bibkey: lenz2010compactness authors: Daniel Lenz, Peter Stollmann, and Daniel Wingert year: 2010 title: Compactness of Schrödinger semigroups doi: 10.1002/mana.200910054 url: https://arxiv.org/abs/0903.0280v2 claim: The relative-compactness and potential criteria, with the actual theta cutoff and prime-tail checks, confine the even mixed operator’s essential spectrum to a bottom of one-half. Constants are the zero-energy space, giving an unspecified positive full-measure Poincare constant. Discrete subthreshold eigenvalues and the RH-strength one-half bound remain unresolved. strata_touched: [] license: bibliographic-reference-only triage: anchor

The mixed theta essential threshold and the unresolved gap

Existing spectral suppliers

The journal reference is Mathematische Nachrichten 283 (2010), 94–103. The inspected primary text is arXiv:0903.0280v2, 23 March 2010, SHA-256 046b461e78774fe31f385cf9fe59f6cb69887a5ed27fdf2390451a030d98da5d. It identifies itself as a pre-peer-reviewed version. The publisher PDF was not retrieved; equality with the journal edition is not asserted.

Theorem 1.3, printed p.3, identifies compactness of a bounded map on a semibounded operator’s form domain with relative resolvent compactness. Theorem 2.1, printed p.4, concerns $H=H_0+V_{+}-V_{-}$ on an arbitrary measure space, where $H_0\ge \gamma$, $V_{+}\ge 0$ is measurable and $V_{-}$ is form small:

$$⟨V_{-}u,u⟩\le q⟨(H_0+V_{+})u,u⟩+C_q\Vert u{\Vert }^2,q<1.$$

If $1_{\{V_{+}<s\}}$ is $(H_0+V_{+})$-relatively compact, its conclusion is ${\sigma }_{\mathrm{ess}}(H)\subseteq [(1-q)(\gamma +s)-C_q,\infty )$. The application below has bounded $V_{+}$ and $V_{-}=0$, so its form domain is dense and $\gamma =q=C_q=0$ is admissible.

For the final perturbation use Gerald Teschl, Mathematical Methods in Quantum Mechanics, authorized first-edition online text dated 12 February 2009, Theorem 6.19, printed p.146, and the relative-compact perturbation discussion on pp.147–148. The inspected PDF SHA-256 is 8dc8de0b58aa0a3fedfe594a345f9b5875322e5526ea581cb640a98d55b82818. That theorem preserves essential spectrum under compact resolvent difference. These generic results are reused; the model checks follow.

Exact prime diagonal and bounded off-diagonal operator

Retain the original minimal realization, $d\nu =\rho dx=2\Phi \cosh (x/2)dx$, and all weights $w_n=\Lambda (n)/\sqrt{n}$. Define the outgoing prime rate

$$a_p(x)=\frac{{\sum }_{n\ge 2}{w}_n(\Phi (x+\log n)+\Phi (x-\log n))}{2\cosh (x/2)}.$$

It is continuous, even and nonnegative. Uniform convergence of its series on compact sets follows from the theta tails. The already reviewed ordinary PNT and Chebyshev suppliers give $a_p(x)\to 1/2$ as $|x|\to \infty$, without RH. The parameter map is explicit: for $x>0$, $X=e^x$ and $f(y)=y^{-1/2}\Phi (\log y)$, its incoming contribution is

$$a_{p,-}(x)=\frac{1}{1+X^{-1}}\frac{1}{X}\sum _{n\ge 2}\Lambda (n)f(n/X).$$

With $\Psi (t)={\sum }_{2\le n\le t}\Lambda (n)$, Stieltjes integration by parts writes the last scaled sum as $-{\int }_0^{\infty }\Psi (Xy)f^{\prime }(y)dy/X$. The source bounds $\Psi (t)\le Bt$ and $\Psi (t)/t\to 1$ permit dominated convergence since $\int y|f^{\prime }(y)|dy=\int e^{r/2}|{\Phi }^{\prime }(r)-\Phi (r)/2|dr<\infty$. The limit is $\int f=\int e^{r/2}\Phi (r)dr=1/2$; the outgoing $x+\log n$ contribution tends to zero by the theta tail. Continuity and this limit give $a_p\in L^{\infty }$. This reuses the completed rate argument, rather than a new PNT or a short-interval prime estimate.

Under the unitary $Uh=v=\sqrt{\rho }h$ from $L^2(\nu )$ to $L^2(dx)$, put $t_n=\log n$, $({\tau }_{n}v)(x)=v(x+t_n)$ and

$$b_n(x)=\frac{w_n\sqrt{\Phi (x)\Phi (x+t_n)}}{2\sqrt{\cosh (x/2)\cosh ((x+t_n)/2)}},S_n=M_{b_n}{\tau }_n.$$

The original tail gives $\Phi (x)\le C{e}^{-c{e}^{2|x|}}$ with $c>0$. Since $e^{2|x|}+e^{2|x+t_n|}\ge 2n$, it follows that $\Vert b_n{\Vert }_{\infty }\le C{w}_n{e}^{-cn}$ and ${\sum }_n\Vert b_n{\Vert }_{\infty }<\infty$. Therefore $B={\sum }_{n\ge 2}(S_n+S_n^{\ast })$ converges in operator norm and is bounded self-adjoint. Write $B_p=U^{-1}BU$. The complete prime energy is exactly

$$D_p(h)=⟨M_{a_p}h,h{⟩}_{\nu }-⟨B_{p}h,h{⟩}_{\nu },0\le D_p(h)\le 2\Vert a_p{\Vert }_{\infty }\Vert h{\Vert }_{\nu }^2.\text{\hspace{1em}(BD)}$$

The inequality follows from the actual symmetric edge measure. It extends the identity to all $L^2(\nu )$, including every prime power and both directions. Thus the Gamma and mixed form norms on their common compact smooth core are equivalent. Their minimal form domains agree; no equality with an arbitrary maximal domain is used.

Let $A_{\Gamma }$ be the full minimal Gamma operator and $C=A_{\Gamma }+M_{a_p}$. The original full mixed minimal operator is $A=C-B_p$, with $D(A)=D(C)=D(A_{\Gamma })$ by bounded self-adjoint perturbation. Both $A$ and $C$ are nonnegative.

Two-sided spatial tails and relative compactness

For compact smooth $0\le {\chi }_R\le 1$, one on $[-R,R]$, the exact adjoint coefficient gives

$$\begin{array}{rl}\Vert (1-{\chi }_R)S_n\Vert & \le \underset{|x|>R}{\sup }{b}_n(x),\\ \Vert (1-{\chi }_R)S_n^{\ast }\Vert & \le \underset{|x|>R}{\sup }{b}_n(x-t_n).\end{array}$$

For each fixed $n$ both suprema tend to zero, and the common summable global majorant permits summing the limits. Hence $\Vert (1-{\chi }_R)B\Vert \to 0$. Self-adjointness gives $\Vert B(1-{\chi }_R)\Vert \to 0$, and the same assertions hold for $B_p$ because $U$ commutes with scalar cutoffs. The shifted adjoint coefficient is necessary for this operator norm statement. Individual weighted translations are not asserted compact.

The theta cutoff comparison and Jarohs–Weth theorem give compact restriction on the Gamma form domain. The bounded potential preserves that domain and its norm equivalence, so ${\chi }_R(C+1{)}^{-1}$ is compact. In

$$B_p(C+1{)}^{-1}=B_p{\chi }_R(C+1{)}^{-1}+B_p(1-{\chi }_R)(C+1{)}^{-1},$$

the first term is compact and the second tends to zero in norm. Thus $B_p(C+1{)}^{-1}$ is compact, while all arithmetic edges remain present.

For every $0<s<1/2$, $\{a_p<s\}$ lies in a compact interval. Its indicator is compact on the form domain of $C$. Apply the source’s Theorems 1.3 and 2.1 with $H_0=A_{\Gamma }$, $V_{+}=a_p$, $V_{-}=0$ and $\gamma =q=C_q=0$. This yields ${\sigma }_{\mathrm{ess}}(C)\subseteq [s,\infty )$ for every such $s$, hence ${\sigma }_{\mathrm{ess}}(C)\subseteq [1/2,\infty )$.

Finally, the resolvent difference

$$(A+1{)}^{-1}-(C+1{)}^{-1}=(A+1{)}^{-1}{B}_p(C+1{)}^{-1}$$

is compact. Teschl’s Weyl theorem gives ${\sigma }_{\mathrm{ess}}(A)={\sigma }_{\mathrm{ess}}(C)\subseteq [1/2,\infty )$.

Reflection preserves the measure, the two conductance measures and the core, so it commutes with the minimal resolvents. The even space reduces $A$; its spectral projections strictly below $1/2-\epsilon$ are restrictions of full-space finite-rank projections. The already reviewed infinite critical eigenfamily puts $1/2$ in the even essential spectrum. Therefore

$$\inf {\sigma }_{\mathrm{ess}}(A_{\mathrm{even}})=\frac{1}{2}.\text{\hspace{1em}(ET)}$$

This identifies its bottom, not all essential spectrum above it or the complete one-half eigenspace.

A weaker actual-measure Poincare bound

If $D(h)=0$, positivity of the Gamma conductance off the diagonal forces $h(y)=h(x)$ for almost every pair. Fubini and equivalence of $dx$ and $\nu$ make $h$ constant almost everywhere. Conversely the constant $1$ belongs to the minimal domain and has zero energy. Hence $\ker A=\mathbb{C}1$.

By (ET), zero is a simple discrete eigenvalue of the even operator and is isolated. The spectral theorem on its centered reducing space gives an unspecified constant

$$0<c_{\ast }:=\inf \sigma (A_{\mathrm{even}}{|}_{1^{\perp }})\le \frac{1}{2},D(h)\ge c_{\ast }{\mathrm{Var}}_{\nu }(h)(h\in {\mathcal{F}}_{\mathrm{even}}).\text{\hspace{1em}(PG)}$$

The upper bound uses the known critical eigenvectors. This is the original pole probability measure and complete mixed energy. No explicit numerical value or effective approximation rate for $c_{\ast }$ is obtained, and $c_{\ast }=1/2$ remains unproved.

For the even operator, the remaining possible eigenvalues $0<\lambda <1/2$ are isolated and have finite multiplicities. Their eigenvectors are orthogonal to constants and the known critical family, hence lie in the previously defined remainder. There are finitely many below each fixed $1/2-\epsilon$, but there may be infinitely many accumulating at $1/2$. For example the abstract positive operator with diagonal $1/2-1/(k+3)$ on one ${\ell }^2$ summand and $\frac{1}{2}I$ on another has exactly this behavior. Thus (ET) and (PG) do not exclude the remaining subthreshold spectrum or establish RH and full Robin. The target stays the exact one-half bound.

Persson and support boundaries

Lenz–Stollmann, arXiv:1705.10398v2, Definition 2.1, printed p.5, and Theorem 3.2, printed p.8, use all measurable sets of finite speed measure in their Persson hypothesis. On this probability space it would require the whole semigroup to be compact, incompatible with the known infinite one-half eigenspace. Compact-set local embedding does not verify it. The compact-set version in BenAmor–Güneysu–Stollmann, Essential Spectrum and Feller Type Properties, Theorem 5.5, DOI 10.1007/s00020-023-02732-9, requires weak Feller, which is not verified here. Neither direct Persson identity is invoked in the preceding supplier chain.

For an exterior-supported function the actual energy retains jumps into the removed interval as killing contributions. It is not the censored kernel restricted to two exterior endpoints. Likewise a compact FIB test and its noncompact projection onto the critical remainder are different sources. A finite support estimate alone does not exclude eigenvalues arbitrarily close to the threshold.

The model-specific estimates and spectral applications above are paper deductions from the inspected sources, without new Lean certification or an originality claim. They advance an actual weaker lower bound and essential-threshold identification; the global RH-strength lower bound remains unresolved.

Fixed-gap eigenfunction tails and FIB approximation

Qualitative uniform tails on each spectral subspace strictly below one-half already follow from the finite-rank projections above. The following model estimate supplies explicit inputs for the actual noncompact vectors. It reuses the nonlocal IMS calculation and the complete prime decomposition, without a new generic theorem or originality claim.

For $R\ge 4$, choose even smooth $0\le {\eta }_R\le 1$, zero on $[-R,R]$, one outside $[-R-2,R+2]$, with $|{\eta }_R^{\prime }|\le 1$. Put

$$\begin{array}{rl}{e}_R(x)& =\frac{1}{2\cosh (x/2)}\int \Phi (y)\psi (|x-y|)|{\eta }_R(y)-{\eta }_R(x){|}^{2}dy,\\ E_R& =\Vert e_R{\Vert }_{\infty },b_R=\Vert M_{{\eta }_R}{B}_p\Vert ,\\ {\delta }_R& =\underset{|x|\ge R}{\sup }(\frac{1}{2}-a_p(x){)}_{+}.\end{array}$$

The first expression is exactly the Gamma conductance row divided by the original speed density $\rho$, with the squared cutoff difference. Let $k(t)=\psi (|t|)\min (1,t^2)$ for $t\ne 0$, $k(0)=0$, $K=\Vert k{\Vert }_{\infty }<\infty$, $P=\int \Phi$, $T(r)={\int }_{|y|>r}\Phi (y)dy$ and $d=(1-e^{-2}{)}^{-1}$. Then

$$E_R\le \frac{1}{2}(dP{e}^{-R/8}+KT(R/4))⟶0.\text{\hspace{1em}(TR)}$$

For $|x|\ge R/2$, the cutoff difference is bounded by $\min (1,|x-y|)$. Split the integral at $|y|=R/4$: on the inner part $|x-y|\ge R/4\ge 1$ gives $k(x-y)\le d{e}^{-R/8}$; on the outer part $k\le K$. Divide by $2\cosh (x/2)\ge 2$. For $|x|<R/2$, ${\eta }_R(x)=0$ and a nonzero difference requires $|y|>R$. This gives the smaller bound $d{e}^{-R/4}T(R)/2$. Thus (TR) concerns the actual Gamma kernel, rather than a fractional-kernel replacement.

On the compact smooth core, the product estimate gives $D_{\Gamma }({\eta }_{R}h)\le 2D_{\Gamma }(h)+E_R\Vert h{\Vert }_{\nu }^2$. Core approximants extend this bounded multiplier to the Gamma minimal domain. The same argument applies to ${\eta }_R^2$, whose cutoff rate is at most $4e_R$. The common minimal domain and (BD) give legality in the original mixed form domain. No maximal-domain identification is used. The exact ordered-jump identity is

$$\begin{array}{rl}{D}_{\Gamma }({\eta }_{R}h)-\Re D_{\Gamma }(h,{\eta }_R^{2}h)& =\frac{1}{2}\iint |{\eta }_R(y)-{\eta }_R(x){|}^2\Re (h(x)\overline{h(y)})J_{\Gamma }(dx,dy),\\ \left|D_{\Gamma }({\eta }_{R}h)-\Re D_{\Gamma }(h,{\eta }_R^{2}h)\right|& \le \frac{1}{2}{E}_R\Vert h{\Vert }_{\nu }^2.\end{array}\text{\hspace{1em}(GI)}$$

Here $J_{\Gamma }(dx,dy)=\Phi (x)\Phi (y)\psi (|x-y|)dxdy$. The factor one-half uses both ordered endpoints. Symmetry and $|\Re (h(x)\overline{h(y)})|\le (|h(x){|}^2+|h(y){|}^2)/2$ give the bound; form approximation extends the identity.

Let $Ah=\lambda h$, $\Vert h{\Vert }_{\nu }=1$, and $t_R=\Vert {\eta }_{R}h{\Vert }_{\nu }$. Testing the original eigen-equation with ${\eta }_R^{2}h$, using (GI), gives

$$(\frac{1}{2}-{\delta }_R-\lambda )t_R^2\le b_R{t}_R+\frac{1}{2}{E}_R.\text{\hspace{1em}(EL)}$$

Indeed the left diagonal is at least $(1/2-{\delta }_R)t_R^2$, $D_{\Gamma }({\eta }_{R}h)\ge 0$, and the full prime off-diagonal contribution has modulus at most $\Vert {\eta }_R{B}_{p}h{\Vert }_{\nu }{t}_R\le b_R{t}_R$. All crossing prime-power edges and shifted adjoints remain present. For $\lambda \le 1/2-\epsilon$ and ${\delta }_R\le \epsilon /2$, the quadratic inequality yields

$$\Vert 1_{\{|x|\ge R+2\}}h{\Vert }_{\nu }\le t_R\le \frac{b_R+\sqrt{b_R^2+\epsilon E_R}}{\epsilon }⟶0\text{for fixed }\epsilon >0.\text{\hspace{1em}(LT)}$$

The approved potential limit gives ${\delta }_R\to 0$ and the complete prime operator-tail argument gives $b_R\to 0$. An upper input for $b_R$ is

$$\sum _{n\ge 2}\left(\underset{|x|>R}{\sup }{b}_n(x)+\underset{|x|>R}{\sup }{b}_n(x-\log n)\right).$$

A finite prefix and the existing summable global theta majorant bound this expression. The effective original-series estimates below give a specified convergence rate without discarding the shifted adjoints.

The same equation, $a_p\ge 0$ and (BD) also give

$$D({\eta }_{R}h)\le (\frac{1}{2}+2\Vert a_p{\Vert }_{\infty })t_R^2+b_R{t}_R+\frac{1}{2}{E}_R.\text{\hspace{1em}(FE)}$$

Thus the compactly supported form-domain vector $h_R=(1-{\eta }_R)h$ approximates the actual eigenvector in the original form norm:

$$\Vert h-h_R{\Vert }_{\nu }^2+D(h-h_R)\le (\frac{3}{2}+2\Vert a_p{\Vert }_{\infty })t_R^2+b_R{t}_R+\frac{1}{2}{E}_R.\text{\hspace{1em}(FN)}$$

No smoothness of $h_R$ or effective interior discretization is asserted. If $h\in \mathcal{R}$, cutting it off need not preserve that remainder. Its established reducing projection $P_{\mathcal{R}}$ is a contraction in both the Hilbert and nonnegative form norms, so $P_{\mathcal{R}}{h}_R$ approximates $h$ with at most the same error. The projected vector generally has noncompact support; a support-limited positivity certificate does not thereby become a remainder certificate.

These fixed-$\epsilon$ paper estimates provide a finite-support source for approximating actual low eigenvectors. Interior discretization and complete spectral certification remain separate obligations. The $1/\epsilon$ loss is uncontrolled as $\epsilon \to 0$; threshold accumulation and subthreshold eigenvectors are not excluded. The RH-strength exact-half bound remains unproved.

Effective original-measure cutoff inputs

Reuse the original theta series majorants and the explicit prime-diagonal transfer. The following bounds evaluate the inputs to (LT) and (FN) for the same individual eigenvector, measure, minimal form and cutoff. They do not construct a finite spectral certificate.

Full prime off-diagonal tail

Let $C_{\Phi }=144/5$. From the theta majorant (ST), the exact $b_n$ coefficient and $\cosh \ge 1$,

$$b_n(x)\le \frac{C_{\Phi }}{2}{w}_n\exp \left[-\frac{3}{4}(e^{2|x|}+e^{2|x+\log n|})\right].$$

The quantity in parentheses is at least $2n$, by the arithmetic-geometric mean and $|x|+|x+\log n|\ge \log n$. For $|x|>R$ it is also at least $e^{2R}$. Its mean lower bound is thus $(e^{2R}+2n)/2$. Exactly the same two bounds hold for the shifted adjoint $b_n(x-\log n)$: its pair is $x-\log n,x$, so the exterior endpoint is still retained. Using $w_n\le n$, $e^{-3/4}<1/2$ and ${\sum }_{n\ge 2}n{2}^{-n}=3/2$ in the full two-direction norm sum gives

$$\boxed{b_R\le \frac{216}{5}\exp \left(-\frac{3}{8}{e}^{2R}\right)}(R\ge 0).\text{\hspace{1em}(BT)}$$

Both prime powers and crossing edges occur in this bound. Because $(3/8)e^{2R}\ge R$ for $R\ge 1$, a simpler consequence is

$$b_R\le 44e^{-R}(R\ge 1).\text{\hspace{1em}(BR)}$$

The nonlinear inequality follows at $R=1$ from $e^2>7$ and thereafter from its positive derivative. No individual translation is claimed compact.

Gamma cutoff error

For $t>0$ the elementary bound $1-e^{-2t}\ge 2t/(1+2t)$ gives $\psi (t)t^2\le 3/2$ for $0<t\le 1$. For $t\ge 1$ one has $\psi (t)\le d{e}^{-t/2}$ with $d=(1-e^{-2}{)}^{-1}<7/6$. Thus $k(t)=\psi (t)\min (1,t^2)\le 3/2$ everywhere.

For $|x|\ge R$, divide the cutoff row by $2\cosh (x/2)\ge e^{|x|/2}$ and split at $|x-y|=1$. On long jumps, use $e^{-|x-y|/2}\le e^{-|x|/2}{e}^{|y|/2}$ and the original normalization $\int \Phi (y)e^{|y|/2}dy\le 1$. On short jumps, the Lipschitz cutoff bound applies and $|y|>R-1$. Hence

$$e_R(x)\le d{e}^{-R}+\frac{3}{2}{e}^{-R/2}T(R-1)(|x|\ge R).$$

For $|x|\le R$, ${\eta }_R(x)=0$; a nonzero row term requires $|y|>R$. The same $k$ bound and $2\cosh (x/2)\ge 2$ give $e_R(x)\le (3/4)T(R)$. Combining the two ranges,

$$E_R\le d{e}^{-R}+\frac{9}{4}T(R-1)(R\ge 1).$$

The theta integrated tail (IT) gives $T(R-1)\le e^{-R}$ for $R\ge 4$. Indeed, its ratio to $e^{-R}$ is at most $(96/5)e^{2-R}\exp [-(3/2)e^{2(R-1)}]$, which decreases; at $R=4$ it is less than $24/35$ using $e^2>7$ and $e^{3/2}>4$. Therefore the actual ordered Gamma cutoff error satisfies

$$\boxed{E_R\le 4e^{-R}(R\ge 4).}\text{\hspace{1em}(ER)}$$

This sharpens the coarse rate (TR) for these cutoffs. It keeps the singular Gamma kernel and all endpoint interactions; it is not a fractional-kernel substitution or a censored exterior energy.

A specified fixed-gap form-approximation radius

Fix $0<\epsilon \le 1/2$ and a tolerance $0<\tau \le 1$. Use $R_{\mathrm{PNT}}(\epsilon )$ from (PR) in the linked prime-diagonal note, and take

$$R\ge \max \left\{R_{\mathrm{PNT}}(\epsilon ),\log \frac{11264}{\epsilon \tau },2\log \frac{256}{\sqrt{\epsilon }\tau },\log \frac{8}{{\tau }^2}\right\}.\text{\hspace{1em}(FR)}$$

For any actual normalized eigenvector $Ah=\lambda h$ with $\lambda \le 1/2-\epsilon$, (LT), (BR) and (ER) give

$$t_R\le \frac{2b_R}{\epsilon }+\sqrt{\frac{E_R}{\epsilon }}\le \frac{88e^{-R}}{\epsilon }+\frac{2e^{-R/2}}{\sqrt{\epsilon }}\le \frac{\tau }{64}.$$

Let $h_R=(1-{\eta }_R)h$. With $\Vert a_p{\Vert }_{\infty }<432$ from (AC), the three terms in (FN) are at most $(1731/8192){\tau }^2$, $(11/128){\tau }^3$ and ${\tau }^2/4$, respectively. The middle estimate uses (BR) and the last radius in (FR). Thus

$$\boxed{\Vert h-h_R{\Vert }_{\nu }^2+D(h-h_R)\le \frac{1121}{2048}{\tau }^2<{\tau }^2.}\text{\hspace{1em}(FA)}$$

The reducing projection $P_{\mathcal{R}}$ retains at most this error for $h\in \mathcal{R}$; its image generally remains noncompact. These are coarse analytic constants, with no directed numerical computation, new Lean certification or originality claim.

The conclusion concerns each individual actual eigenvector under the stated fixed gap. The whole-projector estimate below uses a separate bounded commutator and separated-spectra argument for mixtures of different eigenvalues. Interior discretization, a complete lower spectral certificate and control uniform as $\epsilon \downarrow 0$ remain missing. The radius in (FR) diverges with shrinking gap; threshold accumulation, RH and full Robin remain unresolved.

Full fixed-gap low-spectral-subspace cutoff

The effective scalar and cutoff bounds also yield a uniform estimate for all vectors in the low spectral subspace. This is a model transfer of the standard separated-spectra integral argument; it is not a new general spectral criterion, interior discretization or complete spectral exclusion.

Bounded Gamma cutoff commutator

For the same even cutoff define

$$q_R(x)=\frac{1}{2\cosh (x/2)}\int \Phi (y)\psi (|x-y|)|{\eta }_R(x)-{\eta }_R(y)|dy,Q_R=\Vert q_R{\Vert }_{\infty }.$$

This row uses the first power of the cutoff difference. The short-jump bound is now $\psi (t)t\le 3/2$ for $0<t\le 1$, still following from $1-e^{-2t}\ge 2t/(1+2t)$. On long jumps use the same $d{e}^{-t/2}$ bound. The two spatial ranges used for (ER) therefore give, with no squared-row substitution,

$$Q_R\le d{e}^{-R}+\frac{9}{4}T(R-1)\le 4e^{-R}(R\ge 4).\text{\hspace{1em}(QC)}$$

On the compact smooth core, the Gamma commutator is

$$K_{R}h(x):=[A_{\Gamma },M_{{\eta }_R}]h(x)=\frac{1}{2\cosh (x/2)}\int \Phi (y)\psi (|x-y|)({\eta }_R(x)-{\eta }_R(y))h(y)dy.$$

The absolute kernel is symmetric relative to $\nu$ and has row mass $q_R$. The weighted Schur estimate gives $\Vert K_R\Vert \le Q_R$ on $L^2(\nu )$, including the singular Gamma neighborhood. The existing cutoff multiplier bound and core approximation extend

$$D_{\Gamma }({\eta }_{R}h,k)-D_{\Gamma }(h,{\eta }_{R}k)=⟨K_{R}h,k{⟩}_{\nu }$$

to the Gamma minimal form domain, with the inner product linear in its first argument. For $h\in D(A_{\Gamma })$ the operator representation of this identity shows ${\eta }_{R}h\in D(A_{\Gamma })$ and $A_{\Gamma }{\eta }_{R}h={\eta }_R{A}_{\Gamma }h+K_{R}h$. The bounded potential commutes with the cutoff, so the same domain and commutator statement holds for $C=A_{\Gamma }+M_{a_p}$.

The killed exterior operator and separated spectra

Work in the even Hilbert space. Fix $0<\epsilon \le 1/2$, put $a=1/2-\epsilon$, and let $P_{\epsilon }=1_{[0,a]}(A_{\mathrm{even}})$. Write $\mathcal{L}=\mathrm{ran}{P}_{\epsilon }$ and $A_{\mathcal{L}}=A{|}_{\mathcal{L}}$; it is bounded with $0\le A_{\mathcal{L}}\le a$.

Let ${\mathcal{H}}_R$ consist of even functions zero on $[-R,R]$. Restrict the global closed form of $C$ to its form-domain intersection with ${\mathcal{H}}_R$, and let $C_R$ be the associated operator on ${\mathcal{H}}_R$. The restricted form is closed, and smooth even tests supported outside $[-R,R]$ give density. It keeps the Gamma jumps into the removed interval as killing terms. Hence

$$C_R\ge b:=1/2-{\delta }_R.$$

For $R\ge R_{\mathrm{PNT}}(\epsilon )$, $b-a\ge \epsilon /2>0$. Let $T_R=M_{{\eta }_R}{|}_{\mathcal{L}}:\mathcal{L}\to {\mathcal{H}}_R$. The commutator domain statement implies $T_R\mathcal{L}\subset D(C_R)$. Indeed, restriction of the global operator identity to exterior form tests gives

$$C_R{T}_R-T_R{A}_{\mathcal{L}}=F_R,F_R=1_{\{|x|>R\}}(K_R+M_{{\eta }_R}{B}_p){|}_{\mathcal{L}},\Vert F_R\Vert \le Q_R+b_R.\text{\hspace{1em}(SY)}$$

Every prime power and shifted adjoint still occurs in $B_p$. For each vector in $\mathcal{L}$, differentiation of $e^{-t{C}_R}{T}_R{e}^{t{A}_{\mathcal{L}}}$ and integration on $t\ge 0$ give the strong-operator identity

$$T_R={\int }_0^{\infty }{e}^{-t{C}_R}{F}_R{e}^{t{A}_{\mathcal{L}}}dt.$$

Its endpoint at infinity vanishes in operator norm, since the separated spectra give exponential decay by $e^{-(b-a)t}$. Thus

$$\boxed{\Vert M_{{\eta }_R}{P}_{\epsilon }\Vert \le \frac{Q_R+b_R}{\epsilon -{\delta }_R}\le \frac{96}{\epsilon }{e}^{-R}=:s_R.}\text{\hspace{1em}(LP)}$$

This proves an operator-norm bound for the whole low spectral subspace; it does not assign a scalar eigenvalue to a mixture of eigenvectors.

Uniform form approximation on that subspace

For a unit vector $h\in \mathcal{L}$, put $h_R=(1-{\eta }_R)h$. Use the global identity $C{\eta }_{R}h={\eta }_{R}Ah+(K_R+{\eta }_R{B}_p)h$. Because $Ah\in \mathcal{L}$ and $\Vert Ah\Vert \le a$, (LP) gives $\Vert {\eta }_{R}Ah\Vert \le a{s}_R$. Therefore

$$D_{\Gamma }({\eta }_{R}h)\le ⟨C{\eta }_{R}h,{\eta }_{R}h{⟩}_{\nu }\le a{s}_R^2+(Q_R+b_R)s_R.$$

The complete prime energy bound (BD) gives

$$\Vert h-h_R{\Vert }_{\nu }^2+D(h-h_R)\le (3/2+2\Vert a_p{\Vert }_{\infty })s_R^2+(Q_R+b_R)s_R\le 866s_R^2.\text{\hspace{1em}(UF)}$$

For the last step use $\Vert a_p{\Vert }_{\infty }<432$, $Q_R+b_R\le 48e^{-R}=\epsilon s_R/2$ and $\epsilon \le 1/2$. Consequently, for $0<\tau \le 1$, the radius

$$R\ge \max \left\{R_{\mathrm{PNT}}(\epsilon ),\log \frac{2880}{\epsilon \tau }\right\}\text{\hspace{1em}(UR)}$$

ensures, simultaneously for every unit $h\in \mathcal{L}$,

$$\Vert h-h_R{\Vert }_{\nu }^2+D(h-h_R)\le \frac{433}{450}{\tau }^2<{\tau }^2.$$

For $h\in \mathcal{L}\cap \mathcal{R}$, the reducing projection $P_{\mathcal{R}}$ retains at most the same form error, while its output generally has noncompact support. The bound concerns the original operator’s whole fixed-gap spectral subspace. The quantitative interior construction provides a finite-rank map with uniform error in the original form norm, including a prescribed even translated-kernel family indexed by the existing legal FIB interval mesh. Verified finite matrix signs, a complete lower spectral certificate and control uniform as $\epsilon \downarrow 0$ remain unresolved. Subthreshold accumulation is not excluded. That construction also gives a conditional complete-window exclusion from PSD at $c_{\epsilon }=1/2-\epsilon /2$, with its uniform form error chosen below $\epsilon /8$; the matrix condition is not verified. RH and full Robin remain unresolved. These are paper-level model deductions, without new Lean certification or an originality claim.

The directed assembly interface retains the same complete Gamma/prime matrix, regularizes the trial kernels’ jumps, and supplies explicit positive-tail and full-mean correction bounds. Validated quadrature and a simultaneous coefficient-error allowance can produce a Loewner lower matrix. Its entries and sign remain uncomputed; this interface does not discharge the fixed-window matrix hypothesis or the cofinal-window requirement.

Same-form theta exterior bound and low-projector cutoff

Reuse the mixed nullity and ground-state identity, the complete compact Weil formula, and the original minimal operator and complete prime off-diagonal operator above, with the closed-form realization. All statements below concern the actual even minimal realization in $L^2(\nu )$, $d\nu =2\Phi \cosh (x/2)dx$, $\nu (\mathbb{R})=1$. These are paper-level model deductions from the cited identities, with no new generic theorem, originality or Lean-certification claim.

Exact pole cancellation on the same test

Write

$$G_{\mathrm{flat}}(f)={\int }_0^{\infty }\psi (t)\Vert f(\cdot +t)-f{\Vert }_{L^2(dx)}^{2}dt,c_{\Gamma }=\mathrm{digamma}(1/4)-\log \pi ,\psi (t)=\frac{e^{-t/2}}{1-e^{-2t}}.$$

For even complex compact smooth $h$, $f=\Phi h$ is the same admissible test in the two existing identities. They give

$$Q(\Phi h)=D(h)-\frac{1}{2}{\mathrm{Var}}_{\nu }(h)=P(\Phi h)+G_{\mathrm{flat}}(\Phi h)+c_{\Gamma }\Vert \Phi h{\Vert }_2^2-⟨B_{p}h,h{⟩}_{\nu }.$$

Here $B_p$ is the bounded self-adjoint full prime graph off-diagonal operator. Its quadratic form is the complete prime correlation sum, $2{\sum }_{n\ge 2}{w}_n\Re {\mathrm{Corr}}_{\Phi h}(\log n)$, with every prime power. Evenness gives $P(\Phi h)=2|\int e^{x/2}\Phi h|^2 =\tfrac12|\int h,d\nu|^2$. Therefore the pole and variance mean terms cancel exactly:

$$\boxed{D(h)=\frac{1}{2}\Vert h{\Vert }_{\nu }^2+G_{\mathrm{flat}}(\Phi h)+c_{\Gamma }\Vert \Phi h{\Vert }_2^2-⟨B_{p}h,h{⟩}_{\nu }.}\text{\hspace{1em}(JE)}$$

$G_{\mathrm{flat}}(\Phi h)$ is a nonnegative flat translation form on the weighted test $\Phi h$. It is distinct from $E_{\Gamma }(h)$, and (JE) retains the original complete $D=E_{\Gamma }+E_p$.

Extension through the actual minimal form closure

The multiplier $h\mapsto \Phi h$ is bounded from $L^2(\nu )$ to $L^2(dx)$, because ${\Phi }^2/\rho =\Phi /(2\cosh (x/2))\le 9/5$ by the existing global theta bound. Also $c_{\Gamma }<0$ and $|c_{\Gamma }|<8$: the classical quarter digamma value gives $-c_{\Gamma }=\gamma +\pi /2+3\log 2+\log \pi <8$.

Applied to differences of core approximants, (JE) yields

$$G_{\mathrm{flat}}(\Phi u)\le D(u)+(8\cdot 9/5+\Vert B_p\Vert )\Vert u{\Vert }_{\nu }^2.$$

The flat jump form has its standard closed Fourier realization, whose multiplier is $2{\int }_0^{\infty }\psi (t)(1-\cos (\xi t))dt\ge 0$. Thus a Cauchy sequence in the actual minimal $D$ form norm maps to a Cauchy sequence in that closed flat form norm. All bounded terms in (JE) converge and the identity extends to the actual even minimal form domain. No equality with a maximal domain or operator-core assertion is used.

A complete exterior lower bound from theta tails

Let $E_R=\{|x|>R\}$, $R\ge 0$, and let $p_R={\sup }_{E_R}\Phi (x)/(2\cosh (x/2))$. Reuse the original-series bounds

$$p_R\le \frac{72}{5}{e}^{-(3/2)e^{2R}},\Vert 1_{E_R}{B}_p\Vert \le \frac{216}{5}{e}^{-(3/8)e^{2R}}=:b_R^{\ast }.$$

The second is the existing full two-direction prime estimate, including its shifted adjoint. For every minimal-form $h$ zero on $[-R,R]$, nonnegativity of the flat form in (JE) and self-adjointness of $B_p$ give

$$\boxed{D(h)\ge (\frac{1}{2}-d_R)\Vert h{\Vert }_{\nu }^2,d_R=8p_R+b_R^{\ast }\le 160e^{-(3/8)e^{2R}}.}\text{\hspace{1em}(JL)}$$

This controls the complete energy, including Gamma edges crossing the removed interval and all prime powers. It uses the joint formula before estimating its two bounded negative terms. The quantitative prime-counting remainder used to control $a_p$ separately is not needed for (JL).

A Hilbert-Schmidt bound for the full cutoff commutator

Use the existing real even cutoff $0\le {\eta }_R\le 1$, zero on $[-R,R]$, one outside $[-R-2,R+2]$, with Lipschitz constant at most one. Conjugating its Gamma commutator by $Uh=\sqrt{\rho }h$ gives the kernel

$${\tilde{K}}_R(x,y)=\frac{\sqrt{\Phi (x)\Phi (y)}}{2\sqrt{\cosh (x/2)\cosh (y/2)}}\psi (|x-y|)({\eta }_R(x)-{\eta }_R(y)).$$

A nonzero cutoff difference requires at least one endpoint in $E_R$. The existing short- and long-jump bounds give $\psi (t)\min (t,1)\le 3/2$ for all $t>0$. With $P_{\Phi }=\int \Phi \le 1/2$ and $\mathcal{T}(R)={\int }_{E_R}\Phi$, therefore

$$\Vert K_R{\Vert }^2\le \Vert {\tilde{K}}_R{\Vert }_{\mathrm{HS}}^2\le \frac{9}{16}\iint \Phi (x)\Phi (y)1_{\{x\in E_R\text{ or }y\in E_R\}}dxdy\le \frac{9}{16}\mathcal{T}(R).$$

The singular Gamma neighborhood is retained: its cutoff difference pays the factor $t$. Reuse $\mathcal{T}(R)\le (96/5)e^{-2R}{e}^{-(3/2)e^{2R}}$ to obtain

$$\Vert K_R\Vert \le \frac{3}{4}\sqrt{\mathcal{T}(R)}\le 4e^{-(3/4)e^{2R}}.$$

The diagonal prime multiplier commutes with ${\eta }_R$. Its off-diagonal commutator has norm at most $2b_R^{\ast }$, because both $\Vert M_{{\eta }_R}{B}_p\Vert$ and $\Vert B_p{M}_{{\eta }_R}\Vert$ have the same bound. The prior minimal-domain cutoff argument extends the full identity to $D(A)$, giving

$$[A,M_{{\eta }_R}]=K_R-[B_p,M_{{\eta }_R}],\Vert [A,M_{{\eta }_R}]\Vert \le k_R:=91e^{-(3/8)e^{2R}}.\text{\hspace{1em}(JK)}$$

This is a norm bound for the complete commutator. No scalar eigenvalue is assigned to a mixture of low spectral vectors.

Uniform whole-low-subspace form approximation

Fix $0<\epsilon \le 1/2$, $a=1/2-\epsilon$, and $P_{\epsilon }=1_{[0,a]}(A_{\mathrm{even}})$. Restrict the original closed $D$ form to even functions zero on $[-R,R]$ and call its killed operator $A_R$. Formula (JL) gives $A_R\ge 1/2-d_R$.

The already established operator-domain cutoff statement puts $T_R=M_{{\eta }_R}{|}_{\mathrm{ran}{P}_{\epsilon }}$ in $D(A_R)$ and yields the same separated-spectra equation, now for the full $A_R$:

$$A_R{T}_R-T_{R}A{|}_{\mathrm{ran}{P}_{\epsilon }}=1_{E_R}[A,M_{{\eta }_R}]{|}_{\mathrm{ran}{P}_{\epsilon }}.$$

Reuse the standard decaying-semigroup integral from (LP). If $d_R<\epsilon$, then

$$\boxed{\Vert M_{{\eta }_R}{P}_{\epsilon }\Vert \le \frac{k_R}{\epsilon -d_R}.}\text{\hspace{1em}(JLP)}$$

For every unit $h$ in that entire spectral subspace, write $s_R=k_R/(\epsilon -d_R)$. Since $Ah$ remains in that same subspace, (JLP) gives $\Vert {\eta }_{R}Ah{\Vert }_{\nu }\le a{s}_R$. The full operator identity gives

$$D({\eta }_{R}h)\le a{s}_R^2+k_R{s}_R,\Vert {\eta }_{R}h{\Vert }_{\mathcal{F}}^2\le (1+a)s_R^2+k_R{s}_R.\text{\hspace{1em}(JF)}$$

For $0<\tau \le 1$, a single explicit sufficient radius condition is

$$R\ge 2,e^{2R}\ge \frac{8}{3}\log \frac{728}{\epsilon \tau }.\text{\hspace{1em}(JR)}$$

It gives $d_R<\epsilon /2$, $k_R\le \epsilon \tau /8$ and $s_R\le \tau /4$. Since $a\le 1/2$ and $\epsilon \le 1/2$, $\Vert {\eta }_{R}h{\Vert }_{\mathcal{F}}^2\le 7{\tau }^2/64<{\tau }^2$ uniformly, so $(1-{\eta }_R)P_{\epsilon }$ is the required compact-support form approximation to the whole fixed-gap subspace.

When this cutoff feeds the previously prescribed translated-kernel/FIB mesh, retain that mesh transfer’s independent interval-margin and floor conditions. The new radius replaces its quantitative prime-remainder condition; it does not remove the minimum-density factors still present in the smoothing estimate or establish a practical matrix rank.

The center matrix signs and a cofinal $\epsilon \downarrow 0$ certificate remain missing. Neither this complete exterior bound nor the scalar assembly pilot proves the critical-half inequality, RH or Robin.

Weighted Fourier finite-family interface

The weighted Fourier cutoff and finite cosine construction uses the same actual minimal form, (JE), complete prime operator, and (JF)–(JR). The original theta derivative suppliers provide its global coefficient hypotheses. It gives an explicit finite-rank map with uniform original-form error $<\tau$ on the entire $P_{\epsilon }$ subspace, for every fixed $0<\epsilon \le 1/2$ and $0<\tau \le 1$. Its prescribed cosine generators and bounded Fourier cell-integral coefficients require no unknown eigenbasis.

At $\tau =\epsilon /8$, the existing complete-window matrix transfer can use this family if its complete original-form/full-variance matrix is PSD at $1/2-\epsilon /2$. Matrix signs, useful numerical ranks and cofinal certificates remain missing. This is a paper-level construction, with no new Lean certification or arithmetic benefit from FIB labels.