bibkey: jarohsweth2020local authors: Sven Jarohs and Tobias Weth year: 2020 title: Local compactness and nonvanishing for weakly singular nonlocal quadratic forms doi: 10.1016/j.na.2019.01.021 url: https://arxiv.org/abs/1811.12850v1 claim: The source supplies local compactness and a quantitative averaging estimate for a weakly singular comparison kernel. Original-model cutoff and graph estimates transport them to uniform finite-rank approximation in the mixed minimal form norm, with explicit analytic bounds and a prescribed translated-kernel spanning family, and a directed finite matrix assembly interface, but no computed lower spectral certificate. strata_touched: [] license: bibliographic-reference-only triage: anchor

Local compactness in the actual theta measure

Inspected source and comparison space

The journal record is Nonlinear Analysis 193 (2020), article 111431. The inspected primary text is arXiv:1811.12850v1, with SHA-256 70b2bff55e07835ed9d94c6852e9e2e81b1c14f4b4e6c4e5ec637c07f18648dc. The publisher full text was not retrieved; equality with that edition is not asserted. Theorem 1.1, printed p.3, assumes the even measurable kernel $j\ge 0$ satisfies (A1), printed p.2, and (A2), printed p.3:

$$0<\int (1\wedge |z{|}^2)j(z)dz<\infty ,\int j(z)dz=\infty .$$

For ${\mathcal{E}}_j(g)=\frac{1}{2}\iint |g(y)-g(x){|}^{2}j(y-x)dxdy$, the theorem makes the finite-energy-domain inclusion into $L^2(dx)$ compact after restriction to every compact set. No strictly positive fractional order is required. We reuse this theorem rather than prove a new logarithmic-frequency compactness theorem. Complexification preserves the compactness conclusion.

In one dimension take

$$j_0(z)=\frac{1_{\{0<|z|<1\}}}{|z|}.$$

Its second-moment integral is $1$ and its total integral is infinite, so both source hypotheses hold.

The cutoff map from the theta minimal domain

Use the measure and Gamma energy of the actual minimal mixed realization: $\rho =2\Phi \cosh (x/2)$, $d\nu =\rho dx$ and $\psi (t)=e^{-t/2}/(1-e^{-2t})$. Let ${\mathcal{F}}_{\Gamma }$ be the minimal closure of the full compact smooth core for $D_{\Gamma }=E_{\Gamma }$. The same source construction applies with the prime measure omitted; this defines the Gamma operator, not a replacement for the full energy.

For compact $K$, choose a real $\chi \in C_c^{\infty }$, equal to one near $K$, and let the compact interval $I$ contain $\mathrm{supp}\chi +[-1,1]$. Put

$$m_{\Phi }=\underset{I}{\min }\Phi >0,m_{\rho }=\underset{I}{\min }\rho >0,c_{\psi }=\underset{0<t<1}{\inf }t\psi (t)>0.$$

For a core function $h$, use

$$|\chi (y)h(y)-\chi (x)h(x){|}^2\le 2|\chi (y){|}^2|h(y)-h(x){|}^2+2|\chi (y)-\chi (x){|}^2|h(x){|}^2.$$

For $|x-y|<1$, every nonzero cutoff term has both endpoints in $I$. On these pairs the Gamma conductance is at least $m_{\Phi }^2{c}_{\psi }/|x-y|$. Also $|\chi (y)-\chi (x)|\le \Vert {\chi }^{\prime }{\Vert }_{\infty }|x-y|$ and ${\int }_{|t|<1}|t|dt=1$. The exact one-half energy convention therefore gives

$$\begin{array}{rl}{\mathcal{E}}_{j_0}(\chi h)& \le \frac{2\Vert \chi {\Vert }_{\infty }^2}{m_{\Phi }^2{c}_{\psi }}{D}_{\Gamma }(h)+\frac{\Vert {\chi }^{\prime }{\Vert }_{\infty }^2}{m_{\rho }}\Vert h{\Vert }_{\nu }^2,\\ \Vert \chi h{\Vert }_{L^2(dx)}^2& \le \frac{\Vert \chi {\Vert }_{\infty }^2}{m_{\rho }}\Vert h{\Vert }_{\nu }^2.\end{array}\text{\hspace{1em}(LC)}$$

The comparison domain is closed. Applying (LC) to differences of core approximants extends this bounded cutoff map to ${\mathcal{F}}_{\Gamma }$; its $L^2(dx)$ limit is $\chi h$ by local equivalence of the two measures. The source theorem then gives compact restriction to $K$, and boundedness of $\rho$ there converts the convergence to $L^2(\nu )$. Thus $1_K:{\mathcal{F}}_{\Gamma }\to L^2(\nu )$ is compact.

This does not identify the theta minimal domain with the comparison kernel’s maximal finite-energy space. It does not apply a density-kernel theorem directly to prime atoms or infer global compactness from $\nu (\mathbb{R})=1$. The bounded prime energy in the mixed spectral application transports this local compactness to the original mixed minimal domain.

The cutoff estimate and source application are paper-level model checks, without new Lean certification or an originality claim. They supply no one-half global Poincare constant.

Quantitative finite-rank approximation in the original form norm

This section transports an existing weak-singularity averaging estimate into the original mixed minimal form space. It gives a finite-rank existence construction with an explicit error and rank bound. It does not supply computed singular functions or a lower spectral matrix.

Reuse the source averaging estimate

Jarohs–Weth, Lemma 2.2, printed pp.7–8 of the inspected v1, states

$$\Vert u-w_{\delta }\ast u{\Vert }_2^2\le \frac{2}{\Vert j_{\delta }{\Vert }_1}(\Vert u{\Vert }_2^2+{\mathcal{E}}_j(u)),w_{\delta }=j_{\delta }/\Vert j_{\delta }{\Vert }_1.$$

The source norm includes the $L^2$ term. Reuse this lemma, with $j=j_0=1_{\{0<|t|<1\}}/|t|$, rather than reprove its Jensen or compactness argument. For $\ell >0$, $\delta =e^{-\ell }$,

$$w_{\delta }(t)=\frac{1_{\{\delta <|t|<1\}}}{2\ell |t|},\Vert j_{\delta }{\Vert }_1=2\ell .$$

Write $L_0$ for the whole-line self-adjoint operator of ${\mathcal{E}}_{j_0}$ in $L^2(dx)$, and $\Vert u{\Vert }_{\mathcal{F}}^2=\Vert u{\Vert }_{\nu }^2+D(u)$ for the original mixed minimal form norm. The comparison operator is used only for estimates; the arithmetic energy and original measure are retained.

An operator comparison for compact low-space cuts

Fix $0<\epsilon \le 1/2$, $R\ge R_{\mathrm{PNT}}(\epsilon )$ and a unit vector in the actual even low subspace $h\in \mathrm{ran}{1}_{[0,1/2-\epsilon ]}(A_{\mathrm{even}})$. Let ${\chi }_R=1-{\eta }_R$, $g={\chi }_{R}h$, and put

$$L=R+2,I=[-L,L],J=[-R-4,R+4],z=R+4,$$

$$m=18e^{-4e^{2z}},W=\frac{36}{5}\cosh (z/2),a_{\ast }=\frac{m}{2\cosh (z/2)},B=\frac{4}{m^2}+\frac{1}{m}.$$

The original theta constants give $\rho \le W$ on $J$. The first positive theta summand, used only for a lower estimate, gives for $|x|\le z$

$$\Phi (x)\ge 2\pi (2\pi -3)e^{-\pi e^{2z}}\ge 18e^{-4e^{2z}}=m,\rho (x)\ge 2m.$$

Here evenness, $3<\pi <4$ and $e^{|x|/2}\ge 1$ are used; the remaining positive summands remain in the original operator. Also $t\psi (t)\ge 1/4$ on $0<t<1$. Thus (LC), with $\Vert {\chi }_R{\Vert }_{\infty },\Vert {\chi }_R^{\prime }{\Vert }_{\infty }\le 1$ and $D_{\Gamma }(h)\le D(h)\le 1/2$, yields

$$\Vert g{\Vert }_2^2\le \frac{1}{2m},\Vert g{\Vert }_2^2+{\mathcal{E}}_{j_0}(g)\le B.\text{\hspace{1em}(IB)}$$

The full low-subspace commutator estimate in the mixed spectral note gives $g\in D(A_{\Gamma })$. With $M=\Vert a_p{\Vert }_{\infty }<432$, (BD) implies $-2M\le B_p\le M$ in operator order, so $\Vert B_p\Vert \le 2M$. Since $A_{\Gamma }h=Ah-M_{a_p}h+B_{p}h$ and $\Vert [A_{\Gamma },{\eta }_R]\Vert \le 4e^{-R}\le 4$,

$$\Vert A_{\Gamma }g{\Vert }_{\nu }\le 1301.\text{\hspace{1em}(IG)}$$

To compare operator domains, decompose

$$\psi (|t|)=\frac{1}{2}{j}_0(t)+k(t),\Vert k{\Vert }_1<7.$$

For $0<t<1$, the inequalities $1/(2t)\le (1-e^{-2t}{)}^{-1}\le 1+1/(2t)$ and $1-e^{-t/2}\le t/2$ give $-1/4\le k(t)\le 1$. For $|t|\ge 1$, the previous $d{e}^{-|t|/2}$ bound gives a two-sided integral below $14/3$, hence the claimed $L^1$ bound. Let $L_{k}u=\int k(t)(u-{\tau }_{t}u)dt$, so $\Vert L_k\Vert \le 14$. For $d(x)=2\cosh (x/2)$ and $a(x)=\Phi (x)/d(x)$ define

$$\mathsf{R}u(x)=\frac{1}{d(x)}\int (\Phi (y)-\Phi (x))\psi (|x-y|)(u(x)-u(y))dy.$$

The source bounds $|{\Phi }^{\prime }|\le 60$ and $\Phi \le 18/5$ give $|\Phi (y)-\Phi (x)|\psi (|x-y|)\le 90$ for $0<|x-y|<1$. For longer jumps its full row integral is at most $(36/5)(14/3)=168/5$. Dividing by $d(x)\ge 2$ gives both absolute rows and columns of the off-diagonal kernel at most $534/5$. Its diagonal has the same bound, so the unweighted Schur estimate gives $\Vert \mathsf{R}{\Vert }_{L^2(dx)\to L^2(dx)}\le 1068/5<214$. On the compact smooth core the exact identity is

$$A_{\Gamma }u=a(x)(\frac{1}{2}{L}_{0}u+L_{k}u)+\mathsf{R}u.\text{\hspace{1em}(OI)}$$

For the actual $g$, compactly supported form-core approximants converge in the $j_0$ form norm by (LC). Consequently (OI) passes to distributions; its Gamma side is interpreted through the original form representation, not an operator-core assumption. Dividing by the positive smooth $a$ on $J$ and using $g\in D(A_{\Gamma })$ shows that $L_{0}g$ is locally $L^2$ there. The distribution $L_{0}g$ is supported in $I+[-1,1]$, strictly inside $J$. Equivalently, the standard whole-line Fourier realization of $L_0$ puts $g$ in its operator domain. Since $a\ge a_{\ast }$ on $J$, (IB), (IG) and (OI) give the uniform bound

$$\Vert L_{0}g{\Vert }_2\le G:=\frac{3030/a_{\ast }+28}{\sqrt{2m}}.\text{\hspace{1em}(LG)}$$

The two units of exterior margin in $J$ keep the support/domain argument inside the region where the density lower bound is available.

Transport the averaging error into the mixed form

For $u$ supported in $J$, the original conductance and $\Phi \le 18/5$ give, retaining every Gamma jump,

$$D_{\Gamma }(u)\le \frac{324}{25}(\frac{1}{2}{\mathcal{E}}_{j_0}(u)+14\Vert u{\Vert }_2^2).$$

Together with (BD) and $\rho \le W$ on $J$, this yields

$$\Vert u{\Vert }_{\mathcal{F}}^2\le C_0\Vert u{\Vert }_2^2+7{\mathcal{E}}_{j_0}(u),C_0=865W+182.\text{\hspace{1em}(UC)}$$

Let $T_{\delta }u=w_{\delta }\ast u$ and $v=g-T_{\delta }g$. Its support lies in $I+[-1,1]\subset J$. The reused source lemma gives $\Vert v{\Vert }_2^2\le B/\ell$. The whole-line convolution commutes with $L_0$, while $\Vert I-T_{\delta }\Vert \le 2$. Hence

$${\mathcal{E}}_{j_0}(v)=⟨L_{0}v,v⟩\le 2G\sqrt{B/\ell },\Vert v{\Vert }_{\mathcal{F}}^2\le C_{0}B/\ell +14G\sqrt{B/\ell }.\text{\hspace{1em}(AF)}$$

This uses commutation with the flat comparison operator, not with the state-dependent original Gamma operator.

A finite-rank map into the original minimal form domain

View $K_{\delta }u=T_{\delta }u$, for $u\in L^2(I;dx)$ extended by zero, as a map into $\mathcal{F}$. For its kernel one has

$$\Vert w_{\delta }{\Vert }_2^2=\frac{{\delta }^{-1}-1}{2{\ell }^2},\Vert w_{\delta }{\Vert }_{\infty }=\frac{1}{2\ell \delta },\mathrm{TV}(w_{\delta })=\frac{2}{\ell \delta }.$$

The standard BV translation bound gives $|\tau_tw_\delta-w_\delta|_2^2 \le2|t|/(\ell^2\delta^2)$, hence ${\mathcal{E}}_{j_0}(w_{\delta })\le 2/({\ell }^2{\delta }^2)$. Each translated kernel is supported inside $I+[-1,1]$. Mollification converges in $L^2$ and in the $j_0$ form norm: the squared translation differences are dominated by four times those of the original kernel. On a fixed slightly larger compact interval (UC) then gives convergence in the original form norm. Thus these kernels belong to the original minimal domain, without identifying a maximal domain.

The shifted kernels depend continuously in form norm on their center; they define a measurable Hilbert-space-valued kernel. Its Hilbert–Schmidt bound into $\mathcal{F}$ is therefore

$$\Vert K_{\delta }{\Vert }_{\mathrm{HS}(L^2(I),\mathcal{F})}^2\le H:=\frac{L}{{\ell }^2}\left[C_0(e^{\ell }-1)+28e^{2\ell }\right].\text{\hspace{1em}(HS)}$$

Restrict to the even source and target spaces. Reflection commutes with convolution and the original form, and restriction only decreases the Hilbert–Schmidt bound. The standard singular-value truncation supplies a rank-at-most-$N$ map $K_{\delta ,N}:L_{\mathrm{even}}^2(I)\to {\mathcal{F}}_{\mathrm{even}}$ with

$$\Vert K_{\delta }-K_{\delta ,N}\Vert \le \sqrt{H/(N+1)}.\text{\hspace{1em}(SV)}$$

The singular vectors for nonzero singular values lie in the range of $K_{\delta }$, so the finite-rank output is supported in $I+[-1,1]$. Neither the generic singular-value theorem nor the source averaging lemma is a new project result.

Uniform finite-rank form approximation and a rank bound

Fix $0<\tau <1$. In addition to $R\ge R_{\mathrm{PNT}}(\epsilon )$, take

$$R\ge \log \frac{8640}{\epsilon \tau },\ell \ge \max \left\{1,\frac{18C_{0}B}{{\tau }^2},B{\left(\frac{252G}{{\tau }^2}\right)}^2\right\},N+1\ge \frac{9H}{2m{\tau }^2}.\text{\hspace{1em}(FP)}$$

For example the ceiling of the last right-hand side is a permissible $N$. The original full low-projector form-tail estimate gives $\Vert h-g{\Vert }_{\mathcal{F}}\le \sqrt{866}(96/\epsilon )e^{-R}<\tau /3$. The two terms in (AF) are each at most ${\tau }^2/18$. Finally (IB) and (SV) give $\Vert K_{\delta }g-K_{\delta ,N}g{\Vert }_{\mathcal{F}}\le \tau /3$. For the single finite-rank map $V_N=K_{\delta ,N}{M}_{{\chi }_R}$, these are simultaneous estimates on the whole actual low subspace, so

$$\boxed{\underset{\begin{array}{c}h\in \mathrm{ran}{P}_{\epsilon }\\ \Vert h{\Vert }_{\nu }=1\end{array}}{\sup }\Vert h-V_{N}h{\Vert }_{\mathcal{F}}<\tau ,\mathrm{rank}{P}_{\epsilon }\le N.}\text{\hspace{1em}(FA)}$$

The rank conclusion follows because $\tau <1$ makes $V_N$ injective on that subspace already in $L^2(\nu )$. It includes the constant eigenspace. Projection onto $\mathcal{R}$ contracts this form error for vectors in the remainder, while generally destroying compact support.

The rank and error formulas are analytic upper bounds. The chosen $m$ uses the minimum density on a growing interval; no computational feasibility, computed singular basis or lower spectral matrix is supplied. The passage from $L^2$ to form estimates uses (LG), (UC), (AF) and the form-valued (HS). A complete lower spectral certificate and control uniform as $\epsilon \downarrow 0$ remain missing. These model deductions are paper-level and have no new Lean certification or originality claim; RH and full Robin remain unresolved.

Explicit translated kernels and the legal FIB mesh

The original-form graph transfer and (IB), (UC), (AF), (FP) also give a prescribed finite spanning family, without using unknown singular functions. Fix $0<\epsilon \le 1/2$ and $0<\tau <1$; retain the preceding $R,\ell ,m,C_0,B,G,L,I,J$, using the radius and averaging choices of (FP) but replacing its SVD rank choice by the mesh below. For a unit even $h\in \mathrm{ran}{P}_{\epsilon }$, put $g={\chi }_{R}h$ and $w=w_{\delta }$, $\delta =e^{-\ell }$. Standard BV translation estimates, Hilbert-space kernel estimates and the existing FIB interval tiling are reused here. The model interface is their uniform error estimate in the original mixed minimal form norm.

Translation modulus in the original form norm

Set $A_{\delta }=2/({\ell }^2{\delta }^2)$. The reused BV translation inequality gives $\Vert {\tau }_{s}w-w{\Vert }_2^2\le A_{\delta }|s|$. For $0<s\le 1$ and $v_s={\tau }_{s}w-w$, commuting flat translations and the triangle inequality give

$$\Vert {\tau }_t{v}_s-v_s{\Vert }_2^2\le 4A_{\delta }\min (s,t)(0<t<1).$$

Using the existing one-half energy convention,

$${\mathcal{E}}_{j_0}(v_s)={\int }_0^1\Vert {\tau }_t{v}_s-v_s{\Vert }_2^2\frac{dt}{t}\le 4A_{\delta }s(1+\log (1/s)).\text{\hspace{1em}(TM)}$$

For $y,c\in I$, $|y-c|\le \Delta \le 1$, the kernels lie in the previously proved original minimal domain and are supported in $I+[-1,1]⋐J$. Apply (UC) to their difference. The function $s(1+\log (1/s))$ is nondecreasing on $(0,1]$, hence

$$\Vert w(\cdot -y)-w(\cdot -c){\Vert }_{\mathcal{F}}^2\le A_{\delta }\Delta [C_0+28(1+\log (1/\Delta ))].\text{\hspace{1em}(FM)}$$

At $y=c$ the difference is zero. This bound also supplies continuity and Bochner measurability of the form-valued kernel. It uses full original form upper comparison, not a prime truncation or convolution commutation with $A_{\Gamma }$.

A prescribed symmetric mesh map

Partition $I$ into cells $I_j$ of maximal width $\Delta \le 1$, using half-open cells at endpoints of measure zero, and take their midpoints $c_j$. Define

$$K^{\mathrm{mesh}}u(x)=\sum _{j}w(x-c_j){\int }_{I_j}u(y)dy.$$

The Hilbert-space-valued kernel bound (FM) gives

$$\Vert K_{\delta }-K^{\mathrm{mesh}}{\Vert }_{\mathrm{HS}({\mathrm{L}}^2(\mathrm{I}),\mathcal{F})}^2\le 2L{A}_{\delta }\Delta [C_0+28(1+\log (1/\Delta ))].$$

For the actual $g$, (IB) states $\Vert g{\Vert }_2^2\le 1/(2m)$, so

$$\Vert K_{\delta }g-K^{\mathrm{mesh}}g{\Vert }_{\mathcal{F}}^2\le \frac{L{A}_{\delta }}{m}\Delta [C_0+28(1+\log (1/\Delta ))].\text{\hspace{1em}(ME)}$$

Use $\Delta \le \sqrt{\Delta }$ and $\Delta (1+\log (1/\Delta ))\le 2\sqrt{\Delta }$ to set

$$d_{\ast }:=\min \left\{1,{\left[\frac{m{\tau }^2}{9L{A}_{\delta }(C_0+56)}\right]}^2\right\}.\text{\hspace{1em}(MS)}$$

Every mesh of maximal width at most $d_{\ast }$ makes (ME) at most ${\tau }^2/9$. For a symmetric partition with zero inserted, list its positive cells $I_j^{+}$ and midpoints $c_j>0$. Define the explicit even real functions

$$b_j(x)=w(x-c_j)+w(x+c_j),Vh=\sum _j{b}_j(x){\int }_{I_j^{+}}{\chi }_R(y)h(y)dy.\text{\hspace{1em}(EB)}$$

For even $h$ this equals $K^{\mathrm{mesh}}g$. Each $b_j$ belongs to the original minimal form domain by the already proved kernel membership; no eigenfunction or singular-vector computation is used. The outer error is $<\tau /3$ by (FP); the averaging error is $\le \tau /3$ by (AF); the mesh error is $\le \tau /3$ by (MS). Thus the single prescribed map satisfies

$$\underset{h\in \mathrm{ran}{P}_{\epsilon },\Vert h{\Vert }_{\nu }=1}{\sup }\Vert h-Vh{\Vert }_{\mathcal{F}}<\tau .\text{\hspace{1em}(EA)}$$

The real spanning family need not be linearly independent; matrix PSD on that family is nevertheless equivalent to form nonnegativity on its span. Its size is a rank upper bound; no computed Gram matrix is given.

Reuse the existing FIB five-mode interval partition

Use the existing FIB interval tiling §§150–151 and legal address graph Definition 14.5 and Proposition 14.6. Their graph is $0\to 0:\mathrm{null},2,3$; $0\to 1:25,5$; $1\to 0:\mathrm{null},3$; $1\to 1:5$. Their contraction is $\lambda =-{\varphi }^{-3}$ and state intervals are $K_0=[-1,\varphi ]$, $K_1=[-1,{\varphi }^{-1}]$. Legal cylinder intervals tile with disjoint interiors; shared endpoints have zero length measure.

At depth $n$, the state-zero partition has $p_n=F_{3n+2}$ intervals (the count of no-adjacent-one bit strings of length $3n$), with maximal width ${\varphi }^2{\varphi }^{-3n}$, where $F_0=0,F_1=1$. The affine map

$$y=-L+\frac{2L}{{\varphi }^2}(t+1)$$

sends it to a partition of $I$ of maximal width $2L{\varphi }^{-3n}$. Take the common refinement with its reflection and insert zero. There are at most $2p_n$ cells: the two original endpoint sets share $-L,L$, and insertion of zero adds at most one endpoint. Symmetry pairs all positive and negative cells, so (EB) uses at most $p_n$ functions. Take

$$n\ge \lceil \frac{\log (2L/d_{\ast })}{3\log \varphi }\rceil .\text{\hspace{1em}(FD)}$$

Then (EA) holds and, because $\tau <1$, $V$ is injective on the low subspace in $L^2(\nu )$, giving $\mathrm{rank}{P}_{\epsilon }\le p_n$. FIB labels index this mesh only; they are not prime weights, and no active golden rotation of the single contracting interval is asserted. An ordinary sufficiently fine symmetric mesh gives the same guarantee; the FIB addressing supplies no proved improvement in approximation rate.

A conditional finite matrix target with completeness paid

For the fixed-gap target, use the explicit approximation above with

$$a=\frac{1}{2}-\epsilon ,c_{\epsilon }=\frac{1}{2}-\epsilon /2,\tau =\epsilon /8.$$

Let $Y=\mathrm{span}\{b_j\}$ and ${\mathcal{W}}_c(v)=D(v)-c{\mathrm{Var}}_{\nu }(v)$, where ${\mathrm{Var}}_{\nu }(v)=\Vert v-⟨1,v{⟩}_{\nu }1{\Vert }_{\nu }^2$ and $⟨u,v{⟩}_{\nu }=\int \overline{u}vd\nu$. Because the spanning functions are real and $\nu (\mathbb{R})=1$, its exact Hermitian matrix is

$$T_{ij}^{(\epsilon )}=D(b_i,b_j)-c_{\epsilon }\left[⟨b_i,b_j{⟩}_{\nu }-{\mu }_i{\mu }_j\right],{\mu }_i=\int b_{i}d\nu .\text{\hspace{1em}(MT)}$$

Here $D(\cdot ,\cdot )$ is the sesquilinear polarization of the complete prime-plus-Gamma energy, retaining every prime power and crossing edge. Suppose (MT) is verified positive semidefinite. If a mean-zero unit $h$ were in $\mathrm{ran}{P}_{\epsilon }$, then $D(h)\le a$ and ${\mathcal{W}}_{c_{\epsilon }}(h)\le -\epsilon /2$. For $v=Vh$, (EA), Cauchy–Schwarz for the energy form and the norm-one projection onto constants’ complement give

$$|{\mathcal{W}}_{c_{\epsilon }}(v)-{\mathcal{W}}_{c_{\epsilon }}(h)|\le [2\sqrt{a}+2c_{\epsilon }+(1+c_{\epsilon })\tau ]\tau \le 3\tau (\tau \le 1/3).$$

Thus ${\mathcal{W}}_{c_{\epsilon }}(v)<-\epsilon /8$, contradicting PSD on $Y$. The conditional conclusion is

$$T^{(\epsilon )}⪰0⟹\mathrm{ran}{P}_{\epsilon }\subseteq \mathbb{C}1.\text{\hspace{1em}(PC)}$$

This conclusion covers the whole fixed-gap window because (EA) is uniform on its actual spectral subspace. Generic matrix PSD and form perturbation are reused, not new spectral theorems. The finite hypothesis requires the lower level $c_{\epsilon }<1/2$: on the same span, half-threshold PSD implies this condition since $\mathcal W_{c_\varepsilon}=\mathcal W_{1/2} +(\varepsilon/2)\operatorname{Var}_\nu$. This implication certifies no actual matrix sign or strict separation of signs.

The entries and signs in (MT) are uncomputed; (PC) is not an actual lower spectral certificate. A cofinal sequence $\epsilon \downarrow 0$, or a separate uniform theorem, is still needed to exclude every nonconstant spectral value below one-half. The bounds do not establish computational feasibility. These explicit model transfers are paper-level, without new Lean certification or an originality claim; RH and full Robin remain unresolved.

Directed assembly for the actual mixed theta matrix

Reuse the original-domain even real family $b_1,\dots ,b_q$, $b_j(x)=w_{\delta }(x-c_j)+w_{\delta }(x+c_j)$, $\delta =e^{-\ell }$, and all previous form/domain estimates. Let $S=L+1$, $K=[-S,S]$ and $B(x)=(b_1(x),\dots ,b_q(x){)}^T$, with $B=0$ off $K$. The exact target is $T_{\epsilon }=D-c_{\epsilon }(G-\mu {\mu }^T)$, $c_{\epsilon }=1/2-\epsilon /2$, $\nu (\mathbb{R})=1$.

Exact same-object matrices

$$G={\int }_{K}B(x)B(x{)}^T\rho (x)dx,\mu ={\int }_{K}B(x)\rho (x)dx,\rho =2\Phi \cosh (x/2),$$

$$\begin{array}{rl}{D}_{\Gamma }& ={\int }_{x<y}\Phi (x)\Phi (y)\psi (y-x)[B(y)-B(x)][B(y)-B(x){]}^{T}dxdy,\\ D_p& =\sum _{n\ge 2}{w}_n{\int }_{\mathbb{R}}\Phi (x)\Phi (x+\log n)[B(x+\log n)-B(x)][B(x+\log n)-B(x){]}^{T}dx,w_n=\Lambda (n)/\sqrt{n}.\end{array}\text{\hspace{1em}(DM)}$$

The positive-shift prime integral includes both original shifted graph terms after change of variables, cancelling the one-half prefactor. There is no further factor two. These are the original minimal-domain energies, with every prime power and Gamma crossing.

For $H>S$, $N\ge 2$ an integer, retain the Gamma square $[-H,H{]}^2$ and the complete prime terms $n\le N$. Call their sum $D_{H,N}$. Before expanding differences, each omitted region contributes a positive weight times an outer product, so

$$D-D_{H,N}=R_{\Gamma }+R_p⪰0.\text{\hspace{1em}(DE)}$$

This is a simultaneous coefficient-vector inequality, not entrywise positivity of the energy matrices or of their off-diagonal entries.

Reuse validated analytic quadrature

The Johansson/FLINT integration supplier provides the existing validated Petras method and the bounded-path, holomorphic-callback contract. Reuse its documented one-dimensional Gauss error $M{e}_n(r)$, where $e_n(r)=64/[15(r-1)r^{2n-1}]$. Finite endpoints, bounded integrands and inspection of the returned ball are necessary. The quadrature result and implementation are existing work; the actual-model integration interface follows below.

Remove both smooth-diagonal and jump-corner singularities

Partition $[-H,H]$ at all $\pm c_j\pm \delta$, $\pm c_j\pm 1$ and its endpoints. On each open interval $B$ is a fixed rational branch, with denominators separated from zero on its closure. Use one-sided branches at endpoints; their point values do not affect either Gamma integration or prime integration in Lebesgue $dx$. Put

$$\kappa (t)=t\psi (t)=\frac{t{e}^{-t/2}}{1-e^{-2t}},\kappa (0)=1/2.$$

On a single cell, algebraically factor $b_i(y)-b_i(x)=(y-x)Q_i(x,y)$ with rational regular $Q_i$. Then

$$\psi (y-x)[b_i(y)-b_i(x)][b_j(y)-b_j(x)]=(y-x)\kappa (y-x)Q_i(x,y)Q_j(x,y).\text{\hspace{1em}(SD)}$$

The apparent singularity is removable. A triangular change of variables maps the ordered same-cell region to a unit square with bounded analytic integrand.

For adjacent cells meeting at $z$ with lengths $A,B_{\ast }>0$, set $x=z-s,y=z+t$, $0\le s\le A$, $0\le t\le B_{\ast }$. The jump difference need not vanish at $(s,t)=0$. Split this rectangle along its normalized diagonal and use

$$(s,t)=(Au,B_{\ast }uv),(s,t)=(Auv,B_{\ast }u),0\le u,v\le 1.$$

Both Jacobians are $A{B}_{\ast }u$. Their kernel-times-Jacobian factors become

$$\frac{A{B}_{\ast }\kappa (u(A+B_{\ast }v))}{A+B_{\ast }v},\frac{A{B}_{\ast }\kappa (u(Av+B_{\ast }))}{Av+B_{\ast }}.\text{\hspace{1em}(JC)}$$

They are bounded and holomorphic on a sufficiently small complex neighborhood of the real square; the left and right rational branches retain the jump difference. Nonadjacent cells have positive separation. This transformation pays the jump corner without assuming continuity or deleting any Gamma diagonal strip. Unequal cell lengths may force small analytic neighborhoods and supply no favorable operation count.

Do not evaluate $\kappa$ through unresolved $0/0$ interval arithmetic. For example, with entire $E(z)={\sum }_{k\ge 0}{z}^k/(k+1)!$, $\kappa (t)=e^{-t/2}/(2E(-2t))$. Equivalently, $\kappa (t)=e^{t/2}/(2\mathrm{sinc}(it))$ where $\mathrm{sinc}(z)=\sin z/z$, $\mathrm{sinc}(0)=1$. Nonfinite enclosures still require subdivision or rejection; a removable analytic singularity is not a guarantee that a coarse interval evaluates tightly.

Uniform two-dimensional enclosure and theta callbacks

For a transformed $F$ bounded by $M$ on a product of affinely rescaled Bernstein ellipses with parameters $r,s>1$, tensor Gauss quadrature on $[0,1{]}^2$ has error at most

$$\frac{M}{2}[e_n(r)+e_m(s)].\text{\hspace{1em}(TQ)}$$

Apply the cited one-dimensional error bound one coordinate at a time; each rescaled Gauss rule has positive weights summing to one. This standard tensor application is not a new quadrature theorem. An adaptively calculated inner integral is not automatically a certified holomorphic outer callback: use (TQ), parameter-uniform inner enclosures, or interval box integration instead. For a bounded continuous interval extension, the fallback ${\int }_{Q}F\in |Q|[F(Q)]$ is a directed enclosure after (SD)/(JC). In the analytic callback use polarized products of real branch functions, not complex absolute squares.

The original theta series, not an asymptotic replacement, is

$$\Phi (z)=\sum _{n\ge 1}(4{\pi }^2{n}^4{e}^{9z/2}-6\pi n^2{e}^{5z/2})e^{-\pi n^2{e}^{2z}}.$$

For $a\le \Re z\le b$, $|\Im z|\le v<\pi /4$, let $\lambda =\pi e^{2a}\cos (2v)>0$. After $P$ terms, a uniform modulus remainder is at most

$$4{\pi }^2{e}^{9b/2}{U}_4+6\pi e^{5b/2}{U}_2,U_r=\frac{(P+1{)}^r{e}^{-\lambda (P+1{)}^2}}{1-{\theta }_r},{\theta }_r={\left(1+\frac{1}{P+1}\right)}^r{e}^{-\lambda (2P+3)}<1.\text{\hspace{1em}(TH)}$$

The successive term ratio decreases, giving the geometric majorant. This bounds the actual callback’s omitted theta summands. It is not a certificate for a callback whose complex box violates the strip or ratio condition. Reflection using evenness can improve negative-real panels.

For each retained prime term, split at the original breakpoints and those shifted by $-\log n$, integrating on $K\cup (K-\log n)$. Its pieces are analytic and nonsingular. Endpoints, $\delta$, $\log n$ and the centers must have rigorous enclosures; floating sorting of nearly coincident breakpoints is not a certificate.

Simultaneous Gamma and all-prime omitted-tail bounds

Reuse the original-series spatial and integral majorants, $\Phi (x)\le A_{\Phi }{e}^{-b_{\Phi }{e}^{2|x|}}$ with $A_{\Phi }=144/5$, $b_{\Phi }=3/2$. The cited integral bound is

$$\mathcal{T}(H)={\int }_{|y|>H}\Phi (y)dy\le \frac{96}{5}{e}^{-2H}{e}^{-(3/2)e^{2H}}.$$

For a nonnegative multiplier $f$ write $H_f={\int }_K\Phi (x)f(x)B(x)B(x{)}^{T}dx$. Because $B=0$ outside $K$ and $\psi$ decreases,

$$R_{\Gamma }={\int }_K\Phi (x)B(x)B(x{)}^T{\int }_{|y|>H}\Phi (y)\psi (|x-y|)dydx,0⪯R_{\Gamma }⪯\psi (H-S)\mathcal{T}(H)H_1.\text{\hspace{1em}(GT)}$$

Define the omitted prime row without speed normalization, $a_{>N}(x)={\sum }_{n>N}{w}_n[\Phi (x+\log n)+\Phi (x-\log n)]$. The squared-difference bound gives $0⪯R_p⪯2H_{a_{>N}}$. Using $w_n\le \log n/\sqrt{n}\le 2/e<1$ and the decreasing Gaussian integral bound, put

$${\beta }_{\pm }(x)=b_{\Phi }{e}^{\pm 2x},V_N(x)=\frac{A_{\Phi }}{2N}\left[\frac{e^{-{\beta }_{+}(x)N^2}}{{\beta }_{+}(x)}+\frac{e^{-{\beta }_{-}(x)N^2}}{{\beta }_{-}(x)}\right].$$

Then $a_{>N}\le V_N$ and

$$0⪯R_p⪯2H_{V_N}.\text{\hspace{1em}(PT)}$$

Every omitted prime power is included in the integer majorant. If $\log (N+1)>2S$, all omitted shifted supports are disjoint, so exactly $R_p=H_{a_{>N}}⪯H_{V_N}$. This is a multiplication-potential Gram matrix; it is not generally diagonal in the $b_j$ coefficient basis. In particular

$$D_{H,N}⪯D⪯D_{H,N}+\psi (H-S)\mathcal{T}(H)H_1+2H_{V_N}.\text{\hspace{1em}(DT)}$$

For a lower certificate omit only these positive energy remainders. The upper bounds allow refinement decisions or validation against the full energy; a negative truncated/lower matrix alone is not a full-form negative witness.

Pay variance and entry enclosures in the correct direction

For a rational vector $r$ approximating the full mean $\mu$, define

$$C_r=G-\mu r^T-r{\mu }^T+r{r}^T={\int }_{\mathbb{R}}(B-r)(B-r{)}^{T}d\nu .$$

Then $C_r-(G-\mu {\mu }^T)=(\mu -r)(\mu -r{)}^T$. Thus, for $F_r=D_{H,N}-c_{\epsilon }{C}_r$,

$$T_{\epsilon }-F_r=R_{\Gamma }+R_p+c_{\epsilon }(\mu -r)(\mu -r{)}^T⪰0.\text{\hspace{1em}(MC)}$$

The deliberate centering loss is only quadratic in $\mu -r$. The last term in $C_r$ is $r{r}^T$, not $\nu (K)r{r}^T$: the constant centering vector extends outside the trial functions’ compact support.

Suppose symmetric rational $\hat{F}$ and symmetric nonnegative rational upper bounds $E_{ij}$ enclose $|(F_r{)}_{ij}-{\hat{F}}_{ij}|\le E_{ij}$. Set

$$d_i=\sum _j{E}_{ij},L_{\mathrm{cert}}=\hat{F}-\mathrm{diag}(d_i).$$

From $2|z_i{z}_j|\le |z_i{|}^2+|z_j{|}^2$ for complex coefficients,

$$L_{\mathrm{cert}}⪯F_r⪯T_{\epsilon }.\text{\hspace{1em}(LCERT)}$$

Outward rational rounding supplies the rational error bounds. Certified PSD of this rational lower matrix would certify the precise (MT) condition for the prior whole-low-subspace approximation at $\tau =\epsilon /8$. Failed lower-matrix positivity does not establish indefiniteness of the full target. Entrywise lower rounding is not a substitute for (LCERT).

This is a finite assembly interface, not an assembled or signed matrix. No numerical feasibility, threshold-uniform control, cofinal window certificate, original general quadrature theorem, Lean, RH or full Robin conclusion is supplied. Existing source methods are reused; the new work is their same-object transfer with correct jumps, tails and variance.