\mathcal R_M,\qquad \mathcal S_M=PSP-\sum_{N\leq q<M}\frac{c_qc_q^T}{t_q}. \tag{7.1} $$

All $c_q$ here are columns of the actual post-core operator (5.2).

For interval assembly use the three lists $V_q^2/t_q,U_qV_q/t_q,U_q^2/t_q$. For any such list let

$$ Z_f(k)=\sum_{N\leq q<M}\frac{f_q}{k_q^2-k^2},\qquad Z_{2,f}(k)=\sum_{N\leq q<M}\frac{f_q}{(k_q^2-k^2)^2}. $$

Both are polynomial convolutions of reciprocal or squared reciprocal Toeplitz/Hankel kernels. In particular

$$ \frac1{k_q^2-k^2}=\frac1{2k} \left(\frac1{k_q-k}-\frac1{k_q+k}\right),\quad Z_{2,f}=\frac{T_{2,f}+H_{2,f}-2Z_f}{4k^2}. \tag{7.2} $$

The correction vectors and diagonal are

$$ a_1=-U Z_{VV}+V Z_{UV},\quad b_1=-U Z_{UV}+V Z_{UU},\qquad \operatorname{diag}(CT^{-1}C^*)= U^2Z_{2,VV}-2UVZ_{2,UV}+V^2Z_{2,UU}. \tag{7.3} $$

The finite matrix has displacement rank four, with pairs

$$ (\mathcal U_0,\mathcal V_0)=(U-b_1,V),\qquad (\mathcal U_1,\mathcal V_1)=(a_1,U). \tag{7.4} $$

The assembly uses 192-bit Arb polynomial products. Its total operator error, including conversion to stored binary64 centers, is below $4.382\,10^{-8}$ and $4.924\,10^{-8}$ respectively. For clarity, if $e_d,e_{\mathcal U_j},e_{\mathcal V_j}$ are componentwise errors and $G_{\mathcal U_j},G_{\mathcal V_j}$ bound the stored magnitudes, the bound used is

$$ e_d+2H_*\sum_{j=0}^1 \bigl(e_{\mathcal U_j}G_{\mathcal V_j} +e_{\mathcal V_j}G_{\mathcal U_j} +e_{\mathcal U_j}e_{\mathcal V_j}\bigr),\qquad H_*=\frac{1+\log(N-K)}K. \tag{7.5} $$

Indeed $\sup_i\sum_{j\ne i}|\lambda_i-\lambda_j|^{-1}\leq H_*$. No floating FFT error assertion is needed for this interval assembly.

We bound all $q\geq M$ in (7.1). From (6.2), (6.3), obtain constants $B_A,B_B,B_C,B_D$ such that the four sums in (5.1) have absolute values at most $B_A/k^2,\ldots,B_D/k^2$ on that tail. Put

$$ a_M=1+B_A/k_M^2,\quad b_M=B_B/k_M^2,\quad v_M=1+B_D/k_M^2+B_C(H/k_M+1/k_M^2), \quad H=\tfrac12+\tfrac2\pi\sum_nw_n. \tag{7.6} $$

Then $|U_q|\leq a_M k|h_q|+a_M+b_M$ and $|V_q|\leq v_M$. Let $J_2,J_4$ now denote (6.7) with $M$ replacing $N$, and let $S_U=\sum_{K\leq q<N}U_q^2$, $S_V=\sum_{K\leq q<N}V_q^2$. The complete omitted covariance satisfies

$$ \|\mathcal R_M\|\leq \frac{\left(\sqrt{S_U}\,v_M\sqrt{J_4} +\sqrt{S_V}\,[a_M\sqrt{J_2}+(a_M+b_M)\sqrt{J_4}]\right)^2} {t_M[1-(k_{N-1}/k_M)^2]^2}. \tag{7.7} $$

This is a Hilbert-Schmidt bound on the entire weighted cross block: use $|k_q^2-k_i^2|\geq k_q^2[1-(k_{N-1}/k_M)^2]$, apply Minkowski to its two generator products, and use $t_q\geq t_M$. Every infinite index is included. The interval source supplies $S_U,S_V$. The resulting bounds are

$$ \|\mathcal R_M\|<0.143999\quad\hbox{(even)},\qquad \|\mathcal R_M\|<0.144458\quad\hbox{(odd)}. \tag{7.8} $$

8. Finite positivity through a structured backward-error bound

Run rank-four point LDL on the stored matrix $\mathcal S_{\rm input}-0.20I$. The update at a pivot $p$ is

$$ c_j=\frac{\mathcal U_j\mathcal V_i^T- \mathcal V_j\mathcal U_i^T}{\lambda_j-\lambda_i}, \quad d_j\leftarrow d_j-c_j^2/p,\quad \mathcal U_j\leftarrow\mathcal U_j-c_j\mathcal U_i/p,\quad \mathcal V_j\leftarrow\mathcal V_j-c_j\mathcal V_i/p. \tag{8.1} $$

Both runs have $n=130560$ pivots. The minima are $0.5259823539\ldots$ and $0.5204596488\ldots$. Their positivity alone is not the certificate.

Compile with contraction disabled and without fast-math. The type used has at least 64 significand bits; use the conservative full-spacing relative bound $\mathfrak u=2^{-63}$. The quarter-integer labels and their differences are exact. All intermediate values are covered by the common rational bounds

$$ p>p_0=\tfrac12,\quad |c|<C=4,\quad |d|<D=15,\qquad (|\mathcal U_0|,|\mathcal U_1|,|\mathcal V_0|,|\mathcal V_1|) <(5500000,16.5,7,1100000). \tag{8.2} $$

Parallel updates operate only on distinct remaining rows at a fixed pivot, so they change no dependency or scalar update in this argument.

Here are the residual details. Put $G=2(G_0G_2+G_1G_3)$ for the four generator bounds in (8.2). The exact current off-diagonal $c_j^*$ and its computed value differ by at most $9\mathfrak uG/|\lambda_j-\lambda_i|$. Each rounded generator update has error at most $4\mathfrak uG_h(1+C/p_0)$. Updating exact generators with the computed $c$ produces off-diagonal numerator

$$ (\lambda_j-\lambda_k)A_{jk} -\{c_j(\lambda_i-\lambda_k)c_k^* +c_k(\lambda_j-\lambda_i)c_j^*\}/p. \tag{8.3} $$

Compare this with the numerator of $A_{jk}-c_jc_k/p$. The factors involving $\lambda_i$ cancel the denominators in the $c$ errors. Including generator rounding, a local residual obeys

$$ |(E_i)_{jk}|\leq \frac{64\mathfrak uG(1+C/p_0)}{|\lambda_j-\lambda_k|}, \qquad |(E_i)_{jj}|\leq4\mathfrak u(D+C^2/p_0). \tag{8.4} $$

The pivot row has the smaller $9\mathfrak uG$ bound. The constant 64 covers the $18\mathfrak uG C/p_0$ contribution of the computed cross entries and the $16\mathfrak uG(1+C/p_0)$ generator contribution, including their quadratic rounding terms.

These residuals enter the original matrix by zero extension, not by triangular congruence amplification. At one step the current matrix is

$$ \begin{pmatrix}p&c^T\\c&A_{\rm next}+cc^T/p\end{pmatrix}+E_i; $$

a later error in $A_{\rm next}$ occupies just that lower-right block. Induction therefore reconstructs a positive matrix $LDL^T$ and an error with

$$ \|\mathcal S_{\rm input}-0.20I-LDL^T\| \leq n\left[64\mathfrak uG(1+C/p_0)H_* +4\mathfrak u(D+C^2/p_0)\right] <2.30580\,10^{-5}. \tag{8.5} $$

A further $10^{-12}$ covers shift conversion, subtraction and subnormal effects; the displayed size bounds exclude overflow. Adding (7.5), the total errors are below $2.31018\,10^{-5}$ and $2.31072\,10^{-5}$. Thus $\mathcal S_M$ exceeds $0.20I$ minus these errors. By (7.8),

$$ \mathcal S\succ\frac1{20}I \tag{8.6} $$

in both sectors. The retained lower margins are $0.0559781\ldots$ and $0.0555196\ldots$.

9. Completion of the full-operator proof

The lower comparison for $S$ has blocks $PSP,C,T$, with $T\geq1.2499I$. Its square completion is

$$ \|T^{1/2}y+T^{-1/2}C^*x\|^2+\langle\mathcal Sx,x\rangle. \tag{9.1} $$

The cross block is bounded: the original $A$ is a diagonal operator plus a bounded operator by Section 2, and eliminating a finite positive block adds a bounded finite-rank term. Equations (8.6), (9.1) therefore give a strictly positive uniform lower bound for $S$ on its complete closed form domain.

Finally the exact first square completion is

$$ \langle M_0(x+M_0^{-1}C_{\rm core}y), x+M_0^{-1}C_{\rm core}y\rangle+\langle Sy,y\rangle. \tag{9.2} $$

Here $M_0>0$ by Section 4 and $M_0^{-1}C_{\rm core}$ is bounded. The invertible triangular changes in (9.1), (9.2) imply uniform strict positivity of the full form in both parity sectors. Orthogonal parity decomposition proves it for arbitrary complex tests. The form-domain identification in Section 2 justifies passage from the smooth core. Zero extension from any smaller support proves (1.1).

10. Reproduction, evidence, and limits

The supplement provides the complete sources, retained output enclosures, an environment record, file manifest and ordered regeneration commands. Large coefficient tables and binary matrices are regenerated, so no essential input requires private access. All computation is on CPU. The principal author runs used Python 3.12.3, python-flint 0.9.0, NumPy 2.5.4, SciPy 1.18.1 and GCC with a 64-bit-significand long double.

The finite-core congruence took about 256 and 262 seconds; certified transfer solves took about 17 seconds per sector. Complete source generation through $M-1$ took about 280 seconds per sector. Interval Schur assembly took about 33 seconds, and the four-thread structured factorizations about 101 and 103 seconds. The inexpensive final error checker recomputes (8.5)–(8.6) from the stated rational bounds and retained assembly enclosures. Reviewing that checker alone does not independently verify the earlier source computations; the complete regeneration route is supplied for that purpose.

Independent direct finite sums check selected assembled entries; small FFT/direct comparisons and exploratory finite eigenvalues were used only during development. They are not inputs to (1.1). The proof uses interval source enclosures, exact rational identities, analytic infinite-tail bounds, and the explicit structured residual estimate.

The theorem certifies a fixed support radius. It establishes neither an effective all-scale induction nor a scalable route to RH. In particular the finite-core conditioning remains severe; reporting the minimum elimination pivot as the full spectral gap would be incorrect. The complete inverse and both infinite couplings are indispensable.

References

[1] Hongmin Li (lihongmin). Cutoff-free positivity of localized Weil forms through the (log 11)/2 threshold. rStation, 2026, version 1. https://rstation.ai/problems/riemann_hypothesis/papers/45/v1

[2] Hiroyuki Yoshida. On Hermitian forms attached to zeta functions. Zeta Functions in Geometry, Advanced Studies in Pure Mathematics 21 (1992), 281–325. https://doi.org/10.2969/aspm/02110281

[3] Masatoshi Suzuki. Weil's quadratic form via the screw function. 2026, version 3. https://arxiv.org/abs/2606.09096v3

[4] Xuefeng Zhu. Weil positivity in compact windows: a finite reduction, certified two-sided bounds, and a Landau-Widom decay law. 2026, version 2. https://arxiv.org/abs/2608.24827v2

[5] Vincent Liu. Certified Weil Positivity Beyond the Unit Window: Source-Exact Block-Schur and Tail-Compensation Bounds for the Riemann Zeta Function. Submitted manuscript, 14 September 2026. https://www.alphaxiv.org/abs/2609.weil-positivity-riemann-zeta-bounds Frozen manuscript and source, commit b6cd2183c1e79c6c27a34267812a7b2d73ed1b59: https://github.com/luciferyu666/certified-weil-positivity/tree/b6cd2183c1e79c6c27a34267812a7b2d73ed1b59

[6] Siegfried M. Rump. Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica 19 (2010), 287–449. https://doi.org/10.1017/S096249291000005X