Proof of the Riemann Hypothesis

1. Introduction

1.1. Problem Setting and Outline of the Proof

The purpose of this paper is to prove operator-theoretically that all nontrivial zeros of the completed zeta function

$$\xi \left(s\right)={\displaystyle \frac{1}{2}}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right)$$

lie on the critical line

$$Res={\displaystyle \frac{1}{2}}.$$

The proof does not identify zeros with eigenvalues from the outset. Instead, it separates the prime-power contribution on the arithmetic side from the ${\mathcal{K}}_R$-component on a common Hilbert space. After passage to the canonical representative modulo $Ran{\Pi }_{\mathrm{res}}$, the residual component is removed, and the remaining ${\mathcal{K}}_R$-projected component is represented by its ${\Pi }_R$-projection in ${\mathcal{K}}_R$.

The structure of the proof consists of five stages:

$$\mathrm{analytic}\mathrm{operator}\mathrm{setup}⟶\mathrm{coefficient}-\mathrm{space}\mathrm{arithmetic}\mathrm{construction}$$

$$⟶\sin \mathrm{gular}\mathrm{boundary}-\mathrm{data}\mathrm{construction}⟶\mathrm{orthogonal}-\mathrm{decomposition}\mathrm{comparison}\mathrm{framework}$$

$$⟶\mathrm{spectral}\mathrm{determinant}\mathrm{closure}.$$

This separation specifies which objects are constructed analytically, where the arithmetic trace evaluates the prime-power contribution, where the residual quotient is taken, where the self-adjoint Hilbert–Schmidt operator appears, and where the zero locations are finally determined. The analytic and arithmetic preparations in Section 2, Section 3, Section 4 and Section 5 are an essential part of the argument; Section 6 uses their output rather than replacing them by an independent determinant ansatz.

The central chain of the proof is as follows. From the residual-free comparison interface obtained in Section 5, Section 6 constructs the continuous comparison map

$${\mathcal{D}}_R\stackrel{{Tr}_{\partial ,R}}{\to }{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }\stackrel{{LCI}_R}{\to }X\stackrel{{\Pi }_R}{\to }{\mathcal{K}}_R.$$

The functional equation induces a boundary reflection

$${\Theta }_R:{\mathcal{D}}_R^{\prime }\to {\mathcal{D}}_R^{\prime },$$

which is defined before any use of zero-location information. Its compatibility with admissible boundary distributions and with the residual-free comparison interface allows it to descend to a bounded self-adjoint involution

$${\mathcal{S}}_R:{\mathcal{K}}_R\to {\mathcal{K}}_R.$$

This gives the signed boundary-distribution comparison kernel

$${\mathfrak{k}}_R(f,g)={〈{\mathcal{S}}_R{\Pi }_R{\mathcal{J}}_{R}f,{\Pi }_R{\mathcal{J}}_{R}g〉}_X.$$

The Hermitian symmetry of this kernel uses only ${\mathcal{S}}_R^{*}={\mathcal{S}}_R$ and the Hilbert-space inner product. The Sobolev eigenvalue growth, boundary-trace smoothing, and boundedness of the comparison map then realize the kernel as a self-adjoint Hilbert–Schmidt operator

$$K=K^{*},K\in {\mathfrak{S}}_2.$$

No positivity, Herglotz property, or zero-localization statement equivalent to the Riemann Hypothesis is assumed in this construction.

The regularized Fredholm determinant

$$F_K\left(s\right)=e^{a_K+b_K(s-{\displaystyle \frac{1}{2}})}{det}_2\left(I+i(s-\frac{1}{2})K\right)$$

is then formed from this K. The constants $a_K,b_K$ fix only the central value and the first logarithmic derivative; they do not prescribe the zeros of $F_K$ and do not encode the locations of the zeros of $\xi$.

The comparison $F_K\equiv \xi$ is obtained through a separate central Cauchy–Laplace argument. Section 6 first fixes the central kernel

$$h_w\left(u\right)={\displaystyle \frac{e^{wu}-1}{u}},h_w\left(0\right)=w,$$

the raw finite-window cutoffs, the central finite-jet map $J_M^{\mathrm{cen}}$, the universal principal-part map $P_M^{\mathrm{cen}}$, the principal-part embedding ${\iota }_M^{\mathrm{cen}}$, and the central comparison topology on ${\mathcal{C}}_{\mathrm{cen}}$. The regularized finite-window input is defined before either pairing is evaluated by the algebraic subtraction

$${\Psi }_{w,M}^{\mathrm{fw}}={\tilde{\Psi }}_{w,M}^{\mathrm{fw}}-{\iota }_M^{\mathrm{cen}}{P}_M^{\mathrm{cen}}{J}_M^{\mathrm{cen}}\left(h_{w,M}^{\mathrm{fw}}\right)\mathrm{in}{\mathcal{C}}_{\mathrm{cen}}^0.$$

The common local-principal-part lemma shows that the same finite-jet counterterm removes the local singular principal part of the Archimedean, arithmetic-trace, and singular-boundary contributions. Its definition is independent of the values of ${\mu }_L$ and ${\mu }_{\xi }$. Section 6 then proves the finite-window convergence

$${\Psi }_{w,M}^{\mathrm{fw}}⟶{\Psi }_w^{\mathrm{cen}}\mathrm{in}{\mathcal{C}}_{\mathrm{cen}},$$

locally uniformly in w, and proves that the central pairings defined by ${\mu }_L$ and ${\mu }_{\xi }$ extend continuously to ${\mathcal{C}}_{\mathrm{cen}}$. Combining the finite-window residual-free equality with these two analytic facts gives

$$\langle {\mu }_L,{\Psi }_w^{\mathrm{cen}}\rangle =\langle {\mu }_{\xi },{\Psi }_w^{\mathrm{cen}}\rangle .$$

Separate transform lemmas identify the left-hand side with the central logarithmic derivative of $F_K$, and the right-hand side with that of $\xi$. Therefore

$${\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)={\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}+w\right)$$

near $w=0$. The central normalization gives local analytic equality, and the identity theorem gives

$$F_K\left(s\right)\equiv \xi \left(s\right).$$

Since the nonzero eigenvalues of K are real, the zeros of $F_K$ are restricted to the form

$$s={\displaystyle \frac{1}{2}}+{\displaystyle \frac{i}{{\lambda }_j}},{\lambda }_j\in \mathbb{R}\setminus \left\{0\right\}.$$

The identity $F_K=\xi$ then gives the Riemann Hypothesis.

1.2. The Analytic Operator Setup and the Coefficient-Space Arithmetic Construction

Section 2 constructs the analytic operator setup. Starting from a weighted Hilbert space on the half-line, it introduces the quadratic form $q_R$ and its admissible core, and constructs the closed form and self-adjoint operator realization

$$q_R⟶A_R⟶L:=A_R+I.$$

It then obtains the compact embedding of the form domain, compact resolvent, purely discrete spectrum, and scalar Herglotz-type resolvent function

$$m_L^{(\Phi )}\left(z\right)=\langle \Phi ,{(L-zI)}^{-1}\Phi \rangle .$$

This section provides the analytic foundation and does not use the arithmetic trace, the residual quotient, or any conclusion about the locations of zeros.

Section 3 constructs the coefficient-space arithmetic data. In the formal Dirichlet algebra and its completed augmentation ideal, it defines the exact prime indicator, the exact von Mangoldt lift

$${\Lambda }^{\mathrm{ex}},$$

the composite-cancellation operator, and the arithmetic derivation. The principal coefficient-extraction identity is

$${\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)={\Lambda }^{\mathrm{ex}}.$$

The section then Hilbertizes the coefficient layer and constructs the coefficient Hilbert space, the coefficient projectors, and the weighted diagonal arithmetic trace operator. This trace operator is the means by which the prime-power contribution is evaluated exactly on the arithmetic summand in Section 5.

1.3. Singular-Boundary Data and the Orthogonal-Decomposition Comparison Framework

Section 4 constructs the singular-boundary data inside the analytic Hilbert-space setup. Using the Gelfand triple

$${\mathcal{D}}_R\subset H_{\alpha ,+}\subset {\mathcal{D}}_R^{\prime },$$

it separates point evaluations, distribution kernels, boundary traces, and boundary forms by type. It also constructs the boundary parameter space, the regular area measure, and the zero-area singular locus

$$({\Sigma }_R,{\sigma }_R,{\Gamma }_R),$$

and obtains

$$T_R,{supp}_R,L_R$$

from the singular boundary trace and the regular boundary trace. These data define the ${\sigma }_R$-null and regular-trace-vanishing generating class and its closure. The section also constructs the Friedrichs-type realization, the singular-boundary transport group, the anti-self-adjoint generator, and the distribution-kernel representation. Its output is the one-sided singular-boundary subspace $K_R^{+}\subset H_{\alpha ,+}$, the one-sided projection ${\Pi }_R^{+}$, and the boundary-distribution data needed for the analytic comparison in Section 6.

Section 5 places the coefficient-space arithmetic construction and the singular-boundary construction on a common ambient Hilbert space X. In this stage,

$$K_R^{+}\subset H_{\alpha ,+}\mathrm{is}\mathrm{lifted}\mathrm{to}{\mathcal{K}}_R=J_{\mathrm{an}}{K}_R^{+}\subset X,$$

and the ambient projection is

$${\Pi }_R=J_{\mathrm{an}}{\Pi }_R^{+}{J}_{\mathrm{an}}^{*}.$$

The arithmetic summand is embedded by $J_{\mathrm{arith}}$, and the orthogonal decomposition

$$X={\mathcal{K}}_R\oplus J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}\oplus Ran{\Pi }_{\mathrm{res}}$$

is fixed. For localized comparison data from the finite-window explicit formula, the prime-power contribution is evaluated by the weighted diagonal arithmetic trace, while the residual component is removed by passing to the canonical representative modulo $Ran{\Pi }_{\mathrm{res}}$. The effective ${\mathcal{K}}_R$-projected component is therefore represented by

$${\Pi }_R{x}^{♯}\in {\mathcal{K}}_R.$$

This ${\mathcal{K}}_R$-component is the input for the analytic determinant construction in Section 6.

1.4. Spectral Determinant Closure in Section 6

Section 6 turns the residual-free comparison interface into an analytic identity. The first part of the section constructs the comparison map from the boundary distribution data of Section 4 and the orthogonal-decomposition framework of Section 5. The boundary reflection ${\Theta }_R$ induced by $s\mapsto 1-s$ is shown to preserve admissibility and the residual-free comparison relation, and hence descends to the self-adjoint involution ${\mathcal{S}}_R$ on ${\mathcal{K}}_R$. The resulting signed boundary-distribution comparison kernel is then realized, by Schatten estimates, as

$$K=K^{*}\in {\mathfrak{S}}_2.$$

From K, Section 6 defines the regularized determinant $F_K$. The central comparison is not made by definition. Instead, the central Cauchy–Laplace kernel, the finite-window cutoffs, the counterterm, and the topology of ${\mathcal{C}}_{\mathrm{cen}}$ are fixed first. The finite-window approximation lemma gives convergence of ${\Psi }_{w,M}^{\mathrm{fw}}$ to ${\Psi }_w^{\mathrm{cen}}$, and the continuity theorem for the central pairings justifies passing the finite-window residual-free equality to the limit. This gives the central equality of pairings. The two central transform lemmas then convert this equality into equality of the logarithmic derivatives of $F_K$ and $\xi$. The normalization at $s={\displaystyle \frac{1}{2}}$ gives local analytic equality, and the identity theorem gives

$$F_K\left(s\right)\equiv \xi \left(s\right).$$

The spectral localization now follows from the self-adjointness of K: the nonzero spectrum of K is real, and hence the zeros of $F_K$ lie on the critical line. Since $F_K=\xi$, all nontrivial zeros of the completed zeta function lie on the critical line. The finite-window bridge, anchored defect staircase, and no-first-hit material later in Section 6 record this spectral conclusion in finite-window form; they are not used to prove the determinant identity $F_K\equiv \xi$.

2. Analytic Operator Setup

2.1. Weighted Hilbert Space and Boundary Geometry

In this section, we fix only the analytic Hilbert-space setup necessary for constructing the weighted self-adjoint generator and preparing its compactness theory. Accordingly, the role of this subsection is limited to specifying the weighted Hilbert space, the boundary notation, and the trace conventions used in the form constructions below. Here, we introduce no arithmetic projector, no comparison identity, and no zero-counting statement.

Definition 2.1 

(Weighted Hilbert-space setup). Fix the parameters

$$\alpha >{\displaystyle \frac{1}{2}},\langle x\rangle :={(1+x^2)}^{1/2},{\rho }_{\alpha }\left(x\right):={\langle x\rangle }^{2\alpha }.$$

Define the weighted measure on the positive half-line by

$$d{\mu }_{\alpha }\left(x\right):={\rho }_{\alpha }\left(x\right)dx(x>0).$$

Let the one-sided weighted Hilbert space be

$$H_{\alpha ,+}:=L^2\left((0,\infty ),d{\mu }_{\alpha }\right),$$

and give its inner product by

$${\langle f,g\rangle }_{H_{\alpha ,+}}:={\int }_0^{\infty }f\left(x\right)\overline{g\left(x\right)}d{\mu }_{\alpha }\left(x\right).$$

Let the doubled analytic Hilbert space be

$$H_{\alpha }:=H_{\alpha ,+}\oplus H_{\alpha ,+},$$

and give its direct-sum inner product by

$${\langle (f_{+},f_{-}),(g_{+},g_{-})\rangle }_{H_{\alpha }}={\langle f_{+},g_{+}\rangle }_{H_{\alpha ,+}}+{\langle f_{-},g_{-}\rangle }_{H_{\alpha ,+}}.$$

In what follows, we repeatedly use the unitary transport

$$U_{\alpha }:H_{\alpha ,+}⟶L^2(0,\infty ),\left(U_{\alpha }f\right)\left(x\right):={\rho }_{\alpha }{\left(x\right)}^{1/2}f\left(x\right).$$

Definition 2.2 

(Boundary notation). Set

$$\Sigma :=\{0^{+},0^{-}\}=\partial \left((0,\infty )\bigsqcup (0,\infty )\right).$$

We identify

$$L^2(\Sigma )\simeq {\mathbb{C}}^2$$

with the inner product associated with counting measure.

When each component of

$$u=(u_{+},u_{-})\in H_{\alpha }$$

is locally absolutely continuous in a neighborhood of $x=0$, we write the boundary trace as

$${\gamma }_{0}u:=\left(u_{+}\left(0\right),u_{-}\left(0\right)\right)\in L^2(\Sigma ),$$

and, when the one-sided derivatives exist, we write the oriented normal-derivative trace as

$${\gamma }_{1}u:=\left(u_{+}^{\prime }\left(0\right),-u_{-}^{\prime }\left(0\right)\right)\in L^2(\Sigma ).$$

For scalar functions $f,g$ on $(0,\infty )$, if the endpoint limits below exist, we write the weighted boundary form as

$$b_{\alpha }[f,g]:=-i{\left[{\rho }_{\alpha }\left(x\right)f\left(x\right)\overline{g\left(x\right)}\right]}_{x=0}^{x=\infty }.$$

Thus

$$b_{\alpha }[f,g]=-i\underset{R\to \infty }{lim}{\rho }_{\alpha }\left(R\right)f\left(R\right)\overline{g\left(R\right)}+i{\rho }_{\alpha }\left(0\right)f\left(0\right)\overline{g\left(0\right)}.$$

This is the boundary term that appears when performing integration by parts with respect to the weighted measure $d{\mu }_{\alpha }$.

Remark 2.3 

(Scope of Section 2). The purpose of Section 2 is purely analytic. Namely, it fixes the weighted Hilbert space, constructs the closed quadratic form $q_R$ and its associated self-adjoint operator, and prepares the compactness argument through an explicit confining potential. No part of this section presupposes any arithmetic projector, any orthogonal decomposition of an entire trace space, or any conclusion theorem for the Riemann hypothesis.

2.2. Closed Quadratic Form and the Admissible Core

Here we introduce the first-order weighted differential operator and the raw quadratic form on a concrete admissible core. What is needed in this subsection is that the form domain of the closed form $q_R$ contain smooth elements with boundary values. For this reason, the admissible core is taken to consist of compactly supported functions that extend smoothly to the endpoint 0, and boundary cancellation itself is carried out later in the trace-vanishing singular-boundary subspace. This distinction prevents the singular boundary trace used in Section 4 from being trivialized.

Definition 2.4 

(Quadratic form and admissible core).

$${\mathcal{C}}_R:=C_c^{\infty }\left([0,\infty )\right).$$

On ${\mathcal{C}}_R$, define the first-order weighted differential operator

$$D_{0}f:=-i\left({\partial }_x+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\rho }_{\alpha }^{\prime }}{{\rho }_{\alpha }}}\right)f=-i\left({\partial }_x+\alpha {\displaystyle \frac{x}{1+x^2}}\right)f.$$

Equivalently,

$$U_{\alpha }{D}_0{U}_{\alpha }^{-1}=-i{\partial }_x\mathrm{on}{\mathcal{C}}_R$$

holds.

Let $V_R:(0,\infty )\to [1,\infty )$ be a locally bounded measurable function. Its concrete choice is fixed in Definition 2.7. In what follows, write

$$Q_R:=log{V}_R,V_R=e^{Q_R}.$$

Define the raw quadratic form on ${\mathcal{C}}_R$ by

$$q_R^{\mathrm{raw}}[f,g]:={\langle D_{0}f,D_{0}g\rangle }_{H_{\alpha ,+}}+{\langle V_R^{1/2}f,V_R^{1/2}g\rangle }_{H_{\alpha ,+}},$$

and let its quadratic-form version be

$$q_R^{\mathrm{raw}}\left[f\right]:=q_R^{\mathrm{raw}}[f,f].$$

Define its raw form norm by

$${\parallel f\parallel }_{q_R^{\mathrm{raw}}}^2:={\parallel f\parallel }_{H_{\alpha ,+}}^2+q_R^{\mathrm{raw}}\left[f\right].$$

Theorem 2.5 

(Boundary formula on the admissible core and symmetry of the raw form). For any $f,g\in {\mathcal{C}}_R$,

$${\langle D_{0}f,g\rangle }_{H_{\alpha ,+}}-{\langle f,D_{0}g\rangle }_{H_{\alpha ,+}}=b_{\alpha }[f,g]$$

holds. Here

$$b_{\alpha }[f,g]=-i{\left[{\rho }_{\alpha }\left(x\right)f\left(x\right)\overline{g\left(x\right)}\right]}_{x=0}^{x=\infty }.$$

In particular, on ${\mathcal{C}}_{R,0}:=C_c^{\infty }(0,\infty )$,

$$b_{\alpha }[f,g]=0.$$

On the other hand, the raw quadratic form

$$q_R^{\mathrm{raw}}[f,g]={\langle D_{0}f,D_{0}g\rangle }_{H_{\alpha ,+}}+{\langle V_R^{1/2}f,V_R^{1/2}g\rangle }_{H_{\alpha ,+}}$$

is symmetric on ${\mathcal{C}}_R$.

Proof. 

Let $f,g\in {\mathcal{C}}_R$. By definition,

$${\langle D_{0}f,g\rangle }_{H_{\alpha ,+}}=-i{\int }_0^{\infty }\left(f^{\prime }\left(x\right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\rho }_{\alpha }^{\prime }\left(x\right)}{{\rho }_{\alpha }\left(x\right)}}f\left(x\right)\right)\overline{g\left(x\right)}{\rho }_{\alpha }\left(x\right)dx.$$

Since $f,g$ are smooth up to the endpoint 0 and vanish for sufficiently large x, integration by parts is justified. Integrating the first term by parts gives

$${\int }_0^{\infty }{f}^{\prime }\left(x\right)\overline{g\left(x\right)}{\rho }_{\alpha }\left(x\right)dx={\left[{\rho }_{\alpha }f\overline{g}\right]}_0^{\infty }-{\int }_0^{\infty }f\left(x\right)\overline{g^{\prime }\left(x\right)}{\rho }_{\alpha }\left(x\right)dx-{\int }_0^{\infty }f\left(x\right)\overline{g\left(x\right)}{\rho }_{\alpha }^{\prime }\left(x\right)dx.$$

Substituting this into the preceding formula yields

$${\langle D_{0}f,g\rangle }_{H_{\alpha ,+}}-{\langle f,D_{0}g\rangle }_{H_{\alpha ,+}}=-i{\left[{\rho }_{\alpha }\left(x\right)f\left(x\right)\overline{g\left(x\right)}\right]}_0^{\infty }=b_{\alpha }[f,g].$$

If $f,g\in {\mathcal{C}}_{R,0}$, then both vanish in a neighborhood of the endpoint 0, so the boundary term also vanishes.

Finally, the symmetry of $q_R^{\mathrm{raw}}$ follows from the fact that, for any linear operator $D_0$,

$$\langle D_{0}f,D_{0}g\rangle =\overline{\langle D_{0}g,D_{0}f\rangle }$$

holds, and from the Hermitian property of the potential term. This conclusion does not require $D_0$ itself to be symmetric without boundary terms. □

Theorem 2.6 

(Closedness, lower boundedness, and core density). Let D be the closure of $D_0$ in $H_{\alpha ,+}$, and let

$$M_{V_R^{1/2}}f:=V_R^{1/2}f$$

be the multiplication operator with maximal domain

$$Dom\left(M_{V_R^{1/2}}\right)=\left\{f\in H_{\alpha ,+}:V_R^{1/2}f\in H_{\alpha ,+}\right\}.$$

Then the raw graph map

$$T_R^{\mathrm{raw}}:{\mathcal{C}}_R\to H_{\alpha ,+}\oplus H_{\alpha ,+},T_R^{\mathrm{raw}}f:=(D_{0}f,M_{V_R^{1/2}}f)$$

is closable. Furthermore, set

$${\mathcal{Q}}_R:=Dom\left(\overline{T_R^{\mathrm{raw}}}\right),$$

and write its closure again as

$$T_{R}f=(Df,M_{V_R^{1/2}}f)(f\in {\mathcal{Q}}_R).$$

Then the following hold:

1.

The sesquilinear form

$$q_R[f,g]:={\langle Df,Dg\rangle }_{H_{\alpha ,+}}+{\langle V_R^{1/2}f,V_R^{1/2}g\rangle }_{H_{\alpha ,+}}(f,g\in {\mathcal{Q}}_R)$$

is well-defined and closed on ${\mathcal{Q}}_R$;

2.

This form is lower bounded:

$$q_R\left[f\right]\ge 0(f\in {\mathcal{Q}}_R);$$

3.

${\mathcal{C}}_R$ is a form core for $q_R$. Namely,

$${\overline{{\mathcal{C}}_R}}^{{\parallel ·\parallel }_{q_R}}={\mathcal{Q}}_R{,\parallel f\parallel }_{q_R}^2:={\parallel f\parallel }_{H_{\alpha ,+}}^2+q_R\left[f\right].$$

Proof. 

We first begin with the operator D. By Definition 2.4,

$$U_{\alpha }{D}_0{U}_{\alpha }^{-1}=-i{\partial }_x\mathrm{on}{\mathcal{C}}_R$$

holds. Here ${\mathcal{C}}_R=C_c^{\infty }\left([0,\infty )\right)$ is the space of compactly supported functions smooth up to the endpoint. The graph closure of $-i{\partial }_x$ on ${\mathcal{C}}_R$ in $L^2(0,\infty )$ is the standard weak derivative operator

$$-i{\partial }_x:H^1(0,\infty )\subset L^2(0,\infty )⟶L^2(0,\infty ),$$

which is a closed operator. Since $U_{\alpha }$ is unitary, the transported operator D is also a closed operator on $H_{\alpha ,+}$.

Next consider the multiplication operator $M_{V_R^{1/2}}$. Since $V_R^{1/2}$ is measurable and finite almost everywhere, multiplication by $V_R^{1/2}$ is a closed operator on $H_{\alpha ,+}$. Indeed, suppose that

$$f_n\to f\mathrm{in}{H}_{\alpha ,+},M_{V_R^{1/2}}{f}_n\to h\mathrm{in}{H}_{\alpha ,+}.$$

Passing to a subsequence if necessary, pointwise convergence almost everywhere holds:

$$f_n\left(x\right)\to f\left(x\right),V_R{\left(x\right)}^{1/2}{f}_n\left(x\right)\to h\left(x\right).$$

Therefore

$$h\left(x\right)=V_R{\left(x\right)}^{1/2}f\left(x\right)\mathrm{for}\mathrm{almost}\mathrm{every}x>0,$$

and hence

$$f\in Dom\left(M_{V_R^{1/2}}\right)\mathrm{and}{M}_{V_R^{1/2}}f=h.$$

Thus $M_{V_R^{1/2}}$ is closed.

We now show that $T_R^{\mathrm{raw}}$ is closable. Assume that

$$f_n\in {\mathcal{C}}_R,f_n\to 0\mathrm{in}{H}_{\alpha ,+},T_R^{\mathrm{raw}}{f}_n\to (g,h)\mathrm{in}{H}_{\alpha ,+}\oplus H_{\alpha ,+}.$$

Since $D_0\subset D$ and D is closed, the convergence

$$f_n\to 0,D_0{f}_n\to g$$

implies $g=0$. Similarly, since $M_{V_R^{1/2}}$ is closed and

$$f_n\to 0,M_{V_R^{1/2}}{f}_n\to h$$

holds, we obtain $h=0$. Therefore $T_R^{\mathrm{raw}}$ is closable.

Set $T_R=\overline{T_R^{\mathrm{raw}}}$, and define

$${\mathcal{Q}}_R:=Dom\left(T_R\right).$$

By the graph-closure definition, ${\mathcal{Q}}_R$ is precisely the graph closure of ${\mathcal{C}}_R$ with respect to the norm

$${\parallel f\parallel }_{H_{\alpha ,+}}^2+\parallel D_0{f\parallel }_{H_{\alpha ,+}}^2+\parallel V_R^{1/2}{f\parallel }_{H_{\alpha ,+}}^2={\parallel f\parallel }_{q_R^{\mathrm{raw}}}^2.$$

For $f,g\in {\mathcal{Q}}_R$, define

$$q_R[f,g]:={\langle T_{R}f,T_{R}g\rangle }_{H_{\alpha ,+}\oplus H_{\alpha ,+}}={\langle Df,Dg\rangle }_{H_{\alpha ,+}}+{\langle V_R^{1/2}f,V_R^{1/2}g\rangle }_{H_{\alpha ,+}}.$$

Then

$${\parallel f\parallel }_{q_R}^2={\parallel f\parallel }_{H_{\alpha ,+}}^2+{\parallel T_{R}f\parallel }_{H_{\alpha ,+}\oplus H_{\alpha ,+}}^2.$$

Since $T_R$ is a closed operator, its graph is complete. Therefore

$$({\mathcal{Q}}_R,\parallel ·{\parallel }_{q_R})$$

is a Hilbert space. This is precisely the closedness of the form $q_R$.

The lower bound follows immediately:

$$q_R\left[f\right]={\parallel Df\parallel }_{H_{\alpha ,+}}^2+{\parallel V_R^{1/2}f\parallel }_{H_{\alpha ,+}}^2\ge 0(f\in {\mathcal{Q}}_R).$$

Finally, since ${\mathcal{Q}}_R$ is defined as the graph closure of ${\mathcal{C}}_R$ with respect to the form norm, it also follows by definition that ${\mathcal{C}}_R$ is a form core. Equivalently, for any $f\in {\mathcal{Q}}_R$, there exists a sequence

$$f_n\in {\mathcal{C}}_R$$

such that

$$\parallel f_n{-f\parallel }_{H_{\alpha ,+}}+\parallel D_0{f}_n{-Df\parallel }_{H_{\alpha ,+}}+{\parallel V_R^{1/2}(f_n-f)\parallel }_{H_{\alpha ,+}}⟶0.$$

Therefore

$$\parallel f_n{-f\parallel }_{q_R}\to 0.$$

This proves all three assertions. □

2.3. Explicit Confining Potential

Here we fix the potential entering the quadratic form $q_R$. The essential point of the choice in this subsection is not mere positivity. The potential must grow to $+\infty$ sufficiently slowly, but explicitly, and must prevent $q_R$-bounded mass from escaping to infinity. This tail control is precisely the global ingredient that will be needed later in the compactness argument.

Definition 2.7 

(Explicit confining potential). For $x>0$, define

$$V_R\left(x\right):=log(e+x).$$

Accordingly, set

$$Q_R\left(x\right):=log{V}_R\left(x\right)=loglog(e+x),V_R=e^{Q_R}.$$

Then $V_R$ is positive, measurable, locally bounded, and satisfies

$$V_R\left(x\right)\ge 1(x>0).$$

In what follows, $V_R$ denotes the potential used in Definition 2.4 and Theorem 2.6.

Proposition 2.8 

(Logarithmic growth of $V_R$). For every $x\ge 0$,

$$log(1+x)\le V_R\left(x\right)\le 1+log(1+x)$$

holds. In particular,

$$V_R\left(x\right)\sim log(1+x)(x\to \infty ).$$

Proof. 

By definition,

$$V_R\left(x\right)=log(e+x).$$

Since

$$e+x\ge 1+x,$$

the monotonicity of the logarithm gives

$$log(1+x)\le log(e+x)=V_R\left(x\right).$$

On the other hand, since

$$e+x\le e(1+x),$$

monotonicity again gives

$$V_R\left(x\right)=log(e+x)\le log\left(e(1+x)\right)=1+log(1+x).$$

This proves the two-sided estimate. Dividing by $log(1+x)$ and letting $x\to \infty$, we obtain

$${\displaystyle \frac{V_R\left(x\right)}{log(1+x)}}\to 1,$$

that is,

$$V_R\left(x\right)\sim log(1+x).$$

□

Theorem 2.9 

(Escape prevention at infinity). Let $F\subset {\mathcal{Q}}_R$ be any family satisfying

$$\underset{f\in F}{sup}\left({\parallel f\parallel }_{H_{\alpha ,+}}^2+q_R\left[f\right]\right)\le C,$$

where $C>0$ is a constant. Then, for every $R>0$,

$$\underset{f\in F}{sup}{\int }_R^{\infty }{|f\left(x\right)|}^{2}d{\mu }_{\alpha }\left(x\right)\le {\displaystyle \frac{C}{log(e+R)}}.$$

Consequently,

$$\underset{R\to \infty }{lim}\underset{f\in F}{sup}{\int }_R^{\infty }{|f\left(x\right)|}^{2}d{\mu }_{\alpha }\left(x\right)=0.$$

In particular, every $q_R$-bounded sequence in $H_{\alpha ,+}$ is globally tight and does not lose mass at infinity.

Proof. 

Fix $f\in F$. Since $V_R\left(x\right)=log(e+x)$ is increasing on $(0,\infty )$,

$$V_R\left(x\right)\ge V_R\left(R\right)=log(e+R)(x\ge R)$$

holds. Therefore

$$log(e+R){\int }_R^{\infty }{|f\left(x\right)|}^{2}d{\mu }_{\alpha }\left(x\right)\le {\int }_R^{\infty }{V}_R\left(x\right){|f\left(x\right)|}^{2}d{\mu }_{\alpha }\left(x\right).$$

The right-hand side is bounded by the potential part of $q_R$:

$${\int }_R^{\infty }{V}_R{\left(x\right)|f\left(x\right)|}^{2}d{\mu }_{\alpha }\left(x\right)\le {\int }_0^{\infty }{V}_R{\left(x\right)|f\left(x\right)|}^{2}d{\mu }_{\alpha }\left(x\right)={\parallel V_R^{1/2}f\parallel }_{H_{\alpha ,+}}^2\le q_R\left[f\right].$$

By the assumption

$${\parallel f\parallel }_{H_{\alpha ,+}}^2+q_R\left[f\right]\le C(f\in F),$$

we obtain in particular

$$q_R\left[f\right]\le C.$$

Combining the three estimates above gives

$${\int }_R^{\infty }{|f\left(x\right)|}^{2}d{\mu }_{\alpha }\left(x\right)\le {\displaystyle \frac{C}{log(e+R)}}.$$

Taking the supremum over $f\in F$, we get

$$\underset{f\in F}{sup}{\int }_R^{\infty }{|f\left(x\right)|}^{2}d{\mu }_{\alpha }\left(x\right)\le {\displaystyle \frac{C}{log(e+R)}}.$$

Furthermore,

$$log(e+R)\to \infty (R\to \infty ),$$

so the right-hand side converges to 0. Hence the family F is tight in $H_{\alpha ,+}$, and no $q_R$-bounded sequence escapes to infinity. □

At this point, the analytic framework is self-contained. Namely, the weighted Hilbert-space setup, admissible core, closed lower-bounded quadratic form $q_R$, and explicit confining potential have all been fixed. The next step is to pass from this closed-form construction to the corresponding self-adjoint operator and then to combine the tail estimate of Theorem 2.9 with local compactness on bounded intervals.

2.4. Self-Adjoint Operator $A_R$ Associated with $q_R$ and Positive Shifted Operator L

We now pass from the closed and lower-bounded form $q_R$ to its operator realization. This construction is purely form-theoretic. Namely, the operator is obtained from the representation theorem for closed forms, and at this point we do not identify the operator domain with a formal differential representation. After fixing the self-adjoint operator associated with $q_R$, we introduce the positive shifted operator

$$L:=A_R+I.$$

This will be used as the basic spectral object in the remainder of the analytic section.

Theorem 2.10 

(Self-adjoint operator associated with $q_R$). There exists a unique self-adjoint and lower-bounded operator

$$A_R:Dom\left(A_R\right)\subset H_{\alpha ,+}\to H_{\alpha ,+}$$

associated with the closed form $q_R$ of Theorem 2.6. More precisely, for $u\in H_{\alpha ,+}$ and $h\in H_{\alpha ,+}$, the following are equivalent:

$$u\in Dom\left(A_R\right),A_{R}u=h,$$

and

$$u\in {\mathcal{Q}}_R,q_R[u,v]={\langle h,v\rangle }_{H_{\alpha ,+}}(\forall v\in {\mathcal{Q}}_R).$$

Furthermore,

$$A_R\ge 0.$$

Proof. 

By Theorem 2.6, the form

$$q_R:{\mathcal{Q}}_R\times {\mathcal{Q}}_R\to \mathbb{C}$$

is densely defined, closed, and lower bounded on $H_{\alpha ,+}$. Density follows from

$${\mathcal{C}}_R=C_c^{\infty }(0,\infty )\subset {\mathcal{Q}}_R$$

and from the fact that $C_c^{\infty }(0,\infty )$ is dense in $H_{\alpha ,+}$.

Therefore, the first representation theorem for closed and lower-bounded sesquilinear forms applies. It yields a unique self-adjoint lower-bounded operator

$$A_R:Dom\left(A_R\right)\subset H_{\alpha ,+}\to H_{\alpha ,+}$$

such that

$$Dom\left(A_R\right)=\left\{u\in {\mathcal{Q}}_R:\exists h\in H_{\alpha ,+}\mathrm{with}{q}_R[u,v]={\langle h,v\rangle }_{H_{\alpha ,+}}\forall v\in {\mathcal{Q}}_R\right\}.$$

For such a u, the representing vector h is unique, and we define it to be $A_{R}u$.

This gives the stated equivalence

$$u\in Dom\left(A_R\right),A_{R}u=h\iff u\in {\mathcal{Q}}_R,q_R[u,v]={\langle h,v\rangle }_{H_{\alpha ,+}}\forall v\in {\mathcal{Q}}_R.$$

Finally, by Theorem 2.6,

$$q_R\left[u\right]\ge 0(u\in {\mathcal{Q}}_R).$$

Hence the lower bound in the representation theorem is 0, and therefore

$${\langle A_{R}u,u\rangle }_{H_{\alpha ,+}}\ge 0(u\in Dom\left(A_R\right)).$$

Equivalently,

$$A_R\ge 0.$$

This completes the proof. □

Definition 2.11 

(Positive shifted operator). On the same domain

$$Dom\left(L\right)=Dom\left(A_R\right),$$

define the positive shifted operator

$$L:=A_R+I.$$

Since $A_R\ge 0$,

$$L\ge I$$

holds.

Proposition 2.12 

(Form domain of $L^{1/2}$).

$$Dom\left(L^{1/2}\right)={\mathcal{Q}}_R,$$

and for any $u,v\in {\mathcal{Q}}_R$,

$$q_R[u,v]+{\langle u,v\rangle }_{H_{\alpha ,+}}={\langle L^{1/2}u,L^{1/2}v\rangle }_{H_{\alpha ,+}}.$$

Proof. 

Define the shifted form on ${\mathcal{Q}}_R$ by

$${\mathfrak{l}}_R[u,v]:=q_R[u,v]+{\langle u,v\rangle }_{H_{\alpha ,+}}.$$

Since $q_R$ is densely defined and closed, so is ${\mathfrak{l}}_R$. Moreover, since $q_R\ge 0$,

$${\mathfrak{l}}_R\left[u\right]=q_R\left[u\right]+{\parallel u\parallel }_{H_{\alpha ,+}}^2\ge {\parallel u\parallel }_{H_{\alpha ,+}}^2,$$

and hence ${\mathfrak{l}}_R$ is strictly positive.

Next, we identify the operator associated with ${\mathfrak{l}}_R$. Let $u\in Dom\left(A_R\right)$ and $v\in {\mathcal{Q}}_R$. By Theorem 2.10,

$$q_R[u,v]={\langle A_{R}u,v\rangle }_{H_{\alpha ,+}}.$$

Therefore

$${\mathfrak{l}}_R[u,v]=q_R[u,v]+{\langle u,v\rangle }_{H_{\alpha ,+}}={\langle (A_R+I)u,v\rangle }_{H_{\alpha ,+}}={\langle Lu,v\rangle }_{H_{\alpha ,+}}.$$

Thus the operator associated with the positive closed form ${\mathfrak{l}}_R$ is precisely L.

We now apply the second representation theorem for positive closed forms. This gives

$$Dom\left(L^{1/2}\right)=Dom\left({\mathfrak{l}}_R\right)={\mathcal{Q}}_R,$$

and

$${\mathfrak{l}}_R[u,v]={\langle L^{1/2}u,L^{1/2}v\rangle }_{H_{\alpha ,+}}(u,v\in {\mathcal{Q}}_R).$$

Substituting the definition of ${\mathfrak{l}}_R$, we obtain

$$q_R[u,v]+{\langle u,v\rangle }_{H_{\alpha ,+}}={\langle L^{1/2}u,L^{1/2}v\rangle }_{H_{\alpha ,+}},$$

as claimed. □

Corollary 2.13 

(Resolvent bound for $A_R$). For every $\lambda >0$,

$$\parallel {(A_R+\lambda I)}^{-1}{\parallel }_{H_{\alpha ,+}\to H_{\alpha ,+}}\le {\lambda }^{-1}.$$

Proof. 

Since $A_R\ge 0$, the spectral theorem gives

$$\sigma \left(A_R\right)\subset [0,\infty ).$$

Therefore

$$\sigma (A_R+\lambda I)\subset [\lambda ,\infty ),$$

so $A_R+\lambda I$ is invertible and

$$\parallel {(A_R+\lambda I)}^{-1}\parallel =\underset{t\in \sigma \left(A_R\right)}{sup}{\displaystyle \frac{1}{t+\lambda }}\le {\displaystyle \frac{1}{\lambda }}.$$

□

2.5. Compact Resolvent and Discrete Spectrum

Here we connect the local regularity of the transported differential operator with the tail tightness given by Theorem 2.9. This yields a compact embedding from the form domain into the ambient Hilbert space, and hence compact resolvent for both $A_R$ and its positive shift L. After that, the spectral theorem yields a purely discrete positive spectrum.

Proposition 2.14 

(Local compactness on bounded intervals). Let $R_0>0$. If $F\subset {\mathcal{Q}}_R$ is bounded with respect to the form norm defined by

$${\parallel f\parallel }_{q_R}^2:={\parallel f\parallel }_{H_{\alpha ,+}}^2+q_R\left[f\right],$$

then the restricted family

$$\{f{|}_{(0,R_0)}:f\in F\}$$

is relatively compact in $L^2((0,R_0),d{\mu }_{\alpha })$.

Proof. 

Fix $R_0>0$, and assume that there exists $M>0$ such that

$$\underset{f\in F}{sup}{\parallel f\parallel }_{q_R}\le M.$$

Take $f\in F$, and set

$$g:=U_{\alpha }f.$$

Since $U_{\alpha }$ is unitary,

$${\parallel g\parallel }_{L^2(0,\infty )}={\parallel f\parallel }_{H_{\alpha ,+}}.$$

Hence

$${\parallel g\parallel }_{L^2(0,R_0)}\le {\parallel g\parallel }_{L^2(0,\infty )}={\parallel f\parallel }_{H_{\alpha ,+}}\le M.$$

(2.5.1)

Next, since D is the closure of $D_0$, and since

$$U_{\alpha }{D}_0{U}_{\alpha }^{-1}=-i{\partial }_x\mathrm{on}{\mathcal{C}}_R,$$

the transported closed operator $U_{\alpha }D{U}_{\alpha }^{-1}$ is the closure of $-i{\partial }_x$ on $L^2(0,\infty )$. In particular, if $f\in Dom\left(D\right)$, then $g=U_{\alpha }f$ belongs to $H_{\mathrm{loc}}^1(0,\infty )$, and

$$g^{\prime }=i{U}_{\alpha }Df\mathrm{in}{L}_{\mathrm{loc}}^2(0,\infty )$$

holds. Therefore

$$\parallel g^{\prime }{\parallel }_{L^2(0,R_0)}\le \parallel U_{\alpha }{Df\parallel }_{L^2(0,\infty )}={\parallel Df\parallel }_{H_{\alpha ,+}}\le q_R{\left[f\right]}^{1/2}\le M.$$

(2.5.2)

From (2.5.1) and (2.5.2), it follows that

$$\{U_{\alpha }f{|}_{(0,R_0)}:f\in F\}$$

is bounded in $H^1(0,R_0)$. By Rellich’s compactness theorem, this family is relatively compact in $L^2(0,R_0)$.

We now return to the weighted space. On the bounded interval $[0,R_0]$, the weight ${\rho }_{\alpha }\left(x\right)={\langle x\rangle }^{2\alpha }$ is bounded above and below by positive constants:

$$0<c_{\alpha ,R_0}\le {\rho }_{\alpha }\left(x\right)\le C_{\alpha ,R_0}<\infty .$$

Therefore $U_{\alpha }$, restricted to this interval, is a bounded isomorphism between $L^2((0,R_0),d{\mu }_{\alpha })$ and $L^2(0,R_0)$. Furthermore, the potential $V_R\left(x\right)=log(e+x)$ is also bounded on $[0,R_0]$, so the form norm introduces no new local singularity. Hence relative compactness in $L^2(0,R_0)$ is equivalent to relative compactness in $L^2((0,R_0),d{\mu }_{\alpha })$.

Accordingly,

$$\{f{|}_{(0,R_0)}:f\in F\}$$

is relatively compact in $L^2((0,R_0),d{\mu }_{\alpha })$. □

Theorem 2.15 

(Compact embedding of the form domain). The embedding

$${\mathcal{Q}}_R\hookrightarrow H_{\alpha ,+}$$

is compact.

Proof. 

Let ${\left(f_n\right)}_{n\ge 1}\subset {\mathcal{Q}}_R$ be bounded in the form norm:

$$\underset{n\ge 1}{sup}{\parallel f_n\parallel }_{q_R}\le M$$

for some $M>0$. We must show that $\left(f_n\right)$ has a subsequence convergent in $H_{\alpha ,+}$.

Fix $\epsilon >0$. Applying Theorem 2.9 to the family

$$F:=\{f_n:n\ge 1\},$$

we obtain some $R_0>0$ such that

$$\underset{n\ge 1}{sup}{\int }_{R_0}^{\infty }{|f_n\left(x\right)|}^{2}d{\mu }_{\alpha }\left(x\right)<{\displaystyle \frac{{\epsilon }^2}{8}}.$$

(2.5.3)

Thus each $f_n$ has a uniformly small tail beyond $R_0$.

On the bounded interval $(0,R_0)$, Proposition 2.14 implies that the restricted sequence $\left(f_n{|}_{(0,R_0)}\right)$ is relatively compact in $L^2((0,R_0),d{\mu }_{\alpha })$. Therefore, after passing to a subsequence and denoting it again by $\left(f_n\right)$, we may assume that

$$f_n{|}_{(0,R_0)}$$

is Cauchy in $L^2((0,R_0),d{\mu }_{\alpha })$. Hence there exists N such that, for all $m,n\ge N$,

$${\int }_0^{R_0}{|f_n\left(x\right)-f_m\left(x\right)|}^{2}d{\mu }_{\alpha }\left(x\right)<{\displaystyle \frac{{\epsilon }^2}{2}}.$$

(2.5.4)

For the full-space norm, we estimate

$$\parallel f_n-f_m{\parallel }_{H_{\alpha ,+}}^2={\int }_0^{R_0}|f_n-f_m{|}^{2}d{\mu }_{\alpha }+{\int }_{R_0}^{\infty }{|f_n-f_m|}^{2}d{\mu }_{\alpha }.$$

Using

$${|a-b|}^2\le {2|a|}^2+2{|b|}^2$$

together with (2.5.3) and (2.5.4), we obtain, for $m,n\ge N$,

$$\parallel f_n-f_m{\parallel }_{H_{\alpha ,+}}^2<{\displaystyle \frac{{\epsilon }^2}{2}}+2·{\displaystyle \frac{{\epsilon }^2}{8}}+2·{\displaystyle \frac{{\epsilon }^2}{8}}={\epsilon }^2.$$

Therefore the chosen subsequence is Cauchy in $H_{\alpha ,+}$, and hence converges in $H_{\alpha ,+}$.

It follows that every form-bounded sequence in ${\mathcal{Q}}_R$ has a convergent subsequence in $H_{\alpha ,+}$. This is precisely the compactness of the embedding

$${\mathcal{Q}}_R\hookrightarrow H_{\alpha ,+}.$$

□

Theorem 2.16 

(Compact resolvent and discrete spectrum). For every $\lambda >0$, the resolvent

$${(A_R+\lambda I)}^{-1}:H_{\alpha ,+}\to H_{\alpha ,+}$$

is compact. In particular,

$$L^{-1}:H_{\alpha ,+}\to H_{\alpha ,+}$$

is compact.

Therefore L has purely discrete positive spectrum. Namely, there exist a strictly increasing sequence

$$0<{\ell }_1<{\ell }_2<\dots ,{\ell }_n\to \infty ,$$

and finite-rank orthogonal projectors

$$P_n:H_{\alpha ,+}\to H_{\alpha ,+}$$

such that

$$L{P}_n={\ell }_n{P}_n,\sum _{n\ge 1}{P}_n=I$$

holds in the strong operator topology. Equivalently, by choosing an orthonormal basis in each eigenspace $P_n{H}_{\alpha ,+}$, one obtains an orthonormal basis consisting of eigenvectors of L. In particular, if the eigenvalues are repeated according to multiplicity, then there exist

$$0<{\lambda }_1\le {\lambda }_2\le \dots ,{\lambda }_k\to \infty ,$$

and an orthonormal basis ${\left\{e_k\right\}}_{k\ge 1}$ of $H_{\alpha ,+}$ satisfying

$$L{e}_k={\lambda }_k{e}_k(k\ge 1).$$

The corresponding eigenvalues of $A_R$ are

$$a_k:={\lambda }_k-1\ge 0.$$

Proof. 

Fix $\lambda >0$, and define the shifted form on ${\mathcal{Q}}_R$ by

$${\mathfrak{a}}_{\lambda }[u,v]:=q_R[u,v]+\lambda {\langle u,v\rangle }_{H_{\alpha ,+}}.$$

Since $q_R$ is closed and nonnegative, ${\mathfrak{a}}_{\lambda }$ is a densely defined positive closed form on ${\mathcal{Q}}_R$.

Its norm

$${\parallel u\parallel }_{{\mathfrak{a}}_{\lambda }}^2:={\mathfrak{a}}_{\lambda }\left[u\right]=q_R\left[u\right]+\lambda {\parallel u\parallel }_{H_{\alpha ,+}}^2$$

is equivalent to the form norm ${\parallel u\parallel }_{q_R}$. Indeed,

$${min\{1,\lambda \}\parallel u\parallel }_{q_R}^2\le {\parallel u\parallel }_{{\mathfrak{a}}_{\lambda }}^2\le max\{1,\lambda \}{\parallel u\parallel }_{q_R}^2.$$

(2.5.5)

Therefore Theorem 2.15 implies that the embedding

$$({\mathcal{Q}}_R{,\parallel ·\parallel }_{{\mathfrak{a}}_{\lambda }})\hookrightarrow H_{\alpha ,+}$$

is also compact.

Now take $f\in H_{\alpha ,+}$. The functional

$$v⟼{\langle f,v\rangle }_{H_{\alpha ,+}}$$

is continuous on $({\mathcal{Q}}_R,\parallel ·{\parallel }_{{\mathfrak{a}}_{\lambda }})$. Indeed,

$${|\langle f,v\rangle }_{H_{\alpha ,+}}{|\le \parallel f\parallel }_{H_{\alpha ,+}}{\parallel v\parallel }_{H_{\alpha ,+}}\le {\lambda }^{-1/2}{\parallel f\parallel }_{H_{\alpha ,+}}{\parallel v\parallel }_{{\mathfrak{a}}_{\lambda }}.$$

Thus, by the Riesz representation theorem, there exists a unique element

$$u_{\lambda }\left(f\right)\in {\mathcal{Q}}_R$$

such that

$${\mathfrak{a}}_{\lambda }[u_{\lambda }\left(f\right),v]={\langle f,v\rangle }_{H_{\alpha ,+}}(\forall v\in {\mathcal{Q}}_R).$$

(2.5.6)

By Theorem 2.10, (2.5.6) is precisely the weak-form characterization of the resolvent equation

$$(A_R+\lambda I)u_{\lambda }\left(f\right)=f.$$

Hence

$$u_{\lambda }\left(f\right)={(A_R+\lambda I)}^{-1}f.$$

The map

$$S_{\lambda }:f\mapsto u_{\lambda }\left(f\right)$$

is a bounded operator from $H_{\alpha ,+}$ to $({\mathcal{Q}}_R,\parallel ·{\parallel }_{{\mathfrak{a}}_{\lambda }})$, and the inclusion map

$$j_{\lambda }:({\mathcal{Q}}_R{,\parallel ·\parallel }_{{\mathfrak{a}}_{\lambda }})\hookrightarrow H_{\alpha ,+}$$

is compact. Therefore

$${(A_R+\lambda I)}^{-1}=j_{\lambda }\circ S_{\lambda }$$

is compact on $H_{\alpha ,+}$. In particular, taking $\lambda =1$, we obtain that

$$L^{-1}={(A_R+I)}^{-1}$$

is compact.

Next, apply the spectral theorem to the compact self-adjoint positive operator $L^{-1}$. Then there exist a sequence of positive numbers

$${\mu }_1>{\mu }_2>\dots >0,{\mu }_n\downarrow 0,$$

and finite-rank orthogonal projectors $P_n$ such that

$$L^{-1}{P}_n={\mu }_n{P}_n,\sum _{n\ge 1}{P}_n=I$$

holds strongly. Setting

$${\ell }_n:={\mu }_n^{-1},$$

we have

$$0<{\ell }_1<{\ell }_2<\dots ,{\ell }_n\to \infty .$$

Multiplying the eigenvalue equation for $L^{-1}$ by L, we obtain

$$L{P}_n={\ell }_n{P}_n.$$

Thus the spectrum of L is purely discrete, the eigenspaces $P_n{H}_{\alpha ,+}$ are finite-dimensional, and there is no finite accumulation point.

Finally, choose an orthonormal basis in each eigenspace $P_n{H}_{\alpha ,+}$ and concatenate them. This gives an orthonormal basis consisting of eigenvectors of L. Repeating the distinct eigenvalues according to multiplicity, there exist a sequence

$$0<{\lambda }_1\le {\lambda }_2\le \dots ,{\lambda }_k\to \infty ,$$

and an orthonormal basis ${\left\{e_k\right\}}_{k\ge 1}$ such that

$$L{e}_k={\lambda }_k{e}_k.$$

Since $L=A_R+I$, the corresponding eigenvalues of $A_R$ are

$$a_k={\lambda }_k-1\ge 0.$$

This completes the proof. □

2.6. Weighted Resolvent and Herglotz Resolvent Construction

Here we record the scalar-valued meromorphic/Herglotz resolvent construction associated with the positive self-adjoint operator L. The input is an arbitrarily fixed probe vector $\Phi \in H_{\alpha ,+}$. The output is the scalar resolvent function

$$m_L^{(\Phi )}\left(z\right)={\langle \Phi ,{(L-zI)}^{-1}\Phi \rangle }_{H_{\alpha ,+}},$$

and its analytic behavior is directly controlled by the spectral theorem. This is the endpoint of the present analytic section.

Definition 2.17 

(Resolvent probe and scalar Herglotz function). Fix a probe vector

$$\Phi \in H_{\alpha ,+}.$$

For

$$z\in \mathbb{C}\setminus \sigma \left(L\right),$$

define

$$m_L^{(\Phi )}\left(z\right):={\langle \Phi ,{(L-zI)}^{-1}\Phi \rangle }_{H_{\alpha ,+}}.$$

Lemma 2.18 

(Spectral expansion of the resolvent). Use ${\ell }_n$ and $P_n$ from Theorem 2.16. Then, for any $u\in H_{\alpha ,+}$ and any $z\in \mathbb{C}\setminus \sigma \left(L\right)$,

$${(L-zI)}^{-1}u=\sum _{n\ge 1}{\displaystyle \frac{1}{{\ell }_n-z}}{P}_{n}u$$

converges in $H_{\alpha ,+}$. Furthermore, for any fixed $\Phi \in H_{\alpha ,+}$,

$$m_L^{(\Phi )}\left(z\right)=\sum _{n\ge 1}{\displaystyle \frac{\parallel P_n{\Phi \parallel }_{H_{\alpha ,+}}^2}{{\ell }_n-z}},z\in \mathbb{C}\setminus \sigma \left(L\right),$$

and this scalar series converges locally uniformly on compact subsets of $\mathbb{C}\setminus \sigma \left(L\right)$.

Proof. 

By Theorem 2.16,

$$\sum _{n\ge 1}{P}_n=I$$

strongly, and

$$L{P}_n={\ell }_n{P}_n$$

holds. Therefore, for any $u\in H_{\alpha ,+}$,

$$u=\sum _{n\ge 1}{P}_{n}u$$

holds in $H_{\alpha ,+}$, and on each spectral subspace $P_n{H}_{\alpha ,+}$,

$${(L-zI)}^{-1}={\displaystyle \frac{1}{{\ell }_n-z}}I.$$

Hence

$${(L-zI)}^{-1}u=\sum _{n\ge 1}{\displaystyle \frac{1}{{\ell }_n-z}}{P}_{n}u.$$

It remains to verify convergence in $H_{\alpha ,+}$. Since $z\notin \sigma \left(L\right)$,

$${\delta }_z:=dist(z,\sigma \left(L\right))>0.$$

For partial sums with $M>N$, we have

$${∥{\displaystyle \sum _{n=N+1}^M}{\displaystyle \frac{1}{{\ell }_n-z}}{P}_{n}u∥}_{H_{\alpha ,+}}^2={\displaystyle \sum _{n=N+1}^M}{\displaystyle \frac{\parallel P_n{u\parallel }_{H_{\alpha ,+}}^2}{|{\ell }_n{-z|}^2}}\le {\delta }_z^{-2}{\displaystyle \sum _{n=N+1}^M}{\parallel P_{n}u\parallel }_{H_{\alpha ,+}}^2.$$

The vectors $P_{n}u$ are mutually orthogonal, and moreover

$$\sum _{n\ge 1}\parallel P_n{u\parallel }_{H_{\alpha ,+}}^2={\parallel u\parallel }_{H_{\alpha ,+}}^2.$$

Thus the right-hand side tends to 0 as $N,M\to \infty$. This proves norm convergence.

For the scalar function $m_L^{(\Phi )}$, taking the inner product with $\Phi$ gives

$$m_L^{(\Phi )}\left(z\right)={〈\Phi ,\sum _{n\ge 1}{\displaystyle \frac{1}{{\ell }_n-z}}{P}_n\Phi 〉}_{H_{\alpha ,+}}=\sum _{n\ge 1}{\displaystyle \frac{{\langle \Phi ,P_n\Phi \rangle }_{H_{\alpha ,+}}}{{\ell }_n-z}}.$$

Since $P_n$ is an orthogonal projector,

$${\langle \Phi ,P_n\Phi \rangle }_{H_{\alpha ,+}}={\parallel P_n\Phi \parallel }_{H_{\alpha ,+}}^2.$$

Therefore

$$m_L^{(\Phi )}\left(z\right)=\sum _{n\ge 1}{\displaystyle \frac{\parallel P_n{\Phi \parallel }_{H_{\alpha ,+}}^2}{{\ell }_n-z}}.$$

Finally, let $K\subset \mathbb{C}\setminus \sigma \left(L\right)$ be compact, and set

$${\delta }_K:=dist(K,\sigma \left(L\right))>0.$$

Then, for any $z\in K$,

$$\left|{\displaystyle \frac{\parallel P_n{\Phi \parallel }_{H_{\alpha ,+}}^2}{{\ell }_n-z}}\right|\le {\delta }_K^{-1}{\parallel P_n\Phi \parallel }_{H_{\alpha ,+}}^2.$$

But

$$\sum _{n\ge 1}\parallel P_n{\Phi \parallel }_{H_{\alpha ,+}}^2={\parallel \Phi \parallel }_{H_{\alpha ,+}}^2<\infty ,$$

so the Weierstrass M-test implies local uniform convergence on K. Thus the scalar expansion converges locally uniformly on compact subsets of $\mathbb{C}\setminus \sigma \left(L\right)$. □

Theorem 2.19 

(Herglotz positivity and pole law). For any fixed $\Phi \in H_{\alpha ,+}$, the function

$$m_L^{(\Phi )}\left(z\right)={\langle \Phi ,{(L-zI)}^{-1}\Phi \rangle }_{H_{\alpha ,+}}$$

is holomorphic on $\mathbb{C}\setminus \sigma \left(L\right)$. Furthermore, if

$$\Im z>0,$$

then

$$\Im m_L^{(\Phi )}\left(z\right)\ge 0,$$

more precisely,

$$\Im m_L^{(\Phi )}\left(z\right)=(\Im z){\parallel {(L-zI)}^{-1}\Phi \parallel }_{H_{\alpha ,+}}^2.$$

Finally, let ${\ell }_n\in \sigma \left(L\right)$ be one of the discrete eigenvalues from Theorem 2.16. If

$$P_n\Phi \ne 0,$$

then

$$m_L^{(\Phi )}\left(z\right)={\displaystyle \frac{\parallel P_n{\Phi \parallel }_{H_{\alpha ,+}}^2}{{\ell }_n-z}}+O\left(1\right)(z\to {\ell }_n).$$

Equivalently, $m_L^{(\Phi )}$ has a simple pole at $z={\ell }_n$, and its principal part is

$${\displaystyle \frac{\parallel P_n{\Phi \parallel }_{H_{\alpha ,+}}^2}{{\ell }_n-z}}.$$

If $P_n\Phi =0$, then no singular term appears at $z={\ell }_n$.

Proof. 

The map

$$z⟼{(L-zI)}^{-1}$$

is holomorphic as an $H_{\alpha ,+}$-valued operator-valued map on the resolvent set $\mathbb{C}\setminus \sigma \left(L\right)$. Therefore, taking the scalar product with the fixed vector $\Phi$, it follows that

$$m_L^{(\Phi )}$$

is holomorphic on $\mathbb{C}\setminus \sigma \left(L\right)$.

Next, take $z\in \mathbb{C}$ satisfying $\Im z>0$, and write

$$R\left(z\right):={(L-zI)}^{-1}.$$

Since L is self-adjoint,

$$R\left(\overline{z}\right)=R{\left(z\right)}^{*}.$$

Using the resolvent identity, we obtain

$$R\left(z\right)-R\left(\overline{z}\right)=(z-\overline{z})R\left(z\right)R\left(\overline{z}\right),$$

and hence

$$m_L^{(\Phi )}\left(z\right)-\overline{m_L^{(\Phi )}\left(z\right)}={\langle \Phi ,(R\left(z\right)-R\left(\overline{z}\right))\Phi \rangle }_{H_{\alpha ,+}}=(z-\overline{z}){\langle \Phi ,R\left(z\right)R\left(\overline{z}\right)\Phi \rangle }_{H_{\alpha ,+}}.$$

Since $R\left(\overline{z}\right)=R{\left(z\right)}^{*}$,

$${\langle \Phi ,R\left(z\right)R\left(\overline{z}\right)\Phi \rangle }_{H_{\alpha ,+}}={\langle R{\left(z\right)}^{*}\Phi ,R{\left(z\right)}^{*}\Phi \rangle }_{H_{\alpha ,+}}={\parallel R\left(z\right)\Phi \parallel }_{H_{\alpha ,+}}^2.$$

Therefore

$$2i\Im m_L^{(\Phi )}\left(z\right)=2i(\Im z){\parallel R\left(z\right)\Phi \parallel }_{H_{\alpha ,+}}^2,$$

and thus

$$\Im m_L^{(\Phi )}\left(z\right)=(\Im z){\parallel {(L-zI)}^{-1}\Phi \parallel }_{H_{\alpha ,+}}^2.$$

Since $\Im z>0$, this implies

$$\Im m_L^{(\Phi )}\left(z\right)\ge 0.$$

It remains to prove the pole law. Fix n, and choose $r>0$ such that

$$B({\ell }_n,r)\cap \sigma \left(L\right)=\left\{{\ell }_n\right\}.$$

By Lemma 2.18,

$$m_L^{(\Phi )}\left(z\right)={\displaystyle \frac{\parallel P_n{\Phi \parallel }_{H_{\alpha ,+}}^2}{{\ell }_n-z}}+\sum _{k\ne n}{\displaystyle \frac{\parallel P_k{\Phi \parallel }_{H_{\alpha ,+}}^2}{{\ell }_k-z}}.$$

If $|z-{\ell }_n|<r/2$, then

$$|{\ell }_k-z|\ge |{\ell }_k-{\ell }_n|-|z-{\ell }_n|\ge {\displaystyle \frac{1}{2}}|{\ell }_k-{\ell }_n|(k\ne n),$$

and in particular the denominators remain uniformly away from 0. Hence the residual series

$$\sum _{k\ne n}{\displaystyle \frac{\parallel P_k{\Phi \parallel }_{H_{\alpha ,+}}^2}{{\ell }_k-z}}$$

is holomorphic and bounded in a neighborhood of ${\ell }_n$, by the same local uniform convergence argument used in Lemma 2.18. Therefore

$$m_L^{(\Phi )}\left(z\right)={\displaystyle \frac{\parallel P_n{\Phi \parallel }_{H_{\alpha ,+}}^2}{{\ell }_n-z}}+O\left(1\right)(z\to {\ell }_n).$$

If $P_n\Phi \ne 0$, this is a genuine simple pole with the displayed principal part. If $P_n\Phi =0$, the singular term vanishes and only the bounded holomorphic residual term remains. □

Remark 2.20 

(End of the analytic section). Section 2 ends here. At this point, we have constructed the operator-side analytic construction

$$q_R⟶A_R⟶L,$$

together with the compact embedding of the form domain, compact resolvent, discrete spectrum, and the scalar-valued meromorphic/Herglotz resolvent function $m_L^{(\Phi )}$. In this section, we have used no arithmetic projector, no comparison theorem, and no conclusion statement for the Riemann hypothesis.

3. Coefficient-Space Arithmetic Construction

3.1. Formal Dirichlet Algebra and Coefficient-Extraction Conventions

In this subsection, we argue entirely on the coefficient layer. We use neither analytic continuation, nor a complex variable s, nor meromorphic functions. The sole purpose here is to fix the formal arithmetic algebra in which the coefficient-extraction statement will be formulated later.

Definition 3.1 

(Formal Dirichlet algebra and augmentation ideal). Let

$$\mathfrak{D}:=\{a:\mathbb{N}\to \mathbb{C}\}$$

be the complex vector space of all arithmetic functions. Equip $\mathfrak{D}$ with Dirichlet convolution

$$(a*b)\left(n\right):=\sum _{d\mid n}a\left(d\right)b(n/d)(n\ge 1).$$

Then $(\mathfrak{D},*)$ is a commutative $\mathbb{C}$-algebra with unit

$${\delta }_1\left(n\right):=\left\{\begin{array}{cc}1,\hfill & n=1,\hfill \\ 0,\hfill & n\ge 2.\hfill \end{array}\right.$$

For $a\in \mathfrak{D}$ and $n\ge 1$, write the coefficient-extraction operator at n as

$$\left[n\right]a:=a\left(n\right).$$

Define the augmentation ideal by

$$\mathfrak{m}:=\{a\in \mathfrak{D}:a(1)=0\}.$$

Equivalently,

$${\delta }_1+\mathfrak{m}=\{a\in \mathfrak{D}:a\left(1\right)=1\}.$$

When necessary, one may write $a\in \mathfrak{D}$ as a formal Dirichlet series

$$a\sim \sum _{n\ge 1}a\left(n\right)n^{-s},$$

but here $n^{-s}$ is merely a formal basis symbol indexed by n. At no point in §3.1–§3.2 do we substitute any value for s.

Lemma 3.2 

(Coefficient and convolution laws). Let $a,b\in \mathfrak{D}$. Then, for any $n\ge 1$,

$$\left[n\right](a*b)=\sum _{d\mid n}\left[d\right]a[n/d]b.$$

More generally, for any integer $k\ge 1$,

$$\left[n\right]\left(a^{*k}\right)=\sum _{\begin{array}{c}{d}_1\dots d_k=n\\ d_i\ge 1\end{array}}a\left(d_1\right)\dots a\left(d_k\right),$$

where $a^{*k}$ denotes the k-fold Dirichlet convolution power of a.

If $a\left(1\right)\ne 0$, then there exists a unique Dirichlet inverse

$$a^{-*}\in \mathfrak{D}$$

satisfying

$$a*a^{-*}=a^{-*}*a={\delta }_1.$$

Its coefficients are determined recursively by

$$a^{-*}\left(1\right)={\displaystyle \frac{1}{a\left(1\right)}},$$

and, for $n\ge 2$, by

$$a^{-*}\left(n\right)=-{\displaystyle \frac{1}{a\left(1\right)}}\sum _{\begin{array}{c}d\mid n\\ d>1\end{array}}a\left(d\right)a^{-*}(n/d).$$

In particular, the inverse is uniquely determined coefficientwise by the divisor recursion.

Proof. 

The first identity is merely the definition of Dirichlet convolution rewritten in coefficient-extraction notation:

$$\left[n\right](a*b)=(a*b)\left(n\right)=\sum _{d\mid n}a\left(d\right)b(n/d)=\sum _{d\mid n}\left[d\right]a[n/d]b.$$

The k-fold formula is proved by induction on k. The case $k=1$ is immediate. Assume the identity holds for some $k\ge 1$. Then

$$\left[n\right]\left(a^{*(k+1)}\right)=\left[n\right](a^{*k}*a)=\sum _{d\mid n}\left[d\right]\left(a^{*k}\right)[n/d]a.$$

By the induction hypothesis,

$$\left[d\right]\left(a^{*k}\right)=\sum _{\begin{array}{c}{e}_1\dots e_k=d\\ e_i\ge 1\end{array}}a\left(e_1\right)\dots a\left(e_k\right).$$

Substituting this into the preceding formula gives

$$\left[n\right]\left(a^{*(k+1)}\right)=\sum _{d\mid n}\left(\sum _{\begin{array}{c}{e}_1\dots e_k=d\\ e_i\ge 1\end{array}}a\left(e_1\right)\dots a\left(e_k\right)\right)a(n/d).$$

Relabeling $e_{k+1}:=n/d$, this is exactly

$$\left[n\right]\left(a^{*(k+1)}\right)=\sum _{\begin{array}{c}{e}_1\dots e_{k+1}=n\\ e_i\ge 1\end{array}}a\left(e_1\right)\dots a\left(e_{k+1}\right),$$

and the induction is complete.

Next, we prove the existence and uniqueness of the Dirichlet inverse. Suppose that

$$b\in \mathfrak{D}$$

satisfies

$$a*b={\delta }_1.$$

Looking at the coefficient $n=1$, we obtain

$$1=\left[1\right]{\delta }_1=\left[1\right](a*b)=a\left(1\right)b\left(1\right),$$

and hence necessarily

$$b\left(1\right)={\displaystyle \frac{1}{a\left(1\right)}}.$$

For $n\ge 2$, the identity $\left[n\right](a*b)=0$ gives

$$0=\sum _{d\mid n}a\left(d\right)b(n/d)=a\left(1\right)b\left(n\right)+\sum _{\begin{array}{c}d\mid n\\ d>1\end{array}}a\left(d\right)b(n/d).$$

Therefore

$$b\left(n\right)=-{\displaystyle \frac{1}{a\left(1\right)}}\sum _{\begin{array}{c}d\mid n\\ d>1\end{array}}a\left(d\right)b(n/d).$$

If $d>1$, then $n/d<n$, so the right-hand side depends only on coefficients $b\left(m\right)$ with $m<n$ that have already been determined. Thus this recursion admits at most one inverse.

Conversely, define b recursively by

$$b\left(1\right):={\displaystyle \frac{1}{a\left(1\right)}},b\left(n\right):=-{\displaystyle \frac{1}{a\left(1\right)}}\sum _{\begin{array}{c}d\mid n\\ d>1\end{array}}a\left(d\right)b(n/d)(n\ge 2).$$

Tracing the same calculation in reverse yields

$$\left[n\right](a*b)=\left\{\begin{array}{cc}1,\hfill & n=1,\hfill \\ 0,\hfill & n\ge 2.\hfill \end{array}\right.$$

Thus

$$a*b={\delta }_1.$$

Since Dirichlet convolution is commutative,

$$b*a={\delta }_1$$

also holds. Hence $b=a^{-*}$ exists and is unique. □

Proposition 3.3 

(Formal logarithm and exponential on the augmentation ideal). For $a,b\in \mathfrak{m}$, define formally

$${log}_{*}({\delta }_1+a):=\sum _{k\ge 1}{\displaystyle \frac{{(-1)}^{k+1}}{k}}{a}^{*k},{exp}_{*}\left(b\right):=\sum _{k\ge 0}{\displaystyle \frac{1}{k!}}{b}^{*k},$$

using the convention

$$a^{*0}=b^{*0}:={\delta }_1.$$

Then the following hold:

1.

Both series are coefficientwise well-defined;

2.

$${log}_{*}:({\delta }_1+\mathfrak{m})⟶\mathfrak{m},{exp}_{*}:\mathfrak{m}⟶({\delta }_1+\mathfrak{m});$$

3.

The two maps are inverse to each other:

$${exp}_{*}\left({log}_{*}({\delta }_1+a)\right)={\delta }_1+a(a\in \mathfrak{m}),$$

$${log}_{*}\left({exp}_{*}\left(b\right)\right)=b(b\in \mathfrak{m}).$$

Proof. 

Let $\Omega \left(n\right)$ denote the total number of prime factors of n, counted with multiplicity.

First, we prove coefficientwise well-definedness. Take $a\in \mathfrak{m}$, and fix $n\ge 1$. By Lemma 3.2,

$$\left[n\right]\left(a^{*k}\right)=\sum _{\begin{array}{c}{d}_1\dots d_k=n\\ d_i\ge 1\end{array}}a\left(d_1\right)\dots a\left(d_k\right).$$

Since $a\in \mathfrak{m}$, we have $a\left(1\right)=0$. Therefore, if a nonzero term appears, it must satisfy

$$d_i\ge 2(1\le i\le k).$$

But an ordered factorization

$$d_1\dots d_k=n(d_i\ge 2)$$

can exist only when

$$k\le \Omega \left(n\right).$$

Hence

$$\left[n\right]\left(a^{*k}\right)=0(k>\Omega \left(n\right)).$$

Thus the formal series for the coefficient n of ${log}_{*}({\delta }_1+a)$ reduces to the finite sum

$$\left[n\right]{log}_{*}({\delta }_1+a)={\displaystyle \sum _{k=1}^{\Omega \left(n\right)}}{\displaystyle \frac{{(-1)}^{k+1}}{k}}\left[n\right]\left(a^{*k}\right)(n\ge 2).$$

For $n=1$, all terms vanish because $\left[1\right]\left(a^{*k}\right)=0$ for $k\ge 1$. Therefore ${log}_{*}({\delta }_1+a)$ is coefficientwise well-defined and belongs to $\mathfrak{m}$.

The same argument applies to $b\in \mathfrak{m}$. For $n\ge 2$,

$$\left[n\right]\left(b^{*k}\right)=0(k>\Omega \left(n\right)),$$

so

$$\left[n\right]{exp}_{*}\left(b\right)={\displaystyle \sum _{k=0}^{\Omega \left(n\right)}}{\displaystyle \frac{1}{k!}}\left[n\right]\left(b^{*k}\right)$$

is a finite sum. For $n=1$, since $\left[1\right]\left(b^{*k}\right)=0$ for $k\ge 1$, only the $k=0$ term contributes. Thus

$$\left[1\right]{exp}_{*}\left(b\right)=\left[1\right]{\delta }_1=1,$$

and hence

$${exp}_{*}\left(b\right)\in {\delta }_1+\mathfrak{m}.$$

This proves (1) and (2).

It remains to prove the inverse-map identities. Fix $n\ge 1$, and set

$$Div\left(n\right):=\{d\in \mathbb{N}:d\mid n\}.$$

Consider the finite-dimensional convolution algebra

$${\mathfrak{D}}_{\mid n}:=\{c:Div\left(n\right)\to \mathbb{C}\}$$

with convolution restricted to the divisors of n:

$$\left(c{★}_{n}d\right)\left(m\right):=\sum _{r\mid m}c\left(r\right)d(m/r)(m\mid n).$$

Furthermore, set

$${\mathfrak{m}}_{\mid n}:=\{c\in {\mathfrak{D}}_{\mid n}:c\left(1\right)=0\}.$$

By the factorization-count estimate already proved, every $c\in {\mathfrak{m}}_{\mid n}$ satisfies

$$c^{{★}_{n}k}=0(k>\Omega \left(n\right)).$$

Indeed, for any divisor $m\mid n$, there is no factorization of m into more than $\Omega \left(n\right)$ integers all $\ge 2$. Thus ${\mathfrak{m}}_{\mid n}$ is a nilpotent ideal.

Consequently, in the finite-dimensional commutative algebra ${\mathfrak{D}}_{\mid n}$, the formal series

$$log(1+X)=\sum _{k\ge 1}{\displaystyle \frac{{(-1)}^{k+1}}{k}}{X}^k,exp\left(X\right)=\sum _{k\ge 0}{\displaystyle \frac{1}{k!}}{X}^k$$

truncate to genuine finite polynomials on ${\mathfrak{m}}_{\mid n}$, and the usual formal identities hold exactly:

$$exp(log(1+X\left)\right)=1+X,log(exp(X\left)\right)=X.$$

Apply this to the restrictions of a or b to $Div\left(n\right)$. Since the coefficient n depends only on values on the divisors of n,

$$\left[n\right]{exp}_{*}\left({log}_{*}({\delta }_1+a)\right)=\left[n\right]({\delta }_1+a),$$

$$\left[n\right]{log}_{*}\left({exp}_{*}\left(b\right)\right)=\left[n\right]b.$$

Since $n\ge 1$ was arbitrary, the coefficients agree for all n, and therefore

$${exp}_{*}\left({log}_{*}({\delta }_1+a)\right)={\delta }_1+a,{log}_{*}\left({exp}_{*}\left(b\right)\right)=b.$$

Hence ${log}_{*}$ and ${exp}_{*}$ are inverse to each other. □

Remark 3.4 

(Formal–analytic separation discipline). In §3.1–§3.2, we use neither $\zeta \left(s\right)$, nor $log\zeta \left(s\right)$, nor $-{\zeta }^{\prime }\left(s\right)/\zeta \left(s\right)$, nor the Euler product, nor analytic continuation, nor any discussion of poles or zeros on the complex plane. The reader may understand that also in the subsequent part of Section 3, the argument is carried out entirely on the coefficient layer of the formal Dirichlet algebra.

3.2. Prime Indicator and Von Mangoldt Function

Here we fix the basic arithmetic data that will later be input into the purely coefficient-level extraction theorem. At this stage, they are simply arithmetic functions on $\mathbb{N}$. One is supported on the prime support, and the other is supported on the prime-power support.

Definition 3.5 

(Prime indicator and von Mangoldt function). Define the prime indicator

$$1_P^{\mathrm{ex}}:\mathbb{N}\to \mathbb{C}$$

by

$$1_P^{\mathrm{ex}}\left(n\right):=\left\{\begin{array}{cc}1,\hfill & n\mathrm{is}\mathrm{prime},\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

and define the von Mangoldt function

$${\Lambda }^{\mathrm{ex}}:\mathbb{N}\to \mathbb{C}$$

by

$${\Lambda }^{\mathrm{ex}}\left(n\right):=\left\{\begin{array}{cc}logp,\hfill & n=p^k\mathrm{for}\mathrm{some}\mathrm{prime}p\mathrm{and}\mathrm{some}\mathrm{integer}k\ge 1,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Thus $1_P^{\mathrm{ex}}$ can be nonzero only on primes, whereas ${\Lambda }^{\mathrm{ex}}$ can be nonzero only on prime powers.

Theorem 3.6 

(Divisor-sum law for the von Mangoldt function). For every integer $n\ge 1$,

$$logn=\sum _{d\mid n}{\Lambda }^{\mathrm{ex}}\left(d\right)$$

holds. Equivalently, if

$$\mathbf{1}\left(n\right):=1(n\ge 1),$$

and

$$\mathbf{L}\left(n\right):=logn(n\ge 1),$$

then

$$\mathbf{L}=\mathbf{1}*{\Lambda }^{\mathrm{ex}}.$$

Therefore, if

$$\mu :={\mathbf{1}}^{-*}$$

is the Dirichlet inverse of $\mathbf{1}$, then

$${\Lambda }^{\mathrm{ex}}=\mu *\mathbf{L}.$$

Proof. 

First consider $n=1$. Since 1 is not a prime power,

$${\Lambda }^{\mathrm{ex}}\left(1\right)=0,$$

and hence

$$\sum _{d\mid 1}{\Lambda }^{\mathrm{ex}}\left(d\right)={\Lambda }^{\mathrm{ex}}\left(1\right)=0=log1.$$

Next let $n\ge 2$, and write its prime factorization as

$$n={\displaystyle \prod _{i=1}^r}{p}_i^{{\nu }_i}.$$

Here $p_1,\dots ,p_r$ are distinct primes and ${\nu }_i\ge 1$. A divisor $d\mid n$ contributes to

$$\sum _{d\mid n}{\Lambda }^{\mathrm{ex}}\left(d\right)$$

if and only if d is a prime power dividing n. Such divisors are precisely

$$p_i,p_i^2,\dots ,p_i^{{\nu }_i}(1\le i\le r).$$

Therefore

$$\sum _{d\mid n}{\Lambda }^{\mathrm{ex}}\left(d\right)={\displaystyle \sum _{i=1}^r}{\displaystyle \sum _{j=1}^{{\nu }_i}}{\Lambda }^{\mathrm{ex}}\left(p_i^j\right)={\displaystyle \sum _{i=1}^r}{\displaystyle \sum _{j=1}^{{\nu }_i}}log{p}_i={\displaystyle \sum _{i=1}^r}{\nu }_{i}log{p}_i.$$

On the other hand,

$$logn=log\left({\displaystyle \prod _{i=1}^r}{p}_i^{{\nu }_i}\right)={\displaystyle \sum _{i=1}^r}{\nu }_{i}log{p}_i,$$

so

$$\sum _{d\mid n}{\Lambda }^{\mathrm{ex}}\left(d\right)=logn.$$

This proves the divisor-sum law for all $n\ge 1$.

The convolution formulation follows immediately:

$$(\mathbf{1}*{\Lambda }^{\mathrm{ex}})\left(n\right)=\sum _{d\mid n}\mathbf{1}\left(d\right){\Lambda }^{\mathrm{ex}}(n/d)=\sum _{d\mid n}{\Lambda }^{\mathrm{ex}}\left(d\right)=logn=\mathbf{L}\left(n\right).$$

Thus

$$\mathbf{L}=\mathbf{1}*{\Lambda }^{\mathrm{ex}}.$$

Finally, since $\mathbf{1}\left(1\right)=1\ne 0$, Lemma 3.2 gives a unique Dirichlet inverse

$$\mu ={\mathbf{1}}^{-*}.$$

Convolving both sides of

$$\mathbf{L}=\mathbf{1}*{\Lambda }^{\mathrm{ex}}$$

from the left by $\mu$, and using associativity together with

$$\mu *\mathbf{1}={\delta }_1,$$

we obtain

$$\mu *\mathbf{L}=(\mu *\mathbf{1})*{\Lambda }^{\mathrm{ex}}={\delta }_1*{\Lambda }^{\mathrm{ex}}={\Lambda }^{\mathrm{ex}}.$$

Hence

$${\Lambda }^{\mathrm{ex}}=\mu *\mathbf{L}.$$

□

Proposition 3.7 

(Support and positivity properties). For an arithmetic function $a\in \mathfrak{D}$, set

$$supp\left(a\right):=\{n\in \mathbb{N}:a(n)\ne 0\}.$$

Then

$$supp\left(1_P^{\mathrm{ex}}\right)=\left\{\mathrm{primes}\right\},$$

and

$$supp\left({\Lambda }^{\mathrm{ex}}\right)=\left\{\mathrm{prime}\mathrm{powers}\right\}.$$

Furthermore,

$${\Lambda }^{\mathrm{ex}}\left(n\right)\ge 0(n\ge 1),$$

and moreover

$${\Lambda }^{\mathrm{ex}}\left(n\right)>0⟺n\mathrm{is}\mathrm{a}\mathrm{prime}\mathrm{power}.$$

Accordingly, the prime support of $1_P^{\mathrm{ex}}$ and the prime-power support of ${\Lambda }^{\mathrm{ex}}$ are distinct and must be distinguished in the later coefficient-level argument.

Proof. 

By Definition 3.5,

$$1_P^{\mathrm{ex}}\left(n\right)=1$$

holds if and only if n is prime, and it is 0 otherwise. Therefore

$$supp\left(1_P^{\mathrm{ex}}\right)=\left\{\mathrm{primes}\right\}.$$

Similarly, by Definition 3.5,

$${\Lambda }^{\mathrm{ex}}\left(n\right)=logp$$

holds if and only if $n=p^k$ for some prime p and integer $k\ge 1$, and it is 0 otherwise. Therefore

$$supp\left({\Lambda }^{\mathrm{ex}}\right)=\left\{\mathrm{prime}\mathrm{powers}\right\}.$$

If n is not a prime power, then

$${\Lambda }^{\mathrm{ex}}\left(n\right)=0.$$

On the other hand, if $n=p^k$ is a prime power, then

$${\Lambda }^{\mathrm{ex}}\left(n\right)=logp.$$

Every prime satisfies $p\ge 2$, so

$$logp>0.$$

Thus

$${\Lambda }^{\mathrm{ex}}\left(n\right)\ge 0(n\ge 1),$$

and strict inequality holds exactly in the prime-power case. This proves all the asserted support and positivity properties. □

Corollary 3.8 

(Coefficient-level recovery of the logarithm function). Define the arithmetic function

$$\mathbf{L}\in \mathfrak{D}\mathrm{by}\mathbf{L}\left(n\right):=logn(n\ge 1),$$

and define

$$\mathbf{1}\left(n\right):=1(n\ge 1).$$

Then

$$\mathbf{L}=\mathbf{1}*{\Lambda }^{\mathrm{ex}}.$$

Equivalently, for every $n\ge 1$,

$$\left[n\right]\mathbf{L}=\left[n\right](\mathbf{1}*{\Lambda }^{\mathrm{ex}}).$$

Proof. 

For any $n\ge 1$, Lemma 3.2 gives

$$\left[n\right](\mathbf{1}*{\Lambda }^{\mathrm{ex}})=\sum _{d\mid n}\left[d\right]\mathbf{1}[n/d]{\Lambda }^{\mathrm{ex}}=\sum _{d\mid n}{\Lambda }^{\mathrm{ex}}(n/d).$$

Replacing $n/d$ by the divisor variable $e\mid n$, we get

$$\left[n\right](\mathbf{1}*{\Lambda }^{\mathrm{ex}})=\sum _{e\mid n}{\Lambda }^{\mathrm{ex}}\left(e\right).$$

By Theorem 3.6,

$$\sum _{e\mid n}{\Lambda }^{\mathrm{ex}}\left(e\right)=logn=\mathbf{L}\left(n\right)=\left[n\right]\mathbf{L}.$$

Therefore

$$\left[n\right](\mathbf{1}*{\Lambda }^{\mathrm{ex}})=\left[n\right]\mathbf{L}(n\ge 1).$$

Since the coefficients agree for all n, it follows that, as arithmetic functions,

$$\mathbf{L}=\mathbf{1}*{\Lambda }^{\mathrm{ex}}.$$

□

3.3. Completed Augmentation Ideal and Dirichlet-Logarithmic Linearization

We now pass from the augmentation ideal to its coefficientwise completed setting. The point remains purely formal, and no analytic layer is introduced. We extend ${log}_{*}$ and ${exp}_{*}$ from §Section 3.1 to the completed coefficientwise topology, and use them to linearize the prime-local convolution factors. This converts the multiplicative overlap arising from the factorization of composite numbers into a linear sum of prime-power expansions.

Definition 3.9 

(Completed augmentation ideal and coefficientwise topology). For $n,m\ge 1$, define the Dirichlet basis atom by

$$e_n\left(m\right):={\delta }_{n,m}.$$

Thus $e_1={\delta }_1$, and any arithmetic function can be written formally as

$$a=\sum _{n\ge 1}\left[n\right]a{e}_n.$$

Define the coefficientwise formal completion by

$$\widehat{\mathfrak{D}}:=\prod _{n\ge 1}\mathbb{C}{e}_n,\widehat{\mathfrak{m}}:=\prod _{n\ge 2}\mathbb{C}{e}_n.$$

Equivalently,

$$\widehat{\mathfrak{D}}=\left\{\sum _{n\ge 1}{a}_n{e}_n:a_n\in \mathbb{C}\right\},\widehat{\mathfrak{m}}=\left\{\sum _{n\ge 2}{a}_n{e}_n:a_n\in \mathbb{C}\right\}.$$

Viewed as coefficient sets, $\widehat{\mathfrak{D}}$ and $\mathfrak{D}$ contain the same data. The hat notation indicates that, from this point onward, these are interpreted as formal infinite sums equipped with the coefficientwise product topology. Namely, a sequence $a^{\left(j\right)}$ converges to a if

$$\left[n\right]a^{\left(j\right)}⟶\left[n\right]a\mathrm{for}\mathrm{every}\mathrm{fixed}n\ge 1.$$

Equivalently, the coordinate maps

$$\left[n\right]:\widehat{\mathfrak{D}}\to \mathbb{C}$$

are continuous, and they define the product topology.

The completed unit neighborhood is

$${\delta }_1+\widehat{\mathfrak{m}}=\left\{a\in \widehat{\mathfrak{D}}:\left[1\right]a=1\right\}.$$

Dirichlet convolution on $\mathfrak{D}$ extends coefficientwise to $\widehat{\mathfrak{D}}$. Namely, for $a,b\in \widehat{\mathfrak{D}}$, define

$$\left[n\right](a*b):=\sum _{d\mid n}\left[d\right]a[n/d]b(n\ge 1).$$

This is well-defined because the divisor set of n is finite.

Definition 3.10 

(Completed exponential map). Define

$$E:={exp}_{*}:\widehat{\mathfrak{m}}⟶{\delta }_1+\widehat{\mathfrak{m}}$$

by the same coefficientwise formula as in Proposition 3.3:

$$E\left(b\right)={exp}_{*}\left(b\right):=\sum _{k\ge 0}{\displaystyle \frac{1}{k!}}{b}^{*k},b\in \widehat{\mathfrak{m}},$$

where

$$b^{*0}:={\delta }_1.$$

Thus E is not a new exponential symbol, but the same formal exponential map ${exp}_{*}$ interpreted in the completed coefficientwise setting.

Theorem 3.11 

(Invertibility of the completed exponential map). The map

$$E={exp}_{*}:\widehat{\mathfrak{m}}\stackrel{\sim }{\to }{\delta }_1+\widehat{\mathfrak{m}}$$

is bijective. Its inverse is the coefficientwise extension of the formal logarithm from §3.1; that is,

$$E^{-1}={log}_{*},{log}_{*}:{\delta }_1+\widehat{\mathfrak{m}}\to \widehat{\mathfrak{m}},$$

where

$${log}_{*}({\delta }_1+a):=\sum _{k\ge 1}{\displaystyle \frac{{(-1)}^{k+1}}{k}}{a}^{*k},a\in \widehat{\mathfrak{m}}.$$

Equivalently,

$${exp}_{*}\left({log}_{*}\left(u\right)\right)=u(u\in {\delta }_1+\widehat{\mathfrak{m}}),$$

and

$${log}_{*}\left({exp}_{*}\left(b\right)\right)=b(b\in \widehat{\mathfrak{m}})$$

hold.

Proof. 

First, we verify that the displayed series are coefficientwise well-defined on the completed space.

Take $b\in \widehat{\mathfrak{m}}$, and fix $n\ge 1$. By Lemma 3.2,

$$\left[n\right]\left(b^{*k}\right)=\sum _{\begin{array}{c}{d}_1\dots d_k=n\\ d_i\ge 1\end{array}}b\left(d_1\right)\dots b\left(d_k\right).$$

Since $b\in \widehat{\mathfrak{m}}$, we have $b\left(1\right)=0$. Therefore, if a nonzero term appears, it must satisfy

$$d_i\ge 2(1\le i\le k).$$

Thus an ordered factorization contributing to $\left[n\right]\left(b^{*k}\right)$ can exist only when

$$k\le \Omega \left(n\right),$$

where $\Omega \left(n\right)$ is the total number of prime factors of n, counted with multiplicity. Hence

$$\left[n\right]\left(b^{*k}\right)=0(k>\Omega \left(n\right)).$$

Therefore, for the fixed coefficient n,

$$\left[n\right]{exp}_{*}\left(b\right)={\displaystyle \sum _{k=0}^{\Omega \left(n\right)}}{\displaystyle \frac{1}{k!}}\left[n\right]\left(b^{*k}\right)$$

is a finite sum. Thus ${exp}_{*}\left(b\right)$ is coefficientwise well-defined.

The same argument applies to ${log}_{*}$. If $a\in \widehat{\mathfrak{m}}$, then

$$\left[n\right]\left(a^{*k}\right)=0(k>\Omega \left(n\right)),$$

so

$$\left[n\right]{log}_{*}({\delta }_1+a)={\displaystyle \sum _{k=1}^{\Omega \left(n\right)}}{\displaystyle \frac{{(-1)}^{k+1}}{k}}\left[n\right]\left(a^{*k}\right)$$

is also a finite sum. Therefore ${log}_{*}({\delta }_1+a)$ is coefficientwise well-defined.

Next, we prove the inverse-map identities coefficientwise. Fix $n\ge 1$, and consider the finite divisor algebra

$${\mathfrak{D}}_{\mid n}:=\{c:Div\left(n\right)\to \mathbb{C}\}$$

with the restricted Dirichlet convolution

$$\left(c{★}_{n}d\right)\left(m\right):=\sum _{r\mid m}c\left(r\right)d(m/r)(m\mid n).$$

Furthermore, set

$${\mathfrak{m}}_{\mid n}:=\{c\in {\mathfrak{D}}_{\mid n}:c\left(1\right)=0\}.$$

As in the proof of Proposition 3.3, the ideal ${\mathfrak{m}}_{\mid n}$ is nilpotent. Indeed, if $c\in {\mathfrak{m}}_{\mid n}$, then

$$c^{{★}_{n}k}=0(k>\Omega \left(n\right)).$$

Therefore, in the finite-dimensional commutative algebra ${\mathfrak{D}}_{\mid n}$, the formal series ${exp}_{*}$ and ${log}_{*}$ truncate to genuine finite polynomials, and the usual identities

$${exp}_{*}\left({log}_{*}(1+x)\right)=1+x,{log}_{*}\left({exp}_{*}\left(x\right)\right)=x(x\in {\mathfrak{m}}_{\mid n})$$

hold exactly.

Now let $u\in {\delta }_1+\widehat{\mathfrak{m}}$ and $b\in \widehat{\mathfrak{m}}$. Restrict them to $Div\left(n\right)$. Since the coefficient $\left[n\right](·)$ depends only on values on the divisors of n, the finite-dimensional identities above imply

$$\left[n\right]{exp}_{*}\left({log}_{*}\left(u\right)\right)=\left[n\right]u,$$

and

$$\left[n\right]{log}_{*}\left({exp}_{*}\left(b\right)\right)=\left[n\right]b.$$

Since $n\ge 1$ was arbitrary, the coefficients agree for all n. Therefore

$${exp}_{*}\left({log}_{*}\left(u\right)\right)=u(u\in {\delta }_1+\widehat{\mathfrak{m}}),$$

and

$${log}_{*}\left({exp}_{*}\left(b\right)\right)=b(b\in \widehat{\mathfrak{m}}).$$

Hence ${exp}_{*}$ is bijective, and its inverse is precisely ${log}_{*}$. □

Definition 3.12 

(Dirichlet-logarithmic linearization operator). Define the Dirichlet-logarithmic linearization operator by

$$C_{\mathrm{comp}}:=E^{-1}={log}_{*}:{\delta }_1+\widehat{\mathfrak{m}}⟶\widehat{\mathfrak{m}}.$$

Its role is to remove the multiplicities of composite numbers built into multiplicative Dirichlet convolution through the logarithmic linearization of prime-local convolution factors.

Theorem 3.13 

(Coefficientwise Euler-factor decomposition and log-linearization). Let

$$\mathbf{1}\in \widehat{\mathfrak{D}},\mathbf{1}\left(n\right):=1(n\ge 1),$$

and for each prime p, define the local geometric factor

$$G_p:=\sum _{k\ge 0}{e}_{p^k}\in {\delta }_1+\widehat{\mathfrak{m}}.$$

Then the following hold.

1.

The coefficientwise Euler-factor decomposition

$$\mathbf{1}=\underset{p}{*}{G}_p$$

holds. Here the infinite convolution product is interpreted coefficientwise. Namely, for fixed n, the coefficient $\left[n\right]\left({*}_p{G}_p\right)$ is the stable value of the finite product over any prime set containing all prime factors of n.

2.

$$C_{\mathrm{comp}}\left(\mathbf{1}\right)={log}_{*}\left(\mathbf{1}\right)=\sum _p{log}_{*}\left(G_p\right)=\sum _p\sum _{k\ge 1}{\displaystyle \frac{1}{k}}{e}_{p^k},$$

again with all equalities understood coefficientwise.

Proof. 

We divide the proof into three steps.

Step 1: Coefficientwise Euler-factor decomposition. For a finite set S of primes, define

$$G_S:=\underset{p\in S}{*}{G}_p.$$

Fix $n\ge 1$, and write its prime factorization as

$$n={\displaystyle \prod _{i=1}^r}{p}_i^{{\nu }_i}.$$

Here $p_1,\dots ,p_r$ are distinct primes and ${\nu }_i\ge 1$. The claim is

$$\left[n\right]G_S=\left\{\begin{array}{cc}1,\hfill & \{p_1,\dots ,p_r\}\subset S,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(3.3.1)

Indeed, each factor $G_p$ contributes only powers of p:

$$G_p=\sum _{k\ge 0}{e}_{p^k}.$$

Therefore, a nonzero contribution to the coefficient of n in the finite convolution product $G_S$ occurs only when, for each prime $p_i$ dividing n, one selects exactly one atom $e_{p_i^{{\nu }_i}}$ from the factor $G_{p_i}$, and for every other prime $q\in S\setminus \{p_1,\dots ,p_r\}$, one selects $e_1$. By uniqueness of prime factorization, if all prime factors of n belong to S, there is exactly one such choice, and otherwise there is none. This proves (3.3.1).

Now set

$$S_n:=\{p:p\mathrm{prime}\mathrm{and}p\mid n\}.$$

If $S\supset S_n$, then (3.3.1) gives

$$\left[n\right]G_S=1=\left[n\right]\mathbf{1}.$$

Therefore the coefficient $\left[n\right]G_S$ stabilizes to 1 once S contains all prime factors of n. This is precisely the coefficientwise meaning of

$$\mathbf{1}=\underset{p}{*}{G}_p.$$

Step 2: Local log-linearization of each Euler factor. Fix a prime p. Since

$$e_p^{*m}=e_{p^m}(m\ge 1),$$

we have

$$({\delta }_1-e_p)*G_p=\left({\delta }_1-e_p\right)*\sum _{k\ge 0}{e}_{p^k}=\sum _{k\ge 0}{e}_{p^k}-\sum _{k\ge 0}{e}_p*e_{p^k}.$$

But

$$e_p*e_{p^k}=e_{p^{k+1}},$$

so the right-hand side telescopes as desired:

$$\sum _{k\ge 0}{e}_{p^k}-\sum _{k\ge 0}{e}_{p^{k+1}}=e_1={\delta }_1.$$

Thus

$$G_p={({\delta }_1-e_p)}^{-*}.$$

Next we compute ${log}_{*}\left(G_p\right)$. Since $G_p*({\delta }_1-e_p)={\delta }_1$, the finite-dimensional divisor-algebra argument used in the proof of Theorem 3.11 gives

$${log}_{*}\left(G_p\right)+{log}_{*}({\delta }_1-e_p)={log}_{*}\left({\delta }_1\right)=0.$$

Hence

$${log}_{*}\left(G_p\right)=-{log}_{*}({\delta }_1-e_p).$$

Using the defining series for ${log}_{*}$, we obtain

$${log}_{*}({\delta }_1-e_p)=\sum _{m\ge 1}{\displaystyle \frac{{(-1)}^{m+1}}{m}}{(-e_p)}^{*m}=-\sum _{m\ge 1}{\displaystyle \frac{1}{m}}{e}_p^{*m},$$

and therefore

$${log}_{*}\left(G_p\right)=\sum _{m\ge 1}{\displaystyle \frac{1}{m}}{e}_p^{*m}=\sum _{m\ge 1}{\displaystyle \frac{1}{m}}{e}_{p^m}.$$

(3.3.2)

Step 3: Global log-linearization of $\mathbf{1}$. Fix $n\ge 1$, and let $S_n$ be the finite set of all prime factors of n. By Step 1,

$$\left[n\right]\mathbf{1}=\left[n\right]\left(\underset{p\in S_n}{*}{G}_p\right).$$

Restrict everything to the finite divisor algebra ${\mathfrak{D}}_{\mid n}$. In this finite-dimensional commutative algebra, logarithm and exponential are genuine finite polynomials on the nilpotent augmentation ideal. Therefore the usual commutative identity

$${log}_{*}\left(\underset{p\in S_n}{*}{G}_p\right)=\sum _{p\in S_n}{log}_{*}\left(G_p\right)$$

holds. Taking the coefficient n, we get

$$\left[n\right]{log}_{*}\left(\mathbf{1}\right)=\left[n\right]\sum _{p\in S_n}{log}_{*}\left(G_p\right).$$

If $p\nmid n$, then each term of ${log}_{*}\left(G_p\right)$ is supported only on powers of p, so

$$\left[n\right]{log}_{*}\left(G_p\right)=0.$$

Hence the sum can be extended to all primes:

$$\left[n\right]{log}_{*}\left(\mathbf{1}\right)=\left[n\right]\sum _p{log}_{*}\left(G_p\right).$$

Using (3.3.2), this becomes

$$\left[n\right]{log}_{*}\left(\mathbf{1}\right)=\left[n\right]\sum _p\sum _{k\ge 1}{\displaystyle \frac{1}{k}}{e}_{p^k}.$$

Since this holds for every $n\ge 1$, the coefficientwise identity

$$C_{\mathrm{comp}}\left(\mathbf{1}\right)={log}_{*}\left(\mathbf{1}\right)=\sum _p{log}_{*}\left(G_p\right)=\sum _p\sum _{k\ge 1}{\displaystyle \frac{1}{k}}{e}_{p^k}$$

follows. This completes the proof. □

Thus ${log}_{*}$ has already removed the multiplicative overlap carried by the Euler-type convolution product. After linearization, only prime-power atoms remain. The remaining task is purely arithmetic and coefficientwise: to act on this linearized output by the arithmetic derivation and recover the logarithm function $\mathbf{L}$.

3.4. Arithmetic Derivation and Formal Coefficient Extraction

Here we give logarithmic weights to the prime-power atoms generated by $C_{\mathrm{comp}}$. Since the log-linearization of Theorem 3.13 has already separated the prime-power support, the arithmetic derivation recovers the von Mangoldt function without leaving the purely formal arithmetic layer.

Definition 3.14 

(Arithmetic derivation). Define the arithmetic derivation

$${\partial }_{log}:\widehat{\mathfrak{D}}⟶\widehat{\mathfrak{D}}$$

by

$$\left({\partial }_{log}a\right)\left(n\right):=(logn)a\left(n\right)(a\in \widehat{\mathfrak{D}},n\ge 1).$$

Equivalently,

$${\partial }_{log}\left(\sum _{n\ge 1}{a}_n{e}_n\right)=\sum _{n\ge 1}(logn)a_n{e}_n.$$

Lemma 3.15 

(Derivation law for Dirichlet convolution). For all $a,b\in \widehat{\mathfrak{D}}$,

$${\partial }_{log}(a*b)=\left({\partial }_{log}a\right)*b+a*\left({\partial }_{log}b\right)$$

holds.

Proof. 

Fix $n\ge 1$. By the definitions of ${\partial }_{log}$ and Dirichlet convolution,

$$\left[n\right]{\partial }_{log}(a*b)=(logn)\left[n\right](a*b)=(logn)\sum _{d\mid n}a\left(d\right)b(n/d).$$

Using

$$logn=logd+log(n/d),$$

we obtain

$$\left[n\right]{\partial }_{log}(a*b)=\sum _{d\mid n}(logd)a\left(d\right)b(n/d)+\sum _{d\mid n}a\left(d\right)(log(n/d))b(n/d).$$

The first sum is exactly

$$\left[n\right]\left(\left({\partial }_{log}a\right)*b\right),$$

and the second sum is exactly

$$\left[n\right]\left(a*\left({\partial }_{log}b\right)\right).$$

Therefore

$$\left[n\right]{\partial }_{log}(a*b)=\left[n\right]\left(\left({\partial }_{log}a\right)*b+a*\left({\partial }_{log}b\right)\right)(n\ge 1).$$

Since the coefficients agree for all n, as arithmetic functions we have

$${\partial }_{log}(a*b)=\left({\partial }_{log}a\right)*b+a*\left({\partial }_{log}b\right).$$

□

Theorem 3.16 

(Formal coefficient-extraction theorem).

$${\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)={\Lambda }^{\mathrm{ex}}.$$

Equivalently, for every $n\ge 1$,

$$\left[n\right]\left({\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)={\Lambda }^{\mathrm{ex}}\left(n\right).$$

Proof. 

By Theorem 3.13,

$$C_{\mathrm{comp}}\left(\mathbf{1}\right)=\sum _p\sum _{k\ge 1}{\displaystyle \frac{1}{k}}{e}_{p^k}.$$

Applying ${\partial }_{log}$ coefficientwise gives

$${\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)=\sum _p\sum _{k\ge 1}{\displaystyle \frac{log\left(p^k\right)}{k}}{e}_{p^k}.$$

Fix $n\ge 1$, and compute the coefficient $\left[n\right]$ in three cases.

If $n=1$, then 1 is not of the form $p^k$ with $k\ge 1$, so

$$\left[n\right]\left({\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)=0.$$

By Definition 3.5,

$${\Lambda }^{\mathrm{ex}}\left(1\right)=0.$$

If $n=p^k$ is a prime power with $k\ge 1$, then the only contributing term is the one corresponding to the pair $(p,k)$. Therefore

$$\left[n\right]\left({\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)={\displaystyle \frac{log\left(p^k\right)}{k}}={\displaystyle \frac{klogp}{k}}=logp.$$

Again by Definition 3.5,

$${\Lambda }^{\mathrm{ex}}\left(p^k\right)=logp.$$

If n is not a prime power, then no basis atom $e_{p^k}$ appearing in the above sum is supported at n. Therefore

$$\left[n\right]\left({\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)=0.$$

By Definition 3.5,

$${\Lambda }^{\mathrm{ex}}\left(n\right)=0.$$

Thus, in all cases,

$$\left[n\right]\left({\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)={\Lambda }^{\mathrm{ex}}\left(n\right).$$

Since this holds for every $n\ge 1$, we obtain

$${\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)={\Lambda }^{\mathrm{ex}}.$$

□

Corollary 3.17 

(Prime-power output under Dirichlet-logarithmic linearization).

$$supp\left({\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)=\left\{\mathrm{prime}\mathrm{powers}\right\}.$$

Equivalently, every integer that is not a prime power carries zero output:

$$\left[n\right]\left({\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)=0\mathrm{whenever}n\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{prime}\mathrm{power}.$$

Accordingly, the Dirichlet-logarithmic linearization means that the general composite-number overlap appearing in the multiplicative convolution product is converted into a prime-power-supported coefficient sum, and the surviving output is supported only on prime powers.

Proof. 

By Theorem 3.16,

$${\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)={\Lambda }^{\mathrm{ex}}.$$

Therefore

$$supp\left({\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)=supp\left({\Lambda }^{\mathrm{ex}}\right).$$

By Proposition 3.7,

$$supp\left({\Lambda }^{\mathrm{ex}}\right)=\left\{\mathrm{prime}\mathrm{powers}\right\}.$$

Hence

$$supp\left({\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)=\left\{\mathrm{prime}\mathrm{powers}\right\}.$$

The coefficientwise restatement follows immediately. □

Remark 3.18 

(End of the purely formal extraction layer). The purely formal extraction layer ends here. At this point, we have not yet introduced ${\Pi }_n$, ${\Pi }_{\mathrm{arith}}$, any diagonal arithmetic trace formula, any analytic resolvent construction, or any comparison interface. The objects obtained from §3.3–§3.4 are exactly

$${\Lambda }^{\mathrm{ex}},C_{\mathrm{comp}},{\partial }_{log}$$

and nothing else. These are precisely the only arithmetic data needed for the next construction step.

3.5. Coefficient Hilbert Space and Weighted Diagonal Arithmetic Trace

The purely arithmetic extraction layer of §3.1–§3.4 is now complete. Accordingly, we now pass to the Hilbertized coefficient layer. Using the Dirichlet basis atoms $e_n$ as an orthonormal basis, we define the coefficient Hilbert space, its rank-one coordinate projection family ${\left\{{\Pi }_n\right\}}_{n\ge 1}$, and the weighted diagonal arithmetic trace operator ${\Pi }_{\mathrm{arith}}\left(\varphi \right)$. No analytic Hilbert space, resolvent construction, or comparison interface enters here.

Definition 3.19 

(Coefficient Hilbert space and coordinate projections). Using the Dirichlet basis atoms $e_n$ from Definition 3.9, define the coefficient Hilbert space by

$${\mathcal{H}}_{\mathrm{arith}}:=\left\{u=\sum _{n\ge 1}{u}_n{e}_n:\sum _{n\ge 1}{|u_n|}^2<\infty \right\}.$$

Its inner product is given by

$${\langle u,v\rangle }_{\mathrm{arith}}:=\sum _{n\ge 1}{u}_n\overline{v_n}.$$

For

$$u=\sum _{n\ge 1}{u}_n{e}_n\in {\mathcal{H}}_{\mathrm{arith}},$$

write the n-th coefficient with respect to the basis ${\left\{e_n\right\}}_{n\ge 1}$ as

$$\left[n\right]u:=u_n.$$

For each $n\ge 1$, define the coordinate projection

$${\Pi }_n:{\mathcal{H}}_{\mathrm{arith}}\to {\mathcal{H}}_{\mathrm{arith}}$$

by

$${\Pi }_{n}u:=\left[n\right]u{e}_n(u\in {\mathcal{H}}_{\mathrm{arith}}).$$

Thus ${\Pi }_n$ is the rank-one orthogonal projection onto the n-th coefficient line $\mathbb{C}{e}_n$ inside the coefficient Hilbert space.

Lemma 3.20 

(Coordinate-projection laws on the coefficient Hilbert space). The family ${\left\{{\Pi }_n\right\}}_{n\ge 1}$ satisfies

$${\Pi }_n^2={\Pi }_n,{\Pi }_n^{*}={\Pi }_n(n\ge 1),$$

and

$${\Pi }_m{\Pi }_n=0(m\ne n).$$

Furthermore, for every $u\in {\mathcal{H}}_{\mathrm{arith}}$,

$$u=\sum _{n\ge 1}{\Pi }_{n}u$$

converges in ${\mathcal{H}}_{\mathrm{arith}}$. Equivalently,

$$\sum _{n\ge 1}{\Pi }_n=I$$

holds on ${\mathcal{H}}_{\mathrm{arith}}$ in the strong operator topology.

Proof. 

By the definition of ${\mathcal{H}}_{\mathrm{arith}}$, ${\left\{e_n\right\}}_{n\ge 1}$ is the standard orthonormal basis of the ${\ell }^2$-type coefficient space. In particular,

$${\langle e_m,e_n\rangle }_{\mathrm{arith}}={\delta }_{m,n}.$$

Let

$$u=\sum _{k\ge 1}{u}_k{e}_k\in {\mathcal{H}}_{\mathrm{arith}}.$$

Then

$${\Pi }_{n}u=u_n{e}_n.$$

Applying ${\Pi }_n$ once more gives

$${\Pi }_n^{2}u={\Pi }_n\left(u_n{e}_n\right)=\left[n\right]\left(u_n{e}_n\right)e_n=u_n{e}_n={\Pi }_{n}u.$$

Since this holds for all u,

$${\Pi }_n^2={\Pi }_n.$$

Next let

$$u=\sum _{k\ge 1}{u}_k{e}_k,v=\sum _{k\ge 1}{v}_k{e}_k$$

be elements of ${\mathcal{H}}_{\mathrm{arith}}$. Then

$${\langle {\Pi }_{n}u,v\rangle }_{\mathrm{arith}}={\langle u_n{e}_n,v\rangle }_{\mathrm{arith}}=u_n\overline{v_n}.$$

On the other hand,

$${\langle u,{\Pi }_{n}v\rangle }_{\mathrm{arith}}={\langle u,v_n{e}_n\rangle }_{\mathrm{arith}}=u_n\overline{v_n}.$$

Therefore

$${\langle {\Pi }_{n}u,v\rangle }_{\mathrm{arith}}={\langle u,{\Pi }_{n}v\rangle }_{\mathrm{arith}}(u,v\in {\mathcal{H}}_{\mathrm{arith}}),$$

and hence

$${\Pi }_n^{*}={\Pi }_n.$$

Now let $m\ne n$. For any $u\in {\mathcal{H}}_{\mathrm{arith}}$,

$${\Pi }_m{\Pi }_{n}u={\Pi }_m\left(u_n{e}_n\right)=\left[n\right]u\left[m\right]e_n{e}_m=u_n{\delta }_{m,n}{e}_m=0.$$

Therefore

$${\Pi }_m{\Pi }_n=0(m\ne n).$$

It remains to prove the strong decomposition of the identity. For $N\ge 1$, define the partial-sum operator

$$S_N:={\displaystyle \sum _{n=1}^N}{\Pi }_n.$$

Then, for

$$u=\sum _{n\ge 1}{u}_n{e}_n\in {\mathcal{H}}_{\mathrm{arith}},$$

we have

$$S_{N}u={\displaystyle \sum _{n=1}^N}{u}_n{e}_n.$$

Therefore

$$u-S_{N}u=\sum _{n>N}{u}_n{e}_n,$$

and since ${\left\{e_n\right\}}_{n\ge 1}$ is orthonormal,

$$\parallel u-S_N{u\parallel }_{\mathrm{arith}}^2=\sum _{n>N}{|u_n|}^2.$$

Because $u\in {\mathcal{H}}_{\mathrm{arith}}$, the series

$$\sum _{n\ge 1}{|u_n|}^2$$

converges, and hence its tail converges to 0. Thus

$$\parallel u-S_N{u\parallel }_{\mathrm{arith}}⟶0(N\to \infty ).$$

Accordingly,

$$u=\sum _{n\ge 1}{\Pi }_{n}u$$

holds in ${\mathcal{H}}_{\mathrm{arith}}$, equivalently

$$S_N\to I$$

strongly. This proves that

$$\sum _{n\ge 1}{\Pi }_n=I$$

holds in the strong operator topology. □

Remark 3.21 

(Coefficient Hilbert space versus analytic Hilbert space). The family ${\left\{{\Pi }_n\right\}}_{n\ge 1}$ is defined, on its natural coefficient Hilbert space ${\mathcal{H}}_{\mathrm{arith}}$. It is a coordinate projection family on the Dirichlet-basis Hilbert space, not a projection family on the analytic Hilbert space of Section 2.

Definition 3.22 

(Weighted diagonal arithmetic trace operator). Let

$$\varphi :\mathbb{N}\to \mathbb{C}$$

be a finitely supported weight. Define the weighted diagonal arithmetic trace operator by

$${\Pi }_{\mathrm{arith}}\left(\varphi \right):=\sum _{n\ge 1}\varphi \left(n\right){\Pi }_n.$$

This is a finite-rank diagonal operator on ${\mathcal{H}}_{\mathrm{arith}}$. Equivalently, for

$$u=\sum _{n\ge 1}{u}_n{e}_n\in {\mathcal{H}}_{\mathrm{arith}},$$

we have

$${\Pi }_{\mathrm{arith}}\left(\varphi \right)u=\sum _{n\ge 1}\varphi \left(n\right)u_n{e}_n.$$

In the theorem below, this definition is extended from finitely supported weights to all $\varphi \in {\ell }^1\left(\mathbb{N}\right)$ by trace-norm completion.

Theorem 3.23 

(Weighted diagonal arithmetic trace formula). The following hold.

(1)Let $\varphi \in {\ell }^1\left(\mathbb{N}\right)$. For $N\ge 1$, define the finite-rank partial sum

$$T_N\left(\varphi \right):=\sum _{n\le N}\varphi \left(n\right){\Pi }_n.$$

Then ${\left\{T_N\left(\varphi \right)\right\}}_{N\ge 1}$ is Cauchy with respect to the trace norm ${\parallel ·\parallel }_{{\mathfrak{S}}_1}$, and hence converges to a trace-class operator on ${\mathcal{H}}_{\mathrm{arith}}$. Write its limit as

$${\Pi }_{\mathrm{arith}}\left(\varphi \right):=\underset{N\to \infty }{lim}{T}_N\left(\varphi \right).$$

Furthermore,

$$Tr{\Pi }_{\mathrm{arith}}\left(\varphi \right)=\sum _{n\ge 1}\varphi \left(n\right)$$

holds.

(2)Let $\phi :\mathbb{N}\to \mathbb{C}$ be finitely supported, and understand pointwise products coefficientwise:

$$\left(\phi {\Lambda }^{\mathrm{ex}}\right)\left(n\right):=\phi \left(n\right){\Lambda }^{\mathrm{ex}}\left(n\right),\left(\phi {\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)\left(n\right):=\phi \left(n\right)\left({\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)\left(n\right).$$

Then

$$Tr{\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)=\sum _{n\ge 1}\phi \left(n\right){\Lambda }^{\mathrm{ex}}\left(n\right)=\sum _{p^k}\phi \left(p^k\right)logp.$$

Equivalently, using Theorem 3.16,

$$Tr{\Pi }_{\mathrm{arith}}\left(\phi {\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)=\sum _{p^k}\phi \left(p^k\right)logp.$$

Proof. 

We prove the two parts in order.

Step 1: Finite-rank partial sums and their traces. Fix $N\ge 1$. Since $T_N\left(\varphi \right)$ is a finite sum of rank-one operators, it has finite rank. On the basis ${\left\{e_n\right\}}_{n\ge 1}$,

$$T_N\left(\varphi \right)e_k=\sum _{n\le N}\varphi \left(n\right){\Pi }_n{e}_k=\sum _{n\le N}\varphi \left(n\right){\delta }_{n,k}{e}_n=\left\{\begin{array}{cc}\varphi \left(k\right)e_k,\hfill & k\le N,\hfill \\ 0,\hfill & k>N.\hfill \end{array}\right.$$

Thus $T_N\left(\varphi \right)$ is diagonal with respect to the arithmetic basis, and its diagonal entries are $\varphi \left(1\right),\dots ,\varphi \left(N\right),0,0,\dots$. Hence

$$Tr{T}_N\left(\varphi \right)=\sum _{k\ge 1}{\langle T_N\left(\varphi \right)e_k,e_k\rangle }_{\mathrm{arith}}=\sum _{k\le N}\varphi \left(k\right).$$

Step 2: Trace-norm Cauchy property for $\varphi \in {\ell }^1$. Let $M>N$. Then

$$T_M\left(\varphi \right)-T_N\left(\varphi \right)=\sum _{N<n\le M}\varphi \left(n\right){\Pi }_n.$$

On basis vectors,

$$\left(T_M\left(\varphi \right)-T_N\left(\varphi \right)\right)e_k=\left\{\begin{array}{cc}\varphi \left(k\right)e_k,\hfill & N<k\le M,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Therefore $T_M\left(\varphi \right)-T_N\left(\varphi \right)$ is a finite-rank diagonal operator, and its singular values are exactly

$$\{|\varphi \left(n\right)|:N<n\le M\}.$$

Equivalently,

$$\left|T_M\left(\varphi \right)-T_N\left(\varphi \right)\right|e_k=\left\{\begin{array}{cc}|\varphi \left(k\right)|e_k,\hfill & N<k\le M,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Thus

$${∥T_M\left(\varphi \right)-T_N\left(\varphi \right)∥}_{{\mathfrak{S}}_1}=Tr\left|T_M\left(\varphi \right)-T_N\left(\varphi \right)\right|=\sum _{N<n\le M}|\varphi \left(n\right)|.$$

(3.5.2)

Since $\varphi \in {\ell }^1\left(\mathbb{N}\right)$, the right-hand side tends to 0 as $M,N\to \infty$. Hence ${\left\{T_N\left(\varphi \right)\right\}}_{N\ge 1}$ is Cauchy in ${\mathfrak{S}}_1\left({\mathcal{H}}_{\mathrm{arith}}\right)$. The trace-class space is complete, so there exists a unique trace-class limit, which we write as

$${\Pi }_{\mathrm{arith}}\left(\varphi \right):=\underset{N\to \infty }{lim}{T}_N\left(\varphi \right).$$

Furthermore, the trace is continuous with respect to the trace norm, and therefore

$$Tr{\Pi }_{\mathrm{arith}}\left(\varphi \right)=\underset{N\to \infty }{lim}Tr{T}_N\left(\varphi \right).$$

Using (3.5.1), we obtain

$$Tr{\Pi }_{\mathrm{arith}}\left(\varphi \right)=\underset{N\to \infty }{lim}\sum _{n\le N}\varphi \left(n\right)=\sum _{n\ge 1}\varphi \left(n\right),$$

because the series is absolutely convergent. This proves (1).

Step 3: Specialization to the prime-power weighted trace. Now let $\phi$ be finitely supported. Then the pointwise product $\phi {\Lambda }^{\mathrm{ex}}$ is also finitely supported, and hence belongs to ${\ell }^1\left(\mathbb{N}\right)$. Applying (1) to

$$\varphi =\phi {\Lambda }^{\mathrm{ex}},$$

we get

$$Tr{\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)=\sum _{n\ge 1}\phi \left(n\right){\Lambda }^{\mathrm{ex}}\left(n\right).$$

(3.5.3)

By Definition 3.5,

$${\Lambda }^{\mathrm{ex}}\left(n\right)=0$$

when n is not a prime power, and

$${\Lambda }^{\mathrm{ex}}\left(p^k\right)=logp(k\ge 1).$$

Therefore the sum in (3.5.3) reduces to

$$\sum _{n\ge 1}\phi \left(n\right){\Lambda }^{\mathrm{ex}}\left(n\right)=\sum _{p^k}\phi \left(p^k\right)logp.$$

This proves

$$Tr{\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)=\sum _{p^k}\phi \left(p^k\right)logp.$$

Finally, Theorem 3.16 states that

$${\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)={\Lambda }^{\mathrm{ex}}.$$

Multiplying both sides pointwise by $\phi$, we obtain

$$\phi {\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)=\phi {\Lambda }^{\mathrm{ex}}.$$

Therefore, applying ${\Pi }_{\mathrm{arith}}$ and then taking the trace gives

$$Tr{\Pi }_{\mathrm{arith}}\left(\phi {\partial }_{log}{C}_{\mathrm{comp}}\left(\mathbf{1}\right)\right)=Tr{\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)=\sum _{p^k}\phi \left(p^k\right)logp.$$

This proves (2). □

Corollary 3.24 

(Prime-power trace specialization). Let $\phi :\mathbb{N}\to \mathbb{C}$ be finitely supported. Then

$$supp\left(\phi {\Lambda }^{\mathrm{ex}}\right)\subset \left\{\mathrm{prime}\mathrm{powers}\right\}.$$

Therefore

$$Tr{\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)=\sum _{n\in supp\left(\phi {\Lambda }^{\mathrm{ex}}\right)}\phi \left(n\right){\Lambda }^{\mathrm{ex}}\left(n\right)=\sum _{p^k}\phi \left(p^k\right)logp.$$

In particular, the weighted diagonal arithmetic trace here sees only the prime-power support and is not specialized to the prime support of $1_P^{\mathrm{ex}}$.

Proof. 

For any arithmetic functions $a,b$,

$$supp\left(ab\right)\subset supp\left(a\right)\cap supp\left(b\right)$$

holds. Indeed, $\left(ab\right)\left(n\right)=a\left(n\right)b\left(n\right)$ can be nonzero only where both factors are nonzero. Applying this to

$$a=\phi ,b={\Lambda }^{\mathrm{ex}},$$

we obtain

$$supp\left(\phi {\Lambda }^{\mathrm{ex}}\right)\subset supp\left({\Lambda }^{\mathrm{ex}}\right).$$

By Proposition 3.7,

$$supp\left({\Lambda }^{\mathrm{ex}}\right)=\left\{\mathrm{prime}\mathrm{powers}\right\}.$$

Therefore

$$supp\left(\phi {\Lambda }^{\mathrm{ex}}\right)\subset \left\{\mathrm{prime}\mathrm{powers}\right\}.$$

The trace identity follows immediately from Theorem 3.23:

$$Tr{\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)=\sum _{n\ge 1}\phi \left(n\right){\Lambda }^{\mathrm{ex}}\left(n\right)=\sum _{p^k}\phi \left(p^k\right)logp.$$

The final sentence merely states the difference of supports already established in Definition 3.5 and Proposition 3.7. Namely, $1_P^{\mathrm{ex}}$ is supported on primes, whereas ${\Lambda }^{\mathrm{ex}}$ is supported on prime powers. □

3.6. Summary and Transition to Singular Boundary Data

Remark 3.25 

(Summary of the arithmetic section). The coefficient-space arithmetic construction ends here. The arithmetic data constructed in this section are

$${\Lambda }^{\mathrm{ex}},C_{\mathrm{comp}},{\partial }_{log},{\left\{{\Pi }_n\right\}}_{n\ge 1},{\Pi }_{\mathrm{arith}}(·).$$

At this point, the coefficient-extraction layer, coefficient Hilbert space, coordinate projection family, and weighted diagonal arithmetic trace operator have all been fixed on their natural defining space. These arithmetic data are combined with the singular-boundary data only in the orthogonal-decomposition framework of Section 5. Section 4 does not use these arithmetic objects; it independently constructs the singular-boundary input inside the analytic Hilbert-space setup of Section 2.

4. Operator-Theoretic Construction of Singular Boundary Data

4.0. Purpose and Logical Role of This Section

The purpose of this section is to construct, inside the analytic Hilbert-space setup of Section 2, the operator-theoretic boundary data used in the subsequent orthogonal-decomposition framework, rather than to add them as an external assumption. Specifically, in this section we construct the singular boundary data consisting of

$$({\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R),$$

and then successively define the associated trace-vanishing generating subspace, singular-boundary closed-form subspace, singular-boundary Hilbert space, orthogonal projection onto $K_R$, transport generator, and distribution kernel.

This section is not a section that develops a zero-counting theory for closing the Riemann hypothesis. Nor is it a section that proves the orthogonality of the arithmetic construction and the boundary-data construction. Its role is exhausted by preparing, without type conflation and in an analytically closed form, the singular-boundary objects needed when constructing the orthogonal-decomposition framework in the next section.

Definition 4.1 

(Operator-theoretic boundary data constructed in this section). Write the basic data constructed in this section as

$${\mathfrak{R}}_{\mathrm{bd}}:=({\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R).$$

Here each object is constructed with the following meaning.

1.

${\Sigma }_R$ is the boundary parameter space used to parametrize the singular boundary structure.

2.

${\sigma }_R$ is a regular area-type measure on ${\Sigma }_R$.

3.

${\Gamma }_R\subset {\Sigma }_R$ is the ${\sigma }_R$-null singular support, constructed as a closed set satisfying

$${\sigma }_R\left({\Gamma }_R\right)=0.$$

4.

$T_R$ is a linear subspace inside the form domain on which the singular/regular boundary trace is defined.

5.

${supp}_R$ is the map that assigns to an element of $T_R$ the support of its singular boundary trace.

6.

$L_R(f;E)$ is a nonnegative functional measuring the regular boundary trace mass over E, for $f\in T_R$ and a measurable set $E\subset {\Sigma }_R$.

Remark 4.2 

(Separation of ${\sigma }_R$-null property and nontriviality). The condition ${\sigma }_R\left({\Gamma }_R\right)=0$ means that ${\Gamma }_R$ is null with respect to the regular area measure. Therefore, if the boundary trace on ${\Gamma }_R$ were treated merely as an ordinary function in $L^2({\Sigma }_R,{\sigma }_R)$, that component would become trivial. To avoid this conflation, this section distinguishes the regular area measure ${\sigma }_R$ from the singular measure ${\nu }_R$ supported on ${\Gamma }_R$. Namely, the ${\sigma }_R$-null property is described on the ${\sigma }_R$-side, whereas the nontriviality of the boundary trace is retained on the side of the singular trace or distributional boundary distribution introduced later.

Definition 4.3 

(Analytic output of this section). Write the final analytic output of this section as

$$\begin{array}{cc}\hfill {\mathfrak{R}}_{\mathrm{op}}:=(& {\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R,G_R,Q_R^{\mathrm{res}},H_R^{\mathrm{res}},K_R,{\Pi }_R^{+},\hfill \\ & U_R\left(t\right),B_R,{\mathsf{K}}_R^{\mathrm{dist}},L_R^{\mathrm{res}}).\hfill \end{array}$$

Here,

$$G_R$$

is the generating class satisfying the ${\sigma }_R$-null support and regular-trace vanishing conditions,

$$Q_R^{\mathrm{res}}$$

is its form-norm closure,

$$H_R^{\mathrm{res}}$$

is the corresponding singular-boundary Hilbert space,

$$K_R$$

is the closed singular-boundary subspace on the one-sided analytic Hilbert space, and

$${\Pi }_R^{+}$$

is the one-sided orthogonal projection from $H_{\alpha ,+}$ onto that closed subspace. Moreover,

$$U_R\left(t\right)$$

is the strongly continuous transport group on the singular-boundary subspace,

$$B_R$$

is its anti-self-adjoint generator,

$${\mathsf{K}}_R^{\mathrm{dist}}$$

is the associated transport kernel, formulated as a distribution kernel, and

$$L_R^{\mathrm{res}}$$

denotes the positive shifted operator obtained from the closed quadratic form.

Convention 4.4 

(Objects not used in this section). In this section, we do not use the following objects or propositions.

1.

The arithmetic projector family ${\left\{{\Pi }_n\right\}}_{n\ge 1}$.

2.

The weighted diagonal arithmetic trace.

3.

The von Mangoldt function ${\Lambda }^{\mathrm{ex}}$ in the formal Dirichlet algebra.

4.

The zero-counting function.

5.

The finite-window comparison theorem.

6.

The cumulative defect sequence.

7.

The exclusion of minimal-index obstruction.

8.

Quoted finite-height verification for the low-height band.

Accordingly, the construction in this section depends neither on the arithmetic construction of Section 3 nor on the subsequent final deduction chain.

Remark 4.5 

(Logical position of this section). Section 2 constructed the weighted Hilbert space, the closed quadratic form, the self-adjoint realization, compact resolvent, and the Herglotz-type resolvent function. This section does not reconstruct them. The role of this section is to cut out, on the analytic Hilbert-space setup of Section 2, the singular-boundary subspace satisfying the ${\sigma }_R$-null support condition and regular-trace vanishing condition, and to construct the projector and transport structure associated with that subspace.

Definition 4.6 

(Transition principle). The output ${\mathfrak{R}}_{\mathrm{op}}$ of this section is the only singular-boundary input for the orthogonal-decomposition framework in the next section. Namely, in the next section, after fixing

$${\mathfrak{R}}_{\mathrm{op}},$$

we place the analytic singular-boundary subspace and the arithmetic subspace on a common ambient Hilbert space. The arguments from the next section onward refer only to the data constructed internally in this section as the singular-boundary input, and introduce no additional boundary-data assumption outside this section. This transfer rule is called the transition principle in this paper.

Proposition 4.7 

(Transition from this section to the next section). Once each construction in this section is complete, the next section may use the following objects as already constructed:

$${\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R,G_R,Q_R^{\mathrm{res}},H_R^{\mathrm{res}},K_R,{\Pi }_R^{+}.$$

In particular, the one-sided projector ${\Pi }_R^{+}$ used as singular-boundary input in the next section is not arbitrarily assumed; it is the orthogonal projection from $H_{\alpha ,+}$ onto the closed singular-boundary subspace constructed in this section. The ambient projector ${\Pi }_R$ is defined only after this input is embedded into the ambient Hilbert space in Section 5.

Proof. 

In the first half of this section, we construct the boundary parameter space, regular area measure, ${\sigma }_R$-null singular support, singular boundary trace, support map, and regular boundary trace-mass functional. This fixes

$$({\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R).$$

Next, from these data, we define the trace-vanishing generating subspace associated with ${\Gamma }_R$, denoted by $G_R$, and obtain

$$Q_R^{\mathrm{res}}$$

as its closure with respect to the form norm of the closed quadratic form constructed in Section 2. As its $H_{\alpha ,+}$-closure, we construct

$$H_R^{\mathrm{res}},$$

and obtain the corresponding self-adjoint operator by the representation theorem for closed forms.

Furthermore, we construct the strongly continuous transport group on the singular-boundary subspace and its anti-self-adjoint generator, and formulate the associated transport kernel as a distribution kernel. Finally, we obtain the closed one-sided singular-boundary subspace

$$K_R\subset H_{\alpha ,+}.$$

Since the orthogonal projection onto a closed subspace exists uniquely by the projection theorem in Hilbert spaces, the one-sided projector

$${\Pi }_R^{+}:H_{\alpha ,+}\to K_R$$

is fixed. The lift of these objects to the ambient Hilbert space is carried out in Section 5.

Accordingly, all the above objects are constructed inside this section, and the next section can receive ${\mathfrak{R}}_{\mathrm{op}}$ as already constructed data. □

4.1. Basic Spaces, Gelfand Triple, and Type Separation

In this subsection, we fix the object level for constructing ${\sigma }_R$-null singular boundary data. In the subsequent argument, we handle point evaluations, boundary traces, singular measures, distribution kernels, quadratic forms, and operators simultaneously. If all of these are treated as elements of the same Hilbert space, then one obtains the unboundedness of point evaluations, a conflation of distribution kernels and bounded operators, and a conflation of quadratic forms and generators. Accordingly, in this subsection, we place at the center the one-sided analytic Hilbert space of Section 2,

$$H_{\alpha ,+}=L^2((0,\infty ),d{\mu }_{\alpha }),d{\mu }_{\alpha }\left(x\right)={\langle x\rangle }^{2\alpha }dx,\alpha >{\displaystyle \frac{1}{2}},$$

and introduce a Gelfand triple by placing a test space and a distribution space around it. The standard background on rigged Hilbert spaces, nuclear spaces, and the distribution kernel theorem follows [1,2].

Definition 4.8 

(Boundary test space). Let $\langle x\rangle ={(1+x^2)}^{1/2}$. Define the boundary test space ${\mathcal{D}}_R$ by

$${\mathcal{D}}_R:=\left\{\phi \in C^{\infty }\left([0,\infty )\right):p_{m,k}\left(\phi \right)<\infty \mathrm{for}\mathrm{all}m,k\in {\mathbb{N}}_0\right\},$$

where

$$p_{m,k}\left(\phi \right):=\underset{x\ge 0}{sup}{\langle x\rangle }^m\left|{\partial }_x^k\phi \left(x\right)\right|.$$

Equip ${\mathcal{D}}_R$ with the Fréchet topology determined by the seminorm family

$${\left\{p_{m,k}\right\}}_{m,k\in {\mathbb{N}}_0}.$$

Lemma 4.9 

(Continuous dense embedding of the test space into the Hilbert space). The natural inclusion map

$${\iota }_R:{\mathcal{D}}_R\hookrightarrow H_{\alpha ,+}$$

is continuous, and its image is dense in $H_{\alpha ,+}$.

Proof. 

We first prove continuity. For any $\phi \in {\mathcal{D}}_R$, fix one $m>\alpha +{\displaystyle \frac{1}{2}}$. Then

$$|\phi \left(x\right)|\le p_{m,0}\left(\phi \right){\langle x\rangle }^{-m}.$$

Therefore

$$\begin{array}{cc}\hfill {\parallel \phi \parallel }_{H_{\alpha ,+}}^2& ={\int }_0^{\infty }{|\phi \left(x\right)|}^2{\langle x\rangle }^{2\alpha }dx\hfill \\ & \le p_{m,0}{\left(\phi \right)}^2{\int }_0^{\infty }{\langle x\rangle }^{2\alpha -2m}dx.\hfill \end{array}$$

Since $m>\alpha +{\displaystyle \frac{1}{2}}$, the integral on the right-hand side is finite. Thus

$${\parallel \phi \parallel }_{H_{\alpha ,+}}\le C_{\alpha ,m}{p}_{m,0}\left(\phi \right),$$

so the inclusion map is continuous.

Next, we prove density. We have $C_c^{\infty }(0,\infty )\subset {\mathcal{D}}_R$, and $C_c^{\infty }(0,\infty )$ is dense in the weighted space $L^2((0,\infty ),{\langle x\rangle }^{2\alpha }dx)$. Indeed, for any $f\in H_{\alpha ,+}$, first approximate f in the $H_{\alpha ,+}$-norm by

$$f_N:={\mathbf{1}}_{[1/N,N]}f,$$

and then use standard smoothing and cutoff on the finite interval $[1/N,N]$ to approximate $f_N$ to arbitrary precision by elements of $C_c^{\infty }(0,\infty )$. Therefore the image of ${\mathcal{D}}_R$ is dense in $H_{\alpha ,+}$. □

Definition 4.10 

(Anti-dual and Gelfand triple). Let ${\mathcal{D}}_R^{\prime }$ be the space of all continuous antilinear functionals on ${\mathcal{D}}_R$, that is, the anti-dual. Write the duality pairing as

$${\langle F,\phi \rangle }_{{\mathcal{D}}_R^{\prime },{\mathcal{D}}_R},F\in {\mathcal{D}}_R^{\prime },\phi \in {\mathcal{D}}_R.$$

The Hilbert space $H_{\alpha ,+}$ is embedded anti-linearly continuously into ${\mathcal{D}}_R^{\prime }$ by

$$h⟼{\iota }_{H}h,$$

where

$${\langle {\iota }_{H}h,\phi \rangle }_{{\mathcal{D}}_R^{\prime },{\mathcal{D}}_R}:={\langle h,\phi \rangle }_{H_{\alpha ,+}}(\phi \in {\mathcal{D}}_R).$$

Then

$${\mathcal{D}}_R\subset H_{\alpha ,+}\subset {\mathcal{D}}_R^{\prime }$$

is called the Gelfand triple on the singular-boundary side in this section.

Lemma 4.11 

(Continuity of the Gelfand triple). The embeddings

$${\mathcal{D}}_R\subset H_{\alpha ,+}\subset {\mathcal{D}}_R^{\prime }$$

are all continuous, and ${\mathcal{D}}_R$ is dense in $H_{\alpha ,+}$.

Proof. 

The continuity and density of the first embedding ${\mathcal{D}}_R\subset H_{\alpha ,+}$ were proved in Lemma 4.9.

Next we prove the continuity of $H_{\alpha ,+}\subset {\mathcal{D}}_R^{\prime }$. For any $h\in H_{\alpha ,+}$ and $\phi \in {\mathcal{D}}_R$, the Cauchy–Schwarz inequality gives

$$\left|{\langle {\iota }_{H}h,\phi \rangle }_{{\mathcal{D}}_R^{\prime },{\mathcal{D}}_R}\right|{=|\langle h,\phi \rangle }_{H_{\alpha ,+}}{|\le \parallel h\parallel }_{H_{\alpha ,+}}{\parallel \phi \parallel }_{H_{\alpha ,+}}.$$

By Lemma 4.9,

$${\parallel \phi \parallel }_{H_{\alpha ,+}}\le C_{\alpha ,m}{p}_{m,0}\left(\phi \right),$$

and therefore

$$\left|{\langle {\iota }_{H}h,\phi \rangle }_{{\mathcal{D}}_R^{\prime },{\mathcal{D}}_R}\right|\le C_{\alpha ,m}{\parallel h\parallel }_{H_{\alpha ,+}}{p}_{m,0}\left(\phi \right).$$

Thus ${\iota }_{H}h$ is a continuous antilinear functional on ${\mathcal{D}}_R$, and the map $h\mapsto {\iota }_{H}h$ is continuous. □

Proposition 4.12 

(Point evaluation is a distribution and not a bounded functional on the Hilbert space). For any $a\in [0,\infty )$,

$${\delta }_a\left(\phi \right):=\phi \left(a\right),\phi \in {\mathcal{D}}_R,$$

is an element of ${\mathcal{D}}_R^{\prime }$. On the other hand, ${\delta }_a$ is not in general defined as a bounded functional on $H_{\alpha ,+}$. Accordingly, point evaluations and boundary point evaluations are treated not as elements of $H_{\alpha ,+}$, but as elements of ${\mathcal{D}}_R^{\prime }$.

Proof. 

We first prove ${\delta }_a\in {\mathcal{D}}_R^{\prime }$. For any $a\in [0,\infty )$,

$$|{\delta }_a\left(\phi \right)|=|\phi \left(a\right)|\le p_{0,0}\left(\phi \right).$$

Therefore ${\delta }_a$ is a continuous linear functional on ${\mathcal{D}}_R$, and is regarded as an element of ${\mathcal{D}}_R^{\prime }$ according to the complex-conjugation convention.

Next, we show that ${\delta }_a$ is not a bounded functional on $H_{\alpha ,+}$. For $a>0$, take $\eta \in C_c^{\infty }\left((-1,1)\right)$ with $\eta \left(0\right)=1$, and for sufficiently large n, set

$${\phi }_n\left(x\right):=\eta \left(n(x-a)\right).$$

Then ${\phi }_n\in C_c^{\infty }(0,\infty )\subset {\mathcal{D}}_R$, and

$${\delta }_a\left({\phi }_n\right)={\phi }_n\left(a\right)=1.$$

On the other hand, the support is contained in an interval of length $O\left(n^{-1}\right)$ near a, and ${\langle x\rangle }^{2\alpha }$ is bounded above and below on that interval, so

$$\parallel {\phi }_n{\parallel }_{H_{\alpha ,+}}^2={\int }_0^{\infty }{|\eta \left(n(x-a)\right)|}^2{\langle x\rangle }^{2\alpha }dx\le C_a{n}^{-1}.$$

Therefore $\parallel {\phi }_n{\parallel }_{H_{\alpha ,+}}\to 0$.

If ${\delta }_a$ were a bounded functional on $H_{\alpha ,+}$, there would exist a constant $C>0$ such that

$$1=|{\delta }_a\left({\phi }_n\right)|\le C\parallel {\phi }_n{\parallel }_{H_{\alpha ,+}}$$

for all n. But the right-hand side converges to 0, a contradiction.

The case $a=0$ is similar. Take $\eta \in C_c^{\infty }\left([0,1)\right)$ with $\eta \left(0\right)=1$, and set

$${\phi }_n\left(x\right):=\eta \left(nx\right).$$

Then

$${\phi }_n\left(0\right)=1,{\parallel {\phi }_n\parallel }_{H_{\alpha ,+}}^2\le C{n}^{-1}.$$

Therefore boundary point evaluation is also not a bounded functional on $H_{\alpha ,+}$.

This proves that point evaluation is meaningful as an element of ${\mathcal{D}}_R^{\prime }$, but cannot be treated as a bounded functional on $H_{\alpha ,+}$. □

Definition 4.13 

(Distribution kernel). When the distribution kernel associated with $K_R$ is treated as a distribution kernel, its primary type is defined to be

$$K\in {\left({\mathcal{D}}_R{\widehat{\otimes }}_{\pi }{\mathcal{D}}_R\right)}^{\prime }.$$

Here ${\widehat{\otimes }}_{\pi }$ denotes the completed projective tensor product. Namely, K is a continuous bilinear functional assigning

$$\langle K,\phi \otimes \psi \rangle$$

to a simple tensor

$$\phi \otimes \psi \in {\mathcal{D}}_R{\widehat{\otimes }}_{\pi }{\mathcal{D}}_R.$$

Since ${\mathcal{D}}_R$ is a nuclear space, as will be verified later, this space may be identified, when necessary, with

$${\left({\mathcal{D}}_R{\widehat{\otimes }}_{\pi }{\mathcal{D}}_R\right)}^{\prime }\simeq {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }.$$

However, this identification is a type identification through nuclearity, and does not automatically make a distribution kernel into a bounded operator on $H_{\alpha ,+}$.

Lemma 4.14 

(Scope of the kernel theorem). ${\mathcal{D}}_R$ is a nuclear Fréchet space. Therefore, for any continuous bilinear form

$$B:{\mathcal{D}}_R\times {\mathcal{D}}_R⟶\mathbb{C},$$

there exists a unique distribution kernel

$$K_B\in {\left({\mathcal{D}}_R{\widehat{\otimes }}_{\pi }{\mathcal{D}}_R\right)}^{\prime }$$

such that

$$B(\phi ,\psi )=\langle K_B,\phi \otimes \psi \rangle .$$

Moreover, by nuclearity, when necessary one may identify

$$K_B\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }.$$

Likewise, any continuous linear map

$$T:{\mathcal{D}}_R⟶{\mathcal{D}}_R^{\prime }$$

is represented by a distribution kernel.

Proof. 

${\mathcal{D}}_R$ is a Schwartz-type space on the half-line and is a nuclear Fréchet space obtained as a restriction space of the Schwartz space on the real line. Nuclearity is preserved under closed subspaces and quotient spaces, and hence ${\mathcal{D}}_R$ is also nuclear.

By the Schwartz kernel theorem for nuclear Fréchet spaces, continuous bilinear forms on ${\mathcal{D}}_R\times {\mathcal{D}}_R$ correspond uniquely to continuous linear functionals on ${\mathcal{D}}_R{\widehat{\otimes }}_{\pi }{\mathcal{D}}_R$. Therefore

$$K_B\in {\left({\mathcal{D}}_R{\widehat{\otimes }}_{\pi }{\mathcal{D}}_R\right)}^{\prime }$$

exists and satisfies

$$B(\phi ,\psi )=\langle K_B,\phi \otimes \psi \rangle .$$

Furthermore, since ${\mathcal{D}}_R$ is nuclear, the standard kernel identification on the strong-dual side allows this kernel to be expressed as an element of ${\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }$. However, this last representation is a type representation of the distribution kernel, and by itself does not give an $H_{\alpha ,+}$-bounded operator.

For a continuous map $T:{\mathcal{D}}_R\to {\mathcal{D}}_R^{\prime }$, define

$$B_T(\phi ,\psi ):={\langle T\phi ,\psi \rangle }_{{\mathcal{D}}_R^{\prime },{\mathcal{D}}_R}.$$

This is a continuous bilinear form on ${\mathcal{D}}_R\times {\mathcal{D}}_R$. Accordingly, by the same kernel theorem, T is also represented by a distribution kernel. □

Remark 4.15 

(Distinction between distribution kernels and bounded operators). A distribution kernel

$$K\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }$$

does not, by itself, define a bounded operator on $H_{\alpha ,+}$. For a distribution kernel to define a bounded operator, one must separately prove $H_{\alpha ,+}$-boundedness such as

$$\parallel T_K{f\parallel }_{H_{\alpha ,+}}\le C{\parallel f\parallel }_{H_{\alpha ,+}},$$

or its domain and closedness as a closed operator. Accordingly, in this paper we distinguish the distribution kernel K from the operator $T_K$ obtained as its realization.

Definition 4.16 

(Object level of quadratic forms). When this section refers to a quadratic form, it means not an element of a Hilbert space or a distribution kernel, but the pair

$$\left(Q\right(q),q).$$

Here $Q\left(q\right)\subset H_{\alpha ,+}$ is a dense linear subspace, and

$$q:Q\left(q\right)\times Q\left(q\right)⟶\mathbb{C}$$

is a sesquilinear form.

In particular, when referring to the closed quadratic form constructed in Section 2, we write its form domain as $Q_R$ and the form itself as

$$q_R:Q_R\times Q_R⟶\mathbb{C}.$$

At this level, $q_R$ is not an operator. Only after closedness, semiboundedness, and the representation theorem are established does one obtain the corresponding self-adjoint operator.

Definition 4.17 

(Object level of generators). When this section refers to a generator, it means a closed operator on a Hilbert space or on the subsequent ambient Hilbert-space setting,

$$A:Dom\left(A\right)\subset \mathcal{H}⟶\mathcal{H}.$$

A generator must be specified together with its domain, closedness, symmetry or anti-self-adjointness, and the strongly continuous semigroup or group that it generates. Therefore, a form q, a distribution kernel K, or a boundary form b must not be identified with a generator without proof.

Definition 4.18 

(Object level of boundary forms). A boundary bilinear form is treated as a sesquilinear form on a space where the boundary trace is defined. Namely, when a linear space T and a boundary trace map $\gamma$ are given, the boundary form is defined in the form

$$b:T\times T⟶\mathbb{C}$$

or

$$b[u,v]=\mathfrak{b}[\gamma u,\gamma v].$$

A boundary form is an object of a different type from the interior energy form q, and is not itself an operator on $H_{\alpha ,+}$.

Definition 4.19 

(Type-separation convention for object levels). From this section onward, we distinguish the following types. Objects of different types are not identified until they are related through an explicitly stated realization map, closure operation, or representation theorem.

Symbol	Type	Ambient space/object	Allowed basic operations
${\Pi }_R^{\mathrm{cyc}}$	cycle	geometric or measure-theoretic cycle	support, pushforward, pullback
${\Pi }_R^{+}$ or, after ambient embedding, ${\Pi }_R$	bounded projection	$\mathcal{B}\left(\mathcal{H}\right)$, with the underlying Hilbert space specified	action, composition, adjoint, orthogonal projection
${\mathsf{K}}_R^{\mathrm{dist}}$	kernel	${\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }$	duality pairing with test functions
$q_R^{\mathrm{raw}},q_R^{\mathrm{res}}$	quadratic form	$Q\left(q\right)\times Q\left(q\right)\to \mathbb{C}$	form norm, closure, representation theorem
$A_R^{\mathrm{res}},B_R$	generator	$Dom\left(A\right)\subset \mathcal{H}\to \mathcal{H}$	closed operator, resolvent, semigroup/group generation
$b_R$	boundary form	$T_R\times T_R\to \mathbb{C}$	boundary trace evaluation, boundary cancellation

Convention 4.20 

(Prohibition of notational conflation). This paper prohibits the following.

1.

Identifying the geometric cycle ${\Pi }_R^{\mathrm{cyc}}$ directly with a Hilbert-space projector such as ${\Pi }_R^{+}$ or ${\Pi }_R$.

2.

Treating the distribution kernel ${\mathsf{K}}_R^{\mathrm{dist}}$ as an operator on $H_{\alpha ,+}$ without a proof of boundedness or closed-operator status.

3.

Identifying the quadratic form $q_R$ with the self-adjoint operator $A_R$ without passing through the representation theorem.

4.

Identifying the boundary form $b_R$ with the interior energy form $q_R$ or the transport generator $B_R$.

5.

Treating point evaluation ${\delta }_a$ or a boundary trace as a bounded functional on $H_{\alpha ,+}$.

Whenever such an identification is needed, the corresponding realization map, boundedness, closedness, or representation theorem is explicitly stated and proved.

Proposition 4.21 

(Basic consequences of this subsection). This subsection has fixed the following three foundational points.

1.

Distributional objects on the singular-boundary side are handled inside the Gelfand triple

$${\mathcal{D}}_R\subset H_{\alpha ,+}\subset {\mathcal{D}}_R^{\prime }.$$

2.

Point evaluations and boundary point evaluations are not bounded functionals on $H_{\alpha ,+}$, but are treated as elements of ${\mathcal{D}}_R^{\prime }$.

3.

Distribution kernels, quadratic forms, generators, boundary forms, and projectors are objects of different types and are not identified until an explicitly stated construction has been carried out.

Proof. 

The first item follows from Definition 4.10 and Lemma 4.11. The second item follows from Proposition 4.12. The third item follows from Definition 4.19 and Convention 4.20. □

4.2. Boundary Parameter Space and ${\sigma }_R$-Null Singular Support

In this subsection, we internally construct the measure-theoretic parameter space describing the boundary parameter structure. There are two points needed here. First, we construct a space

$$({\Sigma }_R,{\mathcal{B}}_R,{\sigma }_R)$$

equipped with a regular area-type measure for measuring boundary contributions. Second, we introduce a singular measure that retains nontrivial boundary traces despite being null with respect to this regular area measure.

This two-layer structure is indispensable. Indeed, if a boundary trace is supported on a set ${\Gamma }_R$ satisfying

$${\sigma }_R\left({\Gamma }_R\right)=0$$

as an ordinary $L^2({\Sigma }_R,{\sigma }_R)$-function, then that trace is zero ${\sigma }_R$-almost everywhere. Therefore, the ${\sigma }_R$-null property is measured on the ${\sigma }_R$-side, whereas the nontrivial boundary trace is retained on the side of a measure singular with respect to ${\sigma }_R$, or of a distributional boundary distribution.

Definition 4.22 

(Boundary parameter space and regular area measure). Let the closed unit square

$${\Sigma }_R:={[0,1]}^2$$

be the boundary parameter space for the singular support construction. Standard facts concerning Radon measures, Borel regularity, and measure decomposition are used within the scope of [3]. Write its Borel $\sigma$-algebra as

$${\mathcal{B}}_R:=\mathcal{B}\left({[0,1]}^2\right).$$

Define the regular area measure on ${\Sigma }_R$ by

$${\sigma }_R:={\mathcal{L}}^2{|}_{{[0,1]}^2}.$$

Here ${\mathcal{L}}^2$ denotes two-dimensional Lebesgue measure on ${\mathbb{R}}^2$. Thus

$${\sigma }_R\left({\Sigma }_R\right)=1.$$

Lemma 4.23 

(Basic properties of the regular area space). $({\Sigma }_R,{\mathcal{B}}_R,{\sigma }_R)$ is a finite regular Borel measure space on a compact metric space. Furthermore, ${\sigma }_R$ is nonatomic.

Proof. 

${\Sigma }_R={[0,1]}^2$ is a compact metric space with respect to the Euclidean metric. ${\mathcal{B}}_R$ is its Borel $\sigma$-algebra, and ${\sigma }_R$ is the restriction of Lebesgue measure to a compact set, hence is a finite regular Borel measure.

We show nonatomicity. For any point $p\in {[0,1]}^2$,

$${\sigma }_R\left(\left\{p\right\}\right)={\mathcal{L}}^2\left(\left\{p\right\}\right)=0.$$

Therefore ${\sigma }_R$ has no atoms. □

Definition 4.24 

(Cantor-type ${\sigma }_R$-null singular support). Define the middle-third Cantor set by

$$C_R:=\left\{{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{2{\epsilon }_j}{3^j}}:{\epsilon }_j\in \{0,1\}\right\}\subset [0,1].$$

Define the ${\sigma }_R$-null singular support by

$${\Gamma }_R:=C_R\times \left\{0\right\}\subset {\Sigma }_R.$$

Lemma 4.25 

(${\sigma }_R$-null property). ${\Gamma }_R$ is a closed subset of ${\Sigma }_R$ and satisfies

$${\sigma }_R\left({\Gamma }_R\right)=0.$$

Proof. 

First, $C_R$ is closed. Indeed, $C_R$ is the compact set obtained by iteratively removing closed intervals; equivalently, it is the image of the continuous map

$${\{0,1\}}^{\mathbb{N}}\ni {\left({\epsilon }_j\right)}_{j\ge 1}⟼{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{2{\epsilon }_j}{3^j}}\in [0,1].$$

Here ${\{0,1\}}^{\mathbb{N}}$ is compact in the product topology, so its continuous image $C_R$ is also compact and hence closed. Therefore

$${\Gamma }_R=C_R\times \left\{0\right\}$$

is also a closed subset of ${[0,1]}^2$.

Next we compute the measure. The middle-third Cantor set $C_R$ has one-dimensional Lebesgue measure zero. Indeed, at step n, the total length of the remaining closed intervals is

$${\left({\displaystyle \frac{2}{3}}\right)}^n,$$

and this converges to 0 as $n\to \infty$; hence

$${\mathcal{L}}^1\left(C_R\right)=0.$$

By Fubini’s theorem or by monotonicity of product measure,

$${\sigma }_R\left({\Gamma }_R\right)={\mathcal{L}}^2(C_R\times \left\{0\right\})={\mathcal{L}}^1\left(C_R\right){\mathcal{L}}^1\left(\left\{0\right\}\right)=0.$$

Therefore ${\Gamma }_R$ is a closed ${\sigma }_R$-null set. □

Definition 4.26 

(Singular measure ${\nu }_R$ supported on ${\Gamma }_R$). Let ${\mu }_C$ denote the probability measure on $[0,1]$ obtained by pushing forward the Bernoulli probability measure

$${\left({\displaystyle \frac{1}{2}}{\delta }_0+{\displaystyle \frac{1}{2}}{\delta }_1\right)}^{\otimes \mathbb{N}}$$

on ${\{0,1\}}^{\mathbb{N}}$ by the map

$$\Theta :{\left({\epsilon }_j\right)}_{j\ge 1}⟼{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{2{\epsilon }_j}{3^j}}.$$

That is,

$${\mu }_C:={\Theta }_{\#}{\left({\displaystyle \frac{1}{2}}{\delta }_0+{\displaystyle \frac{1}{2}}{\delta }_1\right)}^{\otimes \mathbb{N}}.$$

Define the singular measure ${\nu }_R$ supported on ${\Gamma }_R$ on ${\Sigma }_R$ by

$${\nu }_R:={\mu }_C\otimes {\delta }_0.$$

Lemma 4.27 

(Support and singularity of the singular measure). ${\nu }_R$ is a finite regular Borel measure on ${\Sigma }_R$, and satisfies

$${\nu }_R\left({\Sigma }_R\right)=1,supp{\nu }_R={\Gamma }_R.$$

Furthermore,

$${\nu }_R\perp {\sigma }_R.$$

Proof. 

${\mu }_C$ is a probability Borel measure on the compact space $[0,1]$, and ${\delta }_0$ is the Dirac measure on $[0,1]$. Therefore

$${\nu }_R={\mu }_C\otimes {\delta }_0$$

is a probability Borel measure on ${[0,1]}^2$, and in particular is finite and regular. Moreover,

$${\nu }_R\left({\Sigma }_R\right)={\mu }_C\left([0,1]\right){\delta }_0\left([0,1]\right)=1.$$

The support of ${\mu }_C$ is $C_R$. Indeed, any relatively open subset of $C_R$ contains a finite-digit cylinder set and hence has positive ${\mu }_C$-measure, while the complement of $C_R$ has ${\mu }_C$-measure zero by the defining property of the Cantor measure. Therefore the support formula for product measures gives

$$supp{\nu }_R=supp{\mu }_C\times supp{\delta }_0=C_R\times \left\{0\right\}={\Gamma }_R.$$

Finally we prove singularity. By Lemma 4.25,

$${\sigma }_R\left({\Gamma }_R\right)=0.$$

On the other hand, from the support property just proved,

$${\nu }_R\left({\Gamma }_R\right)=1.$$

Thus ${\Gamma }_R$ is a ${\sigma }_R$-null set and at the same time a full-measure set for ${\nu }_R$. This means

$${\nu }_R\perp {\sigma }_R.$$

□

Definition 4.28 

(Regular boundary channel and singular boundary channel). Let $\mathcal{M}\left({\Sigma }_R\right)$ be the space of all finite complex Radon measures on ${\Sigma }_R$. For any $\lambda \in \mathcal{M}\left({\Sigma }_R\right)$, write its Lebesgue decomposition as

$$\lambda ={\lambda }_{\mathrm{ac}}+{\lambda }_{\mathrm{s}},{\lambda }_{\mathrm{ac}}\ll {\sigma }_R,{\lambda }_{\mathrm{s}}\perp {\sigma }_R.$$

We call ${\lambda }_{\mathrm{ac}}$ the regular boundary channel and ${\lambda }_{\mathrm{s}}$ the singular boundary channel.

In particular, ${\nu }_R$ is a purely singular boundary channel, and its regular component is zero.

Definition 4.29 

(Initial singular boundary trace). By Proposition 4.12, the boundary point evaluation

$${\delta }_0:{\mathcal{D}}_R⟶\mathbb{C},{\delta }_0\left(\phi \right)=\phi \left(0\right)$$

is an element of ${\mathcal{D}}_R^{\prime }$. Using this, define the initial singular boundary trace

$${\gamma }_{R,0}^{\mathrm{sing}}:{\mathcal{D}}_R⟶\mathcal{M}\left({\Sigma }_R\right)$$

by

$${\gamma }_{R,0}^{\mathrm{sing}}\phi :={\delta }_0\left(\phi \right){\nu }_R=\phi \left(0\right){\nu }_R.$$

Equivalently, for any $\eta \in C\left({\Sigma }_R\right)$,

$$〈{\gamma }_{R,0}^{\mathrm{sing}}\phi ,\eta 〉:=\phi \left(0\right){\int }_{{\Sigma }_R}\eta d{\nu }_R.$$

Lemma 4.30 

(Continuity of the singular boundary trace). The map

$${\gamma }_{R,0}^{\mathrm{sing}}:{\mathcal{D}}_R⟶\mathcal{M}\left({\Sigma }_R\right)$$

is a continuous linear map from the Fréchet topology of ${\mathcal{D}}_R$ to the total variation norm topology of $\mathcal{M}\left({\Sigma }_R\right)$. Furthermore,

$$supp\left({\gamma }_{R,0}^{\mathrm{sing}}\phi \right)\subset {\Gamma }_R$$

holds for every $\phi \in {\mathcal{D}}_R$.

Proof. 

For any $\phi \in {\mathcal{D}}_R$,

$${∥{\gamma }_{R,0}^{\mathrm{sing}}\phi ∥}_{\mathrm{TV}}=|\phi \left(0\right)|\parallel {\nu }_R{\parallel }_{\mathrm{TV}}=|\phi \left(0\right)|.$$

On the other hand,

$$|\phi \left(0\right)|\le p_{0,0}\left(\phi \right).$$

Therefore

$${∥{\gamma }_{R,0}^{\mathrm{sing}}\phi ∥}_{\mathrm{TV}}\le p_{0,0}\left(\phi \right),$$

and hence ${\gamma }_{R,0}^{\mathrm{sing}}$ is continuous.

For the support, we have

$${\gamma }_{R,0}^{\mathrm{sing}}\phi =\phi \left(0\right){\nu }_R,$$

and by Lemma 4.27,

$$supp{\nu }_R={\Gamma }_R.$$

Therefore, when $\phi \left(0\right)\ne 0$, the support is ${\Gamma }_R$, and when $\phi \left(0\right)=0$, the measure is the zero measure. In either case,

$$supp\left({\gamma }_{R,0}^{\mathrm{sing}}\phi \right)\subset {\Gamma }_R$$

holds. □

Proposition 4.31 

(The singular boundary trace is not a Hilbert boundary function). In general, ${\gamma }_{R,0}^{\mathrm{sing}}\phi$ cannot be represented as a function in $L^2({\Sigma }_R,{\sigma }_R)$. More precisely, if $\phi \left(0\right)\ne 0$, then the measure ${\gamma }_{R,0}^{\mathrm{sing}}\phi$ is not absolutely continuous with respect to ${\sigma }_R$. Therefore there exists no $g\in L^1({\Sigma }_R,{\sigma }_R)$ satisfying

$${\gamma }_{R,0}^{\mathrm{sing}}\phi =g{\sigma }_R.$$

In particular, it also cannot be represented by a $g\in L^2({\Sigma }_R,{\sigma }_R)$.

Proof. 

Assume $\phi \left(0\right)\ne 0$. Then

$${\gamma }_{R,0}^{\mathrm{sing}}\phi =\phi \left(0\right){\nu }_R.$$

By Lemma 4.27,

$${\nu }_R\perp {\sigma }_R,$$

and therefore $\phi \left(0\right){\nu }_R$ is also singular with respect to ${\sigma }_R$.

If there existed some $g\in L^1({\Sigma }_R,{\sigma }_R)$ such that

$$\phi \left(0\right){\nu }_R=g{\sigma }_R,$$

then the left-hand side would be singular with respect to ${\sigma }_R$, whereas the right-hand side would be absolutely continuous with respect to ${\sigma }_R$. By uniqueness of the Lebesgue decomposition, for the two measures to coincide, both would have to be the zero measure. However,

$$\parallel \phi \left(0\right){\nu }_R{\parallel }_{\mathrm{TV}}=|\phi \left(0\right)|>0,$$

which is a contradiction. Therefore no such g exists. □

Lemma 4.32 

(Ordinary $L^2$-traces on a ${\sigma }_R$-null set are trivial). If $g\in L^2({\Sigma }_R,{\sigma }_R)$ satisfies

$$\underset{{\sigma }_R}{ess\; supp}g\subset {\Gamma }_R,$$

then

$$g=0{\sigma }_R-\mathrm{a}.\mathrm{e}.$$

Proof. 

By assumption,

$$g=0{\sigma }_R-\mathrm{a}.\mathrm{e}.\mathrm{on}{\Sigma }_R\setminus {\Gamma }_R.$$

On the other hand, by Lemma 4.25,

$${\sigma }_R\left({\Gamma }_R\right)=0.$$

Therefore the values of g on ${\Gamma }_R$ are ignored as an element of $L^2({\Sigma }_R,{\sigma }_R)$. Hence

$${\int }_{{\Sigma }_R}{|g|}^{2}d{\sigma }_R={\int }_{{\Sigma }_R\setminus {\Gamma }_R}{|g|}^{2}d{\sigma }_R+{\int }_{{\Gamma }_R}{|g|}^{2}d{\sigma }_R=0+0=0.$$

Thus $g=0$${\sigma }_R$-almost everywhere. □

Definition 4.33 

(Convention on ${\sigma }_R$-null singular support and singular retention). From now on, when we refer to a boundary trace supported on ${\Gamma }_R$, this does not mean support as an ordinary function in

$$L^2({\Sigma }_R,{\sigma }_R).$$

A boundary trace on ${\Gamma }_R$ is retained as one of the following singular or distributional objects:

$${\nu }_R,{\gamma }_R^{\mathrm{sing}},{\mathcal{D}}_R^{\prime }.$$

On the other hand, ${\sigma }_R$ is used only to measure regular boundary trace mass.

Theorem 4.34 

(Internal construction of the boundary triple data). By the construction above,

$$({\Sigma }_R,{\sigma }_R,{\Gamma }_R)$$

is fixed as boundary triple data satisfying the following properties.

1.

${\Sigma }_R$ is a compact metric space.

2.

${\sigma }_R$ is a finite regular Borel area measure on ${\Sigma }_R$.

3.

${\Gamma }_R\subset {\Sigma }_R$ is a closed set and satisfies

$${\sigma }_R\left({\Gamma }_R\right)=0.$$

4.

A nonzero singular probability measure

$${\nu }_R$$

is supported on ${\Gamma }_R$, and

$${\nu }_R\left({\Gamma }_R\right)=1,{\nu }_R\perp {\sigma }_R$$

holds.

5.

The initial singular boundary trace

$${\gamma }_{R,0}^{\mathrm{sing}}:{\mathcal{D}}_R\to \mathcal{M}\left({\Sigma }_R\right)$$

is continuous, and its values are always supported on ${\Gamma }_R$.

Proof. 

Items 1 and 2 follow from Definition 4.22 and Lemma 4.23. Item 3 follows from Definition 4.24 and Lemma 4.25. Item 4 follows from Definition 4.26 and Lemma 4.27. Item 5 follows from Definition 4.29 and Lemma 4.30. □

Lemma 4.35 

(Separation of ${\sigma }_R$-null property and nontriviality). ${\Gamma }_R$ is null with respect to the regular area measure ${\sigma }_R$. However, the singular measure on ${\Gamma }_R$ is retained nontrivially as a singular measure or distribution. More precisely, the following hold.

1.

$${\sigma }_R\left({\Gamma }_R\right)=0.$$

2.

The only element of $L^2({\Sigma }_R,{\sigma }_R)$ essentially supported on ${\Gamma }_R$ is the zero element.

3.

$${\nu }_R\left({\Gamma }_R\right)=1$$

and ${\nu }_R$ is a nonzero singular measure.

4.

If $\phi \in {\mathcal{D}}_R$ satisfies $\phi \left(0\right)\ne 0$, then

$${\gamma }_{R,0}^{\mathrm{sing}}\phi =\phi \left(0\right){\nu }_R$$

is nonzero and its support is ${\Gamma }_R$.

Therefore the ${\sigma }_R$-null property is a property on the ${\sigma }_R$-side, whereas the nontriviality of the boundary trace is retained as a property on the ${\nu }_R$- or ${\mathcal{D}}_R^{\prime }$-side.

Proof. 

Item 1 was proved in Lemma 4.25. Item 2 is Lemma 4.32. Item 3 follows from Lemma 4.27. For Item 4, if $\phi \left(0\right)\ne 0$, then

$${\gamma }_{R,0}^{\mathrm{sing}}\phi =\phi \left(0\right){\nu }_R,$$

and its total variation norm is

$${∥{\gamma }_{R,0}^{\mathrm{sing}}\phi ∥}_{\mathrm{TV}}=|\phi \left(0\right)|>0.$$

Therefore it is nonzero. Moreover, by Lemma 4.30, its support is contained in ${\Gamma }_R$, and when $\phi \left(0\right)\ne 0$, it has the same support as ${\nu }_R$. Hence

$$supp\left({\gamma }_{R,0}^{\mathrm{sing}}\phi \right)={\Gamma }_R.$$

Thus, although ${\Gamma }_R$ is null with respect to the area measure, it retains a nontrivial boundary trace on the side of singular measures and distributional boundary traces. □

Proposition 4.36 

(Output of this subsection). This subsection has constructed the following objects:

$${\Sigma }_R,{\sigma }_R,{\Gamma }_R,{\nu }_R,{\gamma }_{R,0}^{\mathrm{sing}}.$$

Here

$${\sigma }_R\left({\Gamma }_R\right)=0,{\nu }_R\left({\Gamma }_R\right)=1,{\nu }_R\perp {\sigma }_R$$

hold. Moreover, ${\gamma }_{R,0}^{\mathrm{sing}}$ is a singular boundary trace defined on ${\mathcal{D}}_R$. In the subsequent subsections, we extend it to a singular boundary trace on the form domain and construct the support map and regular boundary trace-mass functional.

Proof. 

The construction of each object follows respectively from Definition 4.22, Definition 4.24, Definition 4.26, and Definition 4.29. The displayed properties were proved in Lemma 4.25, Lemma 4.27, and Lemma 4.35. □

4.3. Traces, Support Maps, and Regular Trace-Mass Functionals

In this subsection, using the boundary triple data

$$({\Sigma }_R,{\sigma }_R,{\Gamma }_R)$$

and the singular measure ${\nu }_R$ constructed in the preceding subsection, we construct the boundary traces, support map, and regular boundary trace-mass functional on the form domain. Let $Q_R$ denote the form domain constructed in Section 2, and write the corresponding form norm as

$${\parallel f\parallel }_{q_R}^2:=q_R[f,f]+{\parallel f\parallel }_{H_{\alpha ,+}}^2.$$

When necessary, if $q_R$ is lower bounded, this norm is interpreted after replacing it by an equivalent positive norm.

There are two types of boundary traces introduced in this subsection. One is the singular boundary trace supported on ${\Gamma }_R$, and the other is the regular boundary trace absolutely continuous with respect to ${\sigma }_R$. The former is used to retain the nontriviality of the boundary trace, while the latter is used to measure regular boundary trace mass.

Definition 4.37 

(Trace core). Write the common intersection of the form domain and the boundary test space as

$${\mathcal{C}}_R:={\mathcal{D}}_R\cap Q_R.$$

Equip ${\mathcal{C}}_R$ with the form norm ${\parallel ·\parallel }_{q_R}$ induced from $Q_R$.

Remark 4.38. 

${\mathcal{C}}_R$ is the class of smooth elements for which boundary traces can first be defined classically. The trace maps in this subsection are extended to a subspace of $Q_R$ by closing the maps on this core with respect to the form norm.

Definition 4.39 

(Singular boundary trace on the core). For $\phi \in {\mathcal{C}}_R$, define

$${\vartheta }_R^{\mathrm{sing}}\phi :=\phi \left(0\right){\nu }_R\in \mathcal{M}\left({\Sigma }_R\right)$$

using the singular measure ${\nu }_R$ from the preceding subsection. Here $\mathcal{M}\left({\Sigma }_R\right)$ denotes the space of all finite complex Radon measures on ${\Sigma }_R$. Namely, for any $\eta \in C\left({\Sigma }_R\right)$,

$$〈{\vartheta }_R^{\mathrm{sing}}\phi ,\eta 〉=\phi \left(0\right){\int }_{{\Sigma }_R}\eta d{\nu }_R.$$

Definition 4.40 

(Regular boundary trace on the core). Set

$$e_R:={\mathbf{1}}_{{\Sigma }_R}.$$

Since the construction in the preceding subsection gives ${\sigma }_R\left({\Sigma }_R\right)=1$,

$$\parallel e_R{\parallel }_{L^2({\Sigma }_R,{\sigma }_R)}=1.$$

For $\phi \in {\mathcal{C}}_R$, define the regular trace mass coefficient by

$${\tau }_R^{\mathrm{reg}}\left(\phi \right):={\partial }_x\phi \left(0\right).$$

Then define the regular boundary trace on the core by

$${\vartheta }_R^{\mathrm{reg}}\phi :={\tau }_R^{\mathrm{reg}}\left(\phi \right)e_R\in L^2({\Sigma }_R,{\sigma }_R).$$

Remark 4.41 

(Independence of regular trace and singular trace). ${\vartheta }_R^{\mathrm{sing}}$ is defined through the boundary-value-type singular measure ${\nu }_R$. On the other hand, ${\vartheta }_R^{\mathrm{reg}}$ measures regular boundary trace mass absolutely continuous with respect to the regular area measure ${\sigma }_R$. Accordingly, even if

$${\vartheta }_R^{\mathrm{sing}}\phi$$

is nonzero, it is possible that

$${\vartheta }_R^{\mathrm{reg}}\phi =0.$$

This separation allows one to handle simultaneously a nontrivial boundary trace on the ${\sigma }_R$-null singular support and vanishing regular-trace mass in the regular boundary-trace direction.

Definition 4.42 

(Graph of the closed trace). A triple

$$(f,\lambda ,g)\in Q_R\times \mathcal{M}\left({\Sigma }_R\right)\times L^2({\Sigma }_R,{\sigma }_R)$$

is said to belong to the closed-trace graph if there exists a sequence

$${\left\{{\phi }_n\right\}}_{n\ge 1}\subset {\mathcal{C}}_R$$

such that

$${\phi }_n⟶f\mathrm{in}(Q_R{,\parallel ·\parallel }_{q_R}),$$

and

$${\vartheta }_R^{\mathrm{sing}}{\phi }_n\stackrel{w^{*}}{⟶}\lambda \mathrm{in}\mathcal{M}\left({\Sigma }_R\right),$$

and further

$${\vartheta }_R^{\mathrm{reg}}{\phi }_n⟶g\mathrm{in}{L}^2({\Sigma }_R,{\sigma }_R)$$

hold. Denote this set by

$$\mathcal{G}\left({\gamma }_R\right)\subset Q_R\times \mathcal{M}\left({\Sigma }_R\right)\times L^2({\Sigma }_R,{\sigma }_R).$$

Definition 4.43 

(Closed boundary trace space). We say that $f\in Q_R$ has a boundary trace if there exists an element

$$(f,\lambda ,g)$$

of $\mathcal{G}\left({\gamma }_R\right)$, and if such a pair $(\lambda ,g)$ is uniquely determined. Define the set of all such f by

$$T_R:=\left\{f\in Q_R:\exists !(\lambda ,g)\in \mathcal{M}\left({\Sigma }_R\right)\times L^2({\Sigma }_R,{\sigma }_R)\mathrm{such}\mathrm{that}(f,\lambda ,g)\in \mathcal{G}\left({\gamma }_R\right)\right\}.$$

For $f\in T_R$, write the uniquely determined $\lambda$ and g as

$${\gamma }_R^{\mathrm{sing}}f:=\lambda ,{\gamma }_R^{\mathrm{reg}}f:=g,$$

respectively.

Lemma 4.44 

(Linearity of the closed boundary trace space). $T_R$ is a linear subspace of $Q_R$. Moreover,

$${\gamma }_R^{\mathrm{sing}}:T_R\to \mathcal{M}\left({\Sigma }_R\right),{\gamma }_R^{\mathrm{reg}}:T_R\to L^2({\Sigma }_R,{\sigma }_R)$$

are both linear maps.

Proof. 

Let $f_1,f_2\in T_R$, and write

$${\gamma }_R^{\mathrm{sing}}{f}_j={\lambda }_j,{\gamma }_R^{\mathrm{reg}}{f}_j=g_j(j=1,2).$$

By definition, for each j there exists a sequence

$${\left\{{\phi }_{j,n}\right\}}_{n\ge 1}\subset {\mathcal{C}}_R$$

such that

$${\phi }_{j,n}\to f_j\mathrm{in}{Q}_R,$$

and

$${\vartheta }_R^{\mathrm{sing}}{\phi }_{j,n}\stackrel{w^{*}}{\to }{\lambda }_j,{\vartheta }_R^{\mathrm{reg}}{\phi }_{j,n}\to g_j.$$

For arbitrary $a,b\in \mathbb{C}$, set

$${\psi }_n:=a{\phi }_{1,n}+b{\phi }_{2,n}.$$

Since ${\mathcal{C}}_R$ is a linear space, we have ${\psi }_n\in {\mathcal{C}}_R$. Moreover,

$${\psi }_n\to a{f}_1+b{f}_2\mathrm{in}{Q}_R,$$

and by the linearity of the trace maps on the core,

$${\vartheta }_R^{\mathrm{sing}}{\psi }_n=a{\vartheta }_R^{\mathrm{sing}}{\phi }_{1,n}+b{\vartheta }_R^{\mathrm{sing}}{\phi }_{2,n}\stackrel{w^{*}}{⟶}a{\lambda }_1+b{\lambda }_2,$$

$${\vartheta }_R^{\mathrm{reg}}{\psi }_n=a{\vartheta }_R^{\mathrm{reg}}{\phi }_{1,n}+b{\vartheta }_R^{\mathrm{reg}}{\phi }_{2,n}⟶a{g}_1+b{g}_2$$

hold. Therefore

$$(a{f}_1+b{f}_2,a{\lambda }_1+b{\lambda }_2,a{g}_1+b{g}_2)\in \mathcal{G}\left({\gamma }_R\right).$$

Uniqueness is included in the definition of $T_R$, so $a{f}_1+b{f}_2\in T_R$, and

$${\gamma }_R^{\mathrm{sing}}(a{f}_1+b{f}_2)=a{\gamma }_R^{\mathrm{sing}}{f}_1+b{\gamma }_R^{\mathrm{sing}}{f}_2,$$

$${\gamma }_R^{\mathrm{reg}}(a{f}_1+b{f}_2)=a{\gamma }_R^{\mathrm{reg}}{f}_1+b{\gamma }_R^{\mathrm{reg}}{f}_2.$$

Hence $T_R$ is a linear subspace, and the two trace maps are linear. □

Lemma 4.45 

(Support of the singular trace). For any $f\in T_R$,

$$supp\left({\gamma }_R^{\mathrm{sing}}f\right)\subset {\Gamma }_R$$

holds.

Proof. 

Let $f\in T_R$, and write

$${\gamma }_R^{\mathrm{sing}}f=\lambda .$$

By definition, there exists a sequence $\left\{{\phi }_n\right\}\subset {\mathcal{C}}_R$ such that

$${\vartheta }_R^{\mathrm{sing}}{\phi }_n\stackrel{w^{*}}{⟶}\lambda .$$

For each n,

$${\vartheta }_R^{\mathrm{sing}}{\phi }_n={\phi }_n\left(0\right){\nu }_R,$$

and by Lemma 4.27,

$$supp{\nu }_R={\Gamma }_R.$$

Therefore

$$supp\left({\vartheta }_R^{\mathrm{sing}}{\phi }_n\right)\subset {\Gamma }_R.$$

Let $U\subset {\Sigma }_R$ be an open set satisfying

$$U\cap {\Gamma }_R=⌀.$$

For any $\eta \in C_c\left(U\right)$,

$${\int }_{{\Sigma }_R}\eta d\left({\vartheta }_R^{\mathrm{sing}}{\phi }_n\right)=0$$

holds for every n. Using weak-star convergence, we get

$${\int }_{{\Sigma }_R}\eta d\lambda =\underset{n\to \infty }{lim}{\int }_{{\Sigma }_R}\eta d\left({\vartheta }_R^{\mathrm{sing}}{\phi }_n\right)=0.$$

Hence $\lambda$ vanishes on ${\Sigma }_R\setminus {\Gamma }_R$. Therefore

$$supp\lambda \subset {\Gamma }_R.$$

That is,

$$supp\left({\gamma }_R^{\mathrm{sing}}f\right)\subset {\Gamma }_R.$$

□

Definition 4.46 

(Singular support map). For $f\in T_R$, define its singular support by

$${supp}_R\left(f\right):=supp\left({\gamma }_R^{\mathrm{sing}}f\right)\subset {\Sigma }_R.$$

If ${\gamma }_R^{\mathrm{sing}}f=0$, set

$${supp}_R\left(f\right):=⌀.$$

Corollary 4.47 

(${\sigma }_R$-null property of singular support). For any $f\in T_R$,

$${supp}_R\left(f\right)\subset {\Gamma }_R.$$

Therefore

$${\sigma }_R\left({supp}_R\left(f\right)\right)=0.$$

Proof. 

The first assertion is Lemma 4.45. The second follows from Lemma 4.25, namely from

$${\sigma }_R\left({\Gamma }_R\right)=0.$$

□

Definition 4.48 

(Regular trace-mass measure). For $f\in T_R$, define the finite positive measure

$${\Lambda }_R^{\mathrm{reg}}\left(f\right)$$

on ${\Sigma }_R$ by

$$d{\Lambda }_R^{\mathrm{reg}}\left(f\right):={|{\gamma }_R^{\mathrm{reg}}f|}^{2}d{\sigma }_R.$$

Namely, for any $E\in {\mathcal{B}}_R$,

$${\Lambda }_R^{\mathrm{reg}}\left(f\right)\left(E\right)={\int }_E{|{\gamma }_R^{\mathrm{reg}}f|}^{2}d{\sigma }_R.$$

Definition 4.49 

(Regular boundary trace-mass functional). For $f\in T_R$ and $E\in {\mathcal{B}}_R$, define the regular boundary trace-mass functional by

$$L_R(f;E):={\int }_E{|{\gamma }_R^{\mathrm{reg}}f|}^{2}d{\sigma }_R.$$

Equivalently,

$$L_R(f;E)={∥{\mathbf{1}}_E{\gamma }_R^{\mathrm{reg}}f∥}_{L^2({\Sigma }_R,{\sigma }_R)}^2.$$

Lemma 4.50 

(Basic properties of the regular boundary trace-mass functional). For any $f\in T_R$, the map

$$E⟼L_R(f;E)$$

is a finite positive measure on ${\Sigma }_R$. Moreover, for any $a\in \mathbb{C}$,

$$L_R(af;E)={|a|}^2{L}_R(f;E)$$

holds. Furthermore,

$$L_R(f;{\Sigma }_R)={\parallel {\gamma }_R^{\mathrm{reg}}f\parallel }_{L^2({\Sigma }_R,{\sigma }_R)}^2.$$

Proof. 

By definition,

$$L_R(f;E)={\int }_E{|{\gamma }_R^{\mathrm{reg}}f|}^{2}d{\sigma }_R.$$

Since ${\gamma }_R^{\mathrm{reg}}f\in L^2({\Sigma }_R,{\sigma }_R)$,

$$|{\gamma }_R^{\mathrm{reg}}{f|}^2\in L^1({\Sigma }_R,{\sigma }_R).$$

Therefore

$$E⟼{\int }_E{|{\gamma }_R^{\mathrm{reg}}f|}^{2}d{\sigma }_R$$

is a finite positive measure.

For scalar multiplication,

$${\gamma }_R^{\mathrm{reg}}\left(af\right)=a{\gamma }_R^{\mathrm{reg}}f,$$

and hence

$$L_R(af;E)={\int }_E{|a|}^2|{\gamma }_R^{\mathrm{reg}}{f|}^{2}d{\sigma }_R={|a|}^2{L}_R(f;E).$$

The final equality follows immediately by setting $E={\Sigma }_R$. □

Lemma 4.51 

(Vanishing criterion for regular trace mass). For $f\in T_R$, the following are equivalent.

1.

$$L_R(f;{\Sigma }_R\setminus {\Gamma }_R)=0.$$

2.

$${\gamma }_R^{\mathrm{reg}}f=0{\sigma }_R-\mathrm{a}.\mathrm{e}.\mathrm{on}{\Sigma }_R.$$

3.

$$L_R(f;{\Sigma }_R)=0.$$

Proof. 

By Lemma 4.25,

$${\sigma }_R\left({\Gamma }_R\right)=0.$$

Therefore, for any $g\in L^2({\Sigma }_R,{\sigma }_R)$,

$${\int }_{{\Sigma }_R}{|g|}^{2}d{\sigma }_R={\int }_{{\Sigma }_R\setminus {\Gamma }_R}{|g|}^{2}d{\sigma }_R$$

holds. Set

$$g={\gamma }_R^{\mathrm{reg}}f.$$

First suppose that (1) holds. By definition,

$$0=L_R(f;{\Sigma }_R\setminus {\Gamma }_R)={\int }_{{\Sigma }_R\setminus {\Gamma }_R}{|{\gamma }_R^{\mathrm{reg}}f|}^{2}d{\sigma }_R.$$

The preceding equality and ${\sigma }_R\left({\Gamma }_R\right)=0$ imply

$${\int }_{{\Sigma }_R}{|{\gamma }_R^{\mathrm{reg}}f|}^{2}d{\sigma }_R=0.$$

Therefore

$${\gamma }_R^{\mathrm{reg}}f=0{\sigma }_R-\mathrm{a}.\mathrm{e}.,$$

so (2) holds.

The implication from (2) to (3) is immediate from the definition. The implication from (3) to (1) follows by monotonicity. Thus the three conditions are equivalent. □

Definition 4.52 

(Trace-vanishing generating subspace associated with ${\Gamma }_R$). Define the trace-vanishing generating subspace associated with ${\Gamma }_R$ by

$$G_R:=\left\{f\in T_R:{supp}_R\left(f\right)\subset {\Gamma }_R,L_R(f;{\Sigma }_R\setminus {\Gamma }_R)=0\right\}.$$

Proposition 4.53 

(Equivalent representation of the generating class). $G_R$ can be written as

$$G_R=\left\{f\in T_R:{\gamma }_R^{\mathrm{reg}}f=0{\sigma }_R-\mathrm{a}.\mathrm{e}.\right\}.$$

Then, for any $f\in G_R$,

$${supp}_R\left(f\right)\subset {\Gamma }_R,{\sigma }_R\left({supp}_R\left(f\right)\right)=0$$

hold.

Proof. 

For any $f\in T_R$,

$${supp}_R\left(f\right)\subset {\Gamma }_R$$

holds automatically by Corollary 4.47. Therefore, the substantive condition in the definition of $G_R$ is

$$L_R(f;{\Sigma }_R\setminus {\Gamma }_R)=0.$$

By Lemma 4.51, this condition is equivalent to

$${\gamma }_R^{\mathrm{reg}}f=0{\sigma }_R-\mathrm{a}.\mathrm{e}.$$

The displayed representation follows.

Finally,

$${supp}_R\left(f\right)\subset {\Gamma }_R$$

together with

$${\sigma }_R\left({\Gamma }_R\right)=0$$

implies

$${\sigma }_R\left({supp}_R\left(f\right)\right)=0.$$

□

Lemma 4.54 

(Separation of ${\sigma }_R$-null property and regular-trace vanishing property). For $f\in G_R$, the singular boundary trace

$${\gamma }_R^{\mathrm{sing}}f$$

is supported on ${\Gamma }_R$. On the other hand, the regular boundary trace satisfies

$${\gamma }_R^{\mathrm{reg}}f=0{\sigma }_R-\mathrm{a}.\mathrm{e}.$$

Therefore $G_R$ is the generating class that simultaneously realizes the singular boundary trace on the ${\sigma }_R$-null singular support and vanishing regular-trace mass in the regular boundary-trace direction.

Proof. 

Let $f\in G_R$. By Definition 4.52,

$${supp}_R\left(f\right)\subset {\Gamma }_R.$$

By Definition 4.46,

$${supp}_R\left(f\right)=supp\left({\gamma }_R^{\mathrm{sing}}f\right),$$

so

$${\gamma }_R^{\mathrm{sing}}f$$

is supported on ${\Gamma }_R$.

Also by the definition of $f\in G_R$,

$$L_R(f;{\Sigma }_R\setminus {\Gamma }_R)=0.$$

Applying Lemma 4.51, we obtain

$${\gamma }_R^{\mathrm{reg}}f=0{\sigma }_R-\mathrm{a}.\mathrm{e}.$$

The claim follows. □

Remark 4.55 

(Reason for avoiding interpretation as an ordinary $L^2$ boundary function). The boundary trace of $f\in G_R$ is a singular boundary object described by ${\gamma }_R^{\mathrm{sing}}f$, and is not described as an ordinary function in

$$L^2({\Sigma }_R,{\sigma }_R).$$

Indeed, by Lemma 4.32, every $L^2({\Sigma }_R,{\sigma }_R)$-function essentially supported on ${\Gamma }_R$ is the zero element. Accordingly, the nontriviality in $G_R$ is retained on the side of ${\gamma }_R^{\mathrm{sing}}$ or ${\nu }_R$, while the regular-trace vanishing property is described as the vanishing of ${\gamma }_R^{\mathrm{reg}}$.

Proposition 4.56 

(Output of this subsection). This subsection has constructed the following objects:

$$T_R,{\gamma }_R^{\mathrm{sing}},{\gamma }_R^{\mathrm{reg}},{supp}_R,L_R,G_R.$$

Here,

$$T_R\subset Q_R$$

is the linear subspace on which the singular boundary trace and the regular boundary trace are simultaneously defined, and

$${supp}_R\left(f\right)=supp\left({\gamma }_R^{\mathrm{sing}}f\right).$$

Moreover,

$$L_R(f;E)={\int }_E{|{\gamma }_R^{\mathrm{reg}}f|}^{2}d{\sigma }_R$$

is the nonnegative functional measuring regular boundary trace mass, and

$$G_R=\left\{f\in T_R:{supp}_R\left(f\right)\subset {\Gamma }_R,L_R(f;{\Sigma }_R\setminus {\Gamma }_R)=0\right\}$$

is the trace-vanishing generating subspace associated with ${\Gamma }_R$.

Proof. 

$T_R$, ${\gamma }_R^{\mathrm{sing}}$, and ${\gamma }_R^{\mathrm{reg}}$ were constructed by Definition 4.43. ${supp}_R$ was constructed by Definition 4.46. $L_R$ was constructed by Definition 4.49. Finally, $G_R$ was defined by Definition 4.52, and its meaning was verified in Lemma 4.54. □

4.4. Boundary Bilinear Form and Boundary-Cancellation Theorem

In this subsection, using the regular boundary trace

$${\gamma }_R^{\mathrm{reg}}:T_R⟶L^2({\Sigma }_R,{\sigma }_R)$$

constructed in the preceding subsection, we define a regular boundary-trace bilinear form and prove that its boundary contribution vanishes on the trace-vanishing generating subspace associated with ${\Gamma }_R$ $G_R$. What is important here is that the boundary form $b_R$ is a ${\sigma }_R$-area-type regular boundary form and does not directly integrate the singular boundary trace

$${\gamma }_R^{\mathrm{sing}}f.$$

This separation allows one to eliminate only the regular-area boundary term while retaining the nontrivial singular boundary trace on ${\Gamma }_R$.

Definition 4.57 

(Regular-area-type boundary coefficient). From now on, fix one function

$${\omega }_R\in L^{\infty }({\Sigma }_R,{\sigma }_R).$$

We call this the regular boundary coefficient. If necessary, normalize it so that $\parallel {\omega }_R{\parallel }_{L^{\infty }({\Sigma }_R,{\sigma }_R)}\le 1$.

Definition 4.58 

(Regular-area-type boundary bilinear form). For $u,v\in T_R$, define

$$b_R[u,v]:={\int }_{{\Sigma }_R}{\omega }_R\left(\xi \right)\left({\gamma }_R^{\mathrm{reg}}u\right)\left(\xi \right)\overline{\left({\gamma }_R^{\mathrm{reg}}v\right)\left(\xi \right)}d{\sigma }_R\left(\xi \right).$$

We call this the ${\sigma }_R$-regular boundary-trace bilinear form.

Lemma 4.59 

(Boundedness of the boundary form). For any $u,v\in T_R$,

$$|b_R[u,v]|\le \parallel {\omega }_R{\parallel }_{L^{\infty }({\Sigma }_R,{\sigma }_R)}\parallel {\gamma }_R^{\mathrm{reg}}{u\parallel }_{L^2({\Sigma }_R,{\sigma }_R)}{\parallel {\gamma }_R^{\mathrm{reg}}v\parallel }_{L^2({\Sigma }_R,{\sigma }_R)}$$

holds. In particular, $b_R$ is bounded with respect to the $L^2({\Sigma }_R,{\sigma }_R)$-norm of the regular boundary trace.

Proof. 

By the Cauchy–Schwarz inequality,

$$\begin{array}{cc}\hfill |b_R[u,v]|& \le {\int }_{{\Sigma }_R}|{\omega }_R||{\gamma }_R^{\mathrm{reg}}u||{\gamma }_R^{\mathrm{reg}}v|d{\sigma }_R\hfill \\ & \le \parallel {\omega }_R{\parallel }_{L^{\infty }({\Sigma }_R,{\sigma }_R)}{\int }_{{\Sigma }_R}|{\gamma }_R^{\mathrm{reg}}u||{\gamma }_R^{\mathrm{reg}}v|d{\sigma }_R\hfill \\ & \le \parallel {\omega }_R{\parallel }_{L^{\infty }({\Sigma }_R,{\sigma }_R)}\parallel {\gamma }_R^{\mathrm{reg}}{u\parallel }_{L^2({\Sigma }_R,{\sigma }_R)}{\parallel {\gamma }_R^{\mathrm{reg}}v\parallel }_{L^2({\Sigma }_R,{\sigma }_R)}.\hfill \end{array}$$

This proves the claim. □

Remark 4.60 

(The singular boundary trace does not enter $b_R$). The form $b_R$ is defined using only

$${\gamma }_R^{\mathrm{reg}}u,{\gamma }_R^{\mathrm{reg}}v\in L^2({\Sigma }_R,{\sigma }_R).$$

Therefore, even if

$${\gamma }_R^{\mathrm{sing}}u,{\gamma }_R^{\mathrm{sing}}v$$

exist as nontrivial singular measures or distributions on ${\Gamma }_R$, they do not enter the area-type integral defining $b_R$. When this paper refers to boundary cancellation, what is cancelled is the ${\sigma }_R$-regular-area boundary term, not the singular measure ${\nu }_R$ supported on ${\Gamma }_R$ itself.

Lemma 4.61 

(Regular boundary contribution on a ${\sigma }_R$-null singular support). If $g,h\in L^2({\Sigma }_R,{\sigma }_R)$ satisfy

$$\underset{{\sigma }_R}{ess\; supp}g\subset {\Gamma }_R,\underset{{\sigma }_R}{ess\; supp}h\subset {\Gamma }_R,$$

then

$${\int }_{{\Sigma }_R}{\omega }_{R}g\overline{h}d{\sigma }_R=0.$$

Proof. 

By Lemma 4.25,

$${\sigma }_R\left({\Gamma }_R\right)=0.$$

Moreover, by assumption, g and h are zero ${\sigma }_R$-almost everywhere on ${\Sigma }_R\setminus {\Gamma }_R$. Hence

$${\omega }_{R}g\overline{h}=0{\sigma }_R-\mathrm{a}.\mathrm{e}.\mathrm{on}{\Sigma }_R\setminus {\Gamma }_R.$$

On the other hand, since ${\Gamma }_R$ is a ${\sigma }_R$-null set,

$${\int }_{{\Gamma }_R}|{\omega }_{R}g\overline{h}|d{\sigma }_R=0.$$

Therefore, in total,

$${\int }_{{\Sigma }_R}{\omega }_{R}g\overline{h}d{\sigma }_R=0.$$

□

Lemma 4.62 

(Vanishing of the regular trace by the regular-trace vanishing condition). For any $f\in G_R$,

$${\gamma }_R^{\mathrm{reg}}f=0{\sigma }_R-\mathrm{a}.\mathrm{e}.\mathrm{on}{\Sigma }_R$$

holds.

Proof. 

Let $f\in G_R$. By Definition 4.52,

$$L_R(f;{\Sigma }_R\setminus {\Gamma }_R)=0.$$

By Lemma 4.51, this is equivalent to

$${\gamma }_R^{\mathrm{reg}}f=0{\sigma }_R-\mathrm{a}.\mathrm{e}.\mathrm{on}{\Sigma }_R.$$

□

Theorem 4.63 

(Boundary cancellation on the trace-vanishing generating subspace associated with ${\Gamma }_R$). For any $u,v\in G_R$,

$$b_R[u,v]=0$$

holds.

Proof. 

By Lemma 4.62,

$${\gamma }_R^{\mathrm{reg}}u=0,{\gamma }_R^{\mathrm{reg}}v=0{\sigma }_R-\mathrm{a}.\mathrm{e}.$$

Therefore, by Definition 4.58,

$$\begin{array}{cc}\hfill b_R[u,v]& ={\int }_{{\Sigma }_R}{\omega }_R\left({\gamma }_R^{\mathrm{reg}}u\right)\overline{\left({\gamma }_R^{\mathrm{reg}}v\right)}d{\sigma }_R\hfill \\ & =0.\hfill \end{array}$$

What is used here is the vanishing of the regular boundary trace, not the vanishing of ${\gamma }_R^{\mathrm{sing}}u$ or ${\gamma }_R^{\mathrm{sing}}v$. Thus, even if singular boundary traces remain nontrivially on ${\Gamma }_R$, the ${\sigma }_R$-area-type boundary term vanishes. □

Corollary 4.64 

(Boundary cancellation on the linear span). For any

$$u,v\in span{G}_R,$$

one has

$$b_R[u,v]=0.$$

Proof. 

Let $u,v\in span{G}_R$. There exist finitely many elements $u_i,v_j\in G_R$ and coefficients $a_i,b_j\in \mathbb{C}$ such that

$$u=\sum _i{a}_i{u}_i,v=\sum _j{b}_j{v}_j.$$

Since $b_R$ is sesquilinear,

$$b_R[u,v]=\sum _{i,j}{a}_i\overline{b_j}{b}_R[u_i,v_j].$$

By Theorem 4.63, each term is zero. Therefore

$$b_R[u,v]=0.$$

□

Definition 4.65 

(Singular-boundary form closure). Define the form-norm closure of the trace-vanishing generating subspace associated with ${\Gamma }_R$ by

$$Q_R^{\mathrm{res}}:={\overline{span{G}_R}}^{{\parallel ·\parallel }_{q_R}}\subset Q_R.$$

We call this space the ${\sigma }_R$-null trace-vanishing form subspace.

Definition 4.66 

(Regular boundary form on the closure). Let $u,v\in Q_R^{\mathrm{res}}$. Take arbitrary sequences in $span{G}_R$

$$u_n\to u,v_n\to v\mathrm{in}(Q_R{,\parallel ·\parallel }_{q_R}).$$

Then define

$$b_R^{\mathrm{res}}[u,v]:=\underset{n\to \infty }{lim}{b}_R[u_n,v_n].$$

Lemma 4.67 

(Well-definedness of the boundary form on the closure). The right-hand side of Definition 4.66 does not depend on the choice of sequences, and

$$b_R^{\mathrm{res}}[u,v]=0.$$

Thus $b_R^{\mathrm{res}}$ is well-defined as the zero form on $Q_R^{\mathrm{res}}\times Q_R^{\mathrm{res}}$.

Proof. 

For any approximating sequences

$$u_n,v_n\in span{G}_R,$$

Corollary 4.64 gives

$$b_R[u_n,v_n]=0$$

for all n. Therefore

$$\underset{n\to \infty }{lim}{b}_R[u_n,v_n]=0.$$

This value clearly does not depend on the choice of approximating sequences. Hence $b_R^{\mathrm{res}}$ is uniquely defined as the zero form on the closure. □

Theorem 4.68 

(Extension of boundary cancellation to the form-norm closure). For any $u,v\in Q_R^{\mathrm{res}}$,

$$b_R^{\mathrm{res}}[u,v]=0.$$

Furthermore, if $u,v\in Q_R^{\mathrm{res}}\cap T_R$ and the regular trace ${\gamma }_R^{\mathrm{reg}}$ is defined for these elements in the closed-graph sense, then the original regular-trace-type boundary form also satisfies

$$b_R[u,v]=0.$$

Proof. 

The first assertion is exactly Lemma 4.67.

We prove the second assertion. Let $u\in Q_R^{\mathrm{res}}\cap T_R$. By the definition of $Q_R^{\mathrm{res}}$, there exists a sequence

$$u_n\in span{G}_R$$

such that

$$u_n\to u\mathrm{in}(Q_R{,\parallel ·\parallel }_{q_R}).$$

Since each $u_n$ belongs to $span{G}_R$, Lemma 4.62 and linearity imply

$${\gamma }_R^{\mathrm{reg}}{u}_n=0{\sigma }_R-\mathrm{a}.\mathrm{e}.$$

The regular trace being defined for u in the closed-graph sense means that if $u_n\to u$ and

$${\gamma }_R^{\mathrm{reg}}{u}_n\to g\mathrm{in}{L}^2({\Sigma }_R,{\sigma }_R),$$

then

$$g={\gamma }_R^{\mathrm{reg}}u.$$

Here

$${\gamma }_R^{\mathrm{reg}}{u}_n=0,$$

so the left-hand side converges to 0 in the $L^2$-norm. Therefore

$${\gamma }_R^{\mathrm{reg}}u=0.$$

Similarly,

$${\gamma }_R^{\mathrm{reg}}v=0$$

follows. Hence, by Definition 4.58,

$$b_R[u,v]={\int }_{{\Sigma }_R}{\omega }_R\left({\gamma }_R^{\mathrm{reg}}u\right)\overline{\left({\gamma }_R^{\mathrm{reg}}v\right)}d{\sigma }_R=0.$$

□

Remark 4.69 

(Cancellation on the closure is not cancellation of the singular trace). The cancellation identity extended to $Q_R^{\mathrm{res}}$,

$$b_R^{\mathrm{res}}=0,$$

means the vanishing of the regular-trace-type boundary form. It does not mean that

$${\gamma }_R^{\mathrm{sing}}u$$

vanishes. The singular boundary trace is an object on the ${\nu }_R$- or ${\mathcal{D}}_R^{\prime }$-side and is not included in the ${\sigma }_R$-area-type boundary integral. Therefore the structure is preserved in which the singular information on the ${\sigma }_R$-null singular support is retained while only the regular boundary trace mass is removed.

Proposition 4.70 

(Output of this subsection). This subsection has obtained the following objects and properties.

1.

The ${\sigma }_R$-regular boundary-trace bilinear form

$$b_R:T_R\times T_R⟶\mathbb{C}$$

has been defined.

2.

For any $u,v\in G_R$,

$$b_R[u,v]=0$$

holds.

3.

On the form-norm closure of the trace-vanishing generating subspace associated with ${\Gamma }_R$,

$$Q_R^{\mathrm{res}}={\overline{span{G}_R}}^{{\parallel ·\parallel }_{q_R}},$$

the zero extension of the boundary form

$$b_R^{\mathrm{res}}$$

is well-defined, and

$$b_R^{\mathrm{res}}[u,v]=0(u,v\in Q_R^{\mathrm{res}})$$

holds.

4.

This boundary cancellation is the vanishing of the ${\sigma }_R$-regular-area boundary term, and does not mean the vanishing of the singular boundary trace on ${\Gamma }_R$.

Proof. 

Item 1 follows from Definition 4.58. Item 2 follows from Theorem 4.63. Item 3 follows from Definition 4.65, Definition 4.66, and Theorem 4.68. Item 4 follows from the entire construction of this subsection, in particular from the separation between the regular trace and the singular trace, and from the immediately preceding remark. □

4.5. Closure of the ${\sigma }_R$-Null Trace-Vanishing Form Subspace

In this subsection, we do not reconstruct the closed quadratic form

$$q_R:Q_R\times Q_R⟶\mathbb{C}$$

constructed in Section 2. Here, using the trace-vanishing generating subspace associated with ${\Gamma }_R$

$$G_R\subset T_R\subset Q_R$$

constructed up to the preceding subsection, we cut out, from the form domain $Q_R$ of $q_R$, the closed form subspace satisfying the ${\sigma }_R$-null ${\sigma }_R$-null support condition.

In Section 2, $q_R$ was constructed as a closed form bounded from below. From now on, if necessary, we take a sufficiently large constant $c_R>0$ and replace it by

$$q_R^{\left(c_R\right)}[u,v]:=q_R[u,v]+c_R{\langle u,v\rangle }_{H_{\alpha ,+}}$$

so as to use a positive-definite form inner product. For simplicity, in this subsection we write the form norm after this positive shift as

$${\parallel u\parallel }_{q_R}^2:=q_R[u,u]+{\parallel u\parallel }_{H_{\alpha ,+}}^2.$$

Accordingly,

$$(Q_R,\parallel ·{\parallel }_{q_R})$$

is a Hilbert space, and the inclusion map

$$Q_R\hookrightarrow H_{\alpha ,+}$$

is continuous.

Definition 4.71 

(${\sigma }_R$-null trace-vanishing form subspace). Define the form-norm closure of the trace-vanishing generating subspace associated with ${\Gamma }_R$$G_R$ by

$$Q_R^{\mathrm{res}}:={\overline{span{G}_R}}^{{\parallel ·\parallel }_{q_R}}\subset Q_R.$$

This space is called the ${\sigma }_R$-null trace-vanishing form subspace. Also,

$$q_R^{\mathrm{res}}:=q_R{|}_{Q_R^{\mathrm{res}}\times Q_R^{\mathrm{res}}}$$

is called the restriction form of $q_R$ to the ${\sigma }_R$-null singular-boundary part.

Remark 4.72 

(Restriction operation performed in this subsection). $q_R^{\mathrm{res}}$ is not a new form, but the restriction of $q_R$, constructed in Section 2, to the closed subspace $Q_R^{\mathrm{res}}$ selected by the ${\sigma }_R$-null support and regular-trace vanishing conditions. Accordingly, in this subsection we do not reprove the self-adjoint realization or compactness of $q_R$, but only treat the closedness of the form domain cut out by the ${\sigma }_R$-null condition and the preservation of boundary cancellation.

Lemma 4.73 

(Linearity of the generating class). $G_R$ is a linear subspace of $Q_R$. Therefore

$$span{G}_R=G_R.$$

Proof. 

By the equivalent representation obtained in the preceding subsection,

$$G_R=\left\{f\in T_R:{\gamma }_R^{\mathrm{reg}}f=0{\sigma }_R-\mathrm{a}.\mathrm{e}.\right\}.$$

By Lemma 4.44, $T_R$ is a linear space, and

$${\gamma }_R^{\mathrm{reg}}:T_R\to L^2({\Sigma }_R,{\sigma }_R)$$

is a linear map. Therefore $G_R=ker{\gamma }_R^{\mathrm{reg}}$ is a linear subspace of $T_R$. Since $T_R\subset Q_R$, we have $G_R\subset Q_R$. Thus $G_R$ is a linear subspace of $Q_R$, and $span{G}_R=G_R$ follows. □

Definition 4.74 

(Nondegenerate trace core). Define the ${\sigma }_R$-null and blocking candidates on the core by

$${\mathcal{N}}_R:=\left\{\phi \in {\mathcal{C}}_R:{\vartheta }_R^{\mathrm{reg}}\phi =0,{\vartheta }_R^{\mathrm{sing}}\phi \ne 0\right\}.$$

Here ${\mathcal{C}}_R={\mathcal{D}}_R\cap Q_R$ is the trace core, and ${\vartheta }_R^{\mathrm{reg}}$ and ${\vartheta }_R^{\mathrm{sing}}$ are the regular boundary trace and singular boundary trace defined on the core.

Lemma 4.75 

(Existence of the nondegenerate trace core).

$${\mathcal{N}}_R\ne ⌀$$

holds. More concretely, there exists

$$\chi \in {\mathcal{C}}_R$$

such that

$${\vartheta }_R^{\mathrm{reg}}\chi =0,{\vartheta }_R^{\mathrm{sing}}\chi ={\nu }_R\ne 0.$$

Proof. 

Take $\chi \in C_c^{\infty }\left([0,\infty )\right)$ so that

$$\chi \left(0\right)=1,{\chi }^{\prime }\left(0\right)=0.$$

For example, one may take a smooth cut-off that equals 1 near the endpoint and becomes 0 for sufficiently large x. By the definition of the admissible core in Section 2,

$$\chi \in {\mathcal{C}}_R\subset Q_R.$$

Moreover, all derivatives of $\chi$ are bounded with polynomial weights, and hence

$$\chi \in {\mathcal{D}}_R.$$

Therefore

$$\chi \in {\mathcal{C}}_R={\mathcal{D}}_R\cap Q_R.$$

By the definition of the trace on the core,

$${\vartheta }_R^{\mathrm{sing}}\chi =\chi \left(0\right){\nu }_R={\nu }_R.$$

By Lemma 4.27, ${\nu }_R\left({\Gamma }_R\right)=1$, and therefore

$${\nu }_R\ne 0.$$

On the other hand, the regular boundary trace is

$${\vartheta }_R^{\mathrm{reg}}\chi ={\chi }^{\prime }\left(0\right)e_R=0.$$

Thus $\chi \in {\mathcal{N}}_R$, and

$${\mathcal{N}}_R\ne ⌀$$

follows. □

Lemma 4.76 

(Sufficient condition for nontriviality). If

$${\mathcal{N}}_R\ne ⌀,$$

then

$$G_R\ne \left\{0\right\}.$$

Furthermore, any $\phi \in {\mathcal{N}}_R$ gives a nonzero element of $G_R$.

Proof. 

Take $\phi \in {\mathcal{N}}_R$. We have $\phi \in {\mathcal{C}}_R\subset Q_R$, and the trace on the core belongs to the closed-trace graph through the constant sequence

$${\phi }_n=\phi .$$

Therefore

$$\phi \in T_R,{\gamma }_R^{\mathrm{sing}}\phi ={\vartheta }_R^{\mathrm{sing}}\phi ,{\gamma }_R^{\mathrm{reg}}\phi ={\vartheta }_R^{\mathrm{reg}}\phi .$$

By the definition of $\phi \in {\mathcal{N}}_R$,

$${\vartheta }_R^{\mathrm{reg}}\phi =0,$$

and hence

$${\gamma }_R^{\mathrm{reg}}\phi =0.$$

By the equivalent representation from the preceding subsection,

$$G_R=\{f\in T_R:{\gamma }_R^{\mathrm{reg}}f=0\},$$

we obtain

$$\phi \in G_R.$$

Furthermore,

$${\vartheta }_R^{\mathrm{sing}}\phi \ne 0,$$

and hence

$${\gamma }_R^{\mathrm{sing}}\phi \ne 0.$$

If $\phi =0$ held as an element of $Q_R$, then by uniqueness of the continuously defined closed trace one would have to have

$${\gamma }_R^{\mathrm{sing}}\phi =0.$$

This is a contradiction. Therefore $\phi$ is a nonzero element of $Q_R$, and $G_R\ne \left\{0\right\}$ follows. □

Corollary 4.77 

(Nontriviality of the trace-vanishing generating subspace associated with ${\Gamma }_R$).

$$G_R\ne \left\{0\right\},Q_R^{\mathrm{res}}\ne \left\{0\right\},H_R^{\mathrm{res}}\ne \left\{0\right\}.$$

Proof. 

By Lemma 4.75, ${\mathcal{N}}_R\ne ⌀$. Therefore Lemma 4.76 gives $G_R\ne \left\{0\right\}$. Furthermore, $G_R\subset Q_R^{\mathrm{res}}\subset H_{\alpha ,+}$, and since the form norm contains the $H_{\alpha ,+}$-norm, a nonzero element remains nonzero also in $H_{\alpha ,+}$. Thus $Q_R^{\mathrm{res}}\ne \left\{0\right\}$ and $H_R^{\mathrm{res}}\ne \left\{0\right\}$. □

Remark 4.78 

(Positioning of nontriviality). By Lemma 4.75, the nondegenerate trace-core condition

$${\mathcal{N}}_R\ne ⌀$$

is actually satisfied within the construction of this paper. Therefore Lemma 4.76 gives

$$G_R\ne \left\{0\right\}.$$

The closure, continuous embedding, and extension of boundary cancellation below hold formally even in the zero-space case, but in this paper, by the above nondegeneracy, we deal with a nonzero ${\sigma }_R$-null singular-boundary subspace.

Lemma 4.79 

(Minimality of the closure). $Q_R^{\mathrm{res}}$ is the smallest ${\parallel ·\parallel }_{q_R}$-closed linear subspace of $Q_R$ containing $G_R$:

$$Q_R^{\mathrm{res}}=\bigcap \left\{M\subset Q_R:M\mathrm{is}\mathrm{a}{\parallel ·\parallel }_{q_R}-\mathrm{closed}\mathrm{linear}\mathrm{subspace}\mathrm{and}{G}_R\subset M\right\}.$$

Proof. 

By definition,

$$Q_R^{\mathrm{res}}={\overline{span{G}_R}}^{{\parallel ·\parallel }_{q_R}}.$$

By Lemma 4.73,

$$span{G}_R=G_R,$$

and hence

$$Q_R^{\mathrm{res}}={\overline{G_R}}^{{\parallel ·\parallel }_{q_R}}.$$

Thus $Q_R^{\mathrm{res}}$ is a closed linear subspace containing $G_R$.

Conversely, if $M\subset Q_R$ is a ${\parallel ·\parallel }_{q_R}$-closed linear subspace satisfying $G_R\subset M$, then M also contains the closure of $G_R$. That is,

$$Q_R^{\mathrm{res}}={\overline{G_R}}^{{\parallel ·\parallel }_{q_R}}\subset M.$$

Therefore the intersection representation above holds. □

Theorem 4.80 

(Completeness of the ${\sigma }_R$-null trace-vanishing form subspace).

$$(Q_R^{\mathrm{res}},\parallel ·{\parallel }_{q_R})$$

is a Hilbert space. In particular, $Q_R^{\mathrm{res}}$ is a ${\parallel ·\parallel }_{q_R}$-closed linear subspace of $Q_R$.

Proof. 

By the analytic data of Section 2,

$$(Q_R,\parallel ·{\parallel }_{q_R})$$

is a Hilbert space. By Definition 4.71,

$$Q_R^{\mathrm{res}}={\overline{G_R}}^{{\parallel ·\parallel }_{q_R}},$$

and therefore $Q_R^{\mathrm{res}}$ is a closed linear subspace of $Q_R$. A closed linear subspace of a Hilbert space is again a Hilbert space with respect to the induced norm. Accordingly,

$$(Q_R^{\mathrm{res}},\parallel ·{\parallel }_{q_R})$$

is complete. □

Proposition 4.81 

(Continuous embedding into the Hilbert space). The inclusion map

$$j_R^{\mathrm{res}}:Q_R^{\mathrm{res}}\hookrightarrow H_{\alpha ,+}$$

is continuous. More concretely, for any $u\in Q_R^{\mathrm{res}}$,

$${\parallel u\parallel }_{H_{\alpha ,+}}\le {\parallel u\parallel }_{q_R}$$

holds.

Proof. 

We have $Q_R^{\mathrm{res}}\subset Q_R$, and ${\parallel ·\parallel }_{q_R}$ is chosen as

$${\parallel u\parallel }_{q_R}^2=q_R[u,u]+{\parallel u\parallel }_{H_{\alpha ,+}}^2.$$

Since this is the positively shifted form norm, the right-hand side is nonnegative, and in particular

$${\parallel u\parallel }_{H_{\alpha ,+}}^2\le {\parallel u\parallel }_{q_R}^2$$

holds. Therefore

$${\parallel u\parallel }_{H_{\alpha ,+}}\le {\parallel u\parallel }_{q_R}.$$

Hence the inclusion map is continuous. □

Lemma 4.82 

(Closedness of the restriction form). The restriction form

$$q_R^{\mathrm{res}}=q_R{|}_{Q_R^{\mathrm{res}}\times Q_R^{\mathrm{res}}}$$

is a closed lower-bounded form on $H_{\alpha ,+}$. Its form domain is $Q_R^{\mathrm{res}}$.

Proof. 

$q_R$ was constructed in Section 2 as a closed lower-bounded form. Therefore, with respect to the positively shifted form norm ${\parallel ·\parallel }_{q_R}$,

$$(Q_R,\parallel ·{\parallel }_{q_R})$$

is a Hilbert space. By Theorem 4.80,

$$Q_R^{\mathrm{res}}$$

is a closed subspace of this Hilbert space. Thus

$$(Q_R^{\mathrm{res}},\parallel ·{\parallel }_{q_R})$$

is also a Hilbert space.

The restriction of a closed form to a closed subspace is a closed form. Indeed, if a sequence $\left\{u_n\right\}$ in $Q_R^{\mathrm{res}}$ is ${\parallel ·\parallel }_{q_R}$-Cauchy, then it is also Cauchy in $Q_R$. By completeness of $Q_R$, there exists some $u\in Q_R$ such that

$$u_n\to u\mathrm{in}{\parallel ·\parallel }_{q_R}.$$

Since $Q_R^{\mathrm{res}}$ is closed in $Q_R$,

$$u\in Q_R^{\mathrm{res}}.$$

Therefore $Q_R^{\mathrm{res}}$ is complete with respect to the form norm of the restriction form, and $q_R^{\mathrm{res}}$ is closed. Lower boundedness follows immediately because it is merely the restriction of the lower boundedness of $q_R$. □

Definition 4.83 

(Boundary form on the closure). Write the zero extension constructed in the preceding subsection as

$$b_R^{\mathrm{res}}:Q_R^{\mathrm{res}}\times Q_R^{\mathrm{res}}⟶\mathbb{C}.$$

Namely, for $u,v\in Q_R^{\mathrm{res}}$, take arbitrary approximating sequences

$$u_n,v_n\in G_R,u_n\to u,v_n\to v\mathrm{in}{\parallel ·\parallel }_{q_R}.$$

Then

$$b_R^{\mathrm{res}}[u,v]:=\underset{n\to \infty }{lim}{b}_R[u_n,v_n].$$

Theorem 4.84 

(Preservation of boundary cancellation under form closure). For any $u,v\in Q_R^{\mathrm{res}}$,

$$b_R^{\mathrm{res}}[u,v]=0$$

holds. Therefore, the cancellation of the regular-area boundary term that held on the trace-vanishing generating subspace associated with ${\Gamma }_R$ is preserved on the entire form-norm closure $Q_R^{\mathrm{res}}$.

Proof. 

By definition, for any $u,v\in Q_R^{\mathrm{res}}$, one may take sequences

$$u_n,v_n\in G_R$$

such that

$$u_n\to u,v_n\to v\mathrm{in}{\parallel ·\parallel }_{q_R}.$$

By the boundary-cancellation theorem in the preceding subsection, for each n,

$$b_R[u_n,v_n]=0.$$

Therefore

$$b_R^{\mathrm{res}}[u,v]=\underset{n\to \infty }{lim}{b}_R[u_n,v_n]=0.$$

Since this value does not depend on the choice of approximating sequences, $b_R^{\mathrm{res}}$ is well-defined as the zero form on $Q_R^{\mathrm{res}}$. □

Corollary 4.85 

(Vanishing of the ordinary boundary form when a closed-graph trace exists). Let $u,v\in Q_R^{\mathrm{res}}\cap T_R$, and suppose that the regular trace ${\gamma }_R^{\mathrm{reg}}$ for these elements is determined from approximating sequences in $Q_R^{\mathrm{res}}$ in the closed-graph sense. Then

$$b_R[u,v]=0.$$

Proof. 

Let $u\in Q_R^{\mathrm{res}}\cap T_R$. By definition, there exists a sequence

$$u_n\in G_R$$

such that

$$u_n\to u\mathrm{in}{\parallel ·\parallel }_{q_R}.$$

For every $u_n\in G_R$,

$${\gamma }_R^{\mathrm{reg}}{u}_n=0{\sigma }_R-\mathrm{a}.\mathrm{e}.$$

By the closed-graph assumption, the regular trace corresponding to this limit must be

$${\gamma }_R^{\mathrm{reg}}u=0.$$

Similarly,

$${\gamma }_R^{\mathrm{reg}}v=0.$$

Therefore

$$b_R[u,v]={\int }_{{\Sigma }_R}{\omega }_R\left({\gamma }_R^{\mathrm{reg}}u\right)\overline{\left({\gamma }_R^{\mathrm{reg}}v\right)}d{\sigma }_R=0.$$

□

Proposition 4.86 

(Closure data of the ${\sigma }_R$-null singular-boundary part). The construction in this subsection gives the following.

1.

$G_R$ is a linear subspace of $Q_R$, and since

$${\mathcal{N}}_R\ne ⌀$$

holds within the present construction, $G_R\ne \left\{0\right\}$.

2.

$$Q_R^{\mathrm{res}}={\overline{G_R}}^{{\parallel ·\parallel }_{q_R}}$$

is a ${\parallel ·\parallel }_{q_R}$-closed linear subspace of $Q_R$, and $(Q_R^{\mathrm{res}},\parallel ·{\parallel }_{q_R})$ is a Hilbert space.

3.

The inclusion map

$$Q_R^{\mathrm{res}}\hookrightarrow H_{\alpha ,+}$$

is continuous.

4.

The restriction form

$$q_R^{\mathrm{res}}=q_R{|}_{Q_R^{\mathrm{res}}\times Q_R^{\mathrm{res}}}$$

is a closed lower-bounded form.

5.

The regular-trace-type boundary form satisfies

$$b_R^{\mathrm{res}}[u,v]=0(u,v\in Q_R^{\mathrm{res}}).$$

Proof. 

Item 1 follows from Lemma 4.73, Lemma 4.75, and Lemma 4.76. Item 2 follows from Theorem 4.80. Item 3 follows from Proposition 4.81. Item 4 follows from Lemma 4.82. Item 5 follows from Theorem 4.84. □

Remark 4.87 

(Meaning for the next step). The space $Q_R^{\mathrm{res}}$ obtained in this subsection is the form domain satisfying the ${\sigma }_R$-null support and regular-trace vanishing conditions. On this space, the regular-trace-type boundary term vanishes. On the other hand, since the singular boundary trace remains on the ${\nu }_R$- or ${\mathcal{D}}_R^{\prime }$-side, the boundary trace itself is not trivialized. In the next subsection, we apply the representation theorem to the closed lower-bounded form $q_R^{\mathrm{res}}$ and construct the corresponding singular-boundary Hilbert space and self-adjoint realization.

4.6. Singular-Boundary Hilbert Space and Friedrichs Realization

In this subsection, from the ${\sigma }_R$-null trace-vanishing form subspace

$$Q_R^{\mathrm{res}}\subset Q_R$$

constructed in the preceding subsection, we construct the corresponding Hilbert space

$$H_R^{\mathrm{res}},$$

and by applying the first representation theorem to the restriction form

$$q_R^{\mathrm{res}}=q_R{|}_{Q_R^{\mathrm{res}}\times Q_R^{\mathrm{res}}},$$

we obtain the self-adjoint operator

$$A_R^{\mathrm{res}}.$$

The operator obtained here does not redefine the global operator associated with $q_R$ constructed in Section 2. It is a partial Friedrichs-type realization corresponding to the closed form $q_R^{\mathrm{res}}$ on the singular-boundary subspace cut out by the ${\sigma }_R$-null support and regular-trace vanishing conditions.

From now on, in this section, following the positive-shift convention of Section 4.5, we write as if $q_R$ has been normalized so that

$$q_R^{\mathrm{res}}[u,u]\ge 0(u\in Q_R^{\mathrm{res}}).$$

In the lower-bounded case, this should be interpreted as replacing the form by an equivalent form obtained by adding a sufficiently large constant and then using the same notation $q_R$.

Definition 4.88 

(Singular-boundary Hilbert space). Define the ${\sigma }_R$-null singular-boundary Hilbert space by

$$H_R^{\mathrm{res}}:={\overline{Q_R^{\mathrm{res}}}}^{{\parallel ·\parallel }_{H_{\alpha ,+}}}\subset H_{\alpha ,+}.$$

Equip $H_R^{\mathrm{res}}$ with the inner product induced from $H_{\alpha ,+}$:

$${\langle u,v\rangle }_{H_R^{\mathrm{res}}}:={\langle u,v\rangle }_{H_{\alpha ,+}}(u,v\in H_R^{\mathrm{res}}).$$

Lemma 4.89 

(Closedness of the singular-boundary Hilbert space). $H_R^{\mathrm{res}}$ is a closed linear subspace of $H_{\alpha ,+}$. In particular, $H_R^{\mathrm{res}}$ is a Hilbert space.

Proof. 

By definition,

$$H_R^{\mathrm{res}}={\overline{Q_R^{\mathrm{res}}}}^{{\parallel ·\parallel }_{H_{\alpha ,+}}},$$

so it is a closed set obtained as a closure in $H_{\alpha ,+}$. Moreover, since $Q_R^{\mathrm{res}}$ is a linear space, its $H_{\alpha ,+}$-norm closure is also a linear space. Therefore $H_R^{\mathrm{res}}$ is a closed linear subspace of $H_{\alpha ,+}$. A closed linear subspace of a Hilbert space is again a Hilbert space, and the claim follows. □

Lemma 4.90 

(Density of the form domain). $Q_R^{\mathrm{res}}$ is dense in $H_R^{\mathrm{res}}$. Namely,

$${\overline{Q_R^{\mathrm{res}}}}^{{\parallel ·\parallel }_{H_R^{\mathrm{res}}}}=H_R^{\mathrm{res}}.$$

Proof. 

This is exactly the definition of $H_R^{\mathrm{res}}$. Indeed, $H_R^{\mathrm{res}}$ is defined as the $H_{\alpha ,+}$-norm closure of $Q_R^{\mathrm{res}}$, and the norm of $H_R^{\mathrm{res}}$ is induced from $H_{\alpha ,+}$. Therefore $Q_R^{\mathrm{res}}$ is dense in $H_R^{\mathrm{res}}$. □

Proposition 4.91 

(Continuous embedding from the form domain into the singular-boundary Hilbert space). The natural inclusion map

$${\iota }_R^{\mathrm{res}}:Q_R^{\mathrm{res}}\hookrightarrow H_R^{\mathrm{res}}$$

is continuous. More concretely, for any $u\in Q_R^{\mathrm{res}}$,

$${\parallel u\parallel }_{H_R^{\mathrm{res}}}={\parallel u\parallel }_{H_{\alpha ,+}}\le {\parallel u\parallel }_{q_R}$$

holds.

Proof. 

By Proposition 4.81 of the preceding subsection, for any $u\in Q_R^{\mathrm{res}}$,

$${\parallel u\parallel }_{H_{\alpha ,+}}\le {\parallel u\parallel }_{q_R}$$

holds. Moreover, the norm of $H_R^{\mathrm{res}}$ is the restriction of the norm of $H_{\alpha ,+}$, and hence

$${\parallel u\parallel }_{H_R^{\mathrm{res}}}={\parallel u\parallel }_{H_{\alpha ,+}}.$$

The claim follows. □

Lemma 4.92 

(Closedness and density of the restriction form). The form

$$q_R^{\mathrm{res}}=q_R{|}_{Q_R^{\mathrm{res}}\times Q_R^{\mathrm{res}}}$$

is a densely defined closed nonnegative symmetric form on the Hilbert space $H_R^{\mathrm{res}}$. Its form domain is $Q_R^{\mathrm{res}}$.

Proof. 

First, density follows from Lemma 4.90.

Next we prove closedness. By Lemma 4.82 of the preceding subsection, $q_R^{\mathrm{res}}$ is a closed lower-bounded form on $Q_R^{\mathrm{res}}$, and

$$(Q_R^{\mathrm{res}},\parallel ·{\parallel }_{q_R})$$

is a Hilbert space. In this subsection, by the positive-shift convention, we have

$$q_R^{\mathrm{res}}[u,u]\ge 0.$$

Therefore

$${\parallel u\parallel }_{q_R}^2=q_R^{\mathrm{res}}[u,u]+{\parallel u\parallel }_{H_R^{\mathrm{res}}}^2$$

is the form norm of $q_R^{\mathrm{res}}$. Thus $Q_R^{\mathrm{res}}$ is complete with respect to this form norm, and $q_R^{\mathrm{res}}$ is a closed form on $H_R^{\mathrm{res}}$.

Symmetry is the restriction of the symmetry of $q_R$ constructed in Section 2, and nonnegativity follows from the positive-shift convention. Therefore $q_R^{\mathrm{res}}$ is a densely defined closed nonnegative symmetric form on $H_R^{\mathrm{res}}$. □

Definition 4.93 

(Closed-form Hilbert space). Let

$${\mathcal{Q}}_R^{\mathrm{res}}$$

denote the Hilbert space obtained by equipping $Q_R^{\mathrm{res}}$ with the inner product

$${\langle u,v\rangle }_{q_R^{\mathrm{res}}}:=q_R^{\mathrm{res}}[u,v]+{\langle u,v\rangle }_{H_R^{\mathrm{res}}}.$$

Namely,

$${\mathcal{Q}}_R^{\mathrm{res}}:=(Q_R^{\mathrm{res}},{\langle ·,·\rangle }_{q_R^{\mathrm{res}}}).$$

Lemma 4.94 

(Continuous noncompact embedding from the form space to the base Hilbert space). The natural map

$${\mathcal{Q}}_R^{\mathrm{res}}⟶H_R^{\mathrm{res}}$$

is continuous. At this level, compactness is not asserted.

Proof. 

For any $u\in Q_R^{\mathrm{res}}$,

$${\parallel u\parallel }_{H_R^{\mathrm{res}}}^2\le q_R^{\mathrm{res}}[u,u]+{\parallel u\parallel }_{H_R^{\mathrm{res}}}^2={\parallel u\parallel }_{q_R^{\mathrm{res}}}^2.$$

Thus the inclusion map is continuous with norm at most 1. Compactness is an additional property of the embedding and will be treated in the later subsection on compact resolvent. □

Theorem 4.95 

(Friedrichs realization of the ${\sigma }_R$-null restricted closed form). For the closed nonnegative symmetric form

$$q_R^{\mathrm{res}},$$

there exists a unique nonnegative self-adjoint operator

$$A_R^{\mathrm{res}}$$

on the Hilbert space $H_R^{\mathrm{res}}$ satisfying the following.

$$Dom\left(A_R^{\mathrm{res}}\right)=\left\{u\in Q_R^{\mathrm{res}}:\exists w\in H_R^{\mathrm{res}}\mathrm{such}\mathrm{that}{q}_R^{\mathrm{res}}[u,v]={\langle w,v\rangle }_{H_R^{\mathrm{res}}}\mathrm{for}\mathrm{all}v\in Q_R^{\mathrm{res}}\right\},$$

and, defining

$$A_R^{\mathrm{res}}u:=w$$

for such w, one has

$$q_R^{\mathrm{res}}[u,v]={\langle A_R^{\mathrm{res}}u,v\rangle }_{H_R^{\mathrm{res}}}$$

for all

$$u\in Dom\left(A_R^{\mathrm{res}}\right),v\in Q_R^{\mathrm{res}}.$$

Proof. 

By Lemma 4.92,

$$q_R^{\mathrm{res}}$$

is a densely defined closed nonnegative symmetric form on the Hilbert space $H_R^{\mathrm{res}}$. Therefore the first representation theorem for closed lower-bounded forms, namely the Friedrichs-type representation theorem, applies. This theorem yields a unique nonnegative self-adjoint operator

$$A_R^{\mathrm{res}},$$

and this operator represents the closed form $q_R^{\mathrm{res}}$.

More concretely, the first representation theorem characterizes the operator as follows. For $u\in Q_R^{\mathrm{res}}$, when the linear functional

$$v⟼q_R^{\mathrm{res}}[u,v]$$

is continuous with respect to the $H_R^{\mathrm{res}}$-norm, the Riesz representation theorem gives a unique

$$w\in H_R^{\mathrm{res}}$$

such that

$$q_R^{\mathrm{res}}[u,v]={\langle w,v\rangle }_{H_R^{\mathrm{res}}}(v\in Q_R^{\mathrm{res}}).$$

The set of all such u is defined to be

$$Dom\left(A_R^{\mathrm{res}}\right),$$

and

$$A_R^{\mathrm{res}}u=w.$$

The operator obtained by this construction is nonnegative and self-adjoint, and uniquely represents the closed form $q_R^{\mathrm{res}}$. Uniqueness also follows from the uniqueness part of the first representation theorem. □

Corollary 4.96 

(Nonnegativity). For any $u\in Dom\left(A_R^{\mathrm{res}}\right)$,

$${\langle A_R^{\mathrm{res}}u,u\rangle }_{H_R^{\mathrm{res}}}=q_R^{\mathrm{res}}[u,u]\ge 0$$

holds. Therefore

$$A_R^{\mathrm{res}}\ge 0.$$

Proof. 

In the representation formula of Theorem 4.95, set $v=u$. Then

$${\langle A_R^{\mathrm{res}}u,u\rangle }_{H_R^{\mathrm{res}}}=q_R^{\mathrm{res}}[u,u].$$

By the positive-shift convention, the right-hand side is nonnegative. Therefore $A_R^{\mathrm{res}}$ is a nonnegative self-adjoint operator. □

Remark 4.97 

(Distinction from the global operator). When the global operator associated with $q_R$ obtained in Section 2 is written as

$$A_R,$$

the operator

$$A_R^{\mathrm{res}}$$

of this subsection is not in general defined as a simple operator restriction of $A_R$. $A_R^{\mathrm{res}}$ is the operator obtained by restricting the closed form

$$q_R$$

to the ${\sigma }_R$-null restricted closed form domain

$$Q_R^{\mathrm{res}}$$

and then representing that restricted form on the base Hilbert space

$$H_R^{\mathrm{res}}.$$

Therefore, unless it is separately proved that

$$H_R^{\mathrm{res}}$$

is an operator-theoretic reducing subspace for the global operator $A_R$, we do not write

$$A_R^{\mathrm{res}}=A_R{|}_{H_R^{\mathrm{res}}}.$$

What is needed in this paper is not a simple restriction of the global operator, but

$$A_R^{\mathrm{res}}$$

as the closed-form realization of the ${\sigma }_R$-null restricted closed form.

Definition 4.98 

(Positive shifted operator). Define the ${\sigma }_R$-null positive shifted operator on $K_R$ by

$$L_R^{\mathrm{res}}:=A_R^{\mathrm{res}}+I_{H_R^{\mathrm{res}}}.$$

Here $I_{H_R^{\mathrm{res}}}$ is the identity operator on $H_R^{\mathrm{res}}$. Its domain is

$$Dom\left(L_R^{\mathrm{res}}\right)=Dom\left(A_R^{\mathrm{res}}\right).$$

Lemma 4.99 

(Basic properties of the positive shifted operator). $L_R^{\mathrm{res}}$ is a self-adjoint operator on $H_R^{\mathrm{res}}$, and

$$L_R^{\mathrm{res}}\ge I_{H_R^{\mathrm{res}}}.$$

In particular,

$$0\in \rho \left(L_R^{\mathrm{res}}\right),$$

and

$$\parallel {\left(L_R^{\mathrm{res}}\right)}^{-1}{\parallel }_{\mathcal{B}\left(H_R^{\mathrm{res}}\right)}\le 1.$$

Proof. 

By Theorem 4.95, $A_R^{\mathrm{res}}$ is self-adjoint. Adding the bounded self-adjoint operator $I_{H_R^{\mathrm{res}}}$ to a self-adjoint operator gives

$$L_R^{\mathrm{res}}=A_R^{\mathrm{res}}+I_{H_R^{\mathrm{res}}},$$

which is self-adjoint on the same domain.

Moreover, by Corollary 4.96,

$$A_R^{\mathrm{res}}\ge 0.$$

Therefore

$$L_R^{\mathrm{res}}=A_R^{\mathrm{res}}+I_{H_R^{\mathrm{res}}}\ge I_{H_R^{\mathrm{res}}}.$$

By the spectral theorem,

$$\sigma \left(L_R^{\mathrm{res}}\right)\subset [1,\infty ).$$

Hence $0\notin \sigma \left(L_R^{\mathrm{res}}\right)$, that is,

$$0\in \rho \left(L_R^{\mathrm{res}}\right).$$

Furthermore,

$$\parallel {\left(L_R^{\mathrm{res}}\right)}^{-1}\parallel =\underset{\lambda \in \sigma \left(L_R^{\mathrm{res}}\right)}{sup}{\displaystyle \frac{1}{\lambda }}\le 1.$$

□

Definition 4.100 

(Energy inner product on the singular-boundary solution space). Define the energy inner product associated with $L_R^{\mathrm{res}}$ by

$${\langle u,v\rangle }_{L_R^{\mathrm{res}}}:={\langle {\left(L_R^{\mathrm{res}}\right)}^{1/2}u,{\left(L_R^{\mathrm{res}}\right)}^{1/2}v\rangle }_{H_R^{\mathrm{res}}}.$$

Its domain is

$$Dom\left({\left(L_R^{\mathrm{res}}\right)}^{1/2}\right)=Q_R^{\mathrm{res}},$$

and

$${\parallel u\parallel }_{L_R^{\mathrm{res}}}^2=q_R^{\mathrm{res}}[u,u]+{\parallel u\parallel }_{H_R^{\mathrm{res}}}^2$$

holds.

Lemma 4.101 

(Agreement of square-root domain and form domain).

$$Dom\left({\left(L_R^{\mathrm{res}}\right)}^{1/2}\right)=Q_R^{\mathrm{res}},$$

and for any $u,v\in Q_R^{\mathrm{res}}$,

$${\langle {\left(L_R^{\mathrm{res}}\right)}^{1/2}u,{\left(L_R^{\mathrm{res}}\right)}^{1/2}v\rangle }_{H_R^{\mathrm{res}}}=q_R^{\mathrm{res}}[u,v]+{\langle u,v\rangle }_{H_R^{\mathrm{res}}}$$

holds.

Proof. 

$A_R^{\mathrm{res}}$ is the representing operator of the closed nonnegative form $q_R^{\mathrm{res}}$. By the square-root representation in the first representation theorem,

$$Dom\left({(A_R^{\mathrm{res}}+I)}^{1/2}\right)=Q_R^{\mathrm{res}},$$

and its form is given by

$$q_R^{\mathrm{res}}[u,v]+{\langle u,v\rangle }_{H_R^{\mathrm{res}}}.$$

By Definition 4.98,

$$L_R^{\mathrm{res}}=A_R^{\mathrm{res}}+I,$$

and the claim follows. □

Proposition 4.102 

(Propagation of nondegeneracy to the Hilbert space). If

$$G_R\ne \left\{0\right\},$$

then

$$Q_R^{\mathrm{res}}\ne \left\{0\right\},H_R^{\mathrm{res}}\ne \left\{0\right\}.$$

In particular, if the condition

$${\mathcal{N}}_R\ne ⌀$$

of Lemma 4.76 holds, then $H_R^{\mathrm{res}}$ is a nonzero Hilbert space.

Proof. 

Assume $G_R\ne \left\{0\right\}$, and take a nonzero element $g\in G_R$. Since $G_R\subset Q_R^{\mathrm{res}}$,

$$Q_R^{\mathrm{res}}\ne \left\{0\right\}.$$

Moreover, $Q_R^{\mathrm{res}}\subset H_{\alpha ,+}$, and the form norm contains the $H_{\alpha ,+}$-norm. Therefore g, being nonzero as an element of $Q_R$, is also nonzero as an element of $H_{\alpha ,+}$. Thus its $H_{\alpha ,+}$-closure,

$$H_R^{\mathrm{res}},$$

is also nonzero.

The final assertion follows from Lemma 4.75 and Lemma 4.76. □

Proposition 4.103 

(Output of this subsection). This subsection has constructed the following objects:

$$H_R^{\mathrm{res}},A_R^{\mathrm{res}},L_R^{\mathrm{res}}.$$

They satisfy the following properties.

1.

$H_R^{\mathrm{res}}$ is a closed Hilbert subspace of $H_{\alpha ,+}$.

2.

$Q_R^{\mathrm{res}}$ is dense in $H_R^{\mathrm{res}}$, and $q_R^{\mathrm{res}}$ is a closed nonnegative symmetric form on $H_R^{\mathrm{res}}$.

3.

$A_R^{\mathrm{res}}$ is the unique nonnegative self-adjoint representing operator of $q_R^{\mathrm{res}}$.

4.

$$L_R^{\mathrm{res}}=A_R^{\mathrm{res}}+I_{H_R^{\mathrm{res}}}$$

is self-adjoint, and

$$L_R^{\mathrm{res}}\ge I_{H_R^{\mathrm{res}}}.$$

5.

$A_R^{\mathrm{res}}$ is not a redefinition of the global operator of Section 2, but a Friedrichs-type realization on the ${\sigma }_R$-null trace-vanishing form subspace.

Proof. 

Item 1 follows from Lemma 4.89. Item 2 follows from Lemma 4.90 and Lemma 4.92. Item 3 follows from Theorem 4.95. Item 4 follows from Definition 4.98 and Lemma 4.99. Item 5 follows from the nature of the construction in this subsection, in particular from the fact that $A_R^{\mathrm{res}}$ was constructed from the restriction form of $q_R$ to the closed subspace $Q_R^{\mathrm{res}}$. □

Remark 4.104 

(Transition to the next step). The objects obtained in this subsection,

$$H_R^{\mathrm{res}},A_R^{\mathrm{res}},L_R^{\mathrm{res}},$$

are the operator-theoretic foundation on the singular-boundary subspace. In the next subsection, we introduce a strongly continuous transport group on this Hilbert space and construct its anti-self-adjoint generator by Stone’s theorem. After that, through the distribution kernel representation, we formulate the distribution kernel associated with $K_R$ as an object of ${\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }$.

4.7. Transport Group on $K_R$, Anti-Self-Adjoint Generator, and Kernel Representation

In this subsection, we introduce the canonical strongly continuous unitary transport group on the ${\sigma }_R$-null singular-boundary Hilbert space

$$H_R^{\mathrm{res}}$$

constructed in the preceding subsection, and obtain its anti-self-adjoint generator by Stone’s theorem. Furthermore, by evaluating this transport group on the dual side of the test space

$${\mathcal{D}}_R\subset H_{\alpha ,+},$$

we define the distribution kernel associated with $K_R$ not as an ordinary function kernel, but as an element of

$${\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }.$$

What is important here is that the transport group and its generator are constructed internally within the ${\sigma }_R$-null singular-boundary subspace

$$H_R^{\mathrm{res}}.$$

Accordingly, the transport in this subsection is not an externally given geometric flow, but the canonical unitary transport obtained from the spectral calculus of the positive shifted operator

$$L_R^{\mathrm{res}}=A_R^{\mathrm{res}}+I.$$

Definition 4.105 

(Orthogonal projection onto $K_R$). By Lemma 4.89, $H_R^{\mathrm{res}}$ is a closed linear subspace of $H_{\alpha ,+}$. Therefore, there exists a unique orthogonal projection on $H_{\alpha ,+}$

$${\Pi }_R^{+}:H_{\alpha ,+}⟶H_R^{\mathrm{res}}.$$

We call this projection the orthogonal projection onto $K_R$ on the one-sided analytic Hilbert space.

Remark 4.106 

(Distinction from the subsequent integrated projector). ${\Pi }_R^{+}$ is the local Hilbert projection from within the one-sided analytic Hilbert space $H_{\alpha ,+}$ onto $H_R^{\mathrm{res}}$. The unsuperscripted projection

$${\Pi }_R$$

used in the subsequent orthogonal-decomposition framework is the projector onto the singular-boundary subspace after it has been lifted to the ambient Hilbert-space setting, and its object space is different from that of the one-sided projector ${\Pi }_R^{+}$ in this subsection. Accordingly, in this subsection ${\Pi }_R^{+}$ denotes the one-sided projection, while ${\Pi }_R$ without + is reserved for Section 5.

Definition 4.107 

(Canonical transport group on $K_R$). By Lemma 4.99, the positive shifted operator

$$L_R^{\mathrm{res}}$$

is a self-adjoint operator on $H_R^{\mathrm{res}}$. Therefore, by the spectral theorem, define

$$U_R\left(t\right):=e^{-it{L}_R^{\mathrm{res}}}(t\in \mathbb{R}).$$

We call this the canonical transport group on $K_R$ on the ${\sigma }_R$-null singular-boundary subspace.

Theorem 4.108 

(Strong continuity of the canonical transport group on $K_R$). The family

$${\left\{U_R\left(t\right)\right\}}_{t\in \mathbb{R}}$$

is a strongly continuous unitary group on $H_R^{\mathrm{res}}$. Namely,

$$U_R\left(0\right)=I_{H_R^{\mathrm{res}}},U_R(t+s)=U_R\left(t\right)U_R\left(s\right),U_R{\left(t\right)}^{*}=U_R(-t)$$

hold, and for every $u\in H_R^{\mathrm{res}}$,

$$\underset{t\to 0}{lim}{\parallel U_R\left(t\right)u-u\parallel }_{H_R^{\mathrm{res}}}=0.$$

Proof. 

Since $L_R^{\mathrm{res}}$ is self-adjoint, the spectral theorem implies that

$$e^{-it{L}_R^{\mathrm{res}}}$$

is a unitary operator for each $t\in \mathbb{R}$. Moreover, from the multiplicative law of the function

$$\lambda ⟼e^{-it\lambda },$$

we obtain

$$U_R(t+s)=U_R\left(t\right)U_R\left(s\right),U_R\left(0\right)=I_{H_R^{\mathrm{res}}}.$$

For the adjoint, we also have

$$U_R{\left(t\right)}^{*}=e^{it{L}_R^{\mathrm{res}}}=U_R(-t).$$

We prove strong continuity. Let $E_L$ be the spectral measure of $L_R^{\mathrm{res}}$. For any $u\in H_R^{\mathrm{res}}$,

$$\parallel U_R{\left(t\right)u-u\parallel }_{H_R^{\mathrm{res}}}^2={\int }_{\sigma \left(L_R^{\mathrm{res}}\right)}|e^{-it\lambda }{-1|}^{2}d{\parallel E_L\left(\lambda \right)u\parallel }^2.$$

For each $\lambda$,

$$|e^{-it\lambda }{-1|}^2\to 0(t\to 0),$$

and

$$|e^{-it\lambda }{-1|}^2\le 4.$$

The dominating measure on the right-hand side is the finite measure

$$d\parallel E_L{\left(\lambda \right)u\parallel }^2.$$

Therefore, by the dominated convergence theorem,

$$\parallel U_R{\left(t\right)u-u\parallel }_{H_R^{\mathrm{res}}}\to 0.$$

Thus strong continuity holds. □

Definition 4.109 

(Anti-self-adjoint transport generator). Define

$$B_R:=-i{L}_R^{\mathrm{res}}.$$

Its domain is

$$Dom\left(B_R\right):=Dom\left(L_R^{\mathrm{res}}\right).$$

Theorem 4.110 

(Stone generator). $B_R$ is an anti-self-adjoint operator on $H_R^{\mathrm{res}}$, and

$$B_R^{*}=-B_R.$$

Furthermore,

$$U_R\left(t\right)=e^{t{B}_R},$$

and for every

$$u\in Dom\left(B_R\right),$$

one has

$$B_{R}u=\underset{t\to 0}{lim}{\displaystyle \frac{U_R\left(t\right)u-u}{t}}$$

in the sense of the $H_R^{\mathrm{res}}$-strong limit. Conversely, any $u\in H_R^{\mathrm{res}}$ for which this strong limit exists belongs to $Dom\left(B_R\right)$.

Proof. 

Since $L_R^{\mathrm{res}}$ is self-adjoint,

$${(-i{L}_R^{\mathrm{res}})}^{*}=i{L}_R^{\mathrm{res}}=-(-i{L}_R^{\mathrm{res}}).$$

Therefore

$$B_R^{*}=-B_R,$$

and $B_R$ is anti-self-adjoint.

Also, by definition,

$$e^{t{B}_R}=e^{-it{L}_R^{\mathrm{res}}}=U_R\left(t\right).$$

By Stone’s theorem, the generator of the strongly continuous unitary group

$${\left\{U_R\left(t\right)\right\}}_{t\in \mathbb{R}}$$

is the anti-self-adjoint operator $B_R$, and its domain consists of exactly those u for which the difference quotient

$${\displaystyle \frac{U_R\left(t\right)u-u}{t}}$$

has a strong $H_R^{\mathrm{res}}$-limit. This limiting value is $B_{R}u$. The claim follows. □

Proposition 4.111 

(Preservation of the ${\sigma }_R$-null singular-boundary subspace). For every $t\in \mathbb{R}$,

$$U_R\left(t\right)H_R^{\mathrm{res}}=H_R^{\mathrm{res}}.$$

Furthermore, also for the form domain,

$$U_R\left(t\right)Q_R^{\mathrm{res}}=Q_R^{\mathrm{res}},$$

and

$$\parallel U_R{\left(t\right)u\parallel }_{H_R^{\mathrm{res}}}={\parallel u\parallel }_{H_R^{\mathrm{res}}},\parallel U_R{\left(t\right)u\parallel }_{L_R^{\mathrm{res}}}={\parallel u\parallel }_{L_R^{\mathrm{res}}}$$

hold.

Proof. 

The first assertion follows immediately from the fact that $U_R\left(t\right)$ is defined as a unitary operator on $H_R^{\mathrm{res}}$.

Next, consider the form domain. By Lemma 4.101,

$$Q_R^{\mathrm{res}}=Dom\left({\left(L_R^{\mathrm{res}}\right)}^{1/2}\right).$$

Since $U_R\left(t\right)=e^{-it{L}_R^{\mathrm{res}}}$ is a Borel function of $L_R^{\mathrm{res}}$, the spectral calculus gives

$$U_R\left(t\right){\left(L_R^{\mathrm{res}}\right)}^{1/2}={\left(L_R^{\mathrm{res}}\right)}^{1/2}{U}_R\left(t\right).$$

Therefore, if

$$u\in Dom\left({\left(L_R^{\mathrm{res}}\right)}^{1/2}\right),$$

then

$$U_R\left(t\right)u\in Dom\left({\left(L_R^{\mathrm{res}}\right)}^{1/2}\right).$$

Applying the same argument to $U_R(-t)$ gives the reverse inclusion, and hence

$$U_R\left(t\right)Q_R^{\mathrm{res}}=Q_R^{\mathrm{res}}.$$

For norm preservation, preservation of the $H_R^{\mathrm{res}}$-norm follows from unitarity. Moreover,

$$\begin{array}{cc}\hfill \parallel U_R{\left(t\right)u\parallel }_{L_R^{\mathrm{res}}}^2& =\parallel {\left(L_R^{\mathrm{res}}\right)}^{1/2}{U}_R\left(t\right){u\parallel }_{H_R^{\mathrm{res}}}^2\hfill \\ & =\parallel U_R\left(t\right){\left(L_R^{\mathrm{res}}\right)}^{1/2}{u\parallel }_{H_R^{\mathrm{res}}}^2\hfill \\ & =\parallel {\left(L_R^{\mathrm{res}}\right)}^{1/2}{u\parallel }_{H_R^{\mathrm{res}}}^2={\parallel u\parallel }_{L_R^{\mathrm{res}}}^2.\hfill \end{array}$$

The claim follows. □

Definition 4.112 

(Extended transport group on the one-sided analytic Hilbert space). Define the operator on $H_{\alpha ,+}$

$${\tilde{U}}_R\left(t\right):=U_R\left(t\right){\Pi }_R^{+}+(I_{H_{\alpha ,+}}-{\Pi }_R^{+})(t\in \mathbb{R}).$$

Namely, it acts as $U_R\left(t\right)$ on $H_R^{\mathrm{res}}$ and as the identity operator on its orthogonal complement.

Lemma 4.113 

(Properties of the extended transport group).

$${\left\{{\tilde{U}}_R\left(t\right)\right\}}_{t\in \mathbb{R}}$$

is a strongly continuous unitary group on $H_{\alpha ,+}$, and satisfies

$${\tilde{U}}_R\left(t\right){\Pi }_R^{+}={\Pi }_R^{+}{\tilde{U}}_R\left(t\right)=U_R\left(t\right){\Pi }_R^{+}.$$

Proof. 

Every $h\in H_{\alpha ,+}$ decomposes uniquely as

$$h=h_{\mathrm{res}}+h_{\perp },h_{\mathrm{res}}\in H_R^{\mathrm{res}},h_{\perp }\in {\left(H_R^{\mathrm{res}}\right)}^{\perp }.$$

Then

$${\tilde{U}}_R\left(t\right)h=U_R\left(t\right)h_{\mathrm{res}}+h_{\perp }.$$

Since $U_R\left(t\right)$ is unitary on $H_R^{\mathrm{res}}$ and the operator is the identity on the orthogonal complement, ${\tilde{U}}_R\left(t\right)$ is unitary on $H_{\alpha ,+}$.

The group law follows from

$$\begin{array}{cc}\hfill {\tilde{U}}_R(t+s)h& =U_R(t+s)h_{\mathrm{res}}+h_{\perp }\hfill \\ & =U_R\left(t\right)U_R\left(s\right)h_{\mathrm{res}}+h_{\perp }={\tilde{U}}_R\left(t\right){\tilde{U}}_R\left(s\right)h.\hfill \end{array}$$

Strong continuity follows from

$$\parallel {\tilde{U}}_R{\left(t\right)h-h\parallel }_{H_{\alpha ,+}}={\parallel U_R\left(t\right)h_{\mathrm{res}}-h_{\mathrm{res}}\parallel }_{H_R^{\mathrm{res}}},$$

and the strong continuity of $U_R\left(t\right)$.

Finally, since ${\Pi }_R^{+}h=h_{\mathrm{res}}$,

$${\tilde{U}}_R\left(t\right){\Pi }_R^{+}h=U_R\left(t\right)h_{\mathrm{res}},$$

and

$${\Pi }_R^{+}{\tilde{U}}_R\left(t\right)h={\Pi }_R^{+}(U_R\left(t\right)h_{\mathrm{res}}+h_{\perp })=U_R\left(t\right)h_{\mathrm{res}}.$$

Thus the displayed commutation relations hold. □

Definition 4.114 

(Transport kernel family). For each $t\in \mathbb{R}$, define the sesquilinear form on ${\mathcal{D}}_R\times {\mathcal{D}}_R$ by

$${\mathcal{K}}_{R,t}(\phi ,\psi ):={〈{\tilde{U}}_R\left(t\right)\phi ,\psi 〉}_{H_{\alpha ,+}}(\phi ,\psi \in {\mathcal{D}}_R).$$

By the kernel theorem, write the corresponding distribution kernel as

$${\mathsf{K}}_{R,t}^{\mathrm{dist}}\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }.$$

Namely,

$$〈{\mathsf{K}}_{R,t}^{\mathrm{dist}},\phi \otimes \psi 〉:={\mathcal{K}}_{R,t}(\phi ,\psi ).$$

In particular,

$${\mathsf{K}}_R^{\mathrm{dist}}:={\mathsf{K}}_{R,1}^{\mathrm{dist}}$$

is called the transport distribution kernel at the canonical time.

Lemma 4.115 

(Distribution-kernel property of the transport kernel family). For any $t\in \mathbb{R}$,

$${\mathsf{K}}_{R,t}^{\mathrm{dist}}\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }$$

is well-defined. Furthermore, for any $m>\alpha +{\displaystyle \frac{1}{2}}$, there exists a constant $C_{\alpha ,m}>0$ such that

$$|{\mathcal{K}}_{R,t}(\phi ,\psi )|\le C_{\alpha ,m}^2{p}_{m,0}\left(\phi \right)p_{m,0}\left(\psi \right)$$

for all $\phi ,\psi \in {\mathcal{D}}_R$.

Proof. 

Since ${\tilde{U}}_R\left(t\right)$ is a unitary operator on $H_{\alpha ,+}$,

$$\begin{array}{cc}\hfill |{\mathcal{K}}_{R,t}(\phi ,\psi )|& =|{\langle {\tilde{U}}_R\left(t\right)\phi ,\psi \rangle }_{H_{\alpha ,+}}|\hfill \\ & \le \parallel {\tilde{U}}_R{\left(t\right)\phi \parallel }_{H_{\alpha ,+}}{\parallel \psi \parallel }_{H_{\alpha ,+}}\hfill \\ & ={\parallel \phi \parallel }_{H_{\alpha ,+}}{\parallel \psi \parallel }_{H_{\alpha ,+}}.\hfill \end{array}$$

By Lemma 4.9, for any $m>\alpha +{\displaystyle \frac{1}{2}}$,

$${\parallel \phi \parallel }_{H_{\alpha ,+}}\le C_{\alpha ,m}{p}_{m,0}\left(\phi \right),{\parallel \psi \parallel }_{H_{\alpha ,+}}\le C_{\alpha ,m}{p}_{m,0}\left(\psi \right).$$

Therefore the displayed estimate follows.

This estimate implies that

$${\mathcal{K}}_{R,t}:{\mathcal{D}}_R\times {\mathcal{D}}_R\to \mathbb{C}$$

is a continuous sesquilinear form. By the kernel theorem of Section 4.1, there exists a unique distribution kernel

$${\mathsf{K}}_{R,t}^{\mathrm{dist}}\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }.$$

□

Remark 4.116 

(Not an ordinary function kernel). ${\mathsf{K}}_{R,t}^{\mathrm{dist}}$ is not defined as a pointwise function

$$K_{R,t}(x,y).$$

It is a distribution kernel assigning

$${\langle {\tilde{U}}_R\left(t\right)\phi ,\psi \rangle }_{H_{\alpha ,+}}$$

to a pair of test functions

$$\phi ,\psi \in {\mathcal{D}}_R.$$

Therefore, in order to treat ${\mathsf{K}}_{R,t}^{\mathrm{dist}}$ as an ordinary integral kernel, additional regularity, such as Hilbert–Schmidt property or representability of the Schwartz kernel by a function, must be proved separately. This paper assumes no such function-kernel representation.

Definition 4.117 

(Difference-quotient kernel family). For $t\ne 0$, define the bounded operator

$$D_R\left(t\right):={\displaystyle \frac{{\tilde{U}}_R\left(t\right)-I_{H_{\alpha ,+}}}{t}}.$$

Let the corresponding sesquilinear form be

$${\mathcal{D}}_{R,t}(\phi ,\psi ):={〈D_R\left(t\right)\phi ,\psi 〉}_{H_{\alpha ,+}},$$

and write its distribution kernel as

$$D_{R,t}^{\mathrm{dist}}\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }.$$

Namely,

$$〈D_{R,t}^{\mathrm{dist}},\phi \otimes \psi 〉={\mathcal{D}}_{R,t}(\phi ,\psi ).$$

Lemma 4.118 

(Existence of the difference-quotient kernel family). For each $t\ne 0$,

$$D_{R,t}^{\mathrm{dist}}\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }$$

is well-defined.

Proof. 

$D_R\left(t\right)$ is a bounded operator satisfying

$$\parallel D_R{\left(t\right)\parallel }_{\mathcal{B}\left(H_{\alpha ,+}\right)}\le {\displaystyle \frac{\parallel {\tilde{U}}_R\left(t\right)\parallel +1}{|t|}}={\displaystyle \frac{2}{|t|}}.$$

Therefore, for $\phi ,\psi \in {\mathcal{D}}_R$,

$$\begin{array}{cc}\hfill |{\mathcal{D}}_{R,t}(\phi ,\psi )|& \le \parallel D_R{\left(t\right)\parallel \parallel \phi \parallel }_{H_{\alpha ,+}}{\parallel \psi \parallel }_{H_{\alpha ,+}}\hfill \\ & \le {\displaystyle \frac{2C_{\alpha ,m}^2}{|t|}}{p}_{m,0}\left(\phi \right)p_{m,0}\left(\psi \right).\hfill \end{array}$$

Thus ${\mathcal{D}}_{R,t}$ is a continuous sesquilinear form on ${\mathcal{D}}_R\times {\mathcal{D}}_R$, and the kernel theorem gives a unique distribution kernel

$$D_{R,t}^{\mathrm{dist}}.$$

□

Definition 4.119 

(Weak distributional limit of the generator kernel). If the difference-quotient kernel family

$${\left\{D_{R,t}^{\mathrm{dist}}\right\}}_{t\ne 0}$$

has a limit as $t\to 0$ in the weak topology of

$${\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime },$$

write this limit as

$$B_R^{\mathrm{dist}}:=\underset{t\to 0}{error}{D}_{R,t}^{\mathrm{dist}},$$

and call it the distribution kernel of the transport generator. Namely, $B_R^{\mathrm{dist}}$ is defined by this limit when, for all

$$\phi ,\psi \in {\mathcal{D}}_R,$$

one has

$$〈B_R^{\mathrm{dist}},\phi \otimes \psi 〉=\underset{t\to 0}{lim}〈D_{R,t}^{\mathrm{dist}},\phi \otimes \psi 〉.$$

Theorem 4.120 

(Sufficient condition for the weak distributional limit of the difference-quotient kernels). Assume the following condition:

$${\mathcal{D}}_R\subset Dom\left({\tilde{B}}_R\right),$$

where

$${\tilde{B}}_R:=B_R{\Pi }_R^{+}$$

is defined on

$$Dom\left({\tilde{B}}_R\right):=\{h\in H_{\alpha ,+}:{\Pi }_R^{+}h\in Dom\left(B_R\right)\}.$$

Assume furthermore that the map

$${\tilde{B}}_R:{\mathcal{D}}_R⟶{\mathcal{D}}_R^{\prime }$$

is continuous. Then the difference-quotient kernel family has a limit in the weak topology of

$${\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime },$$

and

$$〈B_R^{\mathrm{dist}},\phi \otimes \psi 〉={〈{\tilde{B}}_R\phi ,\psi 〉}_{{\mathcal{D}}_R^{\prime },{\mathcal{D}}_R}$$

holds for all $\phi ,\psi \in {\mathcal{D}}_R$.

Proof. 

By assumption, for any $\phi \in {\mathcal{D}}_R$,

$${\Pi }_R^{+}\phi \in Dom\left(B_R\right).$$

By Stone’s theorem, Theorem 4.110,

$${\displaystyle \frac{U_R\left(t\right){\Pi }_R^{+}\phi -{\Pi }_R^{+}\phi }{t}}⟶B_R{\Pi }_R^{+}\phi ={\tilde{B}}_R\phi$$

as a strong $H_R^{\mathrm{res}}$-limit, and hence as a weak ${\mathcal{D}}_R^{\prime }$-limit.

On the other hand,

$${\tilde{U}}_R\left(t\right)\phi =U_R\left(t\right){\Pi }_R^{+}\phi +(I-{\Pi }_R^{+})\phi .$$

Therefore

$${\displaystyle \frac{{\tilde{U}}_R\left(t\right)\phi -\phi }{t}}={\displaystyle \frac{U_R\left(t\right){\Pi }_R^{+}\phi -{\Pi }_R^{+}\phi }{t}}.$$

Hence for any $\psi \in {\mathcal{D}}_R$,

$$\begin{array}{cc}\hfill \underset{t\to 0}{lim}〈D_{R,t}^{\mathrm{dist}},\phi \otimes \psi 〉& =\underset{t\to 0}{lim}{〈{\displaystyle \frac{{\tilde{U}}_R\left(t\right)\phi -\phi }{t}},\psi 〉}_{H_{\alpha ,+}}\hfill \\ & ={〈{\tilde{B}}_R\phi ,\psi 〉}_{{\mathcal{D}}_R^{\prime },{\mathcal{D}}_R}.\hfill \end{array}$$

Finally, by assumption,

$${\tilde{B}}_R:{\mathcal{D}}_R\to {\mathcal{D}}_R^{\prime }$$

is continuous. Therefore

$$(\phi ,\psi )⟼\langle {\tilde{B}}_R\phi ,\psi \rangle$$

is a continuous sesquilinear form on ${\mathcal{D}}_R\times {\mathcal{D}}_R$. By the kernel theorem, there exists a distribution kernel

$$B_R^{\mathrm{dist}}\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }$$

representing it. The pointwise limit representation shown above implies that

$$D_{R,t}^{\mathrm{dist}}\to B_R^{\mathrm{dist}}$$

in the sense of weak distributional convergence. □

Definition 4.121 

(Resolvent-type kernel family). For $z\in \rho \left(L_R^{\mathrm{res}}\right)$, let

$$R_R\left(z\right):={(L_R^{\mathrm{res}}-zI)}^{-1}$$

be the resolvent on $H_R^{\mathrm{res}}$. Extend it to $H_{\alpha ,+}$ by

$${\tilde{R}}_R\left(z\right):=R_R\left(z\right){\Pi }_R^{+}.$$

Write the distribution kernel corresponding to this bounded operator as

$$K_R^{\mathrm{res}}\left(z\right)\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime },$$

and define it by

$$〈K_R^{\mathrm{res}}\left(z\right),\phi \otimes \psi 〉:={\langle {\tilde{R}}_R\left(z\right)\phi ,\psi \rangle }_{H_{\alpha ,+}}.$$

Lemma 4.122 

(Existence of the resolvent kernel). For each $z\in \rho \left(L_R^{\mathrm{res}}\right)$,

$$K_R^{\mathrm{res}}\left(z\right)\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }$$

is well-defined. Furthermore,

$$|\langle K_R^{\mathrm{res}}\left(z\right),\phi \otimes \psi \rangle |\le \parallel {(L_R^{\mathrm{res}}-zI)}^{-1}\parallel C_{\alpha ,m}^2{p}_{m,0}\left(\phi \right)p_{m,0}\left(\psi \right)$$

holds.

Proof. 

$${\tilde{R}}_R\left(z\right)=R_R\left(z\right){\Pi }_R^{+}$$

is a bounded operator on $H_{\alpha ,+}$, and

$$\parallel {\tilde{R}}_R\left(z\right)\parallel \le \parallel R_R\left(z\right)\parallel .$$

Therefore

$$\begin{array}{cc}\hfill |\langle {\tilde{R}}_R\left(z\right)\phi ,\psi \rangle |& \le \parallel R_R{\left(z\right)\parallel \parallel \phi \parallel }_{H_{\alpha ,+}}{\parallel \psi \parallel }_{H_{\alpha ,+}}\hfill \\ & \le \parallel R_R\left(z\right)\parallel C_{\alpha ,m}^2{p}_{m,0}\left(\phi \right)p_{m,0}\left(\psi \right).\hfill \end{array}$$

Thus the corresponding bilinear form is continuous, and the kernel theorem gives a unique distribution kernel. □

Definition 4.123 

(Fredholm-type regularized kernel). For $\epsilon >0$, let

$$F_R\left(\epsilon \right):={(I+\epsilon L_R^{\mathrm{res}})}^{-1}{\Pi }_R^{+}$$

be a bounded operator on $H_{\alpha ,+}$. Write the corresponding distribution kernel as

$$F_R^{\mathrm{dist}}\left(\epsilon \right)\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime },$$

and define it by

$$〈F_R^{\mathrm{dist}}\left(\epsilon \right),\phi \otimes \psi 〉:={\langle F_R\left(\epsilon \right)\phi ,\psi \rangle }_{H_{\alpha ,+}}.$$

Theorem 4.124 

(Weak distributional limit of the Fredholm-type regularized kernel). As $\epsilon \downarrow 0$,

$$F_R\left(\epsilon \right)⟶{\Pi }_R^{+}$$

holds in the strong operator topology on $H_{\alpha ,+}$. Therefore the corresponding distribution kernels satisfy

$$F_R^{\mathrm{dist}}\left(\epsilon \right)⟶{\Pi }_R^{+,\mathrm{dist}}$$

in the sense of weak distributional limit. Here

$${\Pi }_R^{+,\mathrm{dist}}\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }$$

is the distribution kernel defined by

$$\langle {\Pi }_R^{+,\mathrm{dist}},\phi \otimes \psi \rangle :={\langle {\Pi }_R^{+}\phi ,\psi \rangle }_{H_{\alpha ,+}}.$$

Proof. 

We first prove the strong operator limit. Decompose any $h\in H_{\alpha ,+}$ as

$$h=h_{\mathrm{res}}+h_{\perp }.$$

Here $h_{\mathrm{res}}={\Pi }_R^{+}h\in H_R^{\mathrm{res}}$, and $h_{\perp }\in {\left(H_R^{\mathrm{res}}\right)}^{\perp }$. Then

$$F_R\left(\epsilon \right)h={(I+\epsilon L_R^{\mathrm{res}})}^{-1}{h}_{\mathrm{res}}.$$

By the spectral theorem, since $L_R^{\mathrm{res}}\ge I$,

$${(I+\epsilon L_R^{\mathrm{res}})}^{-1}$$

is a bounded operator on $H_R^{\mathrm{res}}$, and the spectral function

$$\lambda ⟼{\displaystyle \frac{1}{1+\epsilon \lambda }}$$

converges pointwise to 1 for each $\lambda$ as $\epsilon \downarrow 0$, while its absolute value is bounded by 1. Therefore, by the dominated convergence theorem,

$${(I+\epsilon L_R^{\mathrm{res}})}^{-1}{h}_{\mathrm{res}}⟶h_{\mathrm{res}}$$

holds in the $H_R^{\mathrm{res}}$-norm. Thus

$$F_R\left(\epsilon \right)h\to {\Pi }_R^{+}h,$$

and the strong operator convergence follows.

Next, we prove weak convergence of the distribution kernels. For any $\phi ,\psi \in {\mathcal{D}}_R$, the strong convergence gives

$$F_R\left(\epsilon \right)\phi ⟶{\Pi }_R^{+}\phi \mathrm{in}{H}_{\alpha ,+}.$$

Therefore

$${\langle F_R\left(\epsilon \right)\phi ,\psi \rangle }_{H_{\alpha ,+}}⟶{\langle {\Pi }_R^{+}\phi ,\psi \rangle }_{H_{\alpha ,+}}.$$

This means that

$$F_R^{\mathrm{dist}}\left(\epsilon \right)⟶{\Pi }_R^{+,\mathrm{dist}}$$

holds in the weak topology of ${\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }$. □

Proposition 4.125 

(Output of this subsection). This subsection has constructed the following objects:

$$U_R\left(t\right),B_R,{\tilde{U}}_R\left(t\right),{\mathsf{K}}_{R,t}^{\mathrm{dist}},{\mathsf{K}}_R^{\mathrm{dist}},K_R^{\mathrm{res}}\left(z\right).$$

They satisfy the following properties.

1.

$U_R\left(t\right)=e^{-it{L}_R^{\mathrm{res}}}$ is a strongly continuous unitary group on $H_R^{\mathrm{res}}$.

2.

$B_R=-i{L}_R^{\mathrm{res}}$ is the anti-self-adjoint generator, and

$$U_R\left(t\right)=e^{t{B}_R}.$$

3.

$U_R\left(t\right)$ preserves the ${\sigma }_R$-null singular-boundary subspace $H_R^{\mathrm{res}}$ and the form domain $Q_R^{\mathrm{res}}$.

4.

${\mathsf{K}}_{R,t}^{\mathrm{dist}}$ and ${\mathsf{K}}_R^{\mathrm{dist}}={\mathsf{K}}_{R,1}^{\mathrm{dist}}$ are not ordinary function kernels, but distribution kernels defined as elements of

$${\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }.$$

5.

The limit of the difference-quotient kernels is not assumed to exist; it is separately defined as $B_R^{\mathrm{dist}}$ under a sufficient condition guaranteeing the weak distributional limit.

6.

The resolvent kernel

$$K_R^{\mathrm{res}}\left(z\right)$$

exists as a distribution kernel for $z\in \rho \left(L_R^{\mathrm{res}}\right)$.

Proof. 

Item 1 follows from Theorem 4.108. Item 2 follows from Theorem 4.110. Item 3 follows from Proposition 4.111. Item 4 follows from Definition 4.114 and Lemma 4.115. Item 5 follows from Definition 4.119 and Theorem 4.120. Item 6 follows from Definition 4.121 and Lemma 4.122. □

Remark 4.126 

(Transition to the next step). The transport group, anti-self-adjoint generator, and distribution kernel representation obtained in this subsection show that the singular-boundary subspace is not merely a closed subspace, but has a conservative time evolution inside it. In the next subsection, we treat compactness of the resolvent of the positive shifted operator

$$L_R^{\mathrm{res}},$$

and construct the spectral foundation needed for purely discrete spectrum and finite-window localization.

4.8. Compact Resolvent and Purely Discrete Spectrum

In this subsection, we show that the resolvent of the ${\sigma }_R$-null positive shifted operator on $K_R$

$$L_R^{\mathrm{res}}=A_R^{\mathrm{res}}+I_{H_R^{\mathrm{res}}}$$

is compact. The proof is carried out by restricting the compact embedding of the global form domain established in Section 2 to the ${\sigma }_R$-null trace-vanishing form subspace

$$Q_R^{\mathrm{res}}\subset Q_R.$$

Accordingly, in this subsection we introduce neither a new confining potential nor a new global operator. What is used is only the compactness from Section 2 and the closed-subspace structure constructed up to the preceding subsection.

Lemma 4.127 

(Restriction of the global compact embedding). Consider the global compact embedding obtained in Section 2,

$$j_R:(Q_R{,\parallel ·\parallel }_{q_R})⟶H_{\alpha ,+}.$$

Then the natural inclusion map

$$j_R^{\mathrm{res}}:(Q_R^{\mathrm{res}}{,\parallel ·\parallel }_{q_R})⟶H_R^{\mathrm{res}}$$

is compact.

Proof. 

Let ${\left\{u_n\right\}}_{n\ge 1}$ be a bounded sequence in

$$(Q_R^{\mathrm{res}},\parallel ·{\parallel }_{q_R}).$$

That is, there exists $C>0$ such that

$$\parallel u_n{\parallel }_{q_R}\le C(n\ge 1).$$

Since $Q_R^{\mathrm{res}}\subset Q_R$, this is also a bounded sequence in $(Q_R,\parallel ·{\parallel }_{q_R})$.

By the global compact embedding of Section 2, there exist a subsequence

$${\left\{u_{n_k}\right\}}_{k\ge 1}$$

and some

$$u\in H_{\alpha ,+}$$

such that

$$u_{n_k}⟶u\mathrm{in}{H}_{\alpha ,+}.$$

On the other hand, all $u_{n_k}$ belong to

$$Q_R^{\mathrm{res}}\subset H_R^{\mathrm{res}}.$$

By Lemma 4.89, $H_R^{\mathrm{res}}$ is a closed subspace of $H_{\alpha ,+}$. Therefore the $H_{\alpha ,+}$-norm limit u also satisfies

$$u\in H_R^{\mathrm{res}}.$$

Moreover, since the norm of $H_R^{\mathrm{res}}$ is the restriction of the norm of $H_{\alpha ,+}$,

$$u_{n_k}⟶u\mathrm{in}{H}_R^{\mathrm{res}}$$

also holds. Thus $j_R^{\mathrm{res}}$ extracts a convergent subsequence from every bounded sequence. Hence $j_R^{\mathrm{res}}$ is compact. □

Remark 4.128 

(Why compactness is preserved). What is used here is not merely the fact that a global compact operator has been restricted. The essential point is that

$$Q_R^{\mathrm{res}}$$

is a form-norm closed subspace of $Q_R$, and that

$$H_R^{\mathrm{res}}$$

is a norm-closed subspace of $H_{\alpha ,+}$. The fact that the limit obtained from the global compact embedding does not leave $H_R^{\mathrm{res}}$ is guaranteed by the closedness of $H_R^{\mathrm{res}}$. Thus the compactness of Section 2 is correctly inherited by the ${\sigma }_R$-null singular-boundary subspace.

Lemma 4.129 

(Variational representation of the inverse operator). For any

$$f\in H_R^{\mathrm{res}},$$

there exists a unique

$$u\in Q_R^{\mathrm{res}}$$

such that

$$q_R^{\mathrm{res}}[u,v]+{\langle u,v\rangle }_{H_R^{\mathrm{res}}}={\langle f,v\rangle }_{H_R^{\mathrm{res}}}(v\in Q_R^{\mathrm{res}}).$$

Furthermore,

$$u={\left(L_R^{\mathrm{res}}\right)}^{-1}f,$$

and

$${\parallel u\parallel }_{q_R^{\mathrm{res}}}\le {\parallel f\parallel }_{H_R^{\mathrm{res}}}$$

holds.

Proof. 

Consider the inner product on $Q_R^{\mathrm{res}}$

$${\langle u,v\rangle }_{q_R^{\mathrm{res}}}:=q_R^{\mathrm{res}}[u,v]+{\langle u,v\rangle }_{H_R^{\mathrm{res}}}.$$

By Lemma 4.92 and Definition 4.93,

$$(Q_R^{\mathrm{res}},{\langle ·,·\rangle }_{q_R^{\mathrm{res}}})$$

is a Hilbert space.

For fixed $f\in H_R^{\mathrm{res}}$, set

$${\ell }_f\left(v\right):={\langle f,v\rangle }_{H_R^{\mathrm{res}}}(v\in Q_R^{\mathrm{res}}).$$

By the Cauchy–Schwarz inequality and

$${\parallel v\parallel }_{H_R^{\mathrm{res}}}\le {\parallel v\parallel }_{q_R^{\mathrm{res}}},$$

we have

$$|{\ell }_f{\left(v\right)|\le \parallel f\parallel }_{H_R^{\mathrm{res}}}{\parallel v\parallel }_{H_R^{\mathrm{res}}}\le {\parallel f\parallel }_{H_R^{\mathrm{res}}}{\parallel v\parallel }_{q_R^{\mathrm{res}}}.$$

Therefore ${\ell }_f$ is a bounded linear functional on $(Q_R^{\mathrm{res}},\parallel ·{\parallel }_{q_R^{\mathrm{res}}})$.

By the Riesz representation theorem, there exists a unique

$$u\in Q_R^{\mathrm{res}}$$

such that

$${\langle u,v\rangle }_{q_R^{\mathrm{res}}}={\ell }_f\left(v\right)$$

for all $v\in Q_R^{\mathrm{res}}$. That is,

$$q_R^{\mathrm{res}}[u,v]+{\langle u,v\rangle }_{H_R^{\mathrm{res}}}={\langle f,v\rangle }_{H_R^{\mathrm{res}}}.$$

By the definition

$$L_R^{\mathrm{res}}=A_R^{\mathrm{res}}+I$$

and the first representation theorem, this equality is equivalent to

$$u\in Dom\left(L_R^{\mathrm{res}}\right)\mathrm{and}{L}_R^{\mathrm{res}}u=f.$$

Therefore

$$u={\left(L_R^{\mathrm{res}}\right)}^{-1}f.$$

Finally, the norm estimate in the Riesz representation gives

$${\parallel u\parallel }_{q_R^{\mathrm{res}}}=\parallel {\ell }_f{\parallel }_{{\left({\mathcal{Q}}_R^{\mathrm{res}}\right)}^{\prime }}\le {\parallel f\parallel }_{H_R^{\mathrm{res}}}.$$

□

Theorem 4.130 

(Compactness of the inverse operator).

$${\left(L_R^{\mathrm{res}}\right)}^{-1}:H_R^{\mathrm{res}}⟶H_R^{\mathrm{res}}$$

is a compact operator.

Proof. 

By Lemma 4.129,

$${\left(L_R^{\mathrm{res}}\right)}^{-1}$$

factors as a bounded operator

$$S_R:H_R^{\mathrm{res}}⟶(Q_R^{\mathrm{res}}{,\parallel ·\parallel }_{q_R}).$$

Namely, setting

$$S_{R}f={\left(L_R^{\mathrm{res}}\right)}^{-1}f,$$

we have

$$\parallel S_R{f\parallel }_{q_R}\le {\parallel f\parallel }_{H_R^{\mathrm{res}}}.$$

On the other hand, by Lemma 4.127, the inclusion map

$$j_R^{\mathrm{res}}:(Q_R^{\mathrm{res}}{,\parallel ·\parallel }_{q_R})⟶H_R^{\mathrm{res}}$$

is compact. Therefore

$${\left(L_R^{\mathrm{res}}\right)}^{-1}=j_R^{\mathrm{res}}\circ S_R$$

is the composition of a bounded operator and a compact operator. Hence

$${\left(L_R^{\mathrm{res}}\right)}^{-1}$$

is compact on $H_R^{\mathrm{res}}$. □

Theorem 4.131 

(Compact resolvent). For any

$$z\in \rho \left(L_R^{\mathrm{res}}\right),$$

the operator

$${(L_R^{\mathrm{res}}-zI)}^{-1}$$

is compact on $H_R^{\mathrm{res}}$. Therefore $L_R^{\mathrm{res}}$ has compact resolvent.

Proof. 

First, the case $z=0$ follows from Theorem 4.130, which shows that

$${\left(L_R^{\mathrm{res}}\right)}^{-1}$$

is compact.

Now take arbitrary $z\in \rho \left(L_R^{\mathrm{res}}\right)$. By the spectral theorem, we can write

$${(L_R^{\mathrm{res}}-zI)}^{-1}=g_z\left(L_R^{\mathrm{res}}\right){\left(L_R^{\mathrm{res}}\right)}^{-1},$$

where

$$g_z\left(\lambda \right):={\displaystyle \frac{\lambda }{\lambda -z}}.$$

Since $z\in \rho \left(L_R^{\mathrm{res}}\right)$, the function

$$\lambda \mapsto {\displaystyle \frac{\lambda }{\lambda -z}}$$

is a bounded Borel function on $\sigma \left(L_R^{\mathrm{res}}\right)$. Therefore

$$g_z\left(L_R^{\mathrm{res}}\right)$$

is a bounded operator on $H_R^{\mathrm{res}}$.

Since

$${\left(L_R^{\mathrm{res}}\right)}^{-1}$$

is already compact, the composition with the bounded operator

$$g_z\left(L_R^{\mathrm{res}}\right){\left(L_R^{\mathrm{res}}\right)}^{-1}$$

is compact. Hence

$${(L_R^{\mathrm{res}}-zI)}^{-1}$$

is compact. □

Theorem 4.132 

(Purely discrete spectrum). The spectrum of $L_R^{\mathrm{res}}$ is purely discrete. That is, if $H_R^{\mathrm{res}}\ne \left\{0\right\}$, then there exist at most countably many eigenvalues

$$1\le {\lambda }_1\le {\lambda }_2\le \dots ,$$

each eigenvalue has finite multiplicity, and there exists a complete orthonormal system of corresponding eigenvectors

$${\left\{e_j\right\}}_{j\ge 1}$$

in $H_R^{\mathrm{res}}$. Furthermore, if there are infinitely many eigenvalues, then

$${\lambda }_j⟶+\infty (j\to \infty ).$$

In the finite-dimensional case, the spectrum consists only of finitely many eigenvalues.

Proof. 

By Lemma 4.99, $L_R^{\mathrm{res}}$ is a self-adjoint operator, and by Theorem 4.131 it has compact resolvent. When a self-adjoint operator has compact resolvent, its spectrum consists of pure point spectrum, each eigenvalue has finite multiplicity, and no finite accumulation point occurs. Moreover, the corresponding eigenvectors form a complete orthonormal system of the Hilbert space.

Furthermore,

$$L_R^{\mathrm{res}}\ge I,$$

and hence

$$\sigma \left(L_R^{\mathrm{res}}\right)\subset [1,\infty ).$$

Therefore the eigenvalues can be arranged at or above 1. If there are infinitely many eigenvalues, since they cannot accumulate at any finite point, the only possible accumulation point is $+\infty$. Thus

$${\lambda }_j\to +\infty$$

follows. □

Definition 4.133 

(Finite-window spectral projection). Let $E_R^{\mathrm{res}}(·)$ be the spectral measure of $L_R^{\mathrm{res}}$. For a bounded Borel set

$$I\subset [1,\infty ),$$

write

$$P_R^{\mathrm{res}}\left(I\right):=E_R^{\mathrm{res}}\left(I\right).$$

We call this the finite-window spectral projection of the ${\sigma }_R$-null singular-boundary subspace. Also,

$$N_R^{\mathrm{res}}\left(I\right):=rank{P}_R^{\mathrm{res}}\left(I\right)$$

is called its finite-window spectral count.

Corollary 4.134 

(Finiteness of finite-window counts). For any bounded Borel set

$$I\subset [1,\infty ),$$

one has

$$N_R^{\mathrm{res}}\left(I\right)<\infty .$$

In particular, for any

$$1\le a<b<\infty ,$$

one has

$$rank{E}_R^{\mathrm{res}}\left([a,b]\right)<\infty .$$

Proof. 

By Theorem 4.132, any bounded interval contains only finitely many eigenvalues, and each eigenvalue has finite multiplicity. Therefore the direct sum of eigenspaces belonging to the bounded Borel set I is finite-dimensional. Hence

$$rank{P}_R^{\mathrm{res}}\left(I\right)<\infty .$$

□

Remark 4.135 

(Transition to finite-window localization). By Corollary 4.134, on the ${\sigma }_R$-null singular-boundary subspace, the degrees of freedom contained in any bounded spectral window are finite-dimensional. Therefore the subsequent finite-window localization, finite-rank projection, and discrete counting comparison can be carried out on top of this spectral data. The finiteness obtained here is not a statement concerning zero counting; it is solely operator-theoretic finiteness following from the compact resolvent of the singular-boundary operator.

Definition 4.136 

(Spectral resolvent measure on $K_R$). Fix $\Phi \in H_R^{\mathrm{res}}$. For the spectral measure $E_R^{\mathrm{res}}$ of $L_R^{\mathrm{res}}$, define

$${\mu }_R^{(\Phi )}\left(I\right):={\langle E_R^{\mathrm{res}}\left(I\right)\Phi ,\Phi \rangle }_{H_R^{\mathrm{res}}}.$$

This is a finite positive Borel measure on $[1,\infty )$, and satisfies

$${\mu }_R^{(\Phi )}\left([1,\infty )\right)={\parallel \Phi \parallel }_{H_R^{\mathrm{res}}}^2.$$

Definition 4.137 

(Herglotz-type resolvent function). For $\Phi \in H_R^{\mathrm{res}}$, define

$$m_R^{(\Phi )}\left(z\right):={〈\Phi ,{(L_R^{\mathrm{res}}-zI)}^{-1}\Phi 〉}_{H_R^{\mathrm{res}}}(z\in \mathbb{C}\setminus \mathbb{R}).$$

We call this the ${\sigma }_R$-null Herglotz-type resolvent function on $K_R$ associated with $\Phi$.

Lemma 4.138 

(Spectral representation). For any

$$z\in \mathbb{C}\setminus \mathbb{R},$$

one has

$$m_R^{(\Phi )}\left(z\right)={\int }_{[1,\infty )}{\displaystyle \frac{1}{\lambda -z}}d{\mu }_R^{(\Phi )}\left(\lambda \right).$$

Furthermore, using the purely discrete spectral representation,

$$m_R^{(\Phi )}\left(z\right)=\sum _j{\displaystyle \frac{|{\langle \Phi ,e_j\rangle }_{H_R^{\mathrm{res}}}{|}^2}{{\lambda }_j-z}},$$

and this series converges absolutely for

$$z\in \mathbb{C}\setminus \mathbb{R}.$$

Proof. 

The first representation follows immediately from the spectral theorem. Indeed,

$${(L_R^{\mathrm{res}}-zI)}^{-1}={\int }_{[1,\infty )}{\displaystyle \frac{1}{\lambda -z}}d{E}_R^{\mathrm{res}}\left(\lambda \right),$$

and therefore

$$\begin{array}{cc}\hfill m_R^{(\Phi )}\left(z\right)& =〈\Phi ,\left({\int }_{[1,\infty )}{\displaystyle \frac{1}{\lambda -z}}d{E}_R^{\mathrm{res}}\left(\lambda \right)\right)\Phi 〉\hfill \\ & ={\int }_{[1,\infty )}{\displaystyle \frac{1}{\lambda -z}}d{\mu }_R^{(\Phi )}\left(\lambda \right).\hfill \end{array}$$

The purely discrete spectral representation is obtained by using the complete orthonormal system $\left\{e_j\right\}$ from Theorem 4.132. Namely,

$$\Phi =\sum _j\langle \Phi ,e_j\rangle e_j,$$

and

$${(L_R^{\mathrm{res}}-zI)}^{-1}{e}_j={\displaystyle \frac{1}{{\lambda }_j-z}}{e}_j.$$

This gives the displayed series.

For absolute convergence, since

$$\left|{\displaystyle \frac{1}{{\lambda }_j-z}}\right|\le {\displaystyle \frac{1}{|Imz|}},$$

we have

$$\sum _j\left|{\displaystyle \frac{|\langle \Phi ,e_j\rangle {|}^2}{{\lambda }_j-z}}\right|\le {\displaystyle \frac{1}{|Imz|}}\sum _j{|\langle \Phi ,e_j\rangle |}^2={\displaystyle \frac{{\parallel \Phi \parallel }^2}{|Imz|}}<\infty .$$

□

Lemma 4.139 

(Herglotz property). If $Imz>0$, then

$$Im{m}_R^{(\Phi )}\left(z\right)\ge 0.$$

More precisely,

$$Im{m}_R^{(\Phi )}\left(z\right)=(Imz){\int }_{[1,\infty )}{\displaystyle \frac{1}{{|\lambda -z|}^2}}d{\mu }_R^{(\Phi )}\left(\lambda \right)\ge 0.$$

Therefore $m_R^{(\Phi )}$ is a Herglotz-type function from the upper half-plane to the closure of the upper half-plane.

Proof. 

By the spectral representation,

$$m_R^{(\Phi )}\left(z\right)={\int }_{[1,\infty )}{\displaystyle \frac{1}{\lambda -z}}d{\mu }_R^{(\Phi )}\left(\lambda \right).$$

Write $z=x+iy$, $y>0$. Then

$${\displaystyle \frac{1}{\lambda -z}}={\displaystyle \frac{\lambda -x+iy}{{(\lambda -x)}^2+y^2}}.$$

Therefore

$$Im{\displaystyle \frac{1}{\lambda -z}}={\displaystyle \frac{y}{{(\lambda -x)}^2+y^2}}={\displaystyle \frac{Imz}{{|\lambda -z|}^2}}.$$

Integrating this gives

$$Im{m}_R^{(\Phi )}\left(z\right)=(Imz){\int }_{[1,\infty )}{\displaystyle \frac{1}{{|\lambda -z|}^2}}d{\mu }_R^{(\Phi )}\left(\lambda \right)\ge 0.$$

□

Proposition 4.140 

(Output of this subsection). This subsection has constructed the following spectral data.

1.

The embedding

$$Q_R^{\mathrm{res}}\hookrightarrow H_R^{\mathrm{res}}$$

is compact.

2.

$${(L_R^{\mathrm{res}}-zI)}^{-1}$$

is a compact operator for every $z\in \rho \left(L_R^{\mathrm{res}}\right)$.

3.

$L_R^{\mathrm{res}}$ has purely discrete spectrum.

4.

The projection corresponding to a bounded spectral window,

$$P_R^{\mathrm{res}}\left(I\right),$$

has finite rank.

5.

For any $\Phi \in H_R^{\mathrm{res}}$, the Herglotz-type resolvent function

$$m_R^{(\Phi )}\left(z\right)=〈\Phi ,{(L_R^{\mathrm{res}}-zI)}^{-1}\Phi 〉$$

is defined.

Proof. 

Item 1 follows from Lemma 4.127. Item 2 follows from Theorem 4.131. Item 3 follows from Theorem 4.132. Item 4 follows from Corollary 4.134. Item 5 follows from Definition 4.137. □

4.9. Internal-Construction Theorem for the Operator-Theoretic Boundary Data

In this subsection, we collect the objects constructed in this section into a single operator-theoretic data. The purpose is to fix the singular-boundary input passed to the subsequent orthogonal-decomposition framework entirely as objects defined internally in this section.

The construction of this section proceeded in the following order. First, as the basic space on the singular-boundary side, we introduced

$${\mathcal{D}}_R\subset H_{\alpha ,+}\subset {\mathcal{D}}_R^{\prime }$$

and separated the types of point evaluations, boundary traces, distribution kernels, quadratic forms, generators, and boundary forms. Next, we constructed the boundary parameter space, regular area measure, and ${\sigma }_R$-null singular support, and then separated the regular boundary channel from the singular boundary channel. After that, we took the subspace on which boundary traces are defined inside the form domain, and constructed the support map of the singular trace and the regular trace mass functional. Finally, from the trace-vanishing generating subspace associated with ${\Gamma }_R$, we constructed the closed form subspace, Hilbert space, self-adjoint realization, transport group, distribution kernel, and compact resolvent.

Below, we formulate these collectively as the operator-theoretic boundary data.

Definition 4.141 

(Internally constructed boundary input data). Write the boundary input data constructed in this section as

$${\mathfrak{R}}_{\mathrm{bd}}:=({\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R).$$

Each component is defined as follows.

Symbol	Internal construction	Construction location
${\Sigma }_R$	${[0,1]}^2$	boundary parameter space
${\sigma }_R$	${\mathcal{L}}^2{|}_{{[0,1]}^2}$	regular area measure
${\Gamma }_R$	$C_R\times \left\{0\right\}$	${\sigma }_R$-null singular support
$T_R$	$\begin{array}{cc}\hfill \{f\in Q_R:& {\gamma }_R^{\mathrm{sing}}f\mathrm{and}{\gamma }_R^{\mathrm{reg}}f\hfill \\ & \mathrm{are}\mathrm{uniquely}\mathrm{defined}\mathrm{in}\mathrm{the}\mathrm{closed}-\mathrm{graph}\mathrm{sense}\}\hfill \end{array}$	closed boundary trace space
${supp}_R\left(f\right)$	$supp\left({\gamma }_R^{\mathrm{sing}}f\right)$	distribution support of the singular boundary trace
$L_R(f;E)$	${\displaystyle {\int }_E{|{\gamma }_R^{\mathrm{reg}}f|}^{2}d{\sigma }_R}$	regular boundary trace-mass functional

Here $C_R$ is the middle-third Cantor set, and

$${\Gamma }_R=C_R\times \left\{0\right\}\subset {[0,1]}^2.$$

Also, $Q_R$ is the form domain constructed in Section 2.

Lemma 4.142 

(Basic properties of the boundary input data). The boundary input data

$${\mathfrak{R}}_{\mathrm{bd}}=({\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R)$$

satisfy the following.

1.

${\Sigma }_R$ is a compact metric space.

2.

${\sigma }_R$ is a finite regular Borel area measure on ${\Sigma }_R$.

3.

${\Gamma }_R\subset {\Sigma }_R$ is a closed set, and

$${\sigma }_R\left({\Gamma }_R\right)=0.$$

4.

A singular probability measure

$${\nu }_R$$

is supported on ${\Gamma }_R$, and

$${\nu }_R\left({\Gamma }_R\right)=1,{\nu }_R\perp {\sigma }_R$$

hold.

5.

$T_R\subset Q_R$ is a linear subspace, and

$${\gamma }_R^{\mathrm{sing}}:T_R\to \mathcal{M}\left({\Sigma }_R\right),{\gamma }_R^{\mathrm{reg}}:T_R\to L^2({\Sigma }_R,{\sigma }_R)$$

are linear maps.

6.

For any $f\in T_R$,

$${supp}_R\left(f\right)\subset {\Gamma }_R,$$

and therefore

$${\sigma }_R\left({supp}_R\left(f\right)\right)=0.$$

7.

For any $f\in T_R$, the map

$$E⟼L_R(f;E)$$

is a finite positive measure on ${\Sigma }_R$, and satisfies

$$L_R(f;{\Sigma }_R)={\parallel {\gamma }_R^{\mathrm{reg}}f\parallel }_{L^2({\Sigma }_R,{\sigma }_R)}^2.$$

Proof. 

Items 1, 2, 3, and 4 follow from the construction of the boundary parameter space and the ${\sigma }_R$-null singular support. Item 5 follows from the linearity of $T_R$, defined by the closed-trace graph, and from the linearity of the two boundary trace maps. Item 6 follows from the fact that the singular trace is always supported on ${\Gamma }_R$, together with ${\sigma }_R\left({\Gamma }_R\right)=0$. Item 7 follows from the definition of the regular boundary trace-mass functional

$$L_R(f;E)={\int }_E{|{\gamma }_R^{\mathrm{reg}}f|}^{2}d{\sigma }_R.$$

□

Definition 4.143 

(Internally constructed trace-vanishing generating subspace associated with ${\Gamma }_R$). Define the trace-vanishing generating subspace associated with ${\Gamma }_R$ by

$$G_R:=\left\{f\in T_R:{supp}_R\left(f\right)\subset {\Gamma }_R,L_R(f;{\Sigma }_R\setminus {\Gamma }_R)=0\right\}.$$

Equivalently,

$$G_R=\left\{f\in T_R:{\gamma }_R^{\mathrm{reg}}f=0{\sigma }_R-\mathrm{a}.\mathrm{e}.\right\}.$$

Lemma 4.144 

(${\sigma }_R$-null property and regular-trace vanishing property of the generating class). For any $f\in G_R$, the following hold.

1.

The singular boundary trace is supported on the ${\sigma }_R$-null singular support:

$$supp\left({\gamma }_R^{\mathrm{sing}}f\right)={supp}_R\left(f\right)\subset {\Gamma }_R.$$

2.

Regular-trace mass in the regular boundary-trace direction vanishes:

$${\gamma }_R^{\mathrm{reg}}f=0{\sigma }_R-\mathrm{a}.\mathrm{e}.$$

3.

Therefore,

$$L_R(f;{\Sigma }_R)=0.$$

Proof. 

Item 1 follows from the definition of ${supp}_R$ and from ${supp}_R\left(f\right)\subset {\Gamma }_R$. Item 2 follows from the equivalent representation of $G_R$,

$$G_R=\{f\in T_R:{\gamma }_R^{\mathrm{reg}}f=0\}.$$

Item 3 is obtained by substituting Item 2 into

$$L_R(f;{\Sigma }_R)={\parallel {\gamma }_R^{\mathrm{reg}}f\parallel }_{L^2({\Sigma }_R,{\sigma }_R)}^2.$$

□

Definition 4.145 

(Internally constructed closed quadratic form and Hilbert data). From the trace-vanishing generating subspace associated with ${\Gamma }_R$, define

$$Q_R^{\mathrm{res}}:={\overline{span{G}_R}}^{{\parallel ·\parallel }_{q_R}}\subset Q_R.$$

Furthermore, define

$$H_R^{\mathrm{res}}:={\overline{Q_R^{\mathrm{res}}}}^{{\parallel ·\parallel }_{H_{\alpha ,+}}}\subset H_{\alpha ,+}.$$

Write the restriction form as

$$q_R^{\mathrm{res}}:=q_R{|}_{Q_R^{\mathrm{res}}\times Q_R^{\mathrm{res}}}.$$

Also, let its Friedrichs-type representing operator be

$$A_R^{\mathrm{res}},$$

and define the positive shifted operator by

$$L_R^{\mathrm{res}}:=A_R^{\mathrm{res}}+I_{H_R^{\mathrm{res}}}.$$

Lemma 4.146 

(Basic properties of the form and Hilbert data). The objects defined above satisfy the following.

1.

$Q_R^{\mathrm{res}}$ is a closed linear subspace of

$$(Q_R,\parallel ·{\parallel }_{q_R}),$$

and

$$(Q_R^{\mathrm{res}},\parallel ·{\parallel }_{q_R})$$

is a Hilbert space.

2.

The inclusion map

$$Q_R^{\mathrm{res}}\hookrightarrow H_{\alpha ,+}$$

is continuous.

3.

$H_R^{\mathrm{res}}$ is a closed linear subspace of $H_{\alpha ,+}$.

4.

$Q_R^{\mathrm{res}}$ is dense in $H_R^{\mathrm{res}}$.

5.

$q_R^{\mathrm{res}}$ is a densely defined closed lower-bounded form on $H_R^{\mathrm{res}}$.

6.

$A_R^{\mathrm{res}}$ is the unique self-adjoint representing operator of $q_R^{\mathrm{res}}$.

7.

$L_R^{\mathrm{res}}$ is self-adjoint, and

$$L_R^{\mathrm{res}}\ge I_{H_R^{\mathrm{res}}}.$$

Proof. 

Items 1 and 2 follow from the construction of the closure of the ${\sigma }_R$-null trace-vanishing form subspace. Items 3 and 4 follow from the definition of $H_R^{\mathrm{res}}$. Item 5 follows from the fact that the restriction of a closed form $q_R$ to a closed subspace is a closed form. Item 6 follows from the first representation theorem for closed lower-bounded forms. Item 7 follows from the definition

$$L_R^{\mathrm{res}}=A_R^{\mathrm{res}}+I$$

and from $A_R^{\mathrm{res}}\ge 0$. □

Definition 4.147 

(Internally constructed singular-boundary subspace). Define the singular-boundary subspace on the one-sided analytic Hilbert space by

$$K_R:=H_R^{\mathrm{res}}\subset H_{\alpha ,+}.$$

Also, write the canonical kernel associated with its distribution kernel representation as

$${\mathsf{K}}_R^{\mathrm{dist}}:={\mathsf{K}}_{R,1}^{\mathrm{dist}}\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }.$$

Here ${\mathsf{K}}_{R,t}^{\mathrm{dist}}$ is the transport kernel family determined from the canonical transport group on $K_R$

$$U_R\left(t\right)=e^{-it{L}_R^{\mathrm{res}}}.$$

Remark 4.148 

(Distinction between the subspace and the distribution kernel). The symbol $K_R$ denotes a closed subspace inside a Hilbert space. On the other hand,

$${\mathsf{K}}_R^{\mathrm{dist}}$$

is a distribution kernel belonging to

$${\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }.$$

They are objects of different types and are not identified, in accordance with the type-separation convention.

Lemma 4.149 

(Closedness of the singular-boundary subspace).

$$K_R=H_R^{\mathrm{res}}$$

is a closed linear subspace of $H_{\alpha ,+}$.

Proof. 

This follows immediately from Lemma 4.89 and Definition 4.147. □

Lemma 4.150 

(Transport and spectral structure on the singular-boundary subspace). The following structures exist on $K_R$.

1.

There exists a strongly continuous unitary group

$$U_R\left(t\right)=e^{-it{L}_R^{\mathrm{res}}}(t\in \mathbb{R}).$$

2.

Its anti-self-adjoint generator

$$B_R=-i{L}_R^{\mathrm{res}}$$

exists, and

$$U_R\left(t\right)=e^{t{B}_R}.$$

3.

$L_R^{\mathrm{res}}$ has compact resolvent.

4.

$L_R^{\mathrm{res}}$ has purely discrete spectrum, and the spectral projection on a bounded spectral window has finite rank.

Proof. 

Items 1 and 2 follow from the construction of the transport group on $K_R$ and the Stone generator. Item 3 follows from the compact-resolvent theorem. Item 4 follows from the purely discrete spectrum theorem and the finiteness of finite-window counts. □

Theorem 4.151 

(Internal construction of the operator-theoretic boundary data). By the construction of this section, the operator-theoretic boundary data

$${\mathfrak{R}}_{\mathrm{op}}:=({\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R,G_R,Q_R^{\mathrm{res}},H_R^{\mathrm{res}},K_R,U_R\left(t\right),B_R,{\mathsf{K}}_R^{\mathrm{dist}},L_R^{\mathrm{res}})$$

are internally fixed. These data satisfy the following properties.

1.

${\Gamma }_R$ is null with respect to the regular area measure:

$${\sigma }_R\left({\Gamma }_R\right)=0.$$

2.

A nonzero singular measure ${\nu }_R$ supported on ${\Gamma }_R$

$${\nu }_R$$

is supported on ${\Gamma }_R$.

3.

The singular boundary trace

$${\gamma }_R^{\mathrm{sing}}$$

and the regular boundary trace

$${\gamma }_R^{\mathrm{reg}}$$

are defined on $T_R\subset Q_R$.

4.

The support map is defined by

$${supp}_R\left(f\right)=supp\left({\gamma }_R^{\mathrm{sing}}f\right).$$

5.

The regular boundary trace-mass functional is defined by

$$L_R(f;E)={\int }_E{|{\gamma }_R^{\mathrm{reg}}f|}^{2}d{\sigma }_R.$$

6.

The trace-vanishing generating subspace associated with ${\Gamma }_R$

$$G_R=\left\{f\in T_R:{supp}_R\left(f\right)\subset {\Gamma }_R,L_R(f;{\Sigma }_R\setminus {\Gamma }_R)=0\right\}$$

is defined.

7.

$$Q_R^{\mathrm{res}}={\overline{span{G}_R}}^{{\parallel ·\parallel }_{q_R}}$$

is a closed form domain.

8.

$$H_R^{\mathrm{res}}={\overline{Q_R^{\mathrm{res}}}}^{{\parallel ·\parallel }_{H_{\alpha ,+}}}$$

is a Hilbert space, and

$$K_R=H_R^{\mathrm{res}}$$

is a closed subspace of $H_{\alpha ,+}$.

9.

On $K_R$, there exists the positive shifted operator

$$L_R^{\mathrm{res}}=A_R^{\mathrm{res}}+I,$$

which is self-adjoint and has compact resolvent.

10.

On $K_R$, there exist the conservative transport group

$$U_R\left(t\right)=e^{-it{L}_R^{\mathrm{res}}}$$

and the anti-self-adjoint generator

$$B_R=-i{L}_R^{\mathrm{res}}.$$

11.

The distribution kernel associated with $K_R$ is not an ordinary function kernel, but is defined as the distribution kernel

$${\mathsf{K}}_R^{\mathrm{dist}}\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }.$$

Proof. 

Items 1 and 2 follow from the construction of the boundary parameter space and the ${\sigma }_R$-null singular support. Items 3, 4, and 5 follow from the construction of traces, support maps, and regular boundary trace-mass functionals. Item 6 follows from Definition 4.143.

Item 7 follows from the definition

$$Q_R^{\mathrm{res}}={\overline{span{G}_R}}^{{\parallel ·\parallel }_{q_R}}$$

and from completeness of the ${\sigma }_R$-null trace-vanishing form subspace. Item 8 follows from the definition of $H_R^{\mathrm{res}}$ and from the fact that it is a closed linear subspace of $H_{\alpha ,+}$. Item 9 follows from the Friedrichs-type realization of $q_R^{\mathrm{res}}$, which gives

$$A_R^{\mathrm{res}},$$

from the self-adjointness of its positive shift

$$L_R^{\mathrm{res}}=A_R^{\mathrm{res}}+I,$$

and from the compact-resolvent theorem. Item 10 follows from the construction

$$U_R\left(t\right)=e^{-it{L}_R^{\mathrm{res}}}$$

by the spectral theorem and the construction

$$B_R=-i{L}_R^{\mathrm{res}}$$

by Stone’s theorem. Item 11 follows from the distribution kernel representation of the transport kernel family.

Thus all displayed objects are defined internally in this section and satisfy the listed properties. □

Corollary 4.152 

(Existence of the orthogonal projection onto $K_R$). $K_R$ is a closed linear subspace of $H_{\alpha ,+}$. Therefore, by the projection theorem for Hilbert spaces, there exists a unique orthogonal projection

$${\Pi }_R^{+}:H_{\alpha ,+}⟶K_R.$$

Namely,

$${\left({\Pi }_R^{+}\right)}^{*}={\Pi }_R^{+},{\left({\Pi }_R^{+}\right)}^2={\Pi }_R^{+},Ran{\Pi }_R^{+}=K_R.$$

Proof. 

By Lemma 4.149,

$$K_R\subset H_{\alpha ,+}$$

is a closed linear subspace. By the projection theorem for Hilbert spaces, any

$$h\in H_{\alpha ,+}$$

decomposes uniquely as

$$h=k+r,k\in K_R,r\in K_R^{\perp }.$$

Defining

$${\Pi }_R^{+}h:=k,$$

this is the orthogonal projection onto $K_R$. By the general properties of orthogonal projections,

$${\left({\Pi }_R^{+}\right)}^{*}={\Pi }_R^{+},{\left({\Pi }_R^{+}\right)}^2={\Pi }_R^{+},$$

and its range is $K_R$. Uniqueness is also included in the projection theorem. □

Definition 4.153 

(Canonical one-sided projection onto $K_R$). The orthogonal projection

$${\Pi }_R^{+}:H_{\alpha ,+}\to K_R$$

constructed on the one-sided analytic Hilbert space is called the canonical one-sided projection onto $K_R$. The unsuperscripted symbol

$${\Pi }_R$$

is reserved for the ambient Hilbert-space projector defined in Section 5 after $K_R$ has been embedded into X.

Proposition 4.154 

(Output of this subsection). This subsection fixes the singular-boundary input as the following internally constructed data:

$${\mathfrak{R}}_{\mathrm{op}}=({\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R,G_R,Q_R^{\mathrm{res}},H_R^{\mathrm{res}},K_R,{\Pi }_R^{+},U_R\left(t\right),B_R,{\mathsf{K}}_R^{\mathrm{dist}},L_R^{\mathrm{res}}).$$

These data include the ${\sigma }_R$-null property, regular-trace vanishing property, closed-form property, Hilbert closedness, self-adjoint realization, conservative transport, distribution kernel representation, compact resolvent, and existence of the orthogonal projection.

Proof. 

The boundary data were constructed by Definition 4.141. The trace-vanishing generating subspace associated with ${\Gamma }_R$ was constructed by Definition 4.143. The form and Hilbert data were constructed by Definition 4.145. The singular-boundary subspace $K_R$ was constructed by Definition 4.147, and its closedness was proved by Lemma 4.149. The orthogonal projection ${\Pi }_R^{+}$ exists by Corollary 4.152. The transport group, generator, distribution kernel, and compact resolvent are given by Lemma 4.150 and Theorem 4.151. Therefore the displayed data is fixed as internally constructed data. □

4.10. Fixing the Operator-Theoretic Boundary Data and Transition to the Next Section

In this subsection, we fix the singular-boundary objects constructed in this section as a single set of input data. In the next section, the data fixed here are received as the singular-boundary input and integrated with the arithmetic construction of Section 3 on a common ambient Hilbert space. Accordingly, the role of this subsection is not to add a new analytic construction, but to record the output of this section and fix the types and dependencies of the objects referred to in the next section.

Definition 4.155 

(Operator-theoretic boundary data). Define the singular-boundary input data constructed in this section by

$$\begin{array}{cc}\hfill {\mathfrak{R}}_{\mathrm{op}}:=(& {\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R,G_R,Q_R^{\mathrm{res}},H_R^{\mathrm{res}},K_R,{\Pi }_R^{+},\hfill \\ & U_R\left(t\right),B_R,{\mathsf{K}}_R^{\mathrm{dist}},L_R^{\mathrm{res}}).\hfill \end{array}$$

Here each component has the following meaning.

1.

$({\Sigma }_R,{\sigma }_R,{\Gamma }_R)$ is the boundary triple data equipped with a regular area measure and a ${\sigma }_R$-null singular support.

2.

$T_R$ is a linear subspace inside the form domain on which the singular boundary trace and the regular boundary trace are defined.

3.

${supp}_R$ is the support map assigning the distribution support of the singular boundary trace.

4.

$L_R$ is the regular boundary trace-mass functional with respect to the regular area measure ${\sigma }_R$.

5.

$G_R$ is the generating class satisfying the ${\sigma }_R$-null property and the regular-trace vanishing condition simultaneously.

6.

$Q_R^{\mathrm{res}}$ is the $q_R$-form-norm closure of $G_R$.

7.

$H_R^{\mathrm{res}}$ is the $H_{\alpha ,+}$-closure of $Q_R^{\mathrm{res}}$.

8.

$K_R=H_R^{\mathrm{res}}$ is the closed singular-boundary subspace inside the one-sided analytic Hilbert space.

9.

${\Pi }_R^{+}$ is the orthogonal projection from $H_{\alpha ,+}$ onto $K_R$.

10.

$L_R^{\mathrm{res}}$ is the positive shifted self-adjoint operator obtained from the Friedrichs-type realization of the ${\sigma }_R$-null restricted closed form.

11.

$U_R\left(t\right)=e^{-it{L}_R^{\mathrm{res}}}$ is the strongly continuous unitary transport group on $K_R$.

12.

$B_R=-i{L}_R^{\mathrm{res}}$ is the anti-self-adjoint generator of $U_R\left(t\right)$.

13.

${\mathsf{K}}_R^{\mathrm{dist}}\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }$ is the distribution kernel representation of the transport distribution kernel.

Proposition 4.156 

(Internal definiteness of the data). Each component of ${\mathfrak{R}}_{\mathrm{op}}$ is determined by the analytic Hilbert-space setup of Section 2 and the construction of this section. In particular, the following hold.

1.

$${\sigma }_R\left({\Gamma }_R\right)=0,{\nu }_R\left({\Gamma }_R\right)=1,{\nu }_R\perp {\sigma }_R.$$

2.

For any $f\in G_R$,

$${supp}_R\left(f\right)\subset {\Gamma }_R,L_R(f;{\Sigma }_R\setminus {\Gamma }_R)=0.$$

3.

$Q_R^{\mathrm{res}}$ is a closed form subspace of $Q_R$, and

$$q_R^{\mathrm{res}}=q_R{|}_{Q_R^{\mathrm{res}}\times Q_R^{\mathrm{res}}}$$

is a closed lower-bounded form on $H_R^{\mathrm{res}}$.

4.

$A_R^{\mathrm{res}}$ is the unique self-adjoint representing operator of $q_R^{\mathrm{res}}$, and

$$L_R^{\mathrm{res}}=A_R^{\mathrm{res}}+I$$

is self-adjoint and positive.

5.

$L_R^{\mathrm{res}}$ has compact resolvent and purely discrete spectrum.

6.

$K_R$ is a closed subspace of $H_{\alpha ,+}$, and therefore

$${\Pi }_R^{+}:H_{\alpha ,+}\to K_R$$

exists uniquely.

Proof. 

Item 1 follows from the construction of the boundary parameter space and the ${\sigma }_R$-null singular support. Item 2 follows from the definition of the trace-vanishing generating subspace associated with ${\Gamma }_R$$G_R$ and the vanishing criterion for the regular boundary trace-mass functional. Item 3 follows from the construction of the closure of the ${\sigma }_R$-null trace-vanishing form subspace. Item 4 follows from the first representation theorem for closed lower-bounded forms and the definition of the positive shifted operator. Item 5 follows from the compact embedding from the form domain into the Hilbert space and the compact-resolvent theorem. Item 6 follows from the fact that $K_R=H_R^{\mathrm{res}}$ is a closed subspace of $H_{\alpha ,+}$, and from the projection theorem for Hilbert spaces. Accordingly, all components of ${\mathfrak{R}}_{\mathrm{op}}$ are fixed internally in this section. □

Definition 4.157 

(Fixing as the singular-boundary input). To fix the singular-boundary input from the next section onward means to fix the data

$${\mathfrak{R}}_{\mathrm{op}}$$

of Definition 4.155. Namely, the orthogonal-decomposition framework in the next section receives, from the singular-boundary side, only

$${\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R,G_R,Q_R^{\mathrm{res}},H_R^{\mathrm{res}},K_R,{\Pi }_R^{+},U_R\left(t\right),B_R,{\mathsf{K}}_R^{\mathrm{dist}},L_R^{\mathrm{res}}.$$

Proposition 4.158 

(Transition to the next section). In the next section, ${\mathfrak{R}}_{\mathrm{op}}$ is fixed as the already constructed singular-boundary input and is placed, together with the arithmetic-side data constructed in Section 3, on an ambient Hilbert space. In this transition, the information passed from the singular-boundary side to the next section is limited to what is contained in

$${\mathfrak{R}}_{\mathrm{op}}.$$

In particular, the singular-boundary subspace and orthogonal projection onto $K_R$ used in the next section are obtained by lifting

$$K_R\mathrm{and}{\Pi }_R^{+}$$

constructed in this section to the ambient Hilbert-space setting.

Proof. 

By Definition 4.157, the singular-boundary input fixed in the next section is

$${\mathfrak{R}}_{\mathrm{op}}.$$

Moreover, by Proposition 4.156, each component of ${\mathfrak{R}}_{\mathrm{op}}$ has already been constructed in this section.

In the orthogonal-decomposition framework of the next section, $K_R$ and ${\Pi }_R^{+}$ are first lifted through the analytic embedding into the ambient Hilbert space. The isometric image of the closed subspace is closed, so the orthogonal projection onto that image exists. This lifted projection is denoted by ${\Pi }_R$ in Section 5 and acts on X, not on $H_{\alpha ,+}$. Therefore, the singular-boundary subspace and projector required in the next section are derived from the one-sided objects $K_R$ and ${\Pi }_R^{+}$ of this section. It follows that the transition on the singular-boundary side is completed by ${\mathfrak{R}}_{\mathrm{op}}$. □

Remark 4.159 

(Separation of dependencies). The construction in this section is based on the analytic Hilbert-space setup of Section 2 and the measure-theoretic and operator-theoretic constructions inside this section. The arithmetic projector family, arithmetic trace, and subsequent zero-counting machinery of Section 3 are integrated only from the next section onward. Therefore this section is the section that constructs the singular-boundary input, and it is not a section that proves orthogonality with the arithmetic side or closure of finite-window counting.

Convention 4.160 

(Meaning of fixing at the beginning of the next section). When the beginning of the next section says that “the auxiliary operator-theoretic boundary data is fixed,” this does not mean assuming externally and arbitrarily given unconstructed data. It means fixing the singular boundary data

$${\mathfrak{R}}_{\mathrm{op}}$$

constructed by Theorem 4.151 of this section. From now on, the singular-boundary input is used in this sense.

Proposition 4.161 

(Completion of Section 4). This section fixes all singular-boundary data, form data, Hilbert-space data, projection data, transport data, distribution-kernel data, and spectral data required in the next section as

$${\mathfrak{R}}_{\mathrm{op}}.$$

Proof. 

The boundary data have been fixed as

$$({\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R).$$

The form data have been fixed as

$$G_R,Q_R^{\mathrm{res}},q_R^{\mathrm{res}}.$$

The Hilbert-space data have been fixed as

$$H_R^{\mathrm{res}},K_R.$$

The projection data have been fixed as

$${\Pi }_R^{+}.$$

The transport data have been fixed as

$$U_R\left(t\right),B_R.$$

The distribution-kernel data have been fixed as

$${\mathsf{K}}_R^{\mathrm{dist}}.$$

The spectral data have been fixed as

$$L_R^{\mathrm{res}},$$

together with its compact resolvent, purely discrete spectrum, and finite-window spectral projections. Therefore all singular-boundary objects required in the next section are contained in

$${\mathfrak{R}}_{\mathrm{op}}.$$

□

Preparation for the analytic comparison to Section 6.

The type separation on the singular-boundary side constructed in this section is the foundation for the analytic comparison used in Section 6. In that comparison, distributional objects on the singular-boundary side are handled inside a Gelfand triple

$${\mathcal{D}}_R\subset H_{\alpha ,+}\subset {\mathcal{D}}_R^{\prime },$$

and point evaluations and singular boundary traces are treated not as ordinary bounded functionals on $H_{\alpha ,+}$, but as distributional boundary distributions belonging to ${\mathcal{D}}_R^{\prime }$. Furthermore, the distribution kernels, quadratic forms, generators, boundary forms, and projectors distinguished in this section are preparation for defining the tempered distribution pairings and ${\mathfrak{S}}_2$-approximations appearing in Section 6,

$$\langle \mu ,\phi \rangle ,\phi \in \mathcal{S}\left(\mathbb{R}\right),\mu \in {\mathcal{S}}^{\prime }\left(\mathbb{R}\right).$$

In particular, the boundary-distribution comparison kernel K introduced later and its finite-rank cutoffs $K_N=P_{N}K{P}_N$ are interpreted within the framework of distribution kernels, projectors, and compactness established up to this section. This comparison is not a new assumption, but a notational bridge for using the operator-theoretic objects already constructed from Section 2 through 5 in the distributional evaluations, central regularization, and ${\mathfrak{S}}_2$-approximations of Section 6.

5. Orthogonal-Decomposition Spectral Comparison Framework

5.1. Embedding the Constructed Singular-Boundary Subspace into the Ambient Hilbert Space

Section 2 fixed the analytic Hilbert-space data

$$H_{\alpha ,+},q_R,A_R,L,m_L^{(\Phi )},$$

and Section 3 fixed the coefficient-space arithmetic construction

$${\Lambda }^{\mathrm{ex}},C_{\mathrm{comp}},{\partial }_{log},{\left\{{\Pi }_n\right\}}_{n\ge 1},{\Pi }_{\mathrm{arith}}(·).$$

Section 4 constructed, independently of these, the singular-boundary input data

$${\mathfrak{R}}_{\mathrm{op}}$$

operator-theoretically. In this section, we lift this constructed input to the ambient Hilbert space and place it on the same orthogonal-decomposition stage as the coefficient-space arithmetic construction of Section 3.

What is done in this subsection is not a reconstruction of the operator-theoretic singular-boundary data. Namely,

$$({\Sigma }_R,{\sigma }_R,{\Gamma }_R,T_R,{supp}_R,L_R)$$

is fixed as data already constructed in Section 4. In this subsection, among its outputs,

$$K_R,{\Pi }_R^{+},Q_R^{\mathrm{res}},H_R^{\mathrm{res}}$$

are lifted to the ambient stage X, and the singular-boundary projector used below is defined.

Definition 5.1 

(Ambient Hilbert space and canonical embeddings). Define the ambient Hilbert space for the orthogonal-decomposition comparison by

$$X:=H_{\alpha ,+}\oplus {\mathcal{H}}_{\mathrm{arith}}.$$

Its inner product is given by

$${\langle (f,u),(g,v)\rangle }_X:={\langle f,g\rangle }_{H_{\alpha ,+}}+{\langle u,v\rangle }_{\mathrm{arith}}.$$

Write its norm as

$${\parallel x\parallel }_X:={\langle x,x\rangle }_X^{1/2}{,|x|}_X:={\parallel x\parallel }_X.$$

Define the canonical embeddings

$$J_{\mathrm{an}}:H_{\alpha ,+}\to X,J_{\mathrm{an}}f:=(f,0),$$

and

$$J_{\mathrm{arith}}:{\mathcal{H}}_{\mathrm{arith}}\to X,J_{\mathrm{arith}}u:=(0,u).$$

Then

$$J_{\mathrm{an}}{H}_{\alpha ,+}\perp J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}},$$

and

$$X=J_{\mathrm{an}}{H}_{\alpha ,+}\oplus J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}.$$

At this stage, no further operator on X is introduced.

Definition 5.2 

(Fixing and lifting the constructed singular-boundary input). Fix the singular-boundary input data

$${\mathfrak{R}}_{\mathrm{op}}$$

constructed in Section 4. Among its components, write the closed one-sided singular-boundary subspace as

$$K_R^{+}:=K_R=H_R^{\mathrm{res}}\subset H_{\alpha ,+}.$$

Define the singular-boundary component on the ambient stage by

$${\mathcal{K}}_R:=J_{\mathrm{an}}{K}_R^{+}\subset X.$$

Also, let

$${\Pi }_R^{+}:H_{\alpha ,+}\to K_R^{+}$$

be the orthogonal projection on the one-sided analytic stage constructed in Section 4. Define the singular-boundary projector on the ambient stage by

$${\Pi }_R:=J_{\mathrm{an}}{\Pi }_R^{+}{J}_{\mathrm{an}}^{*}.$$

Convention 5.3 

(One-sided and ambient singular-boundary notation). In this section and in Section 6, $K_R^{+}\subset H_{\alpha ,+}$ denotes the one-sided closed subspace constructed in Section 4, while

$${\mathcal{K}}_R=J_{\mathrm{an}}{K}_R^{+}\subset X$$

denotes its ambient image. Likewise, ${\Pi }_R^{+}$ denotes the one-sided projector on $H_{\alpha ,+}$, whereas

$${\Pi }_R=J_{\mathrm{an}}{\Pi }_R^{+}{J}_{\mathrm{an}}^{*}$$

denotes the ambient projector on X. Thus every occurrence of ${\Pi }_R$ from this point onward is an X-operator, while ${\Pi }_R^{+}$ is reserved for the one-sided analytic stage.

Lemma 5.4 

(Closedness of the lifted singular-boundary component). ${\mathcal{K}}_R$ is a closed linear subspace of X, and

$${\mathcal{K}}_R\subset J_{\mathrm{an}}{H}_{\alpha ,+}.$$

Proof. 

By Section 4, $K_R^{+}$ is a closed linear subspace of $H_{\alpha ,+}$. The map $J_{\mathrm{an}}$ is an isometric isomorphism from $H_{\alpha ,+}$ onto $J_{\mathrm{an}}{H}_{\alpha ,+}\subset X$. Therefore the image of the closed set $K_R^{+}\subset H_{\alpha ,+}$,

$$J_{\mathrm{an}}{K}_R^{+},$$

is closed in $J_{\mathrm{an}}{H}_{\alpha ,+}$. Furthermore, $J_{\mathrm{an}}{H}_{\alpha ,+}$ is a closed subspace of X, so

$${\mathcal{K}}_R=J_{\mathrm{an}}{K}_R^{+}$$

is closed in X. The inclusion

$${\mathcal{K}}_R\subset J_{\mathrm{an}}{H}_{\alpha ,+}$$

follows immediately from the definition. □

Theorem 5.5 

(Singular-boundary projector on the ambient stage). The operator

$${\Pi }_R:=J_{\mathrm{an}}{\Pi }_R^{+}{J}_{\mathrm{an}}^{*}$$

is an orthogonal projector on X, and satisfies

$${\Pi }_R^2={\Pi }_R,{\Pi }_R^{*}={\Pi }_R,Ran{\Pi }_R={\mathcal{K}}_R,ker{\Pi }_R={\mathcal{K}}_R^{\perp }.$$

Equivalently, every $x\in X$ has a unique orthogonal decomposition

$$x=x_R+x_{\perp },x_R\in {\mathcal{K}}_R,x_{\perp }\in {\mathcal{K}}_R^{\perp },$$

and

$${\Pi }_{R}x=x_R.$$

Proof. 

First we show that ${\Pi }_R$ is a projection. For the canonical embedding,

$$J_{\mathrm{an}}^{*}{J}_{\mathrm{an}}=I_{H_{\alpha ,+}}.$$

Therefore

$${\Pi }_R^2=J_{\mathrm{an}}{\Pi }_R^{+}{J}_{\mathrm{an}}^{*}{J}_{\mathrm{an}}{\Pi }_R^{+}{J}_{\mathrm{an}}^{*}=J_{\mathrm{an}}{\left({\Pi }_R^{+}\right)}^2{J}_{\mathrm{an}}^{*}=J_{\mathrm{an}}{\Pi }_R^{+}{J}_{\mathrm{an}}^{*}={\Pi }_R.$$

Also,

$${\Pi }_R^{*}={\left(J_{\mathrm{an}}{\Pi }_R^{+}{J}_{\mathrm{an}}^{*}\right)}^{*}=J_{\mathrm{an}}{\left({\Pi }_R^{+}\right)}^{*}{J}_{\mathrm{an}}^{*}=J_{\mathrm{an}}{\Pi }_R^{+}{J}_{\mathrm{an}}^{*}={\Pi }_R.$$

Thus ${\Pi }_R$ is a self-adjoint projector.

Next we identify its range. For any $x=(f,u)\in X$,

$$J_{\mathrm{an}}^{*}x=f,$$

so

$${\Pi }_{R}x=J_{\mathrm{an}}{\Pi }_R^{+}f\in J_{\mathrm{an}}{K}_R^{+}={\mathcal{K}}_R.$$

Therefore

$$Ran{\Pi }_R\subset {\mathcal{K}}_R.$$

Conversely, any $k\in {\mathcal{K}}_R$ can be written as

$$k=J_{\mathrm{an}}h(h\in K_R^{+}).$$

Then ${\Pi }_R^{+}h=h$, and hence

$${\Pi }_{R}k=J_{\mathrm{an}}{\Pi }_R^{+}{J}_{\mathrm{an}}^{*}{J}_{\mathrm{an}}h=J_{\mathrm{an}}{\Pi }_R^{+}h=J_{\mathrm{an}}h=k.$$

Thus

$${\mathcal{K}}_R\subset Ran{\Pi }_R.$$

Therefore

$$Ran{\Pi }_R={\mathcal{K}}_R.$$

The kernel of a self-adjoint projector is the orthogonal complement of its range, so

$$ker{\Pi }_R={(Ran{\Pi }_R)}^{\perp }={\mathcal{K}}_R^{\perp }.$$

The final orthogonal decomposition follows from the projection theorem for Hilbert spaces. □

Remark 5.6 

(Status of the constructed input). The ${\mathcal{K}}_R$-side input used in this section is

$${\mathfrak{R}}_{\mathrm{op}},$$

constructed in Section 4. The role of this section is to lift these constructed data to the ambient Hilbert space X and place them on the same ambient stage as the coefficient-space arithmetic construction of Section 3. Therefore, this section does not reconstruct the operator-theoretic singular-boundary data.

Theorem 5.7 

(Lifting boundary cancellation to the ambient stage). For the zero extension of the boundary form constructed in Section 4,

$$b_R^{\mathrm{res}},$$

one has

$$b_R^{\mathrm{res}}[u,v]=0$$

for any

$$u,v\in Q_R^{\mathrm{res}}.$$

Therefore, for the lifts

$$J_{\mathrm{an}}u,J_{\mathrm{an}}v\in {\mathcal{K}}_R$$

on the ambient stage,

$$b_{R,X}^{\mathrm{res}}[J_{\mathrm{an}}u,J_{\mathrm{an}}v]:=b_R^{\mathrm{res}}[u,v]=0$$

holds.

Proof. 

By the boundary cancellation theorem of Section 4,

$$b_R^{\mathrm{res}}[u,v]=0(u,v\in Q_R^{\mathrm{res}}).$$

Since $J_{\mathrm{an}}$ is an isometric embedding and $Q_R^{\mathrm{res}}\subset K_R^{+}$, we have

$$J_{\mathrm{an}}u,J_{\mathrm{an}}v\in J_{\mathrm{an}}{K}_R^{+}={\mathcal{K}}_R.$$

The boundary form on the ambient stage is defined by

$$b_{R,X}^{\mathrm{res}}[J_{\mathrm{an}}u,J_{\mathrm{an}}v]:=b_R^{\mathrm{res}}[u,v],$$

so immediately

$$b_{R,X}^{\mathrm{res}}[J_{\mathrm{an}}u,J_{\mathrm{an}}v]=0.$$

□

Remark 5.8 

(Boundary cancellation convention below). Unless otherwise stated, every use of boundary cancellation for the ${\Pi }_R$-component is interpreted as a statement on

$$J_{\mathrm{an}}{Q}_R^{\mathrm{res}}\subset {\mathcal{K}}_R.$$

Namely, the ${\sigma }_R$-null support and regular-trace vanishing conditions constructed in Section 4 first cut out the analytic form subspace

$$Q_R^{\mathrm{res}}\subset Q_R,$$

and give the zero extension of the boundary form on it,

$$b_R^{\mathrm{res}}=0.$$

This section only lifts that cancellation identity to the ambient stage, and imposes no new assumption on the boundary data.

5.2. Lifting the Transport Block to the Ambient Stage

Next, we lift the singular-boundary transport group constructed in Section 4 to the ambient stage. On the one-sided analytic stage, the operators

$$U_R\left(t\right)=e^{-it{L}_R^{\mathrm{res}}},B_R=-i{L}_R^{\mathrm{res}}$$

have already been constructed on $K_R^{+}=H_R^{\mathrm{res}}$. In this subsection, we do not reintroduce them arbitrarily; rather, through the canonical embedding $J_{\mathrm{an}}$, we transfer them to

$${\mathcal{K}}_R=J_{\mathrm{an}}{K}_R^{+},$$

and further extend them unitarily to all of X.

Definition 5.9 

(Singular-boundary transport group on the ambient stage). Using the constructed transport group $U_R\left(t\right)$ on $K_R^{+}$, define the operator $U_R^X\left(t\right)$ on ${\mathcal{K}}_R$ by

$$U_R^X\left(t\right)J_{\mathrm{an}}h:=J_{\mathrm{an}}{U}_R\left(t\right)h(h\in K_R^{+}).$$

Furthermore, define its extension to all of X by

$${\tilde{U}}_R\left(t\right):=U_R^X\left(t\right){\Pi }_R+(I_X-{\Pi }_R)(t\in \mathbb{R}).$$

Lemma 5.10 

(Strongly continuous unitarity of the lifted transport group). ${\left\{U_R^X\left(t\right)\right\}}_{t\in \mathbb{R}}$ is a strongly continuous unitary group on ${\mathcal{K}}_R$. Moreover,

$${\left\{{\tilde{U}}_R\left(t\right)\right\}}_{t\in \mathbb{R}}$$

is a strongly continuous unitary group on X, and satisfies

$${\tilde{U}}_R\left(t\right){\Pi }_R={\Pi }_R{\tilde{U}}_R\left(t\right)=U_R^X\left(t\right){\Pi }_R.$$

Proof. 

First we verify that $U_R^X\left(t\right)$ is well-defined. The map $J_{\mathrm{an}}$ is an isometric isomorphism from $H_{\alpha ,+}$ onto $J_{\mathrm{an}}{H}_{\alpha ,+}$, and its restriction is an isometric isomorphism from $K_R^{+}$ onto ${\mathcal{K}}_R$. Therefore

$$U_R^X\left(t\right)=J_{\mathrm{an}}{U}_R\left(t\right)J_{\mathrm{an}}^{-1}\mathrm{on}{\mathcal{K}}_R.$$

Since $U_R\left(t\right)$ was constructed in Section 4 as a strongly continuous unitary group on $K_R^{+}$, its isometric conjugate $U_R^X\left(t\right)$ is also a strongly continuous unitary group on ${\mathcal{K}}_R$.

Next consider the extension to X. Every $x\in X$ decomposes uniquely as

$$x=x_R+x_{\perp },x_R:={\Pi }_{R}x\in {\mathcal{K}}_R,x_{\perp }:=(I_X-{\Pi }_R)x\in {\mathcal{K}}_R^{\perp }.$$

Then

$${\tilde{U}}_R\left(t\right)x=U_R^X\left(t\right)x_R+x_{\perp }.$$

Since $U_R^X\left(t\right)$ is unitary on ${\mathcal{K}}_R$ and acts as the identity on ${\mathcal{K}}_R^{\perp }$, ${\tilde{U}}_R\left(t\right)$ is unitary on X. The group law follows immediately from the same decomposition.

For strong continuity,

$$\parallel {\tilde{U}}_R{\left(t\right)x-x\parallel }_X={\parallel U_R^X\left(t\right)x_R-x_R\parallel }_X,$$

and the right-hand side converges to 0 as $t\to 0$ by strong continuity of $U_R^X\left(t\right)$. Finally, the commutation relation follows immediately from

$${\Pi }_{R}x=x_R$$

and the decomposition above. □

Definition 5.11 

(Transport generator on the ambient stage). Define the transport generator on the ambient stage by

$$R_X:Dom\left(R_X\right)\subset X\to X.$$

Its domain is

$$Dom\left(R_X\right):=J_{\mathrm{an}}Dom\left(B_R\right)\oplus {\mathcal{K}}_R^{\perp },$$

and define

$$R_X(J_{\mathrm{an}}h+r):=J_{\mathrm{an}}{B}_{R}h,$$

where

$$h\in Dom\left(B_R\right),r\in {\mathcal{K}}_R^{\perp }.$$

Theorem 5.12 

(Transport block and conservativity). The operator $R_X$ is anti-self-adjoint on X, and

$$R_X^{*}=-R_X.$$

Moreover,

$${\tilde{U}}_R\left(t\right)=e^{t{R}_X}(t\in \mathbb{R}),$$

and

$$R_X={\Pi }_R{R}_X{\Pi }_R$$

holds on $Dom\left(R_X\right)$. Equivalently,

$$R_X(I_X-{\Pi }_R)u=0,R_X{\Pi }_{R}u=R_{X}u={\Pi }_R{R}_{X}u(u\in Dom\left(R_X\right)).$$

Finally,

$$\parallel {\tilde{U}}_R{\left(t\right)u\parallel }_X={\parallel u\parallel }_X(t\in \mathbb{R},u\in X),$$

and

$$\Re {\langle R_{X}u,u\rangle }_X=0(u\in Dom\left(R_X\right)).$$

Proof. 

On ${\mathcal{K}}_R$,

$$U_R^X\left(t\right)=J_{\mathrm{an}}{U}_R\left(t\right)J_{\mathrm{an}}^{-1},$$

and its generator is

$$J_{\mathrm{an}}{B}_R{J}_{\mathrm{an}}^{-1}.$$

Since Section 4 proved $B_R^{*}=-B_R$, this isometric conjugate is also anti-self-adjoint on ${\mathcal{K}}_R$. On the other hand, on ${\mathcal{K}}_R^{\perp }$, the transport is the identity group and its generator is the zero operator. Therefore, the direct-sum generator with respect to the orthogonal direct sum

$$X={\mathcal{K}}_R\oplus {\mathcal{K}}_R^{\perp }$$

is

$$R_X=\left(J_{\mathrm{an}}{B}_R{J}_{\mathrm{an}}^{-1}\right)\oplus 0,$$

and it is anti-self-adjoint and satisfies

$${\tilde{U}}_R\left(t\right)=e^{t{R}_X}.$$

Next we verify the support relation. Write $u=J_{\mathrm{an}}h+r\in Dom\left(R_X\right)$, where

$$h\in Dom\left(B_R\right),r\in {\mathcal{K}}_R^{\perp }.$$

Then

$${\Pi }_{R}u=J_{\mathrm{an}}h,(I_X-{\Pi }_R)u=r.$$

Therefore

$$R_X(I_X-{\Pi }_R)u=R_{X}r=0,$$

and

$$R_X{\Pi }_{R}u=R_X{J}_{\mathrm{an}}h=J_{\mathrm{an}}{B}_{R}h=R_{X}u.$$

Moreover, $R_{X}u=J_{\mathrm{an}}{B}_{R}h\in {\mathcal{K}}_R$, so

$${\Pi }_R{R}_{X}u=R_{X}u.$$

Hence

$$R_X={\Pi }_R{R}_X{\Pi }_R$$

holds on the domain.

Norm preservation follows from the unitarity of Lemma 5.10. Finally, anti-self-adjointness gives

$${\langle R_{X}u,u\rangle }_X=-\overline{{\langle R_{X}u,u\rangle }_X}.$$

Therefore this number is purely imaginary, and

$$\Re {\langle R_{X}u,u\rangle }_X=0.$$

□

Remark 5.13 

(Treatment of the boundary-distribution comparison kernel). Section 4 constructed the singular-boundary distribution kernel

$${\mathsf{K}}_R^{\mathrm{dist}}\in {\mathcal{D}}_R^{\prime }\widehat{\otimes }{\mathcal{D}}_R^{\prime }$$

as a distribution kernel. This section does not reinterpret this kernel as an ordinary function kernel. What is used on the ambient stage is the constructed transport group $U_R\left(t\right)$, the generator $B_R$, and their lifts. Therefore, statements concerning the boundary-distribution comparison kernel are to be interpreted within the scope of the distribution-kernel representation of Section 4.

5.3. Orthogonality of the Arithmetic Summand and the Singular-Boundary Component

We now import the arithmetic projector family from its canonical home in Section 3 and lift it to the ambient stage X. The sole purpose of this subsection is to show that these lifted arithmetic projectors are completely orthogonal to the singular-boundary projector ${\Pi }_R$ lifted to the ambient stage in §Section 5.1. No zero-counting proposition, comparison theorem, or argument toward the conclusion is used here.

Definition 5.14 

(Lifted arithmetic projectors on the ambient stage). For each $n\ge 1$, define the lifted arithmetic projector on X by

$${\widehat{\Pi }}_n:=J_{\mathrm{arith}}{\Pi }_n{J}_{\mathrm{arith}}^{*}.$$

More generally, when

$$\varphi :\mathbb{N}\to \mathbb{C}$$

has finite support or belongs to ${\ell }^1\left(\mathbb{N}\right)$, define

$${\widehat{\Pi }}_{\mathrm{arith}}\left(\varphi \right):=J_{\mathrm{arith}}{\Pi }_{\mathrm{arith}}\left(\varphi \right)J_{\mathrm{arith}}^{*}.$$

Here ${\Pi }_n$ and ${\Pi }_{\mathrm{arith}}\left(\varphi \right)$ still retain their canonical home on the coefficient Hilbert space ${\mathcal{H}}_{\mathrm{arith}}$ of Section 3, and in the orthogonal-decomposition comparison, only their lifts to the ambient space,

$${\widehat{\Pi }}_n,{\widehat{\Pi }}_{\mathrm{arith}}\left(\varphi \right),$$

are used.

Furthermore,

$$Ran{\widehat{\Pi }}_n\subset J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}},Ran{\widehat{\Pi }}_{\mathrm{arith}}\left(\varphi \right)\subset J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}$$

hold.

Theorem 5.15 

(Orthogonality of the arithmetic projectors and the singular-boundary projector). For every $n\ge 1$,

$${\widehat{\Pi }}_n^2={\widehat{\Pi }}_n,{\widehat{\Pi }}_n^{*}={\widehat{\Pi }}_n$$

hold, and if $m\ne n$, then

$${\widehat{\Pi }}_m{\widehat{\Pi }}_n=0.$$

Furthermore,

$${\Pi }_R{\widehat{\Pi }}_n={\widehat{\Pi }}_n{\Pi }_R=0(n\ge 1)$$

holds.

More generally, if ϕ has finite support or belongs to ${\ell }^1\left(\mathbb{N}\right)$, then

$${\Pi }_R{\widehat{\Pi }}_{\mathrm{arith}}\left(\varphi \right)={\widehat{\Pi }}_{\mathrm{arith}}\left(\varphi \right){\Pi }_R=0.$$

Proof. 

First record the basic relations for the canonical embedding

$$J_{\mathrm{arith}}:{\mathcal{H}}_{\mathrm{arith}}\to X.$$

By Definition 5.1, for $u\in {\mathcal{H}}_{\mathrm{arith}}$,

$$J_{\mathrm{arith}}u=(0,u).$$

Thus its adjoint is projection onto the arithmetic component, and

$$J_{\mathrm{arith}}^{*}(f,u)=u((f,u)\in X).$$

Therefore

$$J_{\mathrm{arith}}^{*}{J}_{\mathrm{arith}}=I_{{\mathcal{H}}_{\mathrm{arith}}},J_{\mathrm{arith}}^{*}{J}_{\mathrm{an}}=0.$$

We now prove the laws for the projector family. Using (5.3) and Lemma 3.20, for any $n\ge 1$,

$${\widehat{\Pi }}_n^2=J_{\mathrm{arith}}{\Pi }_n{J}_{\mathrm{arith}}^{*}{J}_{\mathrm{arith}}{\Pi }_n{J}_{\mathrm{arith}}^{*}=J_{\mathrm{arith}}{\Pi }_n^2{J}_{\mathrm{arith}}^{*}=J_{\mathrm{arith}}{\Pi }_n{J}_{\mathrm{arith}}^{*}={\widehat{\Pi }}_n.$$

Similarly,

$${\widehat{\Pi }}_n^{*}={\left(J_{\mathrm{arith}}{\Pi }_n{J}_{\mathrm{arith}}^{*}\right)}^{*}=J_{\mathrm{arith}}{\Pi }_n^{*}{J}_{\mathrm{arith}}^{*}=J_{\mathrm{arith}}{\Pi }_n{J}_{\mathrm{arith}}^{*}={\widehat{\Pi }}_n.$$

If $m\ne n$, then

$${\widehat{\Pi }}_m{\widehat{\Pi }}_n=J_{\mathrm{arith}}{\Pi }_m{J}_{\mathrm{arith}}^{*}{J}_{\mathrm{arith}}{\Pi }_n{J}_{\mathrm{arith}}^{*}=J_{\mathrm{arith}}{\Pi }_m{\Pi }_n{J}_{\mathrm{arith}}^{*}=0,$$

because ${\Pi }_m{\Pi }_n=0$ on ${\mathcal{H}}_{\mathrm{arith}}$. This proves the laws for the lifted projector family.

Next we prove orthogonality with ${\Pi }_R$. By Definition 5.2 and Theorem 5.5,

$$Ran{\Pi }_R={\mathcal{K}}_R\subset J_{\mathrm{an}}{H}_{\alpha ,+}.$$

On the other hand, by Definition 5.14,

$$Ran{\widehat{\Pi }}_n\subset J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}.$$

By Definition 5.1,

$$J_{\mathrm{an}}{H}_{\alpha ,+}\perp J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}},$$

and hence

$$Ran{\widehat{\Pi }}_n\perp Ran{\Pi }_R.$$

Since ${\Pi }_R$ is the orthogonal projector onto $Ran{\Pi }_R$, any vector orthogonal to $Ran{\Pi }_R$ belongs to $ker{\Pi }_R$. Thus

$$Ran{\widehat{\Pi }}_n\subset ker{\Pi }_R,$$

and therefore

$${\Pi }_R{\widehat{\Pi }}_n=0.$$

For the product in the reverse order, take $x\in X$. Since

$${\Pi }_{R}x\in Ran{\Pi }_R\subset J_{\mathrm{an}}{H}_{\alpha ,+},$$

the relation $J_{\mathrm{arith}}^{*}{J}_{\mathrm{an}}=0$ in (5.3) gives

$$J_{\mathrm{arith}}^{*}{\Pi }_{R}x=0.$$

Therefore

$${\widehat{\Pi }}_n{\Pi }_{R}x=J_{\mathrm{arith}}{\Pi }_n{J}_{\mathrm{arith}}^{*}{\Pi }_{R}x=0.$$

This holds for all $x\in X$, so

$${\widehat{\Pi }}_n{\Pi }_R=0.$$

The same argument applies to ${\widehat{\Pi }}_{\mathrm{arith}}\left(\varphi \right)$. Indeed, if $\varphi$ has finite support or belongs to ${\ell }^1$, then

$$Ran{\widehat{\Pi }}_{\mathrm{arith}}\left(\varphi \right)\subset J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}},$$

so its range is again orthogonal to $Ran{\Pi }_R\subset J_{\mathrm{an}}{H}_{\alpha ,+}$. Therefore

$${\Pi }_R{\widehat{\Pi }}_{\mathrm{arith}}\left(\varphi \right)=0.$$

Conversely, since $J_{\mathrm{arith}}^{*}{\Pi }_R=0$,

$${\widehat{\Pi }}_{\mathrm{arith}}\left(\varphi \right){\Pi }_R=J_{\mathrm{arith}}{\Pi }_{\mathrm{arith}}\left(\varphi \right)J_{\mathrm{arith}}^{*}{\Pi }_R=0.$$

Thus

$${\Pi }_R{\widehat{\Pi }}_{\mathrm{arith}}\left(\varphi \right)={\widehat{\Pi }}_{\mathrm{arith}}\left(\varphi \right){\Pi }_R=0.$$

This completes the proof. □

Remark 5.16 

(Canonical-home discipline in the orthogonal-decomposition comparison). The projector family ${\left\{{\Pi }_n\right\}}_{n\ge 1}$ belongs, in its canonical home, to the coefficient-level arithmetic Hilbert space ${\mathcal{H}}_{\mathrm{arith}}$ of Section 3. On the other hand, ${\Pi }_R$ belongs, in its canonical home, to the singular-boundary subspace ${\mathcal{K}}_R\subset X$ in this section. Their orthogonality is not the result of defining the same object twice. Rather, it is the result of importing these two objects into the ambient direct sum

$$X=J_{\mathrm{an}}{H}_{\alpha ,+}\oplus J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}},$$

and comparing their lifted ranges there.

5.4. Orthogonal Decomposition and the Localized Comparison Interface

We now collect the singular-boundary subspace and the arithmetic summand as an orthogonal decomposition of the ambient stage. After that, we record the minimal localized arithmetic comparison interface needed later. Namely, the arithmetic trace contribution is available on the ambient stage and has no cross term with the singular-boundary component. Even so, no conclusion theorem arises here.

Definition 5.17 

(Arithmetic-summand projector and residual projector). Define the arithmetic-summand projector on X by

$${\Pi }_{\mathrm{arith}}^X:=J_{\mathrm{arith}}{J}_{\mathrm{arith}}^{*}.$$

By Lemma 3.20, on ${\mathcal{H}}_{\mathrm{arith}}$,

$$\mathrm{s}-\sum _{n\ge 1}{\Pi }_n=I_{{\mathcal{H}}_{\mathrm{arith}}}$$

holds, so its lift to the ambient space satisfies

$${\Pi }_{\mathrm{arith}}^X=J_{\mathrm{arith}}{I}_{{\mathcal{H}}_{\mathrm{arith}}}{J}_{\mathrm{arith}}^{*}=\mathrm{s}-\sum _{n\ge 1}{\widehat{\Pi }}_n$$

on X.

Define the residual projector by

$${\Pi }_{\mathrm{res}}:=I_X-{\Pi }_R-{\Pi }_{\mathrm{arith}}^X.$$

If one wishes to retain the conventional notation ${\Delta }_X$, it is used only as the decomposition notation

$${\Delta }_X:={\Pi }_R+{\Pi }_{\mathrm{arith}}^X+{\Pi }_{\mathrm{res}}=I_X.$$

It is not a new dynamical operator.

Theorem 5.18 

(Orthogonal decomposition of the ambient stage). The operators

$${\Pi }_R,{\Pi }_{\mathrm{arith}}^X,{\Pi }_{\mathrm{res}}$$

are mutually orthogonal self-adjoint projectors on X. More precisely,

$${\Pi }_R^2={\Pi }_R,{\left({\Pi }_{\mathrm{arith}}^X\right)}^2={\Pi }_{\mathrm{arith}}^X,{\Pi }_{\mathrm{res}}^2={\Pi }_{\mathrm{res}},$$

$${\Pi }_R^{*}={\Pi }_R,{\left({\Pi }_{\mathrm{arith}}^X\right)}^{*}={\Pi }_{\mathrm{arith}}^X,{\Pi }_{\mathrm{res}}^{*}={\Pi }_{\mathrm{res}},$$

and

$${\Pi }_R{\Pi }_{\mathrm{arith}}^X={\Pi }_{\mathrm{arith}}^X{\Pi }_R=0,$$

$${\Pi }_R{\Pi }_{\mathrm{res}}={\Pi }_{\mathrm{res}}{\Pi }_R=0,$$

$${\Pi }_{\mathrm{arith}}^X{\Pi }_{\mathrm{res}}={\Pi }_{\mathrm{res}}{\Pi }_{\mathrm{arith}}^X=0.$$

Furthermore,

$$X=Ran{\Pi }_R\oplus J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}\oplus Ran{\Pi }_{\mathrm{res}},$$

and

$$Ran{\Pi }_{\mathrm{res}}=J_{\mathrm{an}}{H}_{\alpha ,+}\ominus {\mathcal{K}}_R.$$

Equivalently,

$$X={\mathcal{K}}_R\oplus J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}\oplus \left(J_{\mathrm{an}}{H}_{\alpha ,+}\ominus {\mathcal{K}}_R\right).$$

Proof. 

We divide the proof into four steps.

Step 1: The projector ${\Pi }_{\mathrm{arith}}^X$. By Definition 5.1, the embedding

$$J_{\mathrm{arith}}:{\mathcal{H}}_{\mathrm{arith}}\to X$$

is an isometry onto the closed subspace

$$J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}\subset X.$$

Therefore

$$J_{\mathrm{arith}}^{*}{J}_{\mathrm{arith}}=I_{{\mathcal{H}}_{\mathrm{arith}}},$$

and hence

$${\left({\Pi }_{\mathrm{arith}}^X\right)}^2=J_{\mathrm{arith}}{J}_{\mathrm{arith}}^{*}{J}_{\mathrm{arith}}{J}_{\mathrm{arith}}^{*}=J_{\mathrm{arith}}{I}_{{\mathcal{H}}_{\mathrm{arith}}}{J}_{\mathrm{arith}}^{*}={\Pi }_{\mathrm{arith}}^X.$$

Also,

$${\left({\Pi }_{\mathrm{arith}}^X\right)}^{*}={\left(J_{\mathrm{arith}}{J}_{\mathrm{arith}}^{*}\right)}^{*}=J_{\mathrm{arith}}{J}_{\mathrm{arith}}^{*}={\Pi }_{\mathrm{arith}}^X.$$

Thus ${\Pi }_{\mathrm{arith}}^X$ is the orthogonal projector onto $J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}$.

Its kernel is the orthogonal complement of the arithmetic direct-sum component, namely

$$ker{\Pi }_{\mathrm{arith}}^X=J_{\mathrm{an}}{H}_{\alpha ,+}.$$

Step 2: Orthogonality of ${\Pi }_R$ and ${\Pi }_{\mathrm{arith}}^X$. By Theorem 5.5,

$$Ran{\Pi }_R={\mathcal{K}}_R\subset J_{\mathrm{an}}{H}_{\alpha ,+}.$$

By Step 1,

$$Ran{\Pi }_{\mathrm{arith}}^X=J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}.$$

But

$$J_{\mathrm{an}}{H}_{\alpha ,+}\perp J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}},$$

so their ranges are orthogonal. Therefore every vector in $Ran{\Pi }_{\mathrm{arith}}^X$ belongs to $ker{\Pi }_R$, and every vector in $Ran{\Pi }_R$ belongs to $ker{\Pi }_{\mathrm{arith}}^X$. Hence

$${\Pi }_R{\Pi }_{\mathrm{arith}}^X={\Pi }_{\mathrm{arith}}^X{\Pi }_R=0.$$

Step 3: The residual projector ${\Pi }_{\mathrm{res}}$. By Definition 5.17,

$${\Pi }_{\mathrm{res}}=I_X-{\Pi }_R-{\Pi }_{\mathrm{arith}}^X.$$

Using (5.4) and

$${\Pi }_R^2={\Pi }_R,{\left({\Pi }_{\mathrm{arith}}^X\right)}^2={\Pi }_{\mathrm{arith}}^X,$$

we compute

$${\Pi }_{\mathrm{res}}^2={(I_X-{\Pi }_R-{\Pi }_{\mathrm{arith}}^X)}^2=I_X-{\Pi }_R-{\Pi }_{\mathrm{arith}}^X={\Pi }_{\mathrm{res}}.$$

Similarly,

$${\Pi }_{\mathrm{res}}^{*}=I_X-{\Pi }_R^{*}-{\left({\Pi }_{\mathrm{arith}}^X\right)}^{*}=I_X-{\Pi }_R-{\Pi }_{\mathrm{arith}}^X={\Pi }_{\mathrm{res}}.$$

Thus ${\Pi }_{\mathrm{res}}$ is a self-adjoint projector.

Next we prove orthogonality between ${\Pi }_{\mathrm{res}}$ and the other two projectors. Using (5.4),

$${\Pi }_R{\Pi }_{\mathrm{res}}={\Pi }_R(I_X-{\Pi }_R-{\Pi }_{\mathrm{arith}}^X)={\Pi }_R-{\Pi }_R^2-{\Pi }_R{\Pi }_{\mathrm{arith}}^X=0.$$

Similarly,

$${\Pi }_{\mathrm{res}}{\Pi }_R=(I_X-{\Pi }_R-{\Pi }_{\mathrm{arith}}^X){\Pi }_R={\Pi }_R-{\Pi }_R^2-{\Pi }_{\mathrm{arith}}^X{\Pi }_R=0.$$

The same computation gives

$${\Pi }_{\mathrm{arith}}^X{\Pi }_{\mathrm{res}}={\Pi }_{\mathrm{res}}{\Pi }_{\mathrm{arith}}^X=0.$$

Step 4: Identification of the range and direct-sum decomposition. The three projectors are mutually orthogonal and satisfy

$${\Pi }_R+{\Pi }_{\mathrm{arith}}^X+{\Pi }_{\mathrm{res}}=I_X.$$

Therefore their ranges give the orthogonal direct-sum decomposition

$$X=Ran{\Pi }_R\oplus Ran{\Pi }_{\mathrm{arith}}^X\oplus Ran{\Pi }_{\mathrm{res}}.$$

Since $Ran{\Pi }_{\mathrm{arith}}^X=J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}$, this becomes

$$X=Ran{\Pi }_R\oplus J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}\oplus Ran{\Pi }_{\mathrm{res}}.$$

It remains to identify $Ran{\Pi }_{\mathrm{res}}$. Let $y\in Ran{\Pi }_{\mathrm{res}}$. Then there exists $x\in X$ such that

$$y={\Pi }_{\mathrm{res}}x.$$

Since

$${\Pi }_{\mathrm{arith}}^X{\Pi }_{\mathrm{res}}=0,$$

we have

$${\Pi }_{\mathrm{arith}}^{X}y=0.$$

By (5.4),

$$y\in J_{\mathrm{an}}{H}_{\alpha ,+}.$$

Also,

$${\Pi }_R{\Pi }_{\mathrm{res}}=0,$$

so

$${\Pi }_{R}y=0.$$

Since ${\Pi }_R$ is the orthogonal projector onto ${\mathcal{K}}_R$, this means that y is orthogonal to ${\mathcal{K}}_R$. Therefore

$$y\in J_{\mathrm{an}}{H}_{\alpha ,+}\ominus {\mathcal{K}}_R.$$

Hence

$$Ran{\Pi }_{\mathrm{res}}\subset J_{\mathrm{an}}{H}_{\alpha ,+}\ominus {\mathcal{K}}_R.$$

Conversely, let

$$y\in J_{\mathrm{an}}{H}_{\alpha ,+}\ominus {\mathcal{K}}_R.$$

Then $y\in J_{\mathrm{an}}{H}_{\alpha ,+}$, so

$${\Pi }_{\mathrm{arith}}^{X}y=0.$$

Also, since $y\perp {\mathcal{K}}_R=Ran{\Pi }_R$,

$${\Pi }_{R}y=0.$$

Therefore

$${\Pi }_{\mathrm{res}}y=(I_X-{\Pi }_R-{\Pi }_{\mathrm{arith}}^X)y=y.$$

Thus

$$y\in Ran{\Pi }_{\mathrm{res}}.$$

Hence

$$Ran{\Pi }_{\mathrm{res}}=J_{\mathrm{an}}{H}_{\alpha ,+}\ominus {\mathcal{K}}_R.$$

This completes the proof. □

Proposition 5.19 

(Localized arithmetic side of the comparison interface). Let

$$\phi :\mathbb{N}\to \mathbb{C}$$

have finite support. Then the lifted arithmetic operator

$${\widehat{\Pi }}_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)$$

is trace-class on X, and

$${Tr}_X{\widehat{\Pi }}_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)=\sum _{n\ge 1}\phi \left(n\right){\Lambda }^{\mathrm{ex}}\left(n\right)=\sum _{p^k}\phi \left(p^k\right)logp.$$

Furthermore, for any $x\in X$,

$${\langle {\Pi }_{R}x,{\widehat{\Pi }}_{\mathrm{arith}}\left(\phi \right)x\rangle }_X=0.$$

Therefore the arithmetic trace contribution and the ${\mathcal{K}}_R$-component contribution have no cross term on the ambient stage.

Proof. 

Since $\phi$ has finite support and ${\Lambda }^{\mathrm{ex}}$ is an arithmetic function, the pointwise product

$$\left(\phi {\Lambda }^{\mathrm{ex}}\right)\left(n\right):=\phi \left(n\right){\Lambda }^{\mathrm{ex}}\left(n\right)$$

has finite support and hence belongs to ${\ell }^1\left(\mathbb{N}\right)$. Therefore

$${\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)$$

is trace-class on ${\mathcal{H}}_{\mathrm{arith}}$ by Theorem 3.23, and its lift to the ambient space,

$${\widehat{\Pi }}_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)=J_{\mathrm{arith}}{\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)J_{\mathrm{arith}}^{*},$$

is also trace-class on X.

Next we compute its trace. Since $J_{\mathrm{arith}}$ and $J_{\mathrm{arith}}^{*}$ are bounded and ${\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)$ is trace-class, cyclicity of the trace gives

$${Tr}_X{\widehat{\Pi }}_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)={Tr}_X\left(J_{\mathrm{arith}}{\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)J_{\mathrm{arith}}^{*}\right)={Tr}_{{\mathcal{H}}_{\mathrm{arith}}}\left({\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)J_{\mathrm{arith}}^{*}{J}_{\mathrm{arith}}\right).$$

But

$$J_{\mathrm{arith}}^{*}{J}_{\mathrm{arith}}=I_{{\mathcal{H}}_{\mathrm{arith}}},$$

so this becomes

$${Tr}_X{\widehat{\Pi }}_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)={Tr}_{{\mathcal{H}}_{\mathrm{arith}}}{\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right).$$

Applying the arithmetic trace formula of Theorem 3.23, we obtain

$${Tr}_{{\mathcal{H}}_{\mathrm{arith}}}{\Pi }_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)=\sum _{n\ge 1}\phi \left(n\right){\Lambda }^{\mathrm{ex}}\left(n\right)=\sum _{p^k}\phi \left(p^k\right)logp.$$

Therefore

$${Tr}_X{\widehat{\Pi }}_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)=\sum _{n\ge 1}\phi \left(n\right){\Lambda }^{\mathrm{ex}}\left(n\right)=\sum _{p^k}\phi \left(p^k\right)logp.$$

It remains to prove the disappearance of the cross term. Take $x\in X$. By Theorem 5.5,

$${\Pi }_{R}x\in Ran{\Pi }_R={\mathcal{K}}_R\subset J_{\mathrm{an}}{H}_{\alpha ,+}.$$

On the other hand, by Definition 5.14,

$${\widehat{\Pi }}_{\mathrm{arith}}\left(\phi \right)x\in J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}.$$

But

$$J_{\mathrm{an}}{H}_{\alpha ,+}\perp J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}},$$

and hence

$${\langle {\Pi }_{R}x,{\widehat{\Pi }}_{\mathrm{arith}}\left(\phi \right)x\rangle }_X=0.$$

Thus the arithmetic trace contribution and the ${\mathcal{K}}_R$-component contribution have no cross term on the ambient stage. This proves the proposition. □

Definition 5.20 

(canonical residual-free representative). For any representative $x_{\phi ,j}\in X$ of localized comparison data obtained from the finite-window explicit formula, define

$$x_{\phi ,j}^{♯}:=({\Pi }_R+{\Pi }_{\mathrm{arith}}^X)x_{\phi ,j}=(I_X-{\Pi }_{\mathrm{res}})x_{\phi ,j}$$

and call it its canonical residual-free representative. When two representatives $x,y\in X$ satisfy

$${\Pi }_{R}x={\Pi }_{R}y,{\Pi }_{\mathrm{arith}}^{X}x={\Pi }_{\mathrm{arith}}^{X}y,$$

we say that they are equivalent as finite-window comparison data.

Definition 5.21 

(quotient finite-window comparison datum). Localized comparison data obtained from the finite-window explicit formula are treated not as representatives $x\in X$ themselves, but as quotient classes

$${\left[x\right]}_{\mathrm{fw}}\in X/Ran{\Pi }_{\mathrm{res}}$$

modulo the residual component. That is, define

$$x{\sim }_{\mathrm{fw}}y⟺x-y\in Ran{\Pi }_{\mathrm{res}}.$$

On this quotient, the arithmetic projection and ${\mathcal{K}}_R$-side projection are defined by

$$\mathcal{A}\left({\left[x\right]}_{\mathrm{fw}}\right):={\Pi }_{\mathrm{arith}}^{X}x,\mathcal{R}\left({\left[x\right]}_{\mathrm{fw}}\right):={\Pi }_{R}x.$$

These are well-defined. Indeed, for $h\in Ran{\Pi }_{\mathrm{res}}$,

$${\Pi }_{\mathrm{arith}}^{X}h=0,{\Pi }_{R}h=0,$$

and hence

$$\mathcal{A}\left({[x+h]}_{\mathrm{fw}}\right)=\mathcal{A}\left({\left[x\right]}_{\mathrm{fw}}\right),\mathcal{R}\left({[x+h]}_{\mathrm{fw}}\right)=\mathcal{R}\left({\left[x\right]}_{\mathrm{fw}}\right).$$

Therefore

$$x^{♯}=({\Pi }_R+{\Pi }_{\mathrm{arith}}^X)x$$

is the standard representative of the quotient class ${\left[x\right]}_{\mathrm{fw}}$, and discarding the residual component does not alter the content of the finite-window comparison datum; it only replaces the representative of the quotient class.

Lemma 5.22 

(residual component is comparison-invisible). Localized comparison data arising from the finite-window explicit formula are treated as the quotient classes ${\left[x\right]}_{\mathrm{fw}}$ of Definition 5.21. Their arithmetic projection and ${\mathcal{K}}_R$-side projection depend only on

$${\Pi }_{\mathrm{arith}}^{X}x,{\Pi }_{R}x,$$

and not on the residual component of the representative x. Therefore, replacing any representative $x_{\phi ,j}$ by its canonical residual-free representative $x_{\phi ,j}^{♯}$ does not change finite-window explicit-formula preservation, exact arithmetic trace evaluation, or ${\mathcal{K}}_R$-projected component.

Proof. 

By Theorem 5.18,

$$X={\mathcal{K}}_R\oplus J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}\oplus Ran{\Pi }_{\mathrm{res}},$$

and the corresponding orthogonal projectors are

$${\Pi }_R,{\Pi }_{\mathrm{arith}}^X,{\Pi }_{\mathrm{res}}.$$

The arithmetic contribution is evaluated from ${\Pi }_{\mathrm{arith}}^{X}x$ by the arithmetic trace of Proposition 5.19. On the other hand, the ${\mathcal{K}}_R$-component is obtained from ${\Pi }_{R}x\in {\mathcal{K}}_R$. Since the residual component ${\Pi }_{\mathrm{res}}x$ is projected onto neither of these two orthogonal components, it contributes to neither the arithmetic trace nor the ${\mathcal{K}}_R$-component. Therefore

$${\left[x\right]}_{\mathrm{fw}}={\left[({\Pi }_R+{\Pi }_{\mathrm{arith}}^X)x\right]}_{\mathrm{fw}}={\left[x^{♯}\right]}_{\mathrm{fw}},$$

and replacement by the canonical residual-free representative is merely a choice of representative in the quotient $X/Ran{\Pi }_{\mathrm{res}}$. Thus the finite-window comparison datum does not change. □

Proposition 5.23 

(Exclusive-complement principle). Let $\phi :\mathbb{N}\to \mathbb{C}$ be a finitely supported weight, and represent the localized comparison data obtained by the finite-window explicit formula by canonical residual-free representatives on the ambient stage X:

$$x_{\phi ,1}^{♯},\dots ,x_{\phi ,N_{\phi }}^{♯}\in X.$$

Assume further that the following two conditions hold.

1.

(Explicit-formula preservation)After calibrating the Archimedean term as a fixed reference term, the total variation of the finite window is preserved as the sum of the prime-power contribution on the arithmetic side and the residual contribution on the zero side.

2.

(Exact arithmetic trace evaluation)The prime-power contribution is evaluated exactly by the arithmetic trace of Proposition 5.19,

$${Tr}_X{\widehat{\Pi }}_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)=\sum _{p^k}\phi \left(p^k\right)logp.$$

Then

$${\Pi }_{\mathrm{res}}{x}_{\phi ,j}^{♯}=0(1\le j\le N_{\phi })$$

holds automatically, and the local residual after removing the arithmetic side is represented uniquely as

$${\Pi }_R{x}_{\phi ,j}^{♯}\in {\mathcal{K}}_R(1\le j\le N_{\phi }).$$

In particular, the prime-power contribution has no projection into the singular-boundary component, and the ${\mathcal{K}}_R$-projected component used in the finite-window comparison is placed on ${\mathcal{K}}_R$ as the exclusive-complement component after exact arithmetic trace evaluation.

Proof. 

By Definition 5.20,

$$x_{\phi ,j}^{♯}=({\Pi }_R+{\Pi }_{\mathrm{arith}}^X)x_{\phi ,j}.$$

Therefore

$${\Pi }_{\mathrm{res}}{x}_{\phi ,j}^{♯}={\Pi }_{\mathrm{res}}{\Pi }_R{x}_{\phi ,j}+{\Pi }_{\mathrm{res}}{\Pi }_{\mathrm{arith}}^X{x}_{\phi ,j}=0,$$

where the last equality follows from the orthogonal projector decomposition of Theorem 5.18. Thus residual vanishing for the canonical representative is not an external assumption; it holds as a property of the canonical representative.

Furthermore,

$$x_{\phi ,j}^{♯}={\Pi }_R{x}_{\phi ,j}^{♯}+{\Pi }_{\mathrm{arith}}^X{x}_{\phi ,j}^{♯},$$

and by Theorem 5.15,

$${\Pi }_R{\Pi }_{\mathrm{arith}}^X={\Pi }_{\mathrm{arith}}^X{\Pi }_R=0.$$

Therefore the prime-power contribution evaluated on the arithmetic side has no projection into ${\mathcal{K}}_R=Ran{\Pi }_R$.

By the exact arithmetic trace evaluation assumption, the prime-power side of the finite-window explicit formula is evaluated exactly as

$${Tr}_X{\widehat{\Pi }}_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right).$$

By Lemma 5.22, replacement by the canonical residual-free representative does not change the comparison data. Thus, under explicit-formula preservation, the effective residual after removing the calibrated reference term and the arithmetic contribution is evaluated as the only remaining effective component in the orthogonal decomposition,

$${\Pi }_R{x}_{\phi ,j}^{♯}\in {\mathcal{K}}_R.$$

Since the orthogonal projector decomposition is unique, this component is also unique. □

Remark 5.24 

(Meaning of the word exclusive). The exclusive complement here does not assert

$$X\ominus J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}={\mathcal{K}}_R.$$

Indeed, by Theorem 5.18, in general there is a residual component

$$J_{\mathrm{an}}{H}_{\alpha ,+}\ominus {\mathcal{K}}_R.$$

Exclusive means that, when finite-window comparison data are evaluated by the canonical residual-free representative of Definition 5.20, the residual projection is removed by the definition of the canonical residual-free representative, and the effective residual component after exact arithmetic trace evaluation is limited to ${\mathcal{K}}_R$. Thus this paper does not identify zeros with $A_R$-eigenvalues in advance, but obtains the ${\mathcal{K}}_R$-component from explicit-formula preservation, exact arithmetic trace evaluation, orthogonality, and canonical residual-free reduction.

Remark 5.25 

(Role of the orthogonal-decomposition comparison framework). This section prepares the orthogonal-decomposition comparison framework used in Section 6. Its role is to place the singular-boundary subspace constructed in Section 4 and the coefficient-space arithmetic construction of Section 3 on the common ambient stage X, prove their orthogonality, and record a localized arithmetic trace interface with no cross term. The closure argument toward the conclusion begins only in Section 6.

6. Analytic Comparison and Finite-Window Closure

6.0. Analytic Comparison Data from the Residual-Free Comparison Interface

In this section, the integrated stage constructed in Section 5 and the canonical residual-free representative is fixed as the input data for passing to the analytic closure argument of Section 6. What is done here is a notational organization for connecting the objects constructed in Section 2 through 5 to distributional equalities, regularized determinants, and finite-window counts, and does not add any new assumption.

All objects used in the analytic closure argument are introduced below by formal definitions or by references to the constructions of Section 2–5. In particular, the residual-free representative, distributional comparison, trace-ideal determinant, global identification, and finite-window counting statements are treated as separate steps. No part of the finite-window record is used as an additional hypothesis in the determinant comparison.

Definition 6.1 

(residual-free comparison data). The residual-free comparison data passed from Section 5 to Section 6 are the following tuple:

$${\mathfrak{H}}_{\mathrm{rf}}:=\left(X,{\Pi }_R,{\Pi }_{\mathrm{arith}}^X,{\Pi }_{\mathrm{res}},{\mathcal{K}}_R,{\Lambda }^{\mathrm{ex}},{\left\{x_{\phi ,j}^{♯}\right\}}_{\phi ,j}\right).$$

Here X is the integrated stage, and by Theorem 5.18 it has the orthogonal decomposition

$$X={\mathcal{K}}_R\oplus J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}\oplus Ran{\Pi }_{\mathrm{res}}.$$

The corresponding orthogonal projections are denoted by

$${\Pi }_R,{\Pi }_{\mathrm{arith}}^X,{\Pi }_{\mathrm{res}}.$$

Moreover, ${\Lambda }^{\mathrm{ex}}$ is the exact von Mangoldt lift fixed in Section 3, and for a finitely supported weight $\phi :\mathbb{N}\to \mathbb{C}$, the arithmetic-side contribution is evaluated as

$${Tr}_X{\widehat{\Pi }}_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right)=\sum _{p^k}\phi \left(p^k\right)logp.$$

Here $\phi$ is a finitely supported weight on the arithmetic side, and it is to be interpreted as notationally distinct from the logarithmic-side test functions used from Section 6.1 onward.

When $x_{\phi ,j}\in X$ denotes a representative of the localized comparison data obtained from the finite-window explicit formula, define

$$x_{\phi ,j}^{♯}:=({\Pi }_R+{\Pi }_{\mathrm{arith}}^X)x_{\phi ,j}=(I_X-{\Pi }_{\mathrm{res}})x_{\phi ,j}$$

as its canonical residual-free representative, in accordance with Definition 5.20. Then

$${\Pi }_{\mathrm{res}}{x}_{\phi ,j}^{♯}=0,{\Pi }_R{x}_{\phi ,j}^{♯}\in {\mathcal{K}}_R$$

holds. Therefore, in Section 6, the finite-window comparison data are represented as quotient classes in the sense of Definition 5.21, and $x_{\phi ,j}^{♯}$ is always used as the representative.

Lemma 6.2 

(source of the analytic comparison data). All data used in the determinant-comparison part of Section 6 are determined by the constructions of Section 2–5, and no additional zero-location input is introduced at the transition to Section 6. More precisely:

1.

Section 2 fixes the analytic Hilbert space $H_{\alpha ,+}$, the dense domain ${\mathcal{D}}_R$, the closed form $q_R$, the associated self-adjoint operator $A_R$, and the compact-resolvent spectral scale used in the Schatten estimates.

2.

Section 3 fixes the coefficient-space arithmetic construction, the exact von Mangoldt lift ${\Lambda }^{\mathrm{ex}}$, and the weighted diagonal arithmetic trace which evaluates the prime-power contribution.

3.

Section 4 fixes the one-sided singular-boundary subspace $K_R^{+}\subset H_{\alpha ,+}$, the one-sided projection ${\Pi }_R^{+}$, the boundary distribution space ${\mathcal{D}}_R^{\prime }$, the singular-boundary trace, and the boundary pairing.

4.

Section 5 fixes the integrated Hilbert space X, the lifted subspace ${\mathcal{K}}_R\subset X$, the ambient projection ${\Pi }_R$, the arithmetic projection ${\Pi }_{\mathrm{arith}}^X$, the residual projection ${\Pi }_{\mathrm{res}}$, and the canonical representative modulo $Ran{\Pi }_{\mathrm{res}}$.

Consequently, the boundary reflection ${\Theta }_R$, its descended involution ${\mathcal{S}}_R$ on ${\mathcal{K}}_R$, the signed boundary-distribution comparison form, the Hilbert–Schmidt operator K, and the central finite-window test inputs of Section 6 are obtained only from these data and from the functional equation $\xi \left(s\right)=\xi (1-s)$. In particular, the construction does not use the location of the zeros of $\xi$, the identity $F_K\equiv \xi$, or any finite-window consequence proved later in Section 6.

Proof. 

The assertions follow by tracing the definitions in Section 2–5. Items (1)–(4) list exactly the objects constructed before Section 6. The map ${LCI}_R$ and the representative $x_{\phi ,j}^{♯}$ use the quotient by $Ran{\Pi }_{\mathrm{res}}$ fixed in Section 5; the arithmetic term has already been evaluated by the trace of Section 3; and the remaining effective component is the ${\mathcal{K}}_R$-component obtained by ${\Pi }_R$. The later definitions of ${\Theta }_R$, ${\mathcal{S}}_R$, ${\mathfrak{k}}_R$, K, and the central finite-window kernels refer only to this list. Thus the determinant-comparison argument starts from the data supplied by Section 2–5 and does not insert a zero-location assumption or a conclusion of the comparison as an input. □

Theorem 6.3 

(continuous extension of the localized comparison interface). The finite-window localized comparison interface of Section 5 is continuously defined on the dense subspace of singular-boundary data generated from finite-window inputs, and extends uniquely to its completion. Namely, there exist a closed subspace

$${\mathcal{D}}_{R,\mathrm{adm}}^{\prime }\subset {\mathcal{D}}_R^{\prime }$$

of ${\mathcal{D}}_R^{\prime }$, and a continuous linear map

$${LCI}_R:{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }⟶X$$

satisfying the following.

1.

The singular boundary trace of Section 4 satisfies

$${Tr}_{\partial ,R}\left({\mathcal{D}}_R\right)\subset {\mathcal{D}}_{R,\mathrm{adm}}^{\prime }.$$

2.

For a singular-boundary test vector $f_{\phi ,j}\in {\mathcal{D}}_R$ arising from a finite-window input, if the comparison representative before residual-free projection is denoted by $x_{\phi ,j}\in X$, then

$${LCI}_R\left({Tr}_{\partial ,R}{f}_{\phi ,j}\right)=x_{\phi ,j}.$$

3.

${LCI}_R$ is a continuous linear map from the locally convex topology of ${\mathcal{D}}_{R,\mathrm{adm}}^{\prime }$ to the Hilbert topology of X.

Proof. 

The localized comparison interface of Section 5 is constructed as a linear map sending the boundary distribution obtained from the finite-window explicit formula to the integrated stage X. By the singular boundary trace of Section 4, the ${\mathcal{K}}_R$-component arising from a finite-window input is represented as a boundary distribution in ${\mathcal{D}}_R^{\prime }$. Writing ${\mathcal{E}}_{R,\mathrm{fw}}$ for the linear subspace spanned by these finite-window boundary distributions, the exact arithmetic trace evaluation and residual-free quotient construction of Section 5 imply that

$${LCI}_{R,0}:{\mathcal{E}}_{R,\mathrm{fw}}⟶X$$

is continuous and linear. Here the continuity follows from the fact that each finite-window comparison functional is uniformly controlled by the boundary-trace seminorms of Section 4 and the X-norm of Section 5.

Define ${\mathcal{D}}_{R,\mathrm{adm}}^{\prime }$ to be the closed subspace consisting of boundary data finite with respect to this uniform estimate inside the ${\mathcal{D}}_R^{\prime }$-closure of ${\mathcal{E}}_{R,\mathrm{fw}}$. Then, by continuity, ${LCI}_{R,0}$ extends uniquely to

$${LCI}_R:{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }\to X.$$

Since the singular boundary trace of Section 4 is constructed so as to send the singular-boundary test space ${\mathcal{D}}_R$ to admissible boundary data, one has

$${Tr}_{\partial ,R}\left({\mathcal{D}}_R\right)\subset {\mathcal{D}}_{R,\mathrm{adm}}^{\prime }.$$

The agreement on finite-window inputs is precisely the defining property of ${LCI}_{R,0}$ before extension. □

Theorem 6.4 

(singular-boundary boundary-distribution comparison realization). Singular-boundary test vectors on the Gelfand triple of Section 4

$${\mathcal{D}}_R\subset H_{\alpha ,+}\subset {\mathcal{D}}_R^{\prime }$$

are realized continuously and linearly as comparison data on the integrated stage X through the localized comparison interface of Section 5. Specifically, define

$${\mathfrak{B}}_R:{\mathcal{D}}_R⟶X$$

by

$${\mathfrak{B}}_R:={LCI}_R\circ {Tr}_{\partial ,R}.$$

Then ${\mathfrak{B}}_R$ is continuous and linear, and for $f_{\phi ,j}\in {\mathcal{D}}_R$ arising from a finite-window input,

$${\mathfrak{B}}_R{f}_{\phi ,j}=x_{\phi ,j}$$

holds.

Proof. 

By Theorem 6.3,

$${LCI}_R:{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }\to X$$

is continuous and linear, and the singular boundary trace of Section 4

$${Tr}_{\partial ,R}:{\mathcal{D}}_R\to {\mathcal{D}}_{R,\mathrm{adm}}^{\prime }$$

is also continuous and linear. Therefore the composition

$${\mathfrak{B}}_R={LCI}_R\circ {Tr}_{\partial ,R}$$

is a continuous linear map from ${\mathcal{D}}_R$ to X. The equality for finite-window inputs $f_{\phi ,j}$ follows from the finite-window compatibility in Theorem 6.3. □

Definition 6.5 

(singular-boundary test-to-comparison map). Apply the canonical residual-free projection to the map ${\mathfrak{B}}_R$ of Theorem 6.4, and define

$${\mathcal{J}}_R:=({\Pi }_R+{\Pi }_{\mathrm{arith}}^X){\mathfrak{B}}_R=(I_X-{\Pi }_{\mathrm{res}}){\mathfrak{B}}_R:{\mathcal{D}}_R⟶X.$$

This is a continuous linear map representing, on the integrated stage X, the residual-free comparison representative corresponding to a singular-boundary test vector $f\in {\mathcal{D}}_R$. In what follows, write

$$x_f^{♯}:={\mathcal{J}}_{R}f.$$

Lemma 6.6 

(continuity and residual-free property of ${\mathcal{J}}_R$). The map

$${\mathcal{J}}_R:{\mathcal{D}}_R\to X$$

is continuous and linear, and for every $f\in {\mathcal{D}}_R$,

$${\Pi }_{\mathrm{res}}{\mathcal{J}}_{R}f=0,{\Pi }_R{\mathcal{J}}_{R}f\in {\mathcal{K}}_R$$

holds. Moreover, the representative $x_{\phi ,j}^{♯}$ obtained from a finite-window weight $\phi$ is read, for a suitable $f_{\phi ,j}\in {\mathcal{D}}_R$, as

$$x_{\phi ,j}^{♯}={\mathcal{J}}_R{f}_{\phi ,j}.$$

Proof. 

By Theorem 6.4, ${\mathfrak{B}}_R:{\mathcal{D}}_R\to X$ is continuous and linear. Since the orthogonal projections ${\Pi }_R,{\Pi }_{\mathrm{arith}}^X,{\Pi }_{\mathrm{res}}$ are bounded on X,

$${\mathcal{J}}_R=({\Pi }_R+{\Pi }_{\mathrm{arith}}^X){\mathfrak{B}}_R$$

is also continuous and linear. Moreover, from

$${\Pi }_{\mathrm{res}}({\Pi }_R+{\Pi }_{\mathrm{arith}}^X)=0$$

one obtains

$${\Pi }_{\mathrm{res}}{\mathcal{J}}_{R}f=0.$$

Furthermore,

$${\Pi }_R{\mathcal{J}}_{R}f\in Ran{\Pi }_R={\mathcal{K}}_R.$$

For $x_{\phi ,j}^{♯}$ arising from the finite-window explicit formula, Theorem 6.4 gives $f_{\phi ,j}\in {\mathcal{D}}_R$ such that

$${\mathfrak{B}}_R{f}_{\phi ,j}=x_{\phi ,j}.$$

Therefore

$${\mathcal{J}}_R{f}_{\phi ,j}=({\Pi }_R+{\Pi }_{\mathrm{arith}}^X)x_{\phi ,j}=x_{\phi ,j}^{♯}.$$

□

Definition 6.7 

(finite-window singular-boundary and zeta functionals). Let ${\mathcal{T}}_{\mathrm{fw}}$ be the initial test algebra generated by the finitely supported weights used in the finite-window explicit formula of Section 5. At this stage, ${\mathcal{T}}_{\mathrm{fw}}$ is treated as a dense input class for the distributional test spaces introduced in Section 6.1.

For $\phi \in {\mathcal{T}}_{\mathrm{fw}}$, define the residual-free singular-boundary functional

$${\mathcal{L}}_R\left(\phi \right)$$

as the local ${\mathcal{K}}_R$-side functional determined by the finite set

$${\left\{{\Pi }_R{x}_{\phi ,j}^{♯}\right\}}_{j=1}^{N_{\phi }}\subset {\mathcal{K}}_R.$$

That is, by Proposition 5.23, ${\mathcal{L}}_R\left(\phi \right)$ is the scalar functional associated with the ${\mathcal{K}}_R$-component remaining after the prime-power contribution evaluated on the arithmetic side has been removed.

On the other hand, let ${\mathcal{E}}_{\xi }^{\mathrm{fw}}\left(\phi \right)$ be the calibrated scalar functional obtained from the finite-window explicit formula for the completed zeta function $\xi \left(s\right)$. Here “calibrated” means that the Archimedean term and the fixed reference term are separated according to the conventions of Section 5, and that the prime-power contribution on the arithmetic side is evaluated, in the notation of arithmetic-side finitely supported weights, as

$$\sum _{p^k}\phi \left(p^k\right)logp.$$

After passing to logarithmic-side test functions, the same contribution is represented as

$$〈{\mu }_{\Lambda }^{\mathrm{ex}},\psi 〉=\sum _{p^k}{\Lambda }^{\mathrm{ex}}\left(p^k\right)\psi (log{p}^k).$$

In what follows, write

$$\langle {\mu }_L,\phi \rangle :={\mathcal{L}}_R\left(\phi \right),\langle {\mu }_{\xi },\phi \rangle :={\mathcal{E}}_{\xi }^{\mathrm{fw}}\left(\phi \right)(\phi \in {\mathcal{T}}_{\mathrm{fw}}).$$

In this definition, ${\mu }_L$ and ${\mu }_{\xi }$ are first introduced as linear functionals on ${\mathcal{T}}_{\mathrm{fw}}$. In Section 6.1, these are realized as continuous functionals on

$$C_c^{\infty }\left(\mathbb{R}\right)$$

for local coefficient identification, and on

$${\mathcal{A}}_{\eta }(0<\eta <log2)$$

for the open-band equality. The test family ${\mathcal{C}}_{\mathrm{cen}}$ for the central logarithmic transform is introduced independently in Section 6.4.

Definition 6.8 

(boundary-distribution comparison kernel and normalized determinant datum). Use the Gelfand triple of Section 4

$${\mathcal{D}}_R\subset H_{\alpha ,+}\subset {\mathcal{D}}_R^{\prime }$$

and the map of Definition 6.5

$${\mathcal{J}}_R:{\mathcal{D}}_R\to X.$$

In Section 6.3, after constructing the signature operator induced by the functional equation,

$${\mathcal{S}}_R:{\mathcal{K}}_R\to {\mathcal{K}}_R,$$

the signed residual-free boundary-distribution comparison kernel is defined by

$${\mathfrak{k}}_R(f,g):={〈{\mathcal{S}}_R{\Pi }_R{\mathcal{J}}_{R}f,{\Pi }_R{\mathcal{J}}_{R}g〉}_X(f,g\in {\mathcal{D}}_R).$$

The operator candidate K corresponding to this kernel is defined on the initial domain ${\mathcal{D}}_R$ by

$${\langle Kf,g\rangle }_{H_{\alpha ,+}}:={\mathfrak{k}}_R(f,g)(f,g\in {\mathcal{D}}_R).$$

At this stage, the construction of ${\mathcal{S}}_R$, the self-adjointness of K, and the Hilbert–Schmidt property are not yet asserted. They are proved in Section 6.3.

After $K=K^{*}\in {\mathfrak{S}}_2$ is established in Section 6.3, define, using the regularized determinant,

$$F_K\left(s\right):=e^{a_K+b_K(s-{\displaystyle \frac{1}{2}})}{det}_2\left(I+i(s-\frac{1}{2})K\right).$$

Here $a_K,b_K\in \mathbb{C}$ are normalization constants, and are fixed in the subsequent trace-ideal determinant theorem so as to satisfy

$$F_K\left({\displaystyle \frac{1}{2}}\right)=\xi \left({\displaystyle \frac{1}{2}}\right),F_K^{\prime }\left({\displaystyle \frac{1}{2}}\right)={\xi }^{\prime }\left({\displaystyle \frac{1}{2}}\right).$$

This definition specifies the type of the comparison function $F_K$, and at this point it does not assert

$$F_K\left(s\right)=\xi \left(s\right).$$

Proposition 6.9 

(comparison data from the residual-free comparison interface). The comparison data of Definition 6.1 satisfies the following three properties.

1.

(residual removal)For the canonical representative of any finite-window comparison datum,

$${\Pi }_{\mathrm{res}}{x}_{\phi ,j}^{♯}=0$$

holds.

2.

(singular-boundary localization)The effective residual after removing the arithmetic side is represented uniquely as

$${\Pi }_R{x}_{\phi ,j}^{♯}\in {\mathcal{K}}_R.$$

3.

(analytic targets)The objects treated in the analytic closure part of Section 6 are

$${\mu }_L,{\mu }_{\xi },K,F_K\left(s\right),$$

and these are respectively defined as the residual-free singular-boundary functional, the explicit-formula functional on the completed-zeta-function side, the boundary-distribution comparison kernel candidate, and its regularized-determinant comparison function.

Proof. 

The first assertion follows immediately from Definition 5.20. Indeed,

$$x_{\phi ,j}^{♯}=({\Pi }_R+{\Pi }_{\mathrm{arith}}^X)x_{\phi ,j},$$

and by the orthogonal decomposition of Theorem 5.18,

$${\Pi }_{\mathrm{res}}{\Pi }_R=0,{\Pi }_{\mathrm{res}}{\Pi }_{\mathrm{arith}}^X=0.$$

Therefore

$${\Pi }_{\mathrm{res}}{x}_{\phi ,j}^{♯}=0.$$

The second assertion is the content of Proposition 5.23. In the finite-window explicit formula, the prime-power contribution on the arithmetic side is evaluated exactly as

$${Tr}_X{\widehat{\Pi }}_{\mathrm{arith}}\left(\phi {\Lambda }^{\mathrm{ex}}\right),$$

and by Lemma 5.22, the residual component does not contribute to the comparison data. Hence the effective residual after removing the calibrated reference term and the arithmetic contribution is evaluated as the only remaining effective component in the orthogonal decomposition,

$${\Pi }_R{x}_{\phi ,j}^{♯}\in {\mathcal{K}}_R.$$

By uniqueness of the orthogonal projection decomposition, this component is also unique.

The third assertion is the notational fixing in Definitions 6.7 and 6.8. Here ${\mu }_L,{\mu }_{\xi }$ are first introduced as linear functionals on the finite-window test algebra, and K is introduced as the boundary-distribution comparison kernel candidate obtained from the distribution-kernel representation of Section 4. Their continuity as distributions, the ${\mathfrak{S}}_2$-realization of K, and the global agreement of $F_K$ with $\xi$ are proved in Section 6.1, 6.3, and 6.4, respectively. Thus this proposition is not a new closure assumption, but records the type consistency of the comparison data that passes the data constructed up to Section 5 to the analytic proof objects of Section 6. □

The purpose of Section 6.1 through 6.4 is to transform this residual-free comparison data into analytic identities containing neither error terms nor residual components. Specifically, one first compares ${\mu }_L$ and ${\mu }_{\xi }$ as continuous functionals on the comparison test classes, then realizes the boundary-distribution comparison kernel K as an ${\mathfrak{S}}_2$-operator, constructs the regularized determinant $F_K$, and finally proves

$$F_K\left(s\right)\equiv \xi \left(s\right)$$

as the global uniqueness theorem of Section 6.4.

6.1. Distributional Comparison Theorem

In this section, the finite-window functionals introduced in the previous section,

$${\mu }_L,{\mu }_{\xi },$$

are realized as continuous functionals on comparison test classes, and it is proved that they agree on small-band test functions. The equality treated here is not a pointwise boundary-value equality, but the distributional equality

$$\langle {\mu }_L,\phi \rangle =\langle {\mu }_{\xi },\phi \rangle .$$

Therefore, this section uses neither a half-value convention, pointwise boundary correction, nor heuristic contribution at the endpoints.

Definition 6.10 

(Fourier convention and open-band Schwartz class). In this section, the Fourier transform is normalized by

$$\widehat{\phi }\left(u\right):={\int }_{\mathbb{R}}\phi \left(t\right)e^{-itu}dt,\phi \left(t\right)={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}\widehat{\phi }\left(u\right)e^{itu}du.$$

For $0<\eta <log2$, define

$${\mathcal{A}}_{\eta }:=\left\{\phi \in \mathcal{S}\left(\mathbb{R}\right):supp\widehat{\phi }\subset (-\eta ,\eta )\right\}.$$

Endow ${\mathcal{A}}_{\eta }$ with the Fréchet topology induced from $\mathcal{S}\left(\mathbb{R}\right)$. That is, when

$$p_{a,b}\left(\phi \right):=\underset{t\in \mathbb{R}}{sup}|t^a{\partial }_t^b\phi \left(t\right)|(a,b\in {\mathbb{Z}}_{\ge 0})$$

are the standard seminorms of $\mathcal{S}\left(\mathbb{R}\right)$, convergence in ${\mathcal{A}}_{\eta }$ is defined by convergence with respect to these seminorms.

Moreover, the dual pairing between ${\mathcal{S}}^{\prime }\left(\mathbb{R}\right)$ and $\mathcal{S}\left(\mathbb{R}\right)$ is denoted by

$$\langle T,\phi \rangle (T\in {\mathcal{S}}^{\prime }\left(\mathbb{R}\right),\phi \in \mathcal{S}\left(\mathbb{R}\right)).$$

Definition 6.11 

(RH comparison test classes). The comparison test classes used in this section are the following two classes:

$$C_c^{\infty }\left(\mathbb{R}\right),{\mathcal{A}}_{\eta }(0<\eta <log2).$$

Here $C_c^{\infty }\left(\mathbb{R}\right)$ is used for local coefficient identification at prime-power positions, and ${\mathcal{A}}_{\eta }$ is used for the open-band residual-free equality. The test family representing the central logarithmic derivative,

$${\mathcal{C}}_{\mathrm{cen}},$$

is introduced independently in Section 6.4. Thus this section does not assume that the central logarithmic derivative is recovered solely from the open-band equality.

Remark 6.12 

(open-band convention). Throughout this section, assume

$$0<\eta <log2.$$

The boundary band

$$\eta =log2$$

is treated separately from the open-band distributional equality, because the Fourier support touches the first prime-power position. The stability of the boundary band is deferred to the endpoint stability theorem of Section 6.2.

Definition 6.13 

(completed von Mangoldt distribution on the logarithmic side). Let ${\Lambda }^{\mathrm{ex}}$ be the exact von Mangoldt lift fixed in Section 3. The corresponding arithmetic distribution on the logarithmic side is denoted by

$${\mu }_{\Lambda }^{\mathrm{ex}}\in {\left(C_c^{\infty }\left(\mathbb{R}\right)\right)}^{\prime }.$$

Namely, for $\psi \in C_c^{\infty }\left(\mathbb{R}\right)$, define

$$〈{\mu }_{\Lambda }^{\mathrm{ex}},\psi 〉:=\sum _{p^k}{\Lambda }^{\mathrm{ex}}\left(p^k\right)\psi (log{p}^k).$$

This sum is finite because $supp\psi$ is compact. Hence ${\mu }_{\Lambda }^{\mathrm{ex}}$ is a distribution on local logarithmic tests. In this paper, this arithmetic singular part is not used as a tempered distribution on the whole of $\mathcal{S}\left(\mathbb{R}\right)$. The open-band equality is defined on ${\mathcal{A}}_{\eta }$, and the central logarithmic transform is defined separately on ${\mathcal{C}}_{\mathrm{cen}}$.

Lemma 6.14 

(coefficient support and von Mangoldt lift). The finite-window explicit-formula distribution ${\mu }_{\xi }$ on the completed-zeta-function side decomposes as

$${\mu }_{\xi }={\mu }_{\xi ,\mathrm{sm}}+{\mu }_{\Lambda }^{\mathrm{ex}}.$$

Here ${\mu }_{\xi ,\mathrm{sm}}\in C^{\infty }\left(\mathbb{R}\right)$ is the smooth tempered distribution consisting of the Archimedean term and the calibrated reference term, and ${\mu }_{\Lambda }^{\mathrm{ex}}$ is the arithmetic distribution of Definition 6.13. Therefore

$$sing\; supp{\mu }_{\xi }\subset \{log{p}^k:p\mathrm{prime},k\ge 1\}.$$

Moreover, the principal delta coefficient at each point $log{p}^k$ is

$${\Lambda }^{\mathrm{ex}}\left(p^k\right).$$

Namely, if $\chi \in C_c^{\infty }\left(\mathbb{R}\right)$ has support in a sufficiently small neighborhood of $log{p}^k$ and contains no other prime-power positions, then

$$\langle {\mu }_{\xi },\chi \rangle -\langle {\mu }_{\xi ,\mathrm{sm}},\chi \rangle ={\Lambda }^{\mathrm{ex}}\left(p^k\right)\chi (log{p}^k).$$

Proof. 

In the explicit formula for the completed zeta function, the calibrated Archimedean term and reference term give a smooth kernel on the logarithmic side. Denote this by ${\mu }_{\xi ,\mathrm{sm}}$. On the other hand, the discrete contribution on the arithmetic side is concentrated at each prime-power position $log{p}^k$ by the exact von Mangoldt lift ${\Lambda }^{\mathrm{ex}}$ fixed in Section 3. Therefore, the arithmetic singular part is

$${\mu }_{\Lambda }^{\mathrm{ex}},$$

and its principal delta coefficient is, by definition, ${\Lambda }^{\mathrm{ex}}\left(p^k\right)$.

Since the smooth term has no singular support,

$$sing\; supp{\mu }_{\xi }=sing\; supp{\mu }_{\Lambda }^{\mathrm{ex}}\subset \{log{p}^k\}.$$

Finally, if $\chi$ is chosen so as to isolate only one prime-power position $log{p}^k$, the other delta components do not act on $\chi$. Hence

$$\langle {\mu }_{\xi },\chi \rangle -\langle {\mu }_{\xi ,\mathrm{sm}},\chi \rangle ={\Lambda }^{\mathrm{ex}}\left(p^k\right)\chi (log{p}^k)$$

follows. □

Theorem 6.15 

(local arithmetic coefficient identification). The arithmetic singular part of the distribution ${\mu }_{\xi }$ on the completed-zeta-function side is given at prime-power positions by the exact von Mangoldt lift. Namely,

$$sing\; supp{\mu }_{\xi }\subset \{log{p}^k:p\mathrm{prime},k\ge 1\},$$

and the principal delta coefficient at each $log{p}^k$ is

$${\Lambda }^{\mathrm{ex}}\left(p^k\right).$$

Proof. 

This is a theorem-level summary of the content of Lemma 6.14. This assertion is coefficient identification that isolates each prime-power position by local test functions, and is used independently of the open-band equality. □

Lemma 6.16 

(continuous realization of the zeta-side functional on the comparison tests). The finite-window functional of the previous section,

$$\phi ⟼\langle {\mu }_{\xi },\phi \rangle (\phi \in {\mathcal{T}}_{\mathrm{fw}}),$$

is realized as a continuous linear functional on the following two types of test classes:

$$C_c^{\infty }\left(\mathbb{R}\right),{\mathcal{A}}_{\eta }(0<\eta <log2).$$

In particular, ${\mu }_{\xi }$ is well-defined on the test spaces required for local coefficient identification and the open-band comparison.

Proof. 

On $C_c^{\infty }\left(\mathbb{R}\right)$, the arithmetic singular part is defined as a finite sum by Definition 6.13. The smooth Archimedean term and calibrated reference term act as ordinary distributions on compact supports, and therefore ${\mu }_{\xi }$ is continuous on $C_c^{\infty }$.

On the open-band class ${\mathcal{A}}_{\eta }$, by the open-band convention of Section 6.1, ${\mu }_{\xi }$ is evaluated as the continuous extension of the calibrated finite-window explicit-formula functional of Section 5. Namely, the functional defined on the initial test algebra ${\mathcal{T}}_{\mathrm{fw}}\cap {\mathcal{A}}_{\eta }$ of Section 6.0 is extended continuously with respect to the Fréchet topology of ${\mathcal{A}}_{\eta }$. This continuity follows from the fact that, on a fixed open band, the Fourier support is compactly restricted, and the Archimedean term and finite-window calibration term are controlled by finitely many seminorms of ${\mathcal{A}}_{\eta }$. □

Lemma 6.17 

(continuous realization of the residual-free singular-boundary functional on the comparison tests). The residual-free singular-boundary functional of the previous section,

$$\phi ⟼\langle {\mu }_L,\phi \rangle ={\mathcal{L}}_R\left(\phi \right)(\phi \in {\mathcal{T}}_{\mathrm{fw}}),$$

is realized as a continuous linear functional on the following two types of test classes:

$$C_c^{\infty }\left(\mathbb{R}\right),{\mathcal{A}}_{\eta }(0<\eta <log2).$$

Proof. 

The residual-free singular-boundary functional is read, through the localized comparison interface of Section 5 and the map ${\mathcal{J}}_R$ of Definition 6.5, as

$$\phi ⟼{\Pi }_R{x}_{\phi }^{♯}\in {\mathcal{K}}_R.$$

The residual-free projection of Section 5 is continuous, and the boundary trace of Section 4 is also continuous with respect to the topologies of local coefficient tests and open-band tests. Therefore ${\mathcal{L}}_R$ acts as a continuous linear functional on $C_c^{\infty }$ and on ${\mathcal{A}}_{\eta }$. Its action on the central logarithmic test family is constructed independently in Section 6.4 after introducing ${\mathcal{R}}_{\mathrm{cen}}$ and the Cauchy–Laplace representation. □

Lemma 6.18 

(continuity of the open-band restriction map). ${\mu }_L$ and ${\mu }_{\xi }$ are restricted as continuous linear functionals on

$${\mathcal{A}}_{\eta }.$$

More generally, for any comparison functional T realized continuously on ${\mathcal{A}}_{\eta }$,

$${T|}_{{\mathcal{A}}_{\eta }}:{\mathcal{A}}_{\eta }\to \mathbb{C}$$

is continuous.

Proof. 

The Fréchet topology of Definition 6.10 is placed on ${\mathcal{A}}_{\eta }$. Lemma 6.16 and Lemma 6.17 state that ${\mu }_{\xi }$ and ${\mu }_L$ act continuously in this topology. The assertion follows. □

Lemma 6.19 

(open-band residual-free comparison). Let $0<\eta <log2$. For every $\phi \in {\mathcal{A}}_{\eta }$, the canonical residual-free comparison data satisfy

$$\langle {\mu }_L,\phi \rangle -\langle {\mu }_{\xi },\phi \rangle =0.$$

Proof. 

First let $\phi \in {\mathcal{T}}_{\mathrm{fw}}\cap {\mathcal{A}}_{\eta }$. By the exclusive-complement principle of Section 5, the total variation of the finite-window explicit formula is preserved as the sum of the calibrated reference term, the arithmetic-side prime-power contribution, and the ${\mathcal{K}}_R$-side residual-free singular-boundary functional. The arithmetic side is evaluated as

$$〈{\mu }_{\Lambda }^{\mathrm{ex}},\phi 〉=\sum _{p^k}{\Lambda }^{\mathrm{ex}}\left(p^k\right)\phi (log{p}^k).$$

For an open-band test, however, this sum is not used as a direct infinite sum, but is interpreted as the pairing continuously extended from the arithmetic trace on finite-window inputs. In this sense the arithmetic side is evaluated exactly, and the residual component does not contribute to the comparison data by

$${\Pi }_{\mathrm{res}}{x}_{\phi ,j}^{♯}=0.$$

Therefore, the local residual after removing the arithmetic side is represented completely by

$${\Pi }_R{x}_{\phi ,j}^{♯}\in {\mathcal{K}}_R.$$

On the other hand, ${\mu }_{\xi }$ is defined as the same calibrated finite-window explicit-formula functional on the completed-zeta-function side. By Lemma 6.14, the coefficients of its arithmetic singular part agree with the exact von Mangoldt lift ${\Lambda }^{\mathrm{ex}}$ of Section 3. Hence the prime-power contribution evaluated by the exact arithmetic trace evaluation of Section 5 and the arithmetic singular part of ${\mu }_{\xi }$ are identical. The remaining calibrated reference term is also fixed commonly by the definition of the previous section, and therefore

$$\langle {\mu }_L,\phi \rangle =\langle {\mu }_{\xi },\phi \rangle (\phi \in {\mathcal{T}}_{\mathrm{fw}}\cap {\mathcal{A}}_{\eta })$$

is obtained.

Next take an arbitrary $\phi \in {\mathcal{A}}_{\eta }$. The space ${\mathcal{T}}_{\mathrm{fw}}\cap {\mathcal{A}}_{\eta }$ is chosen in Section 6.0 to be dense in ${\mathcal{A}}_{\eta }$. Therefore one can take a sequence

$${\phi }_m\in {\mathcal{T}}_{\mathrm{fw}}\cap {\mathcal{A}}_{\eta },{\phi }_m\to \phi \mathrm{in}{\mathcal{A}}_{\eta }.$$

By Lemma 6.16, Lemma 6.17, and Lemma 6.18, ${\mu }_L$ and ${\mu }_{\xi }$ are continuous on ${\mathcal{A}}_{\eta }$. Thus

$$\langle {\mu }_L,\phi \rangle =\underset{m\to \infty }{lim}\langle {\mu }_L,{\phi }_m\rangle =\underset{m\to \infty }{lim}\langle {\mu }_{\xi },{\phi }_m\rangle =\langle {\mu }_{\xi },\phi \rangle .$$

This proves the assertion. □

Theorem 6.20 

(distributional comparison theorem on the open band). Let $0<\eta <log2$. Then ${\mu }_L$ and ${\mu }_{\xi }$ are realized as continuous linear functionals on

$${\mathcal{A}}_{\eta },$$

and for every

$$\phi \in {\mathcal{A}}_{\eta }$$

one has

$$\langle {\mu }_L,\phi \rangle =\langle {\mu }_{\xi },\phi \rangle .$$

Proof. 

The continuous realizations of ${\mu }_{\xi }$ and ${\mu }_L$ on ${\mathcal{A}}_{\eta }$ were shown in Lemma 6.16 and Lemma 6.17. Their restrictions agree by Lemma 6.19. □

Theorem 6.21 

(open-band residual-free equality). For $0<\eta <log2$,

$$\langle {\mu }_L,\phi \rangle =\langle {\mu }_{\xi },\phi \rangle (\phi \in {\mathcal{A}}_{\eta })$$

holds.

Proof. 

This is the equality part of Theorem 6.20. The assertion here is the residual-free equality on open-band test functions whose Fourier support is contained in $(-\eta ,\eta )$, and is used separately from the local coefficient identification of Theorem 6.15. □

Remark 6.22 

(boundary band is not used in the open-band theorem). Theorem 6.20 is the open-band statement for

$$0<\eta <log2.$$

At the boundary value

$$\eta =log2,$$

the Fourier support may touch the first prime-power position, and therefore the proof of this section is not applied as it stands. The boundary band and finite-window endpoint stability are treated independently in Section 6.2.

6.2. Endpoint Stability Theorem

In this section, when connecting the open-band distributional equality obtained in Section 6.1 to the argument principle for finite windows, it is shown that band cutoff and endpoint regularization do not change the integer-valued zero count. What is needed here is not merely an estimate saying that the endpoint error is small, but stability saying that the integer value obtained by the argument principle is invariant. Thus this section is restricted to finite windows whose boundary does not pass through zeros, and endpoint contributions are handled by homotopy invariance.

Definition 6.23 

(admissible finite window). Let G be a holomorphic function on $\Omega \subset \mathbb{C}$. Let $0<T_0<T$, and let $R_{\eta }(T_0,T)⋐\Omega$ be a bounded closed rectangle. Its positively oriented boundary is denoted by

$$\partial R_{\eta }(T_0,T).$$

This finite window $R_{\eta }(T_0,T)$ is said to be G-admissible if

$$G\left(s\right)\ne 0(s\in \partial R_{\eta }(T_0,T))$$

holds. Then, by compactness,

$$m_R\left(G\right):=\underset{s\in \partial R_{\eta }(T_0,T)}{inf}|G\left(s\right)|>0.$$

In what follows, when $R_{\eta }(T_0,T)$ is clear, it is simply denoted by R.

Definition 6.24 

(argument-principle count). Let R be a G-admissible finite window. Define

$$N_{\arg }(G;R):={\displaystyle \frac{1}{2\pi i}}{\int }_{\partial R}{\displaystyle \frac{G^{\prime }\left(s\right)}{G\left(s\right)}}ds.$$

Since G is holomorphic in a neighborhood of R, the argument principle gives

$$N_{\arg }(G;R)=\sum _{\rho \in R^{\circ },G\left(\rho \right)=0}{m}_G\left(\rho \right)\in {\mathbb{Z}}_{\ge 0}.$$

In particular, when $G=\xi$, write

$$N_{\arg }\left(R\right):=N_{\arg }(\xi ;R).$$

Definition 6.25 

(admissible analytic cutoff near a window). Let $R=R_{\eta }(T_0,T)$ be a G-admissible finite window. Let ${\left\{G_{\Lambda }\right\}}_{\Lambda \ge 1}$ be a family of holomorphic functions on an open neighborhood $U_R⋐\Omega$ of R. This family is said to be an admissible analytic cutoff with respect to R if, for some integer $m\ge 3$,

$$G_{\Lambda }⟶G\mathrm{in}{H}^m\left(U_R\right)(\Lambda \to \infty )$$

holds.

Then define the argument count after cutoff by $G_{\Lambda }$ as

$$N_{\arg }^{\mathrm{cutoff}}(G_{\Lambda };R):={\displaystyle \frac{1}{2\pi i}}{\int }_{\partial R}{\displaystyle \frac{G_{\Lambda }^{\prime }\left(s\right)}{G_{\Lambda }\left(s\right)}}ds,$$

provided that the right-hand side is defined, namely that $G_{\Lambda }$ has no zero on $\partial R$.

Definition 6.26 

(endpoint cutoff term). Let R be a G-admissible finite window, and let ${\left\{G_{\Lambda }\right\}}_{\Lambda \ge 1}$ be an admissible analytic cutoff with respect to R. When $G_{\Lambda }$ has no zero on $\partial R$, define the endpoint cutoff term by

$$E_{\Lambda ,\partial R}\left(G\right):={\displaystyle \frac{1}{2\pi i}}{\int }_{\partial R}\left({\displaystyle \frac{G_{\Lambda }^{\prime }\left(s\right)}{G_{\Lambda }\left(s\right)}}-{\displaystyle \frac{G^{\prime }\left(s\right)}{G\left(s\right)}}\right)ds.$$

Namely,

$$E_{\Lambda ,\partial R}\left(G\right)=N_{\arg }^{\mathrm{cutoff}}(G_{\Lambda };R)-N_{\arg }(G;R).$$

Lemma 6.27 

(Sobolev trace control on the boundary). Let $U_R$ be a bounded Lipschitz neighborhood of R, and let $m\ge 3$. If

$$G_{\Lambda }\to G\mathrm{in}{H}^m\left(U_R\right),$$

then

$$G_{\Lambda }{{|}_{\partial R}\to G|}_{\partial R}\mathrm{in}{C}^1(\partial R).$$

In particular,

$$\underset{s\in \partial R}{sup}|G_{\Lambda }\left(s\right)-G\left(s\right)|⟶0.$$

Proof. 

By the Sobolev trace theorem for Lipschitz boundaries, the restriction map

$$H^m\left(U_R\right)⟶H^{m-{\displaystyle \frac{1}{2}}}(\partial R)$$

is continuous. Here $\partial R$ is a one-dimensional piecewise smooth compact curve, and since $m\ge 3$,

$$m-{\displaystyle \frac{1}{2}}>{\displaystyle \frac{3}{2}}.$$

Therefore, by the one-dimensional Sobolev embedding,

$$H^{m-{\displaystyle \frac{1}{2}}}(\partial R)\hookrightarrow C^1(\partial R)$$

is continuous. Hence

$$G_{\Lambda }\to G\mathrm{in}{H}^m\left(U_R\right)$$

implies

$$G_{\Lambda }{{|}_{\partial R}\to G|}_{\partial R}\mathrm{in}{C}^1(\partial R).$$

□

Lemma 6.28 

(boundary non-vanishing stability). Let R be a G-admissible finite window, and let $\left\{G_{\Lambda }\right\}$ be an admissible analytic cutoff with respect to R. Then, for all sufficiently large $\Lambda$,

$$G_{\Lambda }\left(s\right)\ne 0(s\in \partial R)$$

holds. Moreover, for every $u\in [0,1]$,

$$G_{\Lambda ,u}\left(s\right):=(1-u)G\left(s\right)+u{G}_{\Lambda }\left(s\right)$$

satisfies

$$G_{\Lambda ,u}\left(s\right)\ne 0(s\in \partial R).$$

Proof. 

By Definition 6.23,

$$m_R\left(G\right)=\underset{s\in \partial R}{inf}|G\left(s\right)|>0.$$

By Lemma 6.27,

$$\underset{s\in \partial R}{sup}|G_{\Lambda }\left(s\right)-G\left(s\right)|⟶0.$$

Therefore, for sufficiently large $\Lambda$,

$$\underset{s\in \partial R}{sup}|G_{\Lambda }\left(s\right)-G\left(s\right)|<{\displaystyle \frac{1}{2}}{m}_R\left(G\right)$$

holds.

For such $\Lambda$, for $s\in \partial R$ and $u\in [0,1]$,

$$|G_{\Lambda ,u}\left(s\right)|=|G\left(s\right)+u(G_{\Lambda }\left(s\right)-G\left(s\right))|\ge |G\left(s\right)|-|G_{\Lambda }\left(s\right)-G\left(s\right)|>{\displaystyle \frac{1}{2}}{m}_R\left(G\right)>0.$$

The assertion follows. □

Lemma 6.29 

(homotopy invariance of the argument count). Let R be a G-admissible finite window, and let $\left\{G_{\Lambda }\right\}$ be an admissible analytic cutoff with respect to R. For sufficiently large $\Lambda$,

$$N_{\arg }^{\mathrm{cutoff}}(G_{\Lambda };R)=N_{\arg }(G;R)$$

holds. Equivalently,

$$E_{\Lambda ,\partial R}\left(G\right)=0.$$

Proof. 

By Lemma 6.28, for sufficiently large $\Lambda$,

$$G_{\Lambda ,u}=(1-u)G+u{G}_{\Lambda }(0\le u\le 1)$$

has no zero on $\partial R$ for every u. Therefore

$$u⟼{\displaystyle \frac{1}{2\pi i}}{\int }_{\partial R}{\displaystyle \frac{{\partial }_s{G}_{\Lambda ,u}\left(s\right)}{G_{\Lambda ,u}\left(s\right)}}ds$$

is continuous with respect to $u\in [0,1]$. On the other hand, for each u, $G_{\Lambda ,u}$ is holomorphic in a neighborhood of R, and has no zero on $\partial R$, so by the argument principle this quantity is an integer. A continuous integer-valued function is constant on an interval. Therefore the values at $u=0$ and $u=1$ are equal, and

$$N_{\arg }(G;R)=N_{\arg }^{\mathrm{cutoff}}(G_{\Lambda };R).$$

By Definition 6.26,

$$E_{\Lambda ,\partial R}\left(G\right)=0$$

also follows. □

Theorem 6.30 

(endpoint stability theorem). Let $R=R_{\eta }(T_0,T)$ be a G-admissible finite window, and let ${\left\{G_{\Lambda }\right\}}_{\Lambda \ge 1}$ be an admissible analytic cutoff with respect to R. Then, for sufficiently large Λ,

$$N_{\arg }^{\mathrm{cutoff}}(G_{\Lambda };R)=N_{\arg }(G;R).$$

In particular, when $G=\xi$,

$$N_{\arg }^{\mathrm{cutoff}}\left(R_{\eta }(T_0,T)\right)=N_{\arg }\left(R_{\eta }(T_0,T)\right).$$

Proof. 

Applying Lemma 6.29 to G and $\left\{G_{\Lambda }\right\}$ gives, for sufficiently large $\Lambda$,

$$N_{\arg }^{\mathrm{cutoff}}(G_{\Lambda };R)=N_{\arg }(G;R).$$

In the case $G=\xi$, using the notation

$$N_{\arg }\left(R\right)=N_{\arg }(\xi ;R)$$

gives the same conclusion. □

Corollary 6.31 

(integer stability of endpoint cutoff term). Under the assumptions of Theorem 6.30, for sufficiently large Λ,

$$E_{\Lambda ,\partial R}\left(G\right)=0.$$

Therefore endpoint cutoff term does not change the zero count given by the argument principle inside the finite window.

Proof. 

By Definition 6.26,

$$E_{\Lambda ,\partial R}\left(G\right)=N_{\arg }^{\mathrm{cutoff}}(G_{\Lambda };R)-N_{\arg }(G;R).$$

By Theorem 6.30, the right-hand side is zero for sufficiently large $\Lambda$. □

Remark 6.32 

(no zero-counting conclusion in this section). The conclusion of this section is the stability that band cutoff and endpoint regularization do not change the integer-valued count of the argument principle in an admissible finite window. This section does not assert either that the zeros lie on the critical line or that no off-line zero exists. These counting consequences are treated in the subsequent arguments on the finite-window bridge and the defect staircase.

6.3. Trace-Ideal Determinant Theorem

In this section, using the compact-resolvent construction and the distribution-kernel representation of Section 4, we realize the boundary-distribution comparison kernel candidate K introduced in Section 6.0 as a Hilbert–Schmidt operator. After that, we introduce the finite-rank cutoff

$$K_N=P_{N}K{P}_N$$

and prove the local uniform convergence and coefficient transport

$${det}_2(I+z{K}_N)⟶{det}_2(I+zK).$$

The conclusion of this section is the construction of the comparison function $F_K\left(s\right)$ and the stability of its Taylor coefficients, and here we do not yet assert

$$F_K\left(s\right)=\xi \left(s\right).$$

Definition 6.33 

(reference spectral resolution). Write $A_R$ for the compact-resolvent reference operator fixed in Section 4. Let

$${\left\{e_n\right\}}_{n\ge 1}$$

be an orthonormal basis of $H_{\alpha ,+}$ consisting of its eigenfunctions. For each $N\ge 1$, define the corresponding finite-rank orthogonal projection by

$$P_{N}f:={\displaystyle \sum _{n=1}^N}{\langle f,e_n\rangle }_{H_{\alpha ,+}}{e}_n.$$

Then

$$P_N\to I\mathrm{strongly}\mathrm{on}{H}_{\alpha ,+}.$$

Definition 6.34 

(functional-equation boundary reflection). Let

$${\Theta }_R:{\mathcal{D}}_R^{\prime }⟶{\mathcal{D}}_R^{\prime }$$

denote the boundary reflection induced by the functional equation

$$s⟼1-s$$

of the completed function. On the finite-window boundary distributions it exchanges the left and right boundary components and reverses the orientation with respect to the central line. The operator ${\Theta }_R$ is defined at the level of boundary distributions and residual-free comparison data; it is not defined from, and does not use, any information about the location of the zeros of $\xi$.

Lemma 6.35 

(basic properties of the boundary reflection). The boundary reflection ${\Theta }_R$ is a bounded involution on ${\mathcal{D}}_R^{\prime }$. Moreover,

$${\Theta }_R^2=I_{{\mathcal{D}}_R^{\prime }},$$

and it preserves the boundary pairing used in the construction of the singular-boundary component. Equivalently, for admissible boundary distributions $u,v$ for which the boundary pairing is defined,

$${\langle {\Theta }_{R}u,{\Theta }_{R}v\rangle }_{\partial ,R}={\langle u,v\rangle }_{\partial ,R}.$$

Proof. 

The map $s\mapsto 1-s$ is an involution, and applying it twice returns each boundary side and orientation to its original position. Thus ${\Theta }_R^2=I_{{\mathcal{D}}_R^{\prime }}$ on the finite-window boundary distributions. The boundary norm and pairing in Section 4 were constructed from the two reflected boundary components symmetrically, so the same reflection preserves the pairing. By density of the finite-window boundary distributions and continuity of the boundary trace topology, the action extends uniquely to a bounded involution on ${\mathcal{D}}_R^{\prime }$, and the pairing identity extends by continuity. □

Lemma 6.36 

(topological realization of the boundary reflection). Let

$${\mathcal{E}}_{R,\mathrm{fw}}\subset {\mathcal{D}}_R^{\prime }$$

be the finite-window boundary-distribution subspace used to construct the admissible boundary-distribution closure. Then ${\mathcal{E}}_{R,\mathrm{fw}}$ is dense in the boundary-distribution topology of ${\mathcal{D}}_R^{\prime }$, and the reflection ${\Theta }_R$ is bounded with respect to the defining seminorms of that topology. In particular, if $u_n\to u$ in ${\mathcal{D}}_R^{\prime }$, then

$${\Theta }_R{u}_n\to {\Theta }_{R}u\mathrm{in}{\mathcal{D}}_R^{\prime },$$

and the boundary pairing identities verified on ${\mathcal{E}}_{R,\mathrm{fw}}$ extend uniquely to all admissible limits.

Proof. 

In Section 4, ${\mathcal{D}}_R^{\prime }$ is obtained as the distributional completion generated by finite-window boundary distributions subject to the boundary trace estimates and support constraints. Thus ${\mathcal{E}}_{R,\mathrm{fw}}$ is dense by definition of this completion. The reflection $s\mapsto 1-s$ exchanges the two boundary sides and preserves the central weights entering those trace seminorms. Consequently, for every defining seminorm q of the boundary-distribution topology there are a defining seminorm $q^{\prime }$ and a constant $C_q>0$ such that

$$q\left({\Theta }_{R}u\right)\le C_q{q}^{\prime }\left(u\right)(u\in {\mathcal{E}}_{R,\mathrm{fw}}).$$

The estimate extends by density and gives a bounded operator on ${\mathcal{D}}_R^{\prime }$. The boundary pairing is continuous with respect to these seminorms, so the pairing preservation established on finite-window boundary distributions extends to admissible limits. □

Lemma 6.37 

(compatibility with admissible boundary data and residual-free comparison). The boundary reflection preserves the admissible boundary-distribution subspace:

$${\Theta }_R{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }={\mathcal{D}}_{R,\mathrm{adm}}^{\prime }.$$

Moreover, it is compatible with the residual-free comparison interface in the following sense. If $y_1,y_2\in {\mathcal{D}}_{R,\mathrm{adm}}^{\prime }$ have the same ${\mathcal{K}}_R$-projected comparison component,

$${\Pi }_R{LCI}_R\left(y_1\right)={\Pi }_R{LCI}_R\left(y_2\right),$$

then

$${\Pi }_R{LCI}_R\left({\Theta }_R{y}_1\right)={\Pi }_R{LCI}_R\left({\Theta }_R{y}_2\right).$$

Consequently, the rule

$${\Pi }_R{LCI}_R\left(y\right)⟼{\Pi }_R{LCI}_R\left({\Theta }_{R}y\right)$$

is well-defined on the ${\mathcal{K}}_R$-projected comparison range.

Proof. 

On the dense subspace of boundary distributions obtained from finite-window inputs, ${\mathcal{E}}_{R,\mathrm{fw}}\subset {\mathcal{D}}_R^{\prime }$, ${\Theta }_R$ is the left-right boundary reflection determined by the functional equation. The finite-window comparison identity is invariant under this reflection, and the canonical residual-free representative of Section 5 is defined modulo $Ran{\Pi }_{\mathrm{res}}$, which is orthogonal to the ${\mathcal{K}}_R$-component. Therefore equality of the ${\mathcal{K}}_R$-projected components is preserved by applying ${\Theta }_R$.

In Theorem 6.3, ${\mathcal{D}}_{R,\mathrm{adm}}^{\prime }$ was defined as the closed subspace satisfying the uniform boundary estimate in the closure of finite-window boundary inputs. Since ${\Theta }_R$ is bounded in the boundary-distribution topology by Lemma 6.36 and preserves the boundary pairing by Lemma 6.35, it preserves this closure and the same uniform estimate. Hence

$${\Theta }_R{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }={\mathcal{D}}_{R,\mathrm{adm}}^{\prime }.$$

The well-definedness of the displayed rule follows from the first part of the statement and the residual-free quotient compatibility just proved. □

Proposition 6.38 

(descent to the singular-boundary component). The boundary reflection ${\Theta }_R$ descends to a bounded self-adjoint involution

$${\mathcal{S}}_R:{\mathcal{K}}_R⟶{\mathcal{K}}_R$$

on the residual-free ${\mathcal{K}}_R$-component. It satisfies

$${\mathcal{S}}_R^{*}={\mathcal{S}}_R,{\mathcal{S}}_R^2=I_{{\mathcal{K}}_R},\parallel {\mathcal{S}}_R\parallel =1.$$

For every $f\in {\mathcal{D}}_R$,

$${\mathcal{S}}_R{\Pi }_R{\mathcal{J}}_{R}f={\Pi }_R{LCI}_R\left({\Theta }_R{Tr}_{\partial ,R}f\right).$$

Proof. 

By Lemma 6.37, the rule

$${\Pi }_R{LCI}_R\left(y\right)⟼{\Pi }_R{LCI}_R\left({\Theta }_{R}y\right)$$

is well-defined on the projected comparison range. This range is dense in the component ${\mathcal{K}}_R=Ran{\Pi }_R$ generated by the residual-free comparison interface. Because ${\Theta }_R$ preserves the boundary pairing by Lemma 6.35, the induced map is isometric on this dense range. It therefore extends uniquely to a bounded isometry

$${\mathcal{S}}_R:{\mathcal{K}}_R\to {\mathcal{K}}_R.$$

The identity ${\Theta }_R^2=I$ gives ${\mathcal{S}}_R^2=I_{{\mathcal{K}}_R}$. Since an isometric involution satisfies

$${\mathcal{S}}_R^{-1}={\mathcal{S}}_R\mathrm{and}{\mathcal{S}}_R^{-1}={\mathcal{S}}_R^{*},$$

we obtain

$${\mathcal{S}}_R^{*}={\mathcal{S}}_R.$$

Finally, taking

$$y={Tr}_{\partial ,R}f$$

and using

$${\Pi }_R{\mathcal{J}}_{R}f={\Pi }_R{LCI}_R\left({Tr}_{\partial ,R}f\right)$$

gives the displayed formula. □

Definition 6.39 

(${\mathcal{K}}_R$-signature operator). In what follows, the self-adjoint involution

$${\mathcal{S}}_R:{\mathcal{K}}_R\to {\mathcal{K}}_R$$

constructed in Proposition 6.38 is called the ${\mathcal{K}}_R$-signature operator. The operator ${\mathcal{S}}_R$ is not an operator for making the form positive-definite. It is only the self-adjoint involution induced by the functional-equation reflection after passage to the residual-free ${\mathcal{K}}_R$-component.

Definition 6.40 

(signed residual-free ${\mathcal{K}}_R$-quadratic form). For $f\in {\mathcal{D}}_R$, define

$$Q_R^{\mathrm{sgn}}\left(f\right):={〈{\mathcal{S}}_R{\Pi }_R{\mathcal{J}}_{R}f,{\Pi }_R{\mathcal{J}}_{R}f〉}_X.$$

This is not a positive quadratic form, but a Hermitian quadratic form representing the oriented component on the singular-boundary side.

Definition 6.41 

(signed residual-free boundary-distribution comparison kernel). Define the signed residual-free boundary-distribution comparison kernel by

$${\mathfrak{k}}_R(f,g):={〈{\mathcal{S}}_R{\Pi }_R{\mathcal{J}}_{R}f,{\Pi }_R{\mathcal{J}}_{R}g〉}_X(f,g\in {\mathcal{D}}_R).$$

Equivalently, it is the sesquilinear form obtained as the polarization of $Q_R^{\mathrm{sgn}}$. In what follows, the boundary-distribution comparison kernel candidate of Section 6.0 is evaluated as this signed kernel.

Lemma 6.42 

(Hermitian symmetry of the signed boundary-distribution comparison kernel).

$${\mathfrak{k}}_R(f,g)=\overline{{\mathfrak{k}}_R(g,f)}(f,g\in {\mathcal{D}}_R)$$

holds. Moreover,

$${\mathfrak{k}}_R(f,f)=Q_R^{\mathrm{sgn}}\left(f\right)\in \mathbb{R}.$$

In general, ${\mathfrak{k}}_R(f,f)\ge 0$ is not assumed.

Proof. 

By Definition 6.41,

$${\mathfrak{k}}_R(f,g)={〈{\mathcal{S}}_R{\Pi }_R{\mathcal{J}}_{R}f,{\Pi }_R{\mathcal{J}}_{R}g〉}_X.$$

The only structural property used here is the self-adjointness

$${\mathcal{S}}_R^{*}={\mathcal{S}}_R$$

obtained in Proposition 6.38. Therefore

$${〈{\mathcal{S}}_R{\Pi }_R{\mathcal{J}}_{R}f,{\Pi }_R{\mathcal{J}}_{R}g〉}_X=\overline{{〈{\mathcal{S}}_R{\Pi }_R{\mathcal{J}}_{R}g,{\Pi }_R{\mathcal{J}}_{R}f〉}_X}.$$

This gives

$${\mathfrak{k}}_R(f,g)=\overline{{\mathfrak{k}}_R(g,f)}.$$

Taking $f=g$, the value is real. No positivity of $Q_R^{\mathrm{sgn}}$ is used or asserted. □

Remark 6.43 

(non-circularity of the construction of K). The construction of ${\Theta }_R$, ${\mathcal{S}}_R$, the signed boundary-distribution comparison kernel ${\mathfrak{k}}_R$, and the operator K uses only the functional equation

$$\xi \left(s\right)=\xi (1-s),$$

the boundary distribution framework of Section 4–5, and the orthogonal projection structure of X. It does not use any information about the location of the zeros of $\xi$. In particular, no positivity, Herglotz property, or spectral-localization statement equivalent to the Riemann Hypothesis is assumed in the definition of ${\mathcal{S}}_R$, ${\mathfrak{k}}_R$, or K.

Theorem 6.44 

(Sobolev eigenvalue growth of the reference operator). The eigenvalue sequence of the compact-resolvent reference operator $A_R$ constructed in Section 4,

$$A_R{e}_n={\lambda }_n{e}_n,0\le {\lambda }_1\le {\lambda }_2\le \dots ,$$

satisfies, for some constants $c>0$ and $\nu >0$,

$${\lambda }_n\ge c{n}^{\nu }(n\gg 1).$$

Therefore, for any

$$a>{\displaystyle \frac{1}{2\nu }},$$

one has

$$\sum _{n\ge 1}{(1+{\lambda }_n)}^{-2a}<\infty .$$

Proof. 

In the reference Sobolev model of Section 4, $A_R$ is constructed as a compact-resolvent elliptic regularizing operator on $H_{\alpha ,+}$. Its Hilbert scale

$$H_{\alpha ,+}^r:=Dom{(I+A_R)}^{r/2}$$

is the Sobolev scale controlling boundary traces and distribution kernels, and the corresponding resolvent has trace smoothing of sufficiently high order by the compactness estimate of Section 4. By the standard eigenvalue growth estimate for this Sobolev model, there exist $c,\nu >0$ such that

$${\lambda }_n\ge c{n}^{\nu }(n\gg 1)$$

holds.

Then

$${(1+{\lambda }_n)}^{-2a}\le C{(1+n^{\nu })}^{-2a}\le C^{\prime }{n}^{-2a\nu }.$$

If $a>1/\left(2\nu \right)$, then

$$\sum _{n\ge 1}{n}^{-2a\nu }<\infty ,$$

and hence

$$\sum _{n\ge 1}{(1+{\lambda }_n)}^{-2a}<\infty$$

follows. □

Remark 6.45 

(Schatten scale inherited from the Sobolev model). Theorem 6.44 is the only spectral-growth input inherited from the compact-resolvent Sobolev model constructed in Section 4. Namely, the property used below,

$${(I+A_R)}^{-a}\in {\mathfrak{S}}_2\left(H_{\alpha ,+}\right),$$

is not an independent external assumption, but the Schatten smoothing obtained from the Hilbert scale generated by the reference operator $A_R$ of Section 4 and from the eigenvalue growth

$${\lambda }_n\ge c{n}^{\nu }.$$

In the subsequent Hilbert–Schmidt estimates, only this eigenvalue growth and the smoothing factorization of the boundary trace are used.

Theorem 6.46 

(Schatten scale inherited from the reference Sobolev model). The compact-resolvent reference operator $A_R$ of Section 4 has a Schatten smoothing scale on the singular-boundary side. Namely, there exists $a_0>0$ such that, for every $a>a_0$,

$${(I+A_R)}^{-a}\in {\mathfrak{S}}_2\left(H_{\alpha ,+}\right)$$

holds. Equivalently, for the reference spectral resolution $A_R{e}_n={\lambda }_n{e}_n$,

$$\sum _{n\ge 1}{(1+{\lambda }_n)}^{-2a}<\infty (a>a_0).$$

Proof. 

By Theorem 6.44, there exist $c,\nu >0$ such that

$${\lambda }_n\ge c{n}^{\nu }(n\gg 1).$$

Thus, if $a>1/\left(2\nu \right)$, then

$$\sum _{n\ge 1}{(1+{\lambda }_n)}^{-2a}<\infty .$$

Fix one $a_0>1/\left(2\nu \right)$. Then, for every $a>a_0$,

$$\sum _{n\ge 1}{(1+{\lambda }_n)}^{-2a}<\infty ,$$

and this is equivalent to

$${(I+A_R)}^{-a}\in {\mathfrak{S}}_2\left(H_{\alpha ,+}\right).$$

□

Lemma 6.47 

(smoothing nature of the singular-boundary trace). For sufficiently large $a>a_0$, the singular boundary trace of Section 4 factors as

$${Tr}_{\partial ,R}={Tr}_{\partial ,R}^{\left(0\right)}{(I+A_R)}^{-a}.$$

Here

$${Tr}_{\partial ,R}^{\left(0\right)}:H_{\alpha ,+}⟶{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }$$

is a bounded linear map.

Proof. 

The boundary trace of Section 4 is not an ordinary boundary-value map, but is constructed as a singular boundary distribution regularized by the singular-boundary Sobolev scale. Namely, the high-frequency components are estimated by $A_R$-Sobolev smoothing. Taking $a>a_0$ sufficiently large,

$${(I+A_R)}^{-a}:H_{\alpha ,+}\to H_{\alpha ,+}^a$$

provides the regularity required by the boundary trace theorem of Section 4, and the subsequent boundary-distribution map

$${Tr}_{\partial ,R}^{\left(0\right)}:H_{\alpha ,+}\to {\mathcal{D}}_{R,\mathrm{adm}}^{\prime }$$

is bounded. Therefore

$${Tr}_{\partial ,R}={Tr}_{\partial ,R}^{\left(0\right)}{(I+A_R)}^{-a}$$

can be written. □

Lemma 6.48 

(boundary trace smoothing). If $a>a_0$ is taken sufficiently large, then there exists a constant $C_a>0$ such that, for the reference spectral basis $A_R{e}_n={\lambda }_n{e}_n$,

$$\parallel {Tr}_{\partial ,R}{e}_n{\parallel }_{{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }}\le C_a{(1+{\lambda }_n)}^{-a}(n\ge 1).$$

Proof. 

By Lemma 6.47,

$${Tr}_{\partial ,R}={Tr}_{\partial ,R}^{\left(0\right)}{(I+A_R)}^{-a},$$

and ${Tr}_{\partial ,R}^{\left(0\right)}$ is bounded. Therefore

$$\parallel {Tr}_{\partial ,R}{f\parallel }_{{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }}\le \parallel {Tr}_{\partial ,R}^{\left(0\right)}\parallel \parallel {(I+A_R)}^{-a}{f\parallel }_{H_{\alpha ,+}}.$$

Taking $f=e_n$, one has

$$\parallel {(I+A_R)}^{-a}{e}_n{\parallel }_{H_{\alpha ,+}}={(1+{\lambda }_n)}^{-a},$$

and hence

$$\parallel {Tr}_{\partial ,R}{e}_n{\parallel }_{{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }}\le C_a{(1+{\lambda }_n)}^{-a}.$$

□

Lemma 6.49 

(boundedness of the comparison extension). The continuously extended localized comparison interface

$${LCI}_R:{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }\to X$$

is bounded. Namely, there exists a constant $C_{\mathrm{LCI}}>0$ such that

$$\parallel {LCI}_R{y\parallel }_X\le C_{\mathrm{LCI}}{\parallel y\parallel }_{{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }}(y\in {\mathcal{D}}_{R,\mathrm{adm}}^{\prime }).$$

Proof. 

This is the continuity of Theorem 6.3, rewritten in terms of the Hilbertizable boundary norm of ${\mathcal{D}}_{R,\mathrm{adm}}^{\prime }$. □

Theorem 6.50 

(smoothing of the singular-boundary comparison map). Take $a>a_0$ so as to satisfy the condition of Lemma 6.48. Then there exists a constant $C_a^{\prime }>0$ such that

$$\parallel {\Pi }_R{\mathcal{J}}_R{e}_n{\parallel }_X\le C_a^{\prime }{(1+{\lambda }_n)}^{-a}(n\ge 1).$$

More generally, for $f\in {\mathcal{D}}_R$,

$$\parallel {\Pi }_R{\mathcal{J}}_R{f\parallel }_X\le C_a^{\prime }{\parallel {Tr}_{\partial ,R}f\parallel }_{{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }}.$$

Proof. 

By definition,

$${\mathcal{J}}_R=({\Pi }_R+{\Pi }_{\mathrm{arith}}^X){\mathfrak{B}}_R=({\Pi }_R+{\Pi }_{\mathrm{arith}}^X){LCI}_R{Tr}_{\partial ,R}.$$

Therefore

$${\Pi }_R{\mathcal{J}}_R={\Pi }_R{LCI}_R{Tr}_{\partial ,R}.$$

By Lemma 6.49 and the boundedness of ${\Pi }_R$,

$$\parallel {\Pi }_R{\mathcal{J}}_R{f\parallel }_X\le \parallel {\Pi }_R\parallel \parallel {LCI}_R\parallel \parallel {Tr}_{\partial ,R}{f\parallel }_{{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }}.$$

Taking $f=e_n$ and applying Lemma 6.48, one obtains

$$\parallel {\Pi }_R{\mathcal{J}}_R{e}_n{\parallel }_X\le C_a^{\prime }{(1+{\lambda }_n)}^{-a}.$$

□

Theorem 6.51 

(Schatten estimate for the signed boundary-distribution comparison kernel). Take $a>a_0$ so as to satisfy the conditions of Theorem 6.46 and Theorem 6.50. Then, with respect to the reference spectral resolution ${\left\{e_n\right\}}_{n\ge 1}$,

$$|{\mathfrak{k}}_R(e_m,e_n)|\le C_a{(1+{\lambda }_m)}^{-a}{(1+{\lambda }_n)}^{-a}(m,n\ge 1)$$

holds. Therefore

$$\sum _{m,n\ge 1}{|{\mathfrak{k}}_R(e_m,e_n)|}^2<\infty .$$

Proof. 

By definition of the signed kernel,

$${\mathfrak{k}}_R(e_m,e_n)={〈{\mathcal{S}}_R{\Pi }_R{\mathcal{J}}_R{e}_m,{\Pi }_R{\mathcal{J}}_R{e}_n〉}_X.$$

Since ${\mathcal{S}}_R$ is bounded and $\parallel {\mathcal{S}}_R\parallel =1$, the Cauchy–Schwarz inequality and Theorem 6.50 imply

$$|{\mathfrak{k}}_R(e_m,e_n)|\le \parallel {\Pi }_R{\mathcal{J}}_R{e}_m{\parallel }_X{\parallel {\Pi }_R{\mathcal{J}}_R{e}_n\parallel }_X\le C_a^2{(1+{\lambda }_m)}^{-a}{(1+{\lambda }_n)}^{-a}.$$

Renaming the constant as $C_a$, we obtain the asserted pointwise estimate. Furthermore, by Theorem 6.46,

$$\sum _{n\ge 1}{(1+{\lambda }_n)}^{-2a}<\infty .$$

Thus

$$\sum _{m,n\ge 1}{|{\mathfrak{k}}_R(e_m,e_n)|}^2\le C_a^2\left(\sum _{m\ge 1}{(1+{\lambda }_m)}^{-2a}\right)\left(\sum _{n\ge 1}{(1+{\lambda }_n)}^{-2a}\right)<\infty .$$

□

Proposition 6.52 

(Hilbert–Schmidt realization of the boundary-distribution comparison kernel). The boundary-distribution comparison kernel candidate K defined in Section 6.0 closes uniquely as a Hilbert–Schmidt operator on $H_{\alpha ,+}$. Namely,

$$K\in {\mathfrak{S}}_2\left(H_{\alpha ,+}\right).$$

Moreover,

$${\parallel K\parallel }_{{\mathfrak{S}}_2}^2=\sum _{m,n\ge 1}{|{\langle K{e}_n,e_m\rangle }_{H_{\alpha ,+}}|}^2<\infty .$$

Proof. 

By Definition 6.41, the sesquilinear kernel on the initial domain ${\mathcal{D}}_R\subset H_{\alpha ,+}$,

$${\mathfrak{k}}_R(f,g)={\langle Kf,g\rangle }_{H_{\alpha ,+}}(f,g\in {\mathcal{D}}_R),$$

is determined as the polarization of the residual-free ${\mathcal{K}}_R$-quadratic evaluation. By Theorem 6.51, the matrix coefficients with respect to the reference spectral basis $\left\{e_n\right\}$ satisfy

$$\sum _{m,n\ge 1}{|{\mathfrak{k}}_R(e_n,e_m)|}^2<\infty .$$

Therefore the operator $K_0$, initially defined on finite linear combinations by

$$K_0{e}_n:=\sum _{m\ge 1}{\mathfrak{k}}_R(e_n,e_m)e_m,$$

satisfies

$$\sum _{n\ge 1}\parallel K_0{e}_n{\parallel }_{H_{\alpha ,+}}^2=\sum _{m,n\ge 1}{|{\mathfrak{k}}_R(e_n,e_m)|}^2<\infty .$$

Hence $K_0$ extends uniquely to the whole of $H_{\alpha ,+}$ as a Hilbert–Schmidt operator. Writing this extension again as K, one has

$$K\in {\mathfrak{S}}_2\left(H_{\alpha ,+}\right),$$

and

$${\parallel K\parallel }_{{\mathfrak{S}}_2}^2=\sum _{m,n\ge 1}{|{\langle K{e}_n,e_m\rangle }_{H_{\alpha ,+}}|}^2$$

follows. □

Lemma 6.53 

(Hermitian symmetry of the residual-free boundary-distribution comparison kernel). The boundary-distribution comparison kernel of Section 6.0,

$${\mathfrak{k}}_R(f,g)(f,g\in {\mathcal{D}}_R),$$

is Hermitian. That is,

$${\mathfrak{k}}_R(f,g)=\overline{{\mathfrak{k}}_R(g,f)}(f,g\in {\mathcal{D}}_R).$$

Proof. 

This is the content of Lemma 6.42. Namely, since the boundary-distribution comparison kernel of Section 6.0 is evaluated as the polarized kernel of Definition 6.41, it inherits the Hermitian symmetry coming from the inner product of the Hilbert space X. □

Theorem 6.54 

(self-adjoint Hilbert–Schmidt realization). The boundary-distribution comparison kernel candidate K of Section 6.0 is realized as a self-adjoint Hilbert–Schmidt operator satisfying

$$K=K^{*}\mathrm{on}{H}_{\alpha ,+}.$$

Proof. 

By Proposition 6.52, K closes uniquely as a Hilbert–Schmidt operator on $H_{\alpha ,+}$. The symmetry of the initial sesquilinear form is exactly the Hermitian symmetry of Lemma 6.53. That lemma, in turn, uses only the self-adjointness

$${\mathcal{S}}_R^{*}={\mathcal{S}}_R$$

of the ${\mathcal{K}}_R$-involution and the Hilbert-space inner product on X. Thus, for $f,g\in {\mathcal{D}}_R$,

$${\langle Kf,g\rangle }_{H_{\alpha ,+}}={\mathfrak{k}}_R(f,g)=\overline{{\mathfrak{k}}_R(g,f)}={\langle f,Kg\rangle }_{H_{\alpha ,+}}.$$

No positivity of $Q_R^{\mathrm{sgn}}$, Herglotz property, or zero-localization assertion is used in this step. Since ${\mathcal{D}}_R$ is dense in $H_{\alpha ,+}$, and K is bounded, the symmetry extends continuously to all of $H_{\alpha ,+}$. Therefore

$$K=K^{*}$$

holds. □

Definition 6.55 

(finite-rank compressions). Let $P_N$ be the finite-rank projection of Definition 6.33. Define the finite-rank cutoff of the boundary-distribution comparison kernel K by

$$K_N:=P_{N}K{P}_N.$$

Then $K_N$ is a finite-rank operator, and in particular

$$K_N\in {\mathfrak{S}}_1\left(H_{\alpha ,+}\right)\cap {\mathfrak{S}}_2\left(H_{\alpha ,+}\right).$$

Lemma 6.56 

(Hilbert–Schmidt convergence of finite-rank compressions).

$$\parallel K_N{-K\parallel }_{{\mathfrak{S}}_2}⟶0(N\to \infty ).$$

Proof. 

$P_N\to I$ converges strongly on $H_{\alpha ,+}$. For a Hilbert–Schmidt operator K, a bounded strongly convergent sequence of operators $P_N$ satisfies

$$\parallel (I-P_N){K\parallel }_{{\mathfrak{S}}_2}\to 0,{\parallel K(I-P_N)\parallel }_{{\mathfrak{S}}_2}\to 0.$$

Indeed, using the orthonormal basis $\left\{e_n\right\}$,

$$\parallel (I-P_N){K\parallel }_{{\mathfrak{S}}_2}^2=\sum _{j\ge 1}{\parallel (I-P_N)K{e}_j\parallel }_{H_{\alpha ,+}}^2.$$

For each j, $(I-P_N)K{e}_j\to 0$, and moreover

$$\parallel (I-P_N)K{e}_j{\parallel }^2\le {\parallel K{e}_j\parallel }^2$$

and

$$\sum _j\parallel K{e}_j{\parallel }^2={\parallel K\parallel }_{{\mathfrak{S}}_2}^2<\infty .$$

Therefore the first convergence follows by dominated convergence. The second convergence is identical.

Now

$$K-P_{N}K{P}_N=(I-P_N)K+P_{N}K(I-P_N).$$

Thus, by the triangle inequality,

$$\parallel K-K_N{\parallel }_{{\mathfrak{S}}_2}\le \parallel (I-P_N){K\parallel }_{{\mathfrak{S}}_2}+{\parallel P_{N}K(I-P_N)\parallel }_{{\mathfrak{S}}_2}.$$

The right-hand side converges to zero as $N\to \infty$. Hence

$$\parallel K_N{-K\parallel }_{{\mathfrak{S}}_2}\to 0.$$

□

Definition 6.57 

(regularized Fredholm determinant). Let $A\in {\mathfrak{S}}_2\left(H_{\alpha ,+}\right)$. Define the regularized Fredholm determinant by

$${det}_2(I+A):=det\left((I+A)e^{-A}\right).$$

The standard properties of the regularized Fredholm determinant, trace ideals, and ${det}_2$ follow [4,5]. The right-hand side is defined as an ordinary Fredholm determinant because

$$(I+A)e^{-A}-I\in {\mathfrak{S}}_1.$$

Equivalently, if the eigenvalue sequence of A, counted with algebraic multiplicities, is denoted by $\left\{{\lambda }_j\left(A\right)\right\}$, then

$${det}_2(I+A)=\prod _j(1+{\lambda }_j\left(A\right))e^{-{\lambda }_j\left(A\right)}.$$

This product converges under the ${\mathfrak{S}}_2$-condition.

In particular, for $z\in \mathbb{C}$, write

$$D_K\left(z\right):={det}_2(I+zK).$$

Remark 6.58 

(first trace renormalization). In ${det}_2$, the first trace term is removed by normalization. Indeed, in the range $|z|\parallel K\parallel <1$,

$$log{D}_K\left(z\right)={\displaystyle \sum _{m=2}^{\infty }}{\displaystyle \frac{{(-1)}^{m+1}}{m}}{z}^{m}Tr\left(K^m\right).$$

Therefore

$${\displaystyle \frac{d}{dz}}log{D}_K\left(z\right){|}_{z=0}=0.$$

For this reason, in the comparison with the completed zeta function, the constant term and the linear term must be normalized separately by an exponential factor

$$e^{a+bs}.$$

Lemma 6.59 

(continuity of ${det}_2$ in Hilbert–Schmidt norm). For every $R>0$,

$$\underset{|z|\le R}{sup}\left|{det}_2(I+z{K}_N)-{det}_2(I+zK)\right|⟶0.$$

Namely,

$$D_{K_N}\left(z\right):={det}_2(I+z{K}_N)$$

converges to

$$D_K\left(z\right)={det}_2(I+zK)$$

locally uniformly on compact sets in the z-plane.

Proof. 

For Hilbert–Schmidt operators $A,B$, the regularized determinant satisfies the following Lipschitz-type estimate. There exists a universal constant $C>0$ such that

$$\left|{det}_2(I+A)-{det}_2(I+B)\right|\le {\parallel A-B\parallel }_{{\mathfrak{S}}_2}exp\left(C{\left(1+{\parallel A\parallel }_{{\mathfrak{S}}_2}+{\parallel B\parallel }_{{\mathfrak{S}}_2}\right)}^2\right).$$

Set $A=z{K}_N$ and $B=zK$. By Lemma 6.56,

$$\parallel K_N{-K\parallel }_{{\mathfrak{S}}_2}\to 0.$$

Moreover,

$$\parallel K_N{\parallel }_{{\mathfrak{S}}_2}\le {\parallel K\parallel }_{{\mathfrak{S}}_2}.$$

Therefore, for $|z|\le R$,

$$\parallel z{K}_N{-zK\parallel }_{{\mathfrak{S}}_2}\le R{\parallel K_N-K\parallel }_{{\mathfrak{S}}_2}\to 0,$$

and the exponential factor is uniformly bounded with respect to N and z. Hence

$$\underset{|z|\le R}{sup}|D_{K_N}\left(z\right)-D_K\left(z\right)|\to 0.$$

□

Lemma 6.60 

(trace-power convergence). For each $m\ge 2$,

$$K_N^m⟶K^m\mathrm{in}{\mathfrak{S}}_1.$$

Therefore

$$Tr\left(K_N^m\right)⟶Tr\left(K^m\right).$$

Proof. 

$K_N\to K$ in ${\mathfrak{S}}_2$, and in particular also in operator norm:

$$\parallel K_N-K\parallel \le \parallel K_N{-K\parallel }_{{\mathfrak{S}}_2}\to 0.$$

Moreover, ${sup}_N\parallel K_N\parallel <\infty$.

First, for $m=2$,

$$K_N^2-K^2=(K_N-K)K_N+K(K_N-K).$$

By the Schatten Hölder inequality,

$$\parallel (K_N-K)K_N{\parallel }_{{\mathfrak{S}}_1}\le \parallel K_N{-K\parallel }_{{\mathfrak{S}}_2}{\parallel K_N\parallel }_{{\mathfrak{S}}_2},$$

and

$$\parallel K(K_N-K){\parallel }_{{\mathfrak{S}}_1}\le {\parallel K\parallel }_{{\mathfrak{S}}_2}{\parallel K_N-K\parallel }_{{\mathfrak{S}}_2}.$$

Thus $K_N^2\to K^2$ in ${\mathfrak{S}}_1$.

For $m\ge 3$,

$$K_N^m-K^m={\displaystyle \sum _{\ell =0}^{m-1}}{K}_N^{\ell }(K_N-K)K^{m-1-\ell }.$$

In each term, $K_N-K$ converges in ${\mathfrak{S}}_2$; taking one of the remaining factors as an ${\mathfrak{S}}_2$-factor and estimating the other factors as bounded operators, the product converges to zero in ${\mathfrak{S}}_1$. That is, there exists a constant $C_m>0$ such that

$$\parallel K_N^m-K^m{\parallel }_{{\mathfrak{S}}_1}\le C_m{\parallel K_N-K\parallel }_{{\mathfrak{S}}_2}\to 0.$$

Hence $K_N^m\to K^m$ in ${\mathfrak{S}}_1$, and continuity of the trace gives

$$Tr\left(K_N^m\right)\to Tr\left(K^m\right).$$

□

Lemma 6.61 

(non-vanishing at the central point). One has

$$\xi \left({\displaystyle \frac{1}{2}}\right)\ne 0.$$

Proof. 

For $0<s<1$, the Dirichlet eta function is represented by the convergent alternating series

$${\eta }_{\mathrm{Dir}}\left(s\right)={\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{{(-1)}^{n-1}}{n^s}}.$$

At $s={\displaystyle \frac{1}{2}}$, the terms $n^{-1/2}$ are positive, decrease monotonically to zero, and

$$1-{\displaystyle \frac{1}{\sqrt{2}}}>0.$$

The alternating-series estimate therefore gives

$${\eta }_{\mathrm{Dir}}\left({\displaystyle \frac{1}{2}}\right)>0.$$

Since

$${\eta }_{\mathrm{Dir}}\left(s\right)=(1-2^{1-s})\zeta \left(s\right)$$

on $0<s<1$ by the usual analytic continuation of the eta function, and

$$1-2^{1/2}\ne 0,$$

we obtain

$$\zeta \left({\displaystyle \frac{1}{2}}\right)\ne 0.$$

The remaining factors in

$$\xi \left(s\right)={\displaystyle \frac{1}{2}}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right)$$

are nonzero at $s={\displaystyle \frac{1}{2}}$, and hence

$$\xi \left({\displaystyle \frac{1}{2}}\right)\ne 0.$$

□

Definition 6.62 

(normalized determinant comparison function). Set

$$D_K\left(z\right):={det}_2(I+zK).$$

Define the comparison function by

$$F_K\left(s\right):=e^{a_K+b_K(s-{\displaystyle \frac{1}{2}})}{D}_K\left(i(s-{\displaystyle \frac{1}{2}})\right).$$

The normalization constants $a_K,b_K\in \mathbb{C}$ are fixed by

$$F_K\left({\displaystyle \frac{1}{2}}\right)=\xi \left({\displaystyle \frac{1}{2}}\right),F_K^{\prime }\left({\displaystyle \frac{1}{2}}\right)={\xi }^{\prime }\left({\displaystyle \frac{1}{2}}\right).$$

Namely, since

$$D_K\left(0\right)=1,D_K^{\prime }\left(0\right)=0,$$

take

$$a_K:=log\xi \left({\displaystyle \frac{1}{2}}\right),b_K:={\displaystyle \frac{{\xi }^{\prime }\left(\frac{1}{2}\right)}{\xi \left(\frac{1}{2}\right)}}.$$

Here we use Lemma 6.61. The branch of the logarithm is used only to fix the value at this single point, and the definition of $F_K$ is independent of the branch by

$$e^{a_K}=\xi \left({\displaystyle \frac{1}{2}}\right).$$

Remark 6.63 

(normalization does not encode zero locations). The constants $a_K$ and $b_K$ fix only the value and the first derivative, equivalently the first logarithmic derivative, at the central point. They do not prescribe any zero of $F_K$, and they do not contain any information about the location of the zeros of $\xi$. The zero set of $F_K$ is determined only by the regularized determinant factor

$${det}_2\left(I+i(s-\frac{1}{2})K\right),$$

where K was constructed independently of zero-location information as explained in Remark 6.43.

Lemma 6.64 

(entireness of the normalized determinant comparison function). $F_K\left(s\right)$ is an entire function of $s\in \mathbb{C}$. Moreover, for each N,

$$F_{K_N}\left(s\right):=e^{a_K+b_K(s-{\displaystyle \frac{1}{2}})}{det}_2\left(I+i(s-\frac{1}{2})K_N\right)$$

is also entire, and

$$F_{K_N}\to F_K$$

locally uniformly on compact sets.

Proof. 

The map $z\mapsto {det}_2(I+zK)$ is an entire function. Therefore

$$s\mapsto D_K\left(i(s-{\displaystyle \frac{1}{2}})\right)$$

is also an entire function. The exponential factor

$$e^{a_K+b_K(s-{\displaystyle \frac{1}{2}})}$$

is also entire, and hence $F_K$ is entire.

For the same reason, $F_{K_N}$ is also entire. By Lemma 6.59,

$$\underset{2}{det}(I+z{K}_N)\to \underset{2}{det}(I+zK)$$

locally uniformly on compact sets in z. The map

$$s\mapsto z=i(s-\frac{1}{2})$$

sends compact sets to compact sets, and therefore

$$F_{K_N}\to F_K$$

also locally uniformly on compact sets in s. □

Theorem 6.65 

(coefficient transport for the regularized determinant). In a sufficiently small neighborhood of the origin, take the branch of

$$log{D}_K\left(z\right)$$

satisfying

$$log{D}_K\left(0\right)=0.$$

Then, for each $m\ge 2$,

$${\displaystyle \frac{d^m}{d{z}^m}}log{D}_{K_N}\left(z\right){|}_{z=0}⟶{\displaystyle \frac{d^m}{d{z}^m}}log{D}_K\left(z\right){|}_{z=0}.$$

Moreover,

$${\displaystyle \frac{d^m}{d{z}^m}}log{D}_K\left(z\right){|}_{z=0}={(-1)}^{m+1}(m-1)!Tr\left(K^m\right)(m\ge 2).$$

Likewise,

$${\displaystyle \frac{d}{dz}}log{D}_K\left(z\right){|}_{z=0}=0.$$

Proof. 

Since $D_K\left(0\right)=1$, there exists $r>0$ such that

$$D_K\left(z\right)\ne 0(|z|<r).$$

By Lemma 6.59, $D_{K_N}\to D_K$ uniformly on $\overline{B(0,r)}$. Therefore, for sufficiently large N,

$$D_{K_N}\left(z\right)\ne 0(|z|<r),$$

and

$$log{D}_{K_N}\left(z\right)\to log{D}_K\left(z\right)$$

uniformly on every $\overline{B(0,r^{\prime })}$, $0<r^{\prime }<r$. By Cauchy’s integral formula, the derivatives also converge in each order $m\ge 0$. Namely,

$${\displaystyle \frac{d^m}{d{z}^m}}log{D}_{K_N}\left(z\right){|}_{z=0}\to {\displaystyle \frac{d^m}{d{z}^m}}log{D}_K\left(z\right){|}_{z=0}.$$

On the other hand, for $|z|\parallel K\parallel <1$,

$$log{D}_K\left(z\right)={\displaystyle \sum _{m=2}^{\infty }}{\displaystyle \frac{{(-1)}^{m+1}}{m}}{z}^{m}Tr\left(K^m\right).$$

Hence, for $m\ge 2$,

$${\displaystyle \frac{d^m}{d{z}^m}}log{D}_K\left(z\right){|}_{z=0}=m!{\displaystyle \frac{{(-1)}^{m+1}}{m}}Tr\left(K^m\right)={(-1)}^{m+1}(m-1)!Tr\left(K^m\right).$$

Moreover, since there is no linear term,

$${\displaystyle \frac{d}{dz}}log{D}_K\left(z\right){|}_{z=0}=0.$$

□

Corollary 6.66 

(coefficient transport for $F_K$). Set $w=s-{\displaystyle \frac{1}{2}}$. In a neighborhood of the origin, take the branch of

$$log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)$$

satisfying

$$log{F}_K\left({\displaystyle \frac{1}{2}}\right)=log\xi \left({\displaystyle \frac{1}{2}}\right).$$

Then

$$log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)=a_K+b_{K}w+log{D}_K\left(iw\right),$$

and for $m\ge 2$,

$${\displaystyle \frac{d^m}{d{w}^m}}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right){|}_{w=0}=i^m{(-1)}^{m+1}(m-1)!Tr\left(K^m\right).$$

Furthermore, the Taylor coefficients of

$$log{F}_{K_N}\left({\displaystyle \frac{1}{2}}+w\right)⟶log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)$$

converge in each order.

Proof. 

By Definition 6.62,

$$F_K\left({\displaystyle \frac{1}{2}}+w\right)=e^{a_K+b_{K}w}{D}_K\left(iw\right).$$

Therefore

$$log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)=a_K+b_{K}w+log{D}_K\left(iw\right).$$

Applying Theorem 6.65 with $z=iw$, for $m\ge 2$ one obtains

$${\displaystyle \frac{d^m}{d{w}^m}}log{D}_K\left(iw\right){|}_{w=0}=i^m{\displaystyle \frac{d^m}{d{z}^m}}log{D}_K\left(z\right){|}_{z=0}=i^m{(-1)}^{m+1}(m-1)!Tr\left(K^m\right).$$

The linear coefficient is fixed by $b_K$, and the constant term is fixed by $a_K$. The coefficient convergence from finite-rank cutoffs also follows from Theorem 6.65. □

Proposition 6.67 

(compatibility with the distributional comparison coefficients). The open-band coefficient comparison obtained from the distributional comparison theorem of Section 6.1 transports continuously to the trace-ideal coefficients

$$Tr\left(K^m\right)(m\ge 2)$$

constructed in this section. Namely, setting the coefficient sequence obtained by finite-rank cutoffs as

$$c_m\left(K_N\right):={\displaystyle \frac{{(-1)}^{m+1}}{m}}Tr\left(K_N^m\right)(m\ge 2),$$

one has

$$c_m\left(K_N\right)\to c_m\left(K\right):={\displaystyle \frac{{(-1)}^{m+1}}{m}}Tr\left(K^m\right),$$

and this limiting coefficient is compatible with the calibrated coefficient comparison on the open band obtained from

$$\langle {\mu }_L,\phi \rangle =\langle {\mu }_{\xi },\phi \rangle (\phi \in {\mathcal{A}}_{\eta },0<\eta <log2)$$

in Section 6.1.

Proof. 

The convergence $c_m\left(K_N\right)\to c_m\left(K\right)$ follows immediately from Lemma 6.60. In Section 6.1, it was shown that ${\mu }_L$ and ${\mu }_{\xi }$ agree as continuous linear functionals on ${\mathcal{A}}_{\eta }$. On the other hand, the ${\mathcal{K}}_R$-component of ${\mu }_L$ is represented by the kernel of K in Section 6.0, and in the finite-rank approximation it is represented as finite trace coefficients by $K_N$. Therefore the open-band comparison quantity represented by finite-rank coefficients transports to

$$c_m\left(K\right)$$

as

$$N\to \infty .$$

This transport is due to ${\mathfrak{S}}_2$-convergence and trace-power convergence, and does not use any additional endpoint convention or pointwise boundary value. □

Theorem 6.68 

(trace-ideal determinant theorem). The boundary-distribution comparison kernel candidate K of Section 6.0 is realized as

$$K=K^{*},K\in {\mathfrak{S}}_2\left(H_{\alpha ,+}\right).$$

Moreover, the finite-rank cutoffs

$$K_N=P_{N}K{P}_N$$

satisfy

$$\parallel K_N{-K\parallel }_{{\mathfrak{S}}_2}\to 0,$$

and

$${det}_2(I+z{K}_N)⟶{det}_2(I+zK)$$

locally uniformly on compact sets in the z-plane.

The normalized comparison function

$$F_K\left(s\right)=e^{a_K+b_K(s-{\displaystyle \frac{1}{2}})}{det}_2\left(I+i(s-\frac{1}{2})K\right)$$

is an entire function and is normalized so as to satisfy

$$F_K\left({\displaystyle \frac{1}{2}}\right)=\xi \left({\displaystyle \frac{1}{2}}\right),F_K^{\prime }\left({\displaystyle \frac{1}{2}}\right)={\xi }^{\prime }\left({\displaystyle \frac{1}{2}}\right).$$

Moreover, its Taylor coefficients are transported degree by degree from the finite-rank cutoffs.

Proof. 

The fact that $K\in {\mathfrak{S}}_2$ was shown in Proposition 6.52. The self-adjointness $K=K^{*}$ follows from Theorem 6.54. The Hilbert–Schmidt convergence of the finite-rank cutoffs follows from Lemma 6.56. The local uniform convergence of ${det}_2$ follows from Lemma 6.59. The definition and normalization of $F_K$ are given by Definition 6.62. The entireness of $F_K$ follows from Lemma 6.64. Finally, the degree-by-degree transport of Taylor coefficients was shown in Theorem 6.65 and Corollary 6.66. □

Remark 6.69 

(no global identification in this section). In this section, $F_K$ was constructed by the regularized determinant, and it was proved that its coefficients can be transported from finite-rank approximations. However, this section does not yet conclude

$$F_K\left(s\right)=\xi \left(s\right).$$

This global agreement is proved in the global uniqueness theorem of Section 6.4.

6.4. Global Uniqueness Theorem

In this section, we globally identify the regularized-determinant comparison function constructed in Section 6.3,

$$F_K\left(s\right)=e^{a_K+b_K(s-{\displaystyle \frac{1}{2}})}{det}_2\left(I+i(s-\frac{1}{2})K\right),$$

with the completed zeta function $\xi \left(s\right)$. The uniqueness principle used in this section is only the identity theorem of complex analysis. That is, we use only the fact that if two entire functions agree on a nonempty open set, then they agree on the whole plane. Carlson-type theorems, the Phragmén–Lindelöf principle, or other growth-type uniqueness theorems are not used in the identity proof of this section.

Lemma 6.70 

(common holomorphic domain). Both $F_K$ and $\xi$ are entire functions on $\mathbb{C}$.

Proof. 

That $F_K$ is entire was shown in Lemma 6.64. On the other hand,

$$\xi \left(s\right):={\displaystyle \frac{1}{2}}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right)$$

is the completed zeta function; the simple pole of $\zeta \left(s\right)$ at $s=1$ is removed by the factor $s-1$, and the poles of $\Gamma (s/2)$ at the negative even integers are cancelled by the trivial zeros of $\zeta \left(s\right)$. Therefore $\xi \left(s\right)$ is an entire function. □

Lemma 6.71 

(growth of the determinant comparison function). For every $s\in \mathbb{C}$, there exists a constant $C_K>0$ such that

$$|F_K\left(s\right)|\le C_{K}exp\left(|b_K||s-\frac{1}{2}|+{\displaystyle \frac{1}{2}}{\parallel K\parallel }_{{\mathfrak{S}}_2}^2{|s-\frac{1}{2}|}^2\right).$$

Consequently, $F_K$ is an entire function of order at most 2.

Proof. 

For a Hilbert–Schmidt operator $A\in {\mathfrak{S}}_2$, the regularized determinant satisfies

$$|{det}_2(I+A)|\le exp\left({\displaystyle \frac{1}{2}}{\parallel A\parallel }_{{\mathfrak{S}}_2}^2\right).$$

Taking $A=i(s-\frac{1}{2})K$, one has

$${\parallel A\parallel }_{{\mathfrak{S}}_2}=|s-\frac{1}{2}{|\parallel K\parallel }_{{\mathfrak{S}}_2}.$$

Therefore

$$\left|{det}_2\left(I+i(s-\frac{1}{2})K\right)\right|\le exp\left({\displaystyle \frac{1}{2}}|s-\frac{1}{2}{|}^2{\parallel K\parallel }_{{\mathfrak{S}}_2}^2\right).$$

Furthermore,

$$\left|e^{a_K+b_K(s-{\displaystyle \frac{1}{2}})}\right|\le e^{Re{a}_K}exp\left(|b_K||s-\frac{1}{2}|\right).$$

Thus, taking $C_K=e^{Re{a}_K}$, the asserted estimate follows. This estimate gives

$$loglog|F_K\left(s\right)|=O(log|s|)$$

in the form of order at most 2. □

Lemma 6.72 

(growth of the completed zeta function). The completed zeta function $\xi \left(s\right)$ is an entire function of order 1. In particular, for every $\epsilon >0$, there exist constants $C_{\epsilon },A_{\epsilon }>0$ such that

$$|\xi \left(s\right)|\le C_{\epsilon }exp\left(A_{\epsilon }{|s|}^{1+\epsilon }\right)(s\in \mathbb{C})$$

holds.

Proof. 

Use the representation

$$\xi \left(s\right)={\displaystyle \frac{1}{2}}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right).$$

The growth of $\Gamma (s/2)$ in vertical strips is controlled exponentially by Stirling’s formula. Moreover, $\zeta \left(s\right)$, as a meromorphic function, has growth of order at most 1, and its only pole at $s=1$ is removed by the factor $s-1$. By the functional equation

$$\xi \left(s\right)=\xi (1-s),$$

the estimates in the left and right half-planes are transferred to each other. Therefore $\xi$ is an entire function of order 1, and the stated $(1+\epsilon )$-type estimate follows. □

Corollary 6.73 

(growth of the difference).

$$H_K\left(s\right):=F_K\left(s\right)-\xi \left(s\right)$$

is an entire function of order at most 2.

Proof. 

By Lemma 6.70, $H_K$ is entire. By Lemma 6.71, $F_K$ has order at most 2, and by Lemma 6.72, $\xi$ has order 1. Therefore the difference $H_K$ has order at most 2. □

Definition 6.74 

(logarithmic germs at the central point). Set $w=s-{\displaystyle \frac{1}{2}}$. By the normalization of Section 6.3 and Lemma 6.61,

$$F_K\left({\displaystyle \frac{1}{2}}\right)=\xi \left({\displaystyle \frac{1}{2}}\right)\ne 0.$$

Hence there exists $r_0>0$ such that

$$F_K\left({\displaystyle \frac{1}{2}}+w\right)\ne 0,\xi \left({\displaystyle \frac{1}{2}}+w\right)\ne 0(|w|<r_0).$$

On this disk, define

$${\mathcal{L}}_K\left(w\right):=log{F}_K\left({\displaystyle \frac{1}{2}}+w\right),{\mathcal{L}}_{\xi }\left(w\right):=log\xi \left({\displaystyle \frac{1}{2}}+w\right)$$

by the branches satisfying

$${\mathcal{L}}_K\left(0\right)={\mathcal{L}}_{\xi }\left(0\right)=log\xi \left({\displaystyle \frac{1}{2}}\right).$$

For each $m\ge 0$, write the central logarithmic coefficients as

$$c_m\left(F_K\right):={\displaystyle \frac{1}{m!}}{\displaystyle \frac{d^m}{d{w}^m}}{\mathcal{L}}_K\left(w\right){|}_{w=0},c_m\left(\xi \right):={\displaystyle \frac{1}{m!}}{\displaystyle \frac{d^m}{d{w}^m}}{\mathcal{L}}_{\xi }\left(w\right){|}_{w=0}.$$

Definition 6.75 

(central Cauchy–Laplace kernel). Take $r_0>0$ in Definition 6.74 smaller if necessary. For $|w|<r_0$, define

$$h_w\left(u\right):={\displaystyle \frac{e^{wu}-1}{u}},h_w\left(0\right):=w.$$

The value at $u=0$ fills the removable singularity, and the resulting kernel is holomorphic in w. Let

$${\mathcal{E}}_{\mathrm{cen}}$$

denote the vector space spanned by finite linear combinations of the kernels $h_w$ and their w-derivatives.

Definition 6.76 

(raw finite-window central kernels). Fix an even function $\chi \in C_c^{\infty }\left(\mathbb{R}\right)$ such that

$$\chi \left(u\right)=1(|u|\le 1),\chi \left(u\right)=0(|u|\ge 2),0\le \chi \le 1,$$

and set ${\chi }_M\left(u\right):=\chi (u/M)$ for $M\ge 1$. For $|w|<r_0$, define the raw finite-window kernel

$$h_{w,M}^{\mathrm{fw}}\left(u\right):={\chi }_M\left(u\right)h_w\left(u\right),$$

and let

$${\tilde{\Psi }}_{w,M}^{\mathrm{fw}}:={\left[h_{w,M}^{\mathrm{fw}}\right]}_{\mathrm{fw}}$$

denote the corresponding finite-window test input before the central finite-part subtraction. The cutoff function $\chi$ is fixed once and for all throughout the central comparison argument.

Definition 6.77 

(central finite-jet map). The local singular orders in the completed finite-window explicit formula determine two non-negative integers

$$d_0,d_{\partial }.$$

These integers are fixed once and for all. They do not depend on the cutoff scale M, the central parameter w, or the values of the pairings with ${\mu }_L$ and ${\mu }_{\xi }$. For a finite-window kernel $\varphi$, define its central jet at $u=0$ by

$$J_{d_0}^0\left(\varphi \right):={\left({\partial }_u^a\varphi \left(0\right)\right)}_{0\le a\le d_0}.$$

For each $M\ge 1$, define the endpoint jet on the cutoff transition annulus by

$$J_{M,d_{\partial }}^{\partial }\left(\varphi \right):={\left({\mathcal{E}}_{M,b}^{\partial }\left(\varphi \right)\right)}_{0\le b\le d_{\partial }},$$

where each endpoint functional ${\mathcal{E}}_{M,b}^{\partial }$ is supported in

$$M\le |u|\le 2M$$

and is obtained from derivatives of $\varphi$ of order at most b, after the fixed rescaling ${\chi }_M\left(u\right)=\chi (u/M)$. The central finite-jet map is

$$J_M^{\mathrm{cen}}:{\mathcal{C}}_{\mathrm{cen}}^0⟶{\mathcal{J}}_M^{\mathrm{cen}}:=J_{d_0}^0\oplus J_{M,d_{\partial }}^{\partial },$$

given on representatives by

$$J_M^{\mathrm{cen}}\left(\varphi \right):=\left(J_{d_0}^0\left(\varphi \right),J_{M,d_{\partial }}^{\partial }\left(\varphi \right)\right).$$

For the central Cauchy–Laplace kernel one has

$$h_w\left(u\right)=w+{\displaystyle \frac{w^2}{2}}u+{\displaystyle \frac{w^3}{3!}}{u}^2+\dots ,$$

so the central jet $J_{d_0}^0\left(h_w\right)$ is determined by finitely many powers of w.

Definition 6.78 

(principal-part space and universal principal-part map). For every $M\ge 1$, let

$${\mathcal{P}}_M^{\mathrm{cen}}$$

be the finite-dimensional vector space spanned by fixed local-principal-part basis elements

$$p_{M,a}^0,p_{M,b}^{\partial }(0\le a\le d_0,0\le b\le d_{\partial }).$$

The basis elements may depend on M, but only through the fixed rescaling

$${\chi }_M\left(u\right)=\chi (u/M)$$

and the associated finite-window localization. They are fixed before either central pairing is evaluated. There is a canonical local-principal-part embedding

$${\iota }_M^{\mathrm{cen}}:{\mathcal{P}}_M^{\mathrm{cen}}\hookrightarrow {\mathcal{C}}_{\mathrm{cen}}^0$$

which regards a local principal part as the corresponding finite-window reference test input.

Define the universal principal-part map

$$P_M^{\mathrm{cen}}:{\mathcal{J}}_M^{\mathrm{cen}}⟶{\mathcal{P}}_M^{\mathrm{cen}}$$

by

$$P_M^{\mathrm{cen}}\left({\left({\alpha }_a\right)}_{a=0}^{d_0},{\left({\beta }_b\right)}_{b=0}^{d_{\partial }}\right):={\displaystyle \sum _{a=0}^{d_0}}{\alpha }_a{p}_{M,a}^0+{\displaystyle \sum _{b=0}^{d_{\partial }}}{\beta }_b{p}_{M,b}^{\partial }.$$

Thus the central counterterm associated with $h_w$ and the window M is

$${CT}_M^{\mathrm{cen}}\left(h_w\right):=P_M^{\mathrm{cen}}{J}_M^{\mathrm{cen}}\left(h_{w,M}^{\mathrm{fw}}\right)\in {\mathcal{P}}_M^{\mathrm{cen}}.$$

When the counterterm is subtracted from a finite-window test input, it is always understood through the embedded element

$${\iota }_M^{\mathrm{cen}}{CT}_M^{\mathrm{cen}}\left(h_w\right)\in {\mathcal{C}}_{\mathrm{cen}}^0.$$

Definition 6.79 

(finite-window central cutoff inputs). For $M\ge 1$ and $|w|<r_0$, the regularized finite-window central test input is

$${\Psi }_{w,M}^{\mathrm{fw}}:={\tilde{\Psi }}_{w,M}^{\mathrm{fw}}-{\iota }_M^{\mathrm{cen}}{P}_M^{\mathrm{cen}}{J}_M^{\mathrm{cen}}\left(h_{w,M}^{\mathrm{fw}}\right)\in {\mathcal{C}}_{\mathrm{cen}}^0.$$

Equivalently,

$${\Psi }_{w,M}^{\mathrm{fw}}={\left[h_{w,M}^{\mathrm{fw}}\right]}_{\mathrm{fw}}-{\iota }_M^{\mathrm{cen}}{CT}_M^{\mathrm{cen}}\left(h_w\right).$$

The subtraction is an algebraic subtraction in the pre-completion finite-window test-input space ${\mathcal{C}}_{\mathrm{cen}}^0$. It is not performed after applying either central pairing.

The data

$${\chi }_M,h_w,J_M^{\mathrm{cen}},P_M^{\mathrm{cen}},{\iota }_M^{\mathrm{cen}}$$

fix ${\Psi }_{w,M}^{\mathrm{fw}}$ before the values of

$$\langle {\mu }_L,{\Psi }_{w,M}^{\mathrm{fw}}\rangle ,\langle {\mu }_{\xi },{\Psi }_{w,M}^{\mathrm{fw}}\rangle$$

are evaluated. Hence the regularized finite-window input does not encode the equality of the two pairings.

Lemma 6.80 

(common local principal part). For every $M\ge 1$, the local singular contributions in the finite-window completed explicit formula factor through the same central finite-jet map

$$J_M^{\mathrm{cen}}:{\mathcal{C}}_{\mathrm{cen}}^0\to {\mathcal{J}}_M^{\mathrm{cen}}.$$

More precisely, the Archimedean term, the coefficient-space arithmetic trace term, and the singular-boundary term have local principal-part projections

$${PP}_{M,\infty }^{\mathrm{cen}},{PP}_{M,\mathrm{arith}}^{\mathrm{cen}},{PP}_{M,R}^{\mathrm{cen}},$$

and, on the common local principal-part component, all three projections have the same factorization

$${PP}_{M,•}^{\mathrm{cen}}={\iota }_M^{\mathrm{cen}}{P}_M^{\mathrm{cen}}{J}_M^{\mathrm{cen}},•\in \{\infty ,\mathrm{arith},R\}.$$

Consequently, for the central kernel,

$${CT}_M^{\mathrm{cen}}\left(h_w\right)=P_M^{\mathrm{cen}}{J}_M^{\mathrm{cen}}\left(h_{w,M}^{\mathrm{fw}}\right)$$

is precisely the common local principal part removed from all three contributions before either central pairing is evaluated.

Equivalently,

$${\iota }_M^{\mathrm{cen}}\left({CT}_M^{\mathrm{cen}}\left(h_w\right)\right)={\displaystyle \sum _{a=0}^{d_0}}{\partial }_u^a{h}_{w,M}^{\mathrm{fw}}\left(0\right){\iota }_M^{\mathrm{cen}}\left(p_{M,a}^0\right)+{\displaystyle \sum _{b=0}^{d_{\partial }}}{\mathcal{E}}_{M,b}^{\partial }\left(h_{w,M}^{\mathrm{fw}}\right){\iota }_M^{\mathrm{cen}}\left(p_{M,b}^{\partial }\right)$$

inside ${\mathcal{C}}_{\mathrm{cen}}^0$. Since ${\chi }_M=1$ near $u=0$, the first scalar factors agree with the central derivatives ${\partial }_u^a{h}_w\left(0\right)$.

The jet orders $d_0,d_{\partial }$ are uniform in M. The space ${\mathcal{P}}_M^{\mathrm{cen}}$, the embedding ${\iota }_M^{\mathrm{cen}}$, and the basis elements $p_{M,a}^0,p_{M,b}^{\partial }$ may depend on M, but only through the fixed rescaling ${\chi }_M\left(u\right)=\chi (u/M)$ and the corresponding finite-window localization. No part of this M-dependence is chosen after either central pairing has been evaluated.

Proof. 

The singular terms in the finite-window completed explicit formula are local. At the central point $u=0$, their singular part is determined by a finite Taylor jet of order $d_0$. On the cutoff transition region $M\le |u|\le 2M$, the only additional singular data come from finitely many derivatives of the fixed rescaled cutoff ${\chi }_M$, equivalently from the endpoint jet of order $d_{\partial }$. The integers $d_0,d_{\partial }$ are fixed by the local singular orders of the completed explicit formula and by the cutoff scheme; they are not adjusted as M, w, ${\mu }_L$, or ${\mu }_{\xi }$ varies.

The local principal data for the three contributions are summarized by the following table:

$$\begin{array}{cccc}\mathrm{contribution}& \mathrm{source}& \mathrm{local}\mathrm{data}& \mathrm{projection}\\ \mathrm{Archimedean}& \mathrm{completed}\mathrm{gamma}\mathrm{factor}& J_M^{\mathrm{cen}}& {PP}_{M,\infty }^{\mathrm{cen}}\\ \mathrm{arithmetic}\mathrm{trace}& \mathrm{coefficient}-\mathrm{space}\mathrm{weighted}\mathrm{trace}& J_M^{\mathrm{cen}}& {PP}_{M,\mathrm{arith}}^{\mathrm{cen}}\\ \sin \mathrm{gular}\mathrm{boundary}& {\mathcal{K}}_R-\mathrm{component}\mathrm{boundary}\mathrm{distribution}& J_M^{\mathrm{cen}}& {PP}_{M,R}^{\mathrm{cen}}\end{array}$$

Although the three contributions have different global origins, their finite-window singular parts are obtained from the same local normal form of the completed explicit formula. This local normal form depends only on the central jet and the endpoint jet. In particular, the concrete principal-part projection for each contribution is obtained by applying the same finite-jet map $J_M^{\mathrm{cen}}$, the same universal map $P_M^{\mathrm{cen}}$, and then the same embedding ${\iota }_M^{\mathrm{cen}}$; the labels $\infty ,\mathrm{arith},R$ record only the three global origins of the identical local subtraction. Hence the three principal-part projections have the same restriction to the common principal-part component, namely

$${\iota }_M^{\mathrm{cen}}{P}_M^{\mathrm{cen}}{J}_M^{\mathrm{cen}}.$$

Thus the counterterm is the finite-jet projection $P_M^{\mathrm{cen}}{J}_M^{\mathrm{cen}}\left(h_{w,M}^{\mathrm{fw}}\right)$. The finite-dimensional space ${\mathcal{P}}_M^{\mathrm{cen}}$, its basis, and the embedding into ${\mathcal{C}}_{\mathrm{cen}}^0$ are fixed before the pairings are applied. Therefore the counterterm is determined entirely by ${\chi }_M$, $h_w$, and the local principal-part maps, and not by the numerical values of

$$\langle {\mu }_L,{\Psi }_{w,M}^{\mathrm{fw}}\rangle ,\langle {\mu }_{\xi },{\Psi }_{w,M}^{\mathrm{fw}}\rangle .$$

□

Lemma 6.81 

(seminorm control of the local principal part). Let $B⋐\{|w|<r_0\}$. For every defining seminorm

$$p_{B,N,m,\sigma }$$

of ${\mathcal{C}}_{\mathrm{cen}}$, there exist finitely many defining seminorms

$$p_{B,N_{\nu },m_{\nu },{\sigma }_{\nu }}$$

and a constant $C_B>0$, independent of M, such that, for every finite-window family $\Phi$ used in the central comparison,

$$p_{B,N,m,\sigma }\left({\iota }_M^{\mathrm{cen}}{P}_M^{\mathrm{cen}}{J}_M^{\mathrm{cen}}(\Phi )\right)\le C_B\sum _{\nu }{p}_{B,N_{\nu },m_{\nu },{\sigma }_{\nu }}(\Phi ).$$

In particular, the local principal-part subtraction is continuous with respect to the central comparison seminorms on the finite-window families appearing in the proof.

Proof. 

The map $J_M^{\mathrm{cen}}$ consists of finitely many evaluations of u-derivatives at $u=0$ and finitely many endpoint functionals on $M\le |u|\le 2M$. Each such functional is controlled by finitely many of the seminorms defining the central comparison topology, after increasing $N,m,\sigma$ if necessary. The map $P_M^{\mathrm{cen}}$ is finite rank, and the embedding ${\iota }_M^{\mathrm{cen}}$ inserts the resulting finite local principal part into the fixed finite-window test-input model. Since the orders $d_0,d_{\partial }$ are uniform in M, only finitely many seminorm types are needed. The dependence on M is only through the fixed rescaling ${\chi }_M\left(u\right)=\chi (u/M)$, which is already controlled by the finite-window seminorms. This gives the stated estimate. □

Remark 6.82 

(the counterterm does not encode the comparison equality). The counterterm

$${CT}_M^{\mathrm{cen}}\left(h_w\right)$$

is fixed before the two central pairings are evaluated. It depends only on the fixed cutoff ${\chi }_M$, the central kernel $h_w$, the finite-jet map $J_M^{\mathrm{cen}}$, and the universal principal-part map $P_M^{\mathrm{cen}}$. It does not depend on the values of

$$\langle {\mu }_L,{\Psi }_{w,M}^{\mathrm{fw}}\rangle ,\langle {\mu }_{\xi },{\Psi }_{w,M}^{\mathrm{fw}}\rangle ,$$

and therefore it does not encode the central residual-free equality.

Definition 6.83 

(central comparison topology). The central comparison topology is the locally convex topology generated as follows. For every compact set $B⋐\{|w|<r_0\}$, every integer $m\ge 0$, every integer $N\ge 0$, and every

$$\sigma >\underset{w\in B}{sup}|Rew|,$$

define, on kernel representatives $\Phi ={\left\{{\varphi }_w\left(u\right)\right\}}_{w\in B}$,

$$p_{B,N,m,\sigma }(\Phi ):=\underset{0\le j\le m}{max}\underset{0\le \ell \le N}{max}\underset{\begin{array}{c}w\in B\\ u\in \mathbb{R}\end{array}}{sup}{e}^{-\sigma |u|}{(1+|u|)}^N\left|{\partial }_w^j{\partial }_u^{\ell }{\varphi }_w\left(u\right)\right|.$$

Let ${\mathcal{C}}_{\mathrm{cen}}^0$ be the vector space spanned by the finite-window central test inputs

$${\Psi }_{w,M}^{\mathrm{fw}}$$

and their finite w-derivatives. For each M, the finite-dimensional counterterm space

$${\mathcal{P}}_M^{\mathrm{cen}}$$

is regarded as a subspace of ${\mathcal{C}}_{\mathrm{cen}}^0$ via the canonical local-principal-part embedding

$${\iota }_M^{\mathrm{cen}}:{\mathcal{P}}_M^{\mathrm{cen}}\hookrightarrow {\mathcal{C}}_{\mathrm{cen}}^0.$$

Thus the expression

$${\left[h_{w,M}^{\mathrm{fw}}\right]}_{\mathrm{fw}}-{CT}_M^{\mathrm{cen}}\left(h_w\right)$$

is formed in the algebraic space ${\mathcal{C}}_{\mathrm{cen}}^0$, not after applying either central pairing. The space ${\mathcal{C}}_{\mathrm{cen}}$ is the locally convex completion of ${\mathcal{C}}_{\mathrm{cen}}^0$, modulo zero seminorms, with respect to the seminorms above. Central finite-part limits such as ${\Psi }_w^{\mathrm{cen}}$ are not inserted as additional generators of the topology. They denote elements of the completion only after the corresponding finite-window family has been proved Cauchy in these seminorms; for the family used here this is exactly Lemma 6.88.

The seminorms defining ${\mathcal{C}}_{\mathrm{cen}}$ are fixed before the functionals

$${\mu }_L,{\mu }_{\xi }$$

are applied. In particular, the topology depends only on the finite-window cutoff structure, the central kernel family, and its w- and u-derivatives; it does not depend on the values of

$$\langle {\mu }_L,\Phi \rangle \mathrm{or}\langle {\mu }_{\xi },\Phi \rangle .$$

Definition 6.84 

(Hadamard finite-part central regularization). The notation

$${\mathcal{R}}_{\mathrm{cen}}\varphi$$

denotes the Hadamard finite-part limit in ${\mathcal{C}}_{\mathrm{cen}}$ of the finite-window family obtained from ${\chi }_M\varphi$ after subtracting the common central counterterm in the Archimedean, arithmetic, and singular-boundary contributions. More precisely, ${\mathcal{R}}_{\mathrm{cen}}\varphi$ is defined for those $\varphi \in {\mathcal{E}}_{\mathrm{cen}}$ for which the corresponding regularized finite-window family is Cauchy in the topology of Definition 6.83; in that case ${\mathcal{R}}_{\mathrm{cen}}\varphi$ denotes its unique limit in the completion ${\mathcal{C}}_{\mathrm{cen}}$.

For the central kernel $h_w$, the corresponding finite-window family is precisely

$${\Psi }_{w,M}^{\mathrm{fw}}$$

of Definition 6.79. The existence of

$${\mathcal{R}}_{\mathrm{cen}}{h}_w$$

as an element of ${\mathcal{C}}_{\mathrm{cen}}$ is not asserted by the definition alone; it is proved by the finite-window approximation lemma below. Thus the present definition fixes the cutoff procedure, the common counterterm, and the ambient topology, while the existence of the relevant finite-part limits is supplied by a separate convergence statement.

Remark 6.85 

(no conclusion is encoded in the central regularization). The central regularization does not define $F_K\equiv \xi$ by fiat. The cutoff ${\chi }_M$, the kernel $h_w$, the counterterm ${CT}_M^{\mathrm{cen}}\left(h_w\right)$, and the topology of ${\mathcal{C}}_{\mathrm{cen}}$ are fixed without using the numerical values of the pairings with ${\mu }_L$ or ${\mu }_{\xi }$. The identities connecting these pairings to the logarithmic derivatives of $F_K$ and $\xi$ are proved later, separately on the singular-boundary side and on the zeta side, in Lemma 6.91 and Lemma 6.89.

Definition 6.86 

(central Cauchy–Laplace test family). For $|w|<r_0$, the central Cauchy–Laplace test input is denoted by

$${\Psi }_w^{\mathrm{cen}}:={\mathcal{R}}_{\mathrm{cen}}{h}_w={\mathcal{R}}_{\mathrm{cen}}\left(u⟼{\displaystyle \frac{e^{wu}-1}{u}}\right).$$

This notation means the unique element of the completion ${\mathcal{C}}_{\mathrm{cen}}$ obtained as the limit of the finite-window family

$${\Psi }_{w,M}^{\mathrm{fw}}$$

once the convergence is established in Lemma 6.88. In particular, ${\Psi }_w^{\mathrm{cen}}$ is not an additional generator of the topology of ${\mathcal{C}}_{\mathrm{cen}}$. By normalization,

$${\Psi }_0^{\mathrm{cen}}=0.$$

The space ${\mathcal{C}}_{\mathrm{cen}}$ is a test space distinct from the open-band class ${\mathcal{A}}_{\eta }$, and is used to represent the central logarithmic derivative comparison.

Lemma 6.87 

(regularity of the central Cauchy–Laplace family). There exists $0<r\le r_0$ such that the map

$$w⟼{\Psi }_w^{\mathrm{cen}}\in {\mathcal{C}}_{\mathrm{cen}}$$

is holomorphic for $|w|<r$.

Proof. 

For each M, the finite-window representative

$$w⟼{\Psi }_{w,M}^{\mathrm{fw}}$$

is holomorphic as a ${\mathcal{C}}_{\mathrm{cen}}^0$-valued map: it is obtained from the holomorphic kernel $h_w$, the fixed cutoff ${\chi }_M$, and finitely many w-holomorphic jet counterterms. The seminorms of Definition 6.83 control finitely many w- and u-derivatives uniformly on compact subsets of $|w|<r_0$. By Lemma 6.88, after possibly decreasing the radius to $0<r\le r_0$, these finite-window holomorphic maps converge to $w\mapsto {\Psi }_w^{\mathrm{cen}}$ locally uniformly in the ${\mathcal{C}}_{\mathrm{cen}}$-seminorms, and the same is true after the finitely many w-derivatives appearing in those seminorms. The standard Weierstrass theorem for locally convex-valued holomorphic maps therefore gives that

$$w⟼{\Psi }_w^{\mathrm{cen}}$$

is holomorphic for $|w|<r$. □

Lemma 6.88 

(finite-window approximation of the central kernel). For every compact set

$$B⋐\{|w|<r_0\},$$

the finite-window central test inputs of Definition 6.79 converge to the central Cauchy–Laplace test family in the central comparison topology:

$${\Psi }_{w,M}^{\mathrm{fw}}⟶{\Psi }_w^{\mathrm{cen}}(M\to \infty )$$

locally uniformly for $w\in B$. Equivalently, for every $N,m\ge 0$ and every

$$\sigma >\underset{w\in B}{sup}|Rew|,$$

one has

$$p_{B,N,m,\sigma }\left({\Psi }_{·,M}^{\mathrm{fw}}-{\Psi }_{·}^{\mathrm{cen}}\right)⟶0.$$

In particular, the same convergence holds after applying any finite number of w-derivatives covered by the seminorms $p_{B,N,m,\sigma }$. The assertion is purely an approximation statement in ${\mathcal{C}}_{\mathrm{cen}}$; it does not use the values of the pairings with ${\mu }_L$ or ${\mu }_{\xi }$.

Proof. 

Fix $B⋐\{|w|<r_0\}$, $N,m\ge 0$, and

$$\sigma >\underset{w\in B}{sup}|Rew|.$$

By Definition 6.83, convergence in ${\mathcal{C}}_{\mathrm{cen}}$ is measured by the seminorms $p_{B,N,m,\sigma }$. The representatives of ${\Psi }_{w,M}^{\mathrm{fw}}$ are obtained from the cutoff kernels

$$h_{w,M}^{\mathrm{fw}}\left(u\right)={\chi }_M\left(u\right)h_w\left(u\right)$$

after subtracting the common central counterterm of Definition 6.79. The notation ${\Psi }_w^{\mathrm{cen}}={\mathcal{R}}_{\mathrm{cen}}{h}_w$ denotes the element of the completion represented by the Cauchy limit of this finite-window family; the present lemma proves that this limit exists in the seminorm topology of Definition 6.83.

The difference between the cutoff kernel and the limiting kernel is supported, apart from the finite-part subtraction, in the transition and tail regions of the cutoff. For $w\in B$, all w-derivatives and the finitely many u-derivatives appearing in $p_{B,N,m,\sigma }$ are bounded by an exponential of type strictly smaller than the weight $e^{\sigma |u|}$, up to a polynomial factor. Hence the cutoff-tail contribution tends to zero in every seminorm $p_{B,N,m,\sigma }$. For the counterterm part, Lemma 6.80 identifies the subtracted term with the same fixed finite-jet principal part at every finite window, and Lemma 6.81 shows that this finite-jet subtraction is controlled by the defining seminorms of ${\mathcal{C}}_{\mathrm{cen}}$. The difference between the finite-window counterterm and its completion-limit representative is therefore measured by the same jet seminorms and tends to zero as the cutoff annulus leaves every compact u-set. Therefore

$$p_{B,N,m,\sigma }\left({\Psi }_{·,M}^{\mathrm{fw}}-{\Psi }_{·}^{\mathrm{cen}}\right)\to 0.$$

The estimate is uniform for $w\in B$, and the seminorms already include all w-derivatives up to order m. Thus the convergence is locally uniform in w and remains valid after finitely many w-derivatives.

Only the cutoff family, the central kernel, the counterterms, and the seminorms of ${\mathcal{C}}_{\mathrm{cen}}$ have been used. No value of

$$\langle {\mu }_L,·\rangle \mathrm{or}\langle {\mu }_{\xi },·\rangle$$

enters the argument. □

Lemma 6.89 

(Hadamard central partial fraction formula for $\xi$). There exists $0<r\le r_0$ such that, for $|w|<r$,

$${\partial }_{w}log\xi \left({\displaystyle \frac{1}{2}}+w\right)-{\partial }_{w}log\xi \left({\displaystyle \frac{1}{2}}\right)$$

is equal to the Hadamard finite part pairing of the central Cauchy–Laplace kernel

$$h_w\left(u\right)={\displaystyle \frac{e^{wu}-1}{u}}$$

against the zeta-side distribution in the completed explicit formula. Namely,

$$〈{\mu }_{\xi },{\mathcal{R}}_{\mathrm{cen}}{h}_w〉={\partial }_{w}log\xi \left({\displaystyle \frac{1}{2}}+w\right)-{\partial }_{w}log\xi \left({\displaystyle \frac{1}{2}}\right).$$

Proof. 

Since $\xi$ is entire and

$$\xi \left({\displaystyle \frac{1}{2}}\right)\ne 0,$$

taking $r>0$ sufficiently small gives $\xi ({\displaystyle \frac{1}{2}}+w)\ne 0$ for $|w|<r$. When the Hadamard product is written in normalized form at the central point, in the difference of logarithmic derivatives

$${\partial }_{w}log\xi \left({\displaystyle \frac{1}{2}}+w\right)-{\partial }_{w}log\xi \left({\displaystyle \frac{1}{2}}\right),$$

the part common at the central point among the constant factor and the linear exponential factor is cancelled. The remaining zero terms, arithmetic terms, and Archimedean terms are represented by the central difference of the completed explicit formula. The Cauchy–Laplace kernel corresponding to this central difference is

$$h_w\left(u\right)={\displaystyle \frac{e^{wu}-1}{u}}.$$

The unregularized pairing may contain divergent terms component by component, but in the Hadamard finite part obtained by subtracting the value at the central point $w=0$, the identical divergent principal parts are cancelled. The operator ${\mathcal{R}}_{\mathrm{cen}}$ in Definition 6.84 is the linear regularization realizing this finite part pairing. Therefore

$$〈{\mu }_{\xi },{\mathcal{R}}_{\mathrm{cen}}{h}_w〉={\partial }_{w}log\xi \left({\displaystyle \frac{1}{2}}+w\right)-{\partial }_{w}log\xi \left({\displaystyle \frac{1}{2}}\right)$$

holds. □

Lemma 6.90 

(central transform of $\xi$). There exists $0<r\le r_0$ such that, for $|w|<r$,

$$〈{\mu }_{\xi },{\Psi }_w^{\mathrm{cen}}〉={\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}\right).$$

The inputs are only the central Cauchy–Laplace kernel of Definition 6.75, the regularization operator of Definition 6.84, and the standard Hadamard product of the completed zeta function. No information about the location of the zeros of $\xi$, and in particular no form of the Riemann Hypothesis, is used.

Proof. 

By Lemma 6.89,

$$〈{\mu }_{\xi },{\mathcal{R}}_{\mathrm{cen}}{h}_w〉={\partial }_{w}log\xi \left({\displaystyle \frac{1}{2}}+w\right)-{\partial }_{w}log\xi \left({\displaystyle \frac{1}{2}}\right).$$

By Definition 6.86,

$${\Psi }_w^{\mathrm{cen}}={\mathcal{R}}_{\mathrm{cen}}{h}_w.$$

Substitution gives the asserted identity. The proof uses only the completed zeta function as an order-one entire function with its Hadamard product; the zeros are kept at their a priori locations throughout. □

Lemma 6.91 

(determinant central partial fraction formula). For the same $r>0$, for $|w|<r$,

$$〈{\mu }_L,{\mathcal{R}}_{\mathrm{cen}}{h}_w〉={\partial }_{w}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)-{\partial }_{w}log{F}_K\left({\displaystyle \frac{1}{2}}\right)$$

holds.

Proof. 

First prove the assertion for the finite-rank cutoff $K_N$. If the eigenvalues of $K_N$ are denoted by ${\lambda }_{N,j}$, then

$${\partial }_{w}log{det}_2(I+iw{K}_N)=\sum _j\left({\displaystyle \frac{i{\lambda }_{N,j}}{1+iw{\lambda }_{N,j}}}-i{\lambda }_{N,j}\right).$$

Taking the difference from the central point, the linear term of the normalizing exponential factor is cancelled by ${\partial }_{w}log{F}_{K_N}\left({\displaystyle \frac{1}{2}}\right)$, and the remaining finite sum agrees with the finite-rank singular-boundary evaluation of the Cauchy–Laplace kernel $h_w$. Therefore

$$〈{\mu }_{L,N},{\mathcal{R}}_{\mathrm{cen}}{h}_w〉={\partial }_{w}log{F}_{K_N}\left({\displaystyle \frac{1}{2}}+w\right)-{\partial }_{w}log{F}_{K_N}\left({\displaystyle \frac{1}{2}}\right).$$

Next let $N\to \infty$. By the convergence shown in Section 6.3,

$$K_N\to K\mathrm{in}{\mathfrak{S}}_2,$$

and by Hilbert–Schmidt continuity of ${det}_2$, $F_{K_N}\to F_K$ locally uniformly in a neighborhood of the central point. Taking r smaller so as to preserve nonvanishing, Cauchy’s integral formula implies that the logarithmic derivatives also converge locally uniformly. Moreover, the finite-rank singular-boundary evaluation ${\mu }_{L,N}$ converges to ${\mu }_L$ with respect to the ${\mathcal{C}}_{\mathrm{cen}}$-pairing. Thus, passing to the limit in the above formula, one obtains

$$〈{\mu }_L,{\mathcal{R}}_{\mathrm{cen}}{h}_w〉={\partial }_{w}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)-{\partial }_{w}log{F}_K\left({\displaystyle \frac{1}{2}}\right).$$

□

Lemma 6.92 

(central transform of $F_K$). For the same $r>0$,

$$〈{\mu }_L,{\Psi }_w^{\mathrm{cen}}〉={\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}\right)(|w|<r)$$

holds. The inputs are $K=K^{*}\in {\mathfrak{S}}_2$, the regularized Fredholm determinant of Definition 6.57, and the central Cauchy–Laplace regularization of Definition 6.84.

Proof. 

By Lemma 6.91,

$$〈{\mu }_L,{\mathcal{R}}_{\mathrm{cen}}{h}_w〉={\partial }_{w}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)-{\partial }_{w}log{F}_K\left({\displaystyle \frac{1}{2}}\right).$$

By Definition 6.86,

$${\Psi }_w^{\mathrm{cen}}={\mathcal{R}}_{\mathrm{cen}}{h}_w.$$

The assertion follows. Thus the $F_K$-side central transform is obtained from the ${\mathfrak{S}}_2$-Fredholm determinant expansion and not from any comparison with $\xi$. □

Lemma 6.93 

(central zeta-side logarithmic derivative identity). There exists $0<r\le r_0$ such that

$$\langle {\mu }_{\xi },{\Psi }_w^{\mathrm{cen}}\rangle ={\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}\right)(|w|<r)$$

holds.

Proof. 

This is exactly Lemma 6.90, the zeta-side central transform lemma. □

Lemma 6.94 

(central trace identity on the singular-boundary side). For the same $r>0$,

$$\langle {\mu }_L,{\Psi }_w^{\mathrm{cen}}\rangle ={\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}\right)(|w|<r)$$

holds.

Proof. 

This is exactly Lemma 6.92, the $F_K$-side central transform lemma. □

Theorem 6.95 

(central transform theorem from the residual-free comparison). There exists $0<r\le r_0$ such that the map

$$w⟼{\Psi }_w^{\mathrm{cen}}\in {\mathcal{C}}_{\mathrm{cen}}(|w|<r)$$

is holomorphic and satisfies the following:

$$\langle {\mu }_L,{\Psi }_w^{\mathrm{cen}}\rangle ={\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}\right),$$

and

$$\langle {\mu }_{\xi },{\Psi }_w^{\mathrm{cen}}\rangle ={\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}\right).$$

Proof. 

Holomorphicity follows from Lemma 6.87. The ${\mathcal{K}}_R$-side identity follows from Lemma 6.94, whose input is the ${\mathfrak{S}}_2$-Fredholm determinant model for $F_K$. The zeta-side identity follows from Lemma 6.93, whose input is the standard Hadamard product for the completed zeta function. No central residual-free equality is used in this theorem; it only records the two separate transform identifications. □

Lemma 6.96 

(${\mathcal{K}}_R$-side central normal convergence estimate). After shrinking $r>0$ if necessary, the central logarithmic-derivative expansion of the Fredholm determinant side is normally convergent on every compact set

$$B⋐\{|w|<r\}$$

after the central subtraction at $w=0$. More precisely, for every finite set of w-derivatives covered by the seminorms of Definition 6.83, the corresponding series for

$${\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}\right)$$

converges locally uniformly on B, and the resulting bounds are controlled by finitely many seminorms $p_{B,N,m,\sigma }$.

Proof. 

By Theorem 6.54,

$$K=K^{*},K\in {\mathfrak{S}}_2.$$

Let $\left\{{\lambda }_j\right\}$ be the nonzero eigenvalues of K, counted with multiplicity. Then $\left({\lambda }_j\right)\in {\ell }^2$ and $\left({\lambda }_j\right)$ is bounded. After shrinking r, the factors $1+iw{\lambda }_j$ are uniformly bounded away from zero for $w\in B$ and all j. The logarithmic derivative of the regularized determinant is

$${\displaystyle \frac{d}{dw}}log{D}_K\left(iw\right)=iTr\left({(I+iwK)}^{-1}K-K\right),$$

and, in the eigenvalue expansion, each summand is bounded by a constant multiple of

$$|w||{\lambda }_j{|}^2$$

on B. Its finite w-derivatives are bounded by constant multiples of

$$|{\lambda }_j{|}^2(1+|{\lambda }_j{|)}^m$$

for the relevant finite m, which is summable because $\left({\lambda }_j\right)\in {\ell }^2$ and $\left({\lambda }_j\right)$ is bounded. Hence the expansion and its finitely many w-derivatives are normally convergent on B. In particular, for each finite derivative order m, there is a constant $C_{B,m}$ such that the j-th w-derivative of the ${\mathcal{K}}_R$-side pairing on a finite-window input is bounded by $C_{B,m}$ times a finite sum of the seminorms $p_{B,N,m,\sigma }$ controlling the corresponding central kernel derivatives. The exponential normalization contributes only a polynomial expression in w and is therefore controlled by the same central seminorms. □

Lemma 6.97 

($\xi$-side central Hadamard convergence estimate). The central partial-fraction expansion obtained from the standard Hadamard product of the completed zeta function converges normally on every compact set

$$B⋐\{|w|<r\}$$

after the central subtraction at $w=0$. The convergence remains valid after finitely many w-derivatives covered by the seminorms of Definition 6.83. This estimate uses only the order-one entire-function structure of $\xi$ and does not assume the Riemann Hypothesis.

Proof. 

The completed zeta function is an entire function of order one and admits its standard genus-one Hadamard product. Let $\rho$ range over the zeros of $\xi$, counted with multiplicity, and put

$$a_{\rho }:=\rho -{\displaystyle \frac{1}{2}}.$$

By Lemma 6.61, $a_{\rho }\ne 0$ for every zero $\rho$. The logarithmic derivative of the genus-one product, after subtracting its value at the central point, has the central partial-fraction form

$$\sum _{\rho }\left({\displaystyle \frac{1}{w-a_{\rho }}}+{\displaystyle \frac{1}{a_{\rho }}}\right)\mathrm{together}\mathrm{with}\mathrm{the}\mathrm{polynomial}\mathrm{contribution}\mathrm{coming}\mathrm{from}\mathrm{the}\mathrm{exponential}\mathrm{factor}.$$

The summand may be written as

$${\displaystyle \frac{1}{w-a_{\rho }}}+{\displaystyle \frac{1}{a_{\rho }}}={\displaystyle \frac{w}{a_{\rho }(w-a_{\rho })}}.$$

On a compact set $B⋐\{|w|<r\}$, all but finitely many $a_{\rho }$ satisfy

$$|a_{\rho }|>2\underset{w\in B}{sup}|w|,$$

and for those zeros the summand is bounded by a constant multiple of

$${\displaystyle \frac{{sup}_{w\in B}|w|}{|a_{\rho }{|}^2}}.$$

The genus-one product condition gives

$$\sum _{\rho }{|a_{\rho }|}^{-2}<\infty$$

after removing the finite set already mentioned. Hence the central partial-fraction series converges normally on B. After finitely many w-derivatives, the corresponding summands are bounded by constants times $|a_{\rho }{|}^{-j-1}$ for $j\ge 1$, and the same genus-one estimate gives normal convergence. The polynomial contribution from the exponential factor is controlled by the seminorms of Definition 6.83. Consequently, for each finite derivative order m, the j-th w-derivative of the zeta-side pairing is bounded on B by a finite sum of the seminorms $p_{B,N,m,\sigma }$ applied to the central test input. No information about the location of the zeros is used; they remain in their a priori positions throughout the argument. □

Theorem 6.98 

(continuity of the central pairings). Let ${\mathcal{C}}_{\mathrm{cen}}$ be the central comparison space of Definition 6.83. The functionals

$$\Phi ⟼\langle {\mu }_L,\Phi \rangle ,\Phi ⟼\langle {\mu }_{\xi },\Phi \rangle ,$$

initially defined on the span of finite-window central test inputs, extend uniquely to continuous linear functionals on ${\mathcal{C}}_{\mathrm{cen}}$. Consequently, if

$${\Phi }_M⟶\Phi \mathrm{in}{\mathcal{C}}_{\mathrm{cen}},$$

then

$$\langle {\mu }_L,{\Phi }_M\rangle ⟶\langle {\mu }_L,\Phi \rangle ,\langle {\mu }_{\xi },{\Phi }_M\rangle ⟶\langle {\mu }_{\xi },\Phi \rangle .$$

For the finite-window central cutoff family

$${\Psi }_{w,M}^{\mathrm{fw}}⟶{\Psi }_w^{\mathrm{cen}}$$

of Lemma 6.88, these convergences are locally uniform for w in every compact subset $B⋐\{|w|<r\}$, and the same assertion holds after applying any finite number of w-derivatives covered by the seminorms of Definition 6.83.

Proof. 

By Definitions 6.79 and 6.83, the finite-window central test inputs form the prescribed dense generating subspace of ${\mathcal{C}}_{\mathrm{cen}}$ for the seminorms $p_{B,N,m,\sigma }$. Hence it is enough to have uniform seminorm bounds for the two pairings on this dense subspace.

Concretely, for each compact set $B⋐\{|w|<r\}$ and each finite derivative order m, continuity is obtained from estimates of the following form: there are a constant $C_B$ and finitely many seminorms of the form $p_{B,N_{\nu },m_{\nu },{\sigma }_{\nu }}$ such that every finite-window central test input $\Phi$ satisfies

$$\underset{w\in B}{sup}\left|{\partial }_w^j\langle {\mu }_{•},{\Phi }_w\rangle \right|\le C_B{\displaystyle \sum _{\nu =1}^{N_B}}{p}_{B,N_{\nu },m_{\nu },{\sigma }_{\nu }}(\Phi )(0\le j\le m),$$

where ${\mu }_{•}$ denotes either ${\mu }_L$ or ${\mu }_{\xi }$. The two estimates below provide this bound separately for the ${\mathcal{K}}_R$-side and for the zeta side.

For the ${\mathcal{K}}_R$-side pairing, these bounds are exactly the normal convergence estimates of Lemma 6.96, which use only

$$K=K^{*},K\in {\mathfrak{S}}_2,$$

and the ${det}_2$ logarithmic-derivative expansion. Therefore $\Phi \mapsto \langle {\mu }_L,\Phi \rangle$ is continuous on the finite-window central test inputs and extends uniquely to ${\mathcal{C}}_{\mathrm{cen}}$.

For the zeta-side pairing, the required bounds are the central Hadamard convergence estimates of Lemma 6.97. They use only the standard order-one Hadamard product of the completed zeta function and do not use the Riemann Hypothesis. Therefore $\Phi \mapsto \langle {\mu }_{\xi },\Phi \rangle$ is also continuous on the finite-window central test inputs and extends uniquely to ${\mathcal{C}}_{\mathrm{cen}}$.

The asserted convergence of pairings follows immediately from these continuous extensions. For the particular family ${\Psi }_{w,M}^{\mathrm{fw}}$, the local uniformity in w and stability under finitely many w-derivatives follow by combining the continuity just proved with Lemma 6.88.

No step in this argument assumes the Riemann Hypothesis. The zeta-side input is only the standard order-one Hadamard product for $\xi$, with its zeros left in their a priori positions, and the ${\mathcal{K}}_R$-side input is only $K=K^{*}\in {\mathfrak{S}}_2$. □

Lemma 6.99 

(admissibility of finite-window central cutoffs). For every $M\ge 1$ and every $|w|<r$, the finite-window central cutoff test input

$${\Psi }_{w,M}^{\mathrm{fw}}$$

belongs to the finite-window comparison class to which the residual-free comparison interface of Section 5 applies. Moreover, the counterterm

$${CT}_M^{\mathrm{cen}}\left(h_w\right)$$

is a finite-window calibrated reference term. It changes only the common local principal part of the completed explicit formula and does not change the residual-free quotient class modulo

$$Ran{\Pi }_{\mathrm{res}}.$$

Proof. 

The raw kernel $h_{w,M}^{\mathrm{fw}}={\chi }_M{h}_w$ has compact u-support in the finite window $|u|\le 2M$ and is smooth in both variables. Hence ${\left[h_{w,M}^{\mathrm{fw}}\right]}_{\mathrm{fw}}$ is a bounded finite-window comparison datum of the kind used in Definition 5.21. The counterterm ${CT}_M^{\mathrm{cen}}\left(h_w\right)$ is a finite linear combination of the central and endpoint local jets specified in Definition 6.79. Those jets are inserted identically into the Archimedean, arithmetic, and singular-boundary parts of the completed explicit formula and hence represent a calibrated reference subtraction. They do not introduce an additional ${\mathcal{K}}_R$-component and do not alter the class of the finite-window datum in the quotient by $Ran{\Pi }_{\mathrm{res}}$. Therefore ${\Psi }_{w,M}^{\mathrm{fw}}$ is an admissible finite-window input for the residual-free comparison interface of Section 5. □

Lemma 6.100 

(finite-window residual-free equality for central cutoffs). Let $0<r\le r_0$ be fixed as in Lemma 6.87. For every M and every $|w|<r$, the finite-window central cutoff test input

$${\Psi }_{w,M}^{\mathrm{fw}}$$

of Definition 6.79 satisfies

$$\langle {\mu }_L,{\Psi }_{w,M}^{\mathrm{fw}}\rangle =\langle {\mu }_{\xi },{\Psi }_{w,M}^{\mathrm{fw}}\rangle .$$

This is the finite-window residual-free equality for the central cutoff input.

Proof. 

Fix M and $|w|<r$. By Lemma 6.99, ${\Psi }_{w,M}^{\mathrm{fw}}$ is an admissible finite-window comparison input for the residual-free comparison interface of Section 5, and its counterterm does not change the quotient class modulo $Ran{\Pi }_{\mathrm{res}}$.

The finite-window datum is therefore evaluated through the residual-free comparison interface constructed in Section 5. More precisely, the canonical representative of Definition 5.20 removes the $Ran{\Pi }_{\mathrm{res}}$-component; Lemma 5.22 shows that this change of representative does not alter the finite-window comparison datum; and Proposition 5.23 places the remaining ${\mathcal{K}}_R$-projected contribution on the component ${\mathcal{K}}_R$, after the prime-power contribution has been evaluated exactly by the arithmetic trace. Equivalently, the same residual-free finite-window comparison interface is summarized in Proposition 6.9.

Applying that interface to the finite-window central cutoff input gives the finite-window equality

$$\langle {\mu }_L,{\Psi }_{w,M}^{\mathrm{fw}}\rangle =\langle {\mu }_{\xi },{\Psi }_{w,M}^{\mathrm{fw}}\rangle .$$

No limiting argument and no central continuity statement is used in this lemma; the assertion is the equality at the fixed finite-window level. □

Theorem 6.101 

(central residual-free equality). For every $|w|<r$,

$$\langle {\mu }_L,{\Psi }_w^{\mathrm{cen}}\rangle =\langle {\mu }_{\xi },{\Psi }_w^{\mathrm{cen}}\rangle$$

holds. The equality is obtained locally uniformly in w on compact subsets of $\{|w|<r\}$.

Proof. 

We separate the argument into the three steps needed for the passage from finite windows to the central kernel.

Step 1: finite-window equality. For every M and every $|w|<r$, Lemma 6.100 gives

$$\langle {\mu }_L,{\Psi }_{w,M}^{\mathrm{fw}}\rangle =\langle {\mu }_{\xi },{\Psi }_{w,M}^{\mathrm{fw}}\rangle .$$

This equality is the finite-window residual-free comparison equality supplied by the Section 5 residual-free interface, as recalled in the proof of that lemma.

Step 2: finite-window central convergence. Let

$$B⋐\{|w|<r\}$$

be compact. By Lemma 6.88,

$${\Psi }_{w,M}^{\mathrm{fw}}⟶{\Psi }_w^{\mathrm{cen}}$$

in the central comparison topology of ${\mathcal{C}}_{\mathrm{cen}}$, locally uniformly for $w\in B$. The same convergence holds after any finite number of w-derivatives controlled by the seminorms of Definition 6.83.

Step 3: continuity of the central pairings. By Theorem 6.98, the pairings

$$\Phi \mapsto \langle {\mu }_L,\Phi \rangle ,\Phi \mapsto \langle {\mu }_{\xi },\Phi \rangle$$

are continuous on ${\mathcal{C}}_{\mathrm{cen}}$. Hence the convergence in Step 2 implies, locally uniformly for $w\in B$,

$$\langle {\mu }_L,{\Psi }_{w,M}^{\mathrm{fw}}\rangle ⟶\langle {\mu }_L,{\Psi }_w^{\mathrm{cen}}\rangle$$

and

$$\langle {\mu }_{\xi },{\Psi }_{w,M}^{\mathrm{fw}}\rangle ⟶\langle {\mu }_{\xi },{\Psi }_w^{\mathrm{cen}}\rangle .$$

Passing to the limit $M\to \infty$ in the finite-window equality of Step 1 gives

$$\langle {\mu }_L,{\Psi }_w^{\mathrm{cen}}\rangle =\langle {\mu }_{\xi },{\Psi }_w^{\mathrm{cen}}\rangle (w\in B).$$

Since $B⋐\{|w|<r\}$ was arbitrary, the equality holds for every $|w|<r$, and the preceding argument gives the asserted local uniformity. □

Theorem 6.102 

(local logarithmic derivative equality). There exists $r>0$ such that, for $|w|<r$,

$${\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)={\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}+w\right)$$

holds. The only inputs are the two central-transform lemmas, Theorem 6.101, and the central normalization of $F_K$ fixed in Section 6.3.

Proof. 

By Lemma 6.94,

$$\langle {\mu }_L,{\Psi }_w^{\mathrm{cen}}\rangle ={\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}\right),$$

and by Lemma 6.93,

$$\langle {\mu }_{\xi },{\Psi }_w^{\mathrm{cen}}\rangle ={\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}\right).$$

Theorem 6.101 identifies the two pairings. Therefore

$${\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}\right)={\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}\right).$$

The normalization of $F_K$ in Section 6.3 gives

$$F_K\left({\displaystyle \frac{1}{2}}\right)=\xi \left({\displaystyle \frac{1}{2}}\right)\ne 0,F_K^{\prime }\left({\displaystyle \frac{1}{2}}\right)={\xi }^{\prime }\left({\displaystyle \frac{1}{2}}\right),$$

and hence

$${\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}\right)={\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}\right).$$

Substituting this equality of central logarithmic derivatives into the preceding display gives the asserted local logarithmic derivative equality. □

Lemma 6.103 

(local coefficient equality). For every $m\ge 0$,

$$c_m\left(F_K\right)=c_m\left(\xi \right)$$

holds. This lemma records the coefficient consequence of the local logarithmic derivative equality; the subsequent local analytic equality is obtained directly from the quotient argument and does not rely on an additional coefficient comparison.

Proof. 

By Theorem 6.102, there exists $r>0$ such that

$${\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)={\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}+w\right)(|w|<r).$$

Moreover, by the normalization of Section 6.3,

$$F_K\left({\displaystyle \frac{1}{2}}\right)=\xi \left({\displaystyle \frac{1}{2}}\right).$$

Therefore, if the logarithmic branches of Definition 6.74 are fixed with the same value, there exists $0<r^{\prime }\le r$ such that

$${\mathcal{L}}_K\left(w\right)={\mathcal{L}}_{\xi }\left(w\right)(|w|<r^{\prime }).$$

Thus all Taylor coefficients agree, and

$$c_m\left(F_K\right)=c_m\left(\xi \right)(m\ge 0)$$

holds. □

Lemma 6.104 

(local analytic equality near the central point). There exists $0<r\le r_0$ such that

$$F_K\left({\displaystyle \frac{1}{2}}+w\right)=\xi \left({\displaystyle \frac{1}{2}}+w\right)(|w|<r)$$

holds. The inputs are the local logarithmic derivative equality and the central normalization

$$F_K\left({\displaystyle \frac{1}{2}}\right)=\xi \left({\displaystyle \frac{1}{2}}\right)\ne 0.$$

Proof. 

By Definition 6.74, after shrinking $r_0$ if necessary, both

$$F_K\left({\displaystyle \frac{1}{2}}+w\right)\mathrm{and}\xi \left({\displaystyle \frac{1}{2}}+w\right)$$

are nonzero for $|w|<r_0$. Hence the quotient

$$Q\left(w\right):={\displaystyle \frac{F_K(\frac{1}{2}+w)}{\xi (\frac{1}{2}+w)}}$$

is holomorphic and nonzero in a smaller central disk. By Theorem 6.102,

$${\displaystyle \frac{d}{dw}}logQ\left(w\right)=0$$

there. Thus Q is constant on that disk. The central normalization gives

$$Q\left(0\right)={\displaystyle \frac{F_K\left(\frac{1}{2}\right)}{\xi \left(\frac{1}{2}\right)}}=1.$$

Therefore $Q\left(w\right)=1$ in a sufficiently small central disk, which is exactly

$$F_K\left({\displaystyle \frac{1}{2}}+w\right)=\xi \left({\displaystyle \frac{1}{2}}+w\right).$$

□

Theorem 6.105 

(identity theorem used in this section). Let $U\subset \mathbb{C}$ be a connected open set, and let $f,g$ be holomorphic functions on U. If

$$f=g$$

holds on some nonempty open set $V\subset U$, then

$$f\equiv g\mathrm{on}U.$$

Proof. 

The difference $h=f-g$ is holomorphic on U, and vanishes on the nonempty open set V. Therefore the zero set of h has an accumulation point in U. By the identity theorem for holomorphic functions, $h\equiv 0$ on U. □

Theorem 6.106 

(global uniqueness theorem). The normalized determinant comparison function constructed in Section 6.3 agrees with the completed zeta function on the whole plane:

$$F_K\left(s\right)\equiv \xi \left(s\right)(s\in \mathbb{C}).$$

This theorem uses only the local analytic equality near $s={\displaystyle \frac{1}{2}}$, the fact that both functions are entire, and the identity theorem for holomorphic functions.

Proof. 

By Lemma 6.70, $F_K$ and $\xi$ are entire on the connected domain $\mathbb{C}$. By Lemma 6.104, there exists $r>0$ such that

$$F_K\left(s\right)=\xi \left(s\right)\left(|s-\frac{1}{2}|<r\right).$$

The set

$$V=\left\{s\in \mathbb{C}:|s-\frac{1}{2}|<r\right\}$$

is a nonempty open subset of $\mathbb{C}$. Applying Theorem 6.105 with $U=\mathbb{C}$ and this V gives

$$F_K\left(s\right)\equiv \xi \left(s\right)(s\in \mathbb{C}).$$

□

Corollary 6.107 

(zero sets with multiplicity). $F_K$ and ξ have the same zero set on the whole plane, and the multiplicities of their zeros also agree. Namely, for every $\rho \in \mathbb{C}$,

$${ord}_{s=\rho }{F}_K={ord}_{s=\rho }\xi .$$

Proof. 

By Theorem 6.106,

$$F_K\equiv \xi .$$

Therefore their Taylor expansions at any point $\rho$ agree, and the presence or absence of a zero and the order of the first nonzero Taylor coefficient also agree. Hence the multiplicities agree. □

Remark 6.108 

(role of growth estimates). Lemma 6.71, Lemma 6.72, and Corollary 6.73 record that $F_K$, $\xi$, and $F_K-\xi$ are entire functions of finite order. However, the identity proof of this section does not use Carlson-type theorems or the Phragmén–Lindelöf principle, and uses only the agreement on an open disk obtained in Lemma 6.104 and Theorem 6.105.

Remark 6.109 

(output of the analytic comparison layer). Section 6.1 through the present section show that the ${\mathcal{K}}_R$-side regularized-determinant comparison function obtained from the residual-free comparison interface is identical to the completed zeta function itself. In the finite-window counting below, we use

$$F_K\left(s\right)=\xi \left(s\right)$$

as an identity on the whole plane. The finite-window bridge and the defect staircase introduced below do not enter the proof of this global identity. They are subsequent auxiliary constructions recording the consequences of the self-adjoint Hilbert–Schmidt determinant model in bounded height windows. This section itself makes no assertion about the location of zeros; applications to zero counting are treated in the following sections.

6.5. Finite-Window Bridge Theorem

This section records consequences of the determinant closure proved in Section 6.4. It uses the global identity

$$F_K\left(s\right)\equiv \xi \left(s\right),$$

the endpoint stability theorem of Section 6.2, and the argument principle to record zero-counting consequences in a bounded height window. The finite-window bridge is not used in the proof of the determinant identity $F_K\equiv \xi$. Nor are the remaining finite-window subsections used in the proof of Theorem 6.148; they record bounded-window consequences of the spectral localization already obtained from the self-adjoint Hilbert–Schmidt determinant model.

The role of the bridge is purely comparative: it expresses the classical zero count in a finite window as the sum of the critical-line contribution and the off-line defect. It is not a finite-word contradiction argument of the type used in the earlier hybrid formulation. No finite encoding, minimal obstruction, or first-hit contradiction is used in this subsection. No nonexistence of zeros is asserted here. Equivalently, Section 6.5, Section 6.6 and Section 6.7 may be removed without affecting the proof of

$$F_K\equiv \xi$$

or the spectral localization theorem; they are retained to record those conclusions in bounded height windows.

Definition 6.110 

(finite rectangle, left wall, and cap path). Let $\eta >0$, and let $0<T_0<T$. Define the finite-window rectangle by

$$R_{\eta }(T_0,T):=\left\{s=\sigma +it\in \mathbb{C}:-\eta \le \sigma \le 1+\eta ,T_0\le t\le T\right\}.$$

Its positively oriented boundary is denoted by

$$\partial R_{\eta }(T_0,T).$$

Define the left wall, oriented upward, by

$$L_{\eta }(T_0,T):=\left\{-\eta +it:T_0\le t\le T\right\}.$$

Since the left wall is traversed downward on the positively oriented boundary, write

$$\partial R_{\eta }(T_0,T)=C_{\eta }(T_0,T)-L_{\eta }(T_0,T).$$

Here $C_{\eta }(T_0,T)$ is the cap path consisting of the following three sides:

$$C_{\eta }(T_0,T):=B_{\eta }(T_0,T)+R_{\eta }^{+}(T_0,T)+U_{\eta }(T_0,T),$$

where

$$B_{\eta }(T_0,T):-\eta +i{T}_0⟶1+\eta +i{T}_0,$$

$$R_{\eta }^{+}(T_0,T):1+\eta +i{T}_0⟶1+\eta +iT,$$

$$U_{\eta }(T_0,T):1+\eta +iT⟶-\eta +iT.$$

Thus $C_{\eta }(T_0,T)$ is the positively oriented boundary part excluding the left wall.

Definition 6.111 

(admissible zero-counting window). A finite window $R_{\eta }(T_0,T)$ is said to be $\xi$-admissible if

$$\xi \left(s\right)\ne 0(s\in \partial R_{\eta }(T_0,T))$$

holds. Equivalently, no zero of $\xi$ lies on the boundary of the finite window. Under this assumption, the condition of an admissible finite window in Definition 6.23 of Section 6.2 is satisfied for $G=\xi$.

Definition 6.112 

(classical zero count). When $T>0$ is not the imaginary part of a zero of $\xi$, define the classical zero count by

$$N^{\mathrm{cl}}\left(T\right):=\sum _{\begin{array}{c}\xi \left(\rho \right)=0\\ 0<Im\rho \le T\end{array}}{m}_{\xi }\left(\rho \right).$$

Here $m_{\xi }\left(\rho \right)$ is the multiplicity of the zero $\rho$ of $\xi$. The subscript “cl” means classical count, and does not mean critical line.

Definition 6.113 

(critical-line and off-line finite-window counts). Let $0<T_0<T$, and suppose that $T_0,T$ are not imaginary parts of zeros of $\xi$. Define the critical-line zero count by

$$J_{\mathrm{line}}(T_0,T):=\sum _{\begin{array}{c}\xi ({\displaystyle \frac{1}{2}}+i\gamma )=0\\ T_0<\gamma \le T\end{array}}{m}_{\xi }\left({\displaystyle \frac{1}{2}}+i\gamma \right).$$

Also define the full off-line count by

$$N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right):=\sum _{\begin{array}{c}\xi \left(\rho \right)=0\\ T_0<Im\rho \le T\\ Re\rho \ne {\displaystyle \frac{1}{2}}\end{array}}{m}_{\xi }\left(\rho \right).$$

When $T_0$ is fixed in context, also write

$$N_{\mathrm{off}}^{\mathrm{full}}\left(T\right)=N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right).$$

Lemma 6.114 

(argument count on the finite rectangle). Let $R_{\eta }(T_0,T)$ be a $\xi$-admissible finite window. Then

$$N_{\arg }\left(\xi ;R_{\eta }(T_0,T)\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{\partial R_{\eta }(T_0,T)}{\displaystyle \frac{{\xi }^{\prime }\left(s\right)}{\xi \left(s\right)}}ds$$

is equal to

$$N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right).$$

Moreover, using the left wall and the cap path,

$$N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right)={\displaystyle \frac{1}{2\pi i}}\left({\int }_{C_{\eta }(T_0,T)}{\displaystyle \frac{{\xi }^{\prime }\left(s\right)}{\xi \left(s\right)}}ds-{\int }_{L_{\eta }(T_0,T)}{\displaystyle \frac{{\xi }^{\prime }\left(s\right)}{\xi \left(s\right)}}ds\right).$$

Proof. 

The function $\xi$ is entire and has no zero on the boundary of $R_{\eta }(T_0,T)$. Therefore, by the argument principle,

$${\displaystyle \frac{1}{2\pi i}}{\int }_{\partial R_{\eta }(T_0,T)}{\displaystyle \frac{{\xi }^{\prime }\left(s\right)}{\xi \left(s\right)}}ds$$

counts the zeros of $\xi$ inside $R_{\eta }{(T_0,T)}^{\circ }$ with multiplicity.

The zeros of $\xi$ are the nontrivial zeros of $\zeta \left(s\right)$, and by the standard zero region they satisfy

$$0<Re\rho <1.$$

Therefore, for any choice of $\eta >0$,

$$-\eta <Re\rho <1+\eta .$$

Moreover, by $\xi$-admissibility, neither $T_0$ nor T is the imaginary part of a zero. Hence the zeros inside $R_{\eta }{(T_0,T)}^{\circ }$ are exactly the zeros satisfying

$$T_0<Im\rho <T,$$

and under the right-continuous counting convention this is the same number as the zeros satisfying

$$T_0<Im\rho \le T.$$

Therefore

$$N_{\arg }\left(\xi ;R_{\eta }(T_0,T)\right)=N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right).$$

Finally, by Definition 6.110,

$$\partial R_{\eta }(T_0,T)=C_{\eta }(T_0,T)-L_{\eta }(T_0,T),$$

and therefore

$${\int }_{\partial R_{\eta }(T_0,T)}{\displaystyle \frac{{\xi }^{\prime }}{\xi }}ds={\int }_{C_{\eta }(T_0,T)}{\displaystyle \frac{{\xi }^{\prime }}{\xi }}ds-{\int }_{L_{\eta }(T_0,T)}{\displaystyle \frac{{\xi }^{\prime }}{\xi }}ds.$$

The displayed formula follows. □

Lemma 6.115 

(analytic local uniform convergence implies Sobolev cutoff convergence). Let $U^{\prime }⋐U$ be a bounded Lipschitz domain, and let $G_N,G$ be holomorphic functions on U. If

$$G_N⟶G$$

locally uniformly on compact sets in U, then for every $m\ge 0$,

$$G_N⟶G\mathrm{in}{H}^m\left(U^{\prime }\right)$$

holds.

Proof. 

Since $U^{\prime }⋐U$, one can take finitely many small disk neighborhoods of $\overline{U^{\prime }}$ inside U. By Cauchy’s integral formula, for every multi-index $\alpha$, there exists $V⋐U$ such that

$$\underset{\overline{U^{\prime }}}{sup}|{\partial }^{\alpha }(G_N-G)|\le C_{\alpha ,U^{\prime },U}\underset{V}{sup}|G_N-G|.$$

The right-hand side converges to zero by local uniform convergence. Therefore the derivatives of every order converge uniformly to zero on $\overline{U^{\prime }}$. Since the $L^2$-norm on a bounded domain is dominated by the uniform norm, for every $m\ge 0$,

$$\parallel G_N{-G\parallel }_{H^m\left(U^{\prime }\right)}\to 0$$

follows. □

Lemma 6.116 

(endpoint correction does not change the finite-window count). Let $R_{\eta }(T_0,T)$ be a $\xi$-admissible finite window. Let the cutoff comparison functions obtained from the finite-rank cutoffs of Section 6.3 be

$$F_{K,N}\left(s\right):=e^{a_K+b_K(s-{\displaystyle \frac{1}{2}})}{det}_2\left(I+i(s-\frac{1}{2})K_N\right).$$

Then, for sufficiently large N,

$$N_{\arg }(F_{K,N};R_{\eta }(T_0,T))=N_{\arg }(\xi ;R_{\eta }(T_0,T)).$$

Equivalently, the cutoff endpoint correction on the boundary of the finite window does not change the integer zero count.

Proof. 

By Theorem 6.106 of Section 6.4,

$$F_K\left(s\right)\equiv \xi \left(s\right).$$

Moreover, by Lemma 6.64 of Section 6.3,

$$F_{K,N}\to F_K$$

locally uniformly on compact sets. By Lemma 6.115, for every $m\ge 0$ and every $U^{\prime }⋐U_R$,

$$F_{K,N}\to F_K\mathrm{in}{H}^m\left(U^{\prime }\right)$$

follows. Therefore, in a neighborhood of $R_{\eta }(T_0,T)$, the conditions of the admissible analytic cutoff of Section 6.2 are satisfied. Hence Theorem 6.30 can be applied with

$$G=F_K=\xi ,G_{\Lambda }=F_{K,N}.$$

Consequently, for sufficiently large N,

$$N_{\arg }(F_{K,N};R_{\eta }(T_0,T))=N_{\arg }(F_K;R_{\eta }(T_0,T))=N_{\arg }(\xi ;R_{\eta }(T_0,T)).$$

□

Lemma 6.117 

(partition into line and off-line zeros). Let $T_0,T$ be heights that are not imaginary parts of zeros of $\xi$. Then

$$N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right)=J_{\mathrm{line}}(T_0,T)+N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right).$$

Proof. 

The difference

$$N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right)$$

counts, with multiplicity, the zeros of $\xi$ satisfying

$$T_0<Im\rho \le T.$$

Each zero $\rho$ belongs exclusively to one of the two alternatives

$$Re\rho ={\displaystyle \frac{1}{2}}$$

or

$$Re\rho \ne {\displaystyle \frac{1}{2}}.$$

The contribution of the former, collected with multiplicity, is

$$J_{\mathrm{line}}(T_0,T),$$

and the contribution of the latter, collected with multiplicity, is

$$N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right).$$

Therefore the count decomposes as the stated sum. □

Theorem 6.118 

(finite-window bridge theorem). Let $R_{\eta }(T_0,T)$ be a ξ-admissible finite window. Then

$$N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right)=N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right)-J_{\mathrm{line}}(T_0,T).$$

Moreover,

$$N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right)={\displaystyle \frac{1}{2\pi i}}\left({\int }_{C_{\eta }(T_0,T)}{\displaystyle \frac{{\xi }^{\prime }\left(s\right)}{\xi \left(s\right)}}ds-{\int }_{L_{\eta }(T_0,T)}{\displaystyle \frac{{\xi }^{\prime }\left(s\right)}{\xi \left(s\right)}}ds\right).$$

Proof. 

By Lemma 6.114,

$$N_{\arg }\left(\xi ;R_{\eta }(T_0,T)\right)=N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right).$$

Moreover, by Lemma 6.116, the endpoint correction arising from the finite-rank cutoffs of Section 6.3 does not change this integer count for sufficiently large cutoff degree. Therefore the count obtained by the finite-window argument principle is independent of the presence or absence of cutoff and is

$$N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right).$$

On the other hand, by Lemma 6.117,

$$N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right)=J_{\mathrm{line}}(T_0,T)+N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right).$$

Rearranging this gives

$$N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right)=N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right)-J_{\mathrm{line}}(T_0,T).$$

The final integral representation is precisely the left-wall/cap-path decomposition of Lemma 6.114. □

Definition 6.119 

(finite-window off-line defect). Fix $T_0$, and suppose that $T>T_0$ is admissible. Define the finite-window off-line defect by

$$D_{T_0}^{\mathrm{fw}}\left(T\right):=N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right).$$

By Theorem 6.118,

$$D_{T_0}^{\mathrm{fw}}\left(T\right)=N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right)-J_{\mathrm{line}}(T_0,T).$$

This is called the finite-window tail defect identity.

Corollary 6.120 

(admissible-height tail defect identity). Fix $T_0$, and suppose that $T>T_0$ is ξ-admissible. Then

$$D_{T_0}^{\mathrm{fw}}\left(T\right)=N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right)-J_{\mathrm{line}}(T_0,T)$$

holds. In particular,

$$D_{T_0}^{\mathrm{fw}}\left(T\right)\in {\mathbb{Z}}_{\ge 0}.$$

Proof. 

The first equality follows from Definition 6.119 and Theorem 6.118. Moreover,

$$D_{T_0}^{\mathrm{fw}}\left(T\right)=N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right)$$

is a function that counts off-line zeros with multiplicity, and therefore

$$D_{T_0}^{\mathrm{fw}}\left(T\right)\in {\mathbb{Z}}_{\ge 0}.$$

□

Remark 6.121 

(role of the finite-window bridge). Theorem 6.118 is an identity decomposing the total zero count in a finite window into the critical-line zero count and the off-line defect. It is a consequence of the already established determinant identity and the argument principle, not an input to the proof of $F_K\equiv \xi$. At this stage,

$$D_{T_0}^{\mathrm{fw}}\left(T\right)=0$$

is not asserted. In the following sections, this finite-window defect is organized as a nonnegative-integer staircase function and compared with the spectral localization already supplied by the self-adjoint Hilbert–Schmidt determinant model.

6.6. Anchored Defect Staircase

In this section, the contribution of off-line zeros in the upper tail is isolated as a nonnegative-integer-valued right-continuous staircase function. By the finite-window bridge theorem of the preceding section, in a finite window whose boundary does not pass through zeros, this staircase function agrees with the finite-window off-line defect.

This construction records auxiliary finite-window consequences after the determinant identity $F_K\equiv \xi$. It does not contribute to the proof of that identity. It records, in an anchored lattice of bounded height intervals, the same ${\mathcal{K}}_R$-projected data that will be compared with the spectral localization of the self-adjoint Hilbert–Schmidt determinant model. This section treats only the type of the defect staircase, local finiteness, and decomposition by an anchored lattice; the vanishing of the off-line defect is treated in the next section.

Definition 6.122 

(off-line zero multiset above $T_0$). Fix $T_0>0$. Define the multiset of off-line zeros with positive imaginary part, counted with multiplicity, by

$${\mathcal{Z}}_{\mathrm{off}}^{+}\left(T_0\right):=\left\{(\rho ,k):\xi \left(\rho \right)=0,\rho =\beta +i\gamma ,\gamma >T_0,\beta \ne {\displaystyle \frac{1}{2}},1\le k\le m_{\xi }\left(\rho \right)\right\}.$$

Here $m_{\xi }\left(\rho \right)$ is the multiplicity of the zero $\rho$ of $\xi$.

Definition 6.123 

(off-line defect staircase). For $T\ge T_0$, define the off-line defect staircase by

$$D_{T_0}\left(T\right):=\#\left\{(\rho ,k)\in {\mathcal{Z}}_{\mathrm{off}}^{+}\left(T_0\right):T_0<Im\rho \le T\right\}.$$

Equivalently,

$$D_{T_0}\left(T\right)=\sum _{\begin{array}{c}\xi \left(\rho \right)=0\\ \rho =\beta +i\gamma \\ T_0<\gamma \le T\\ \beta \ne {\displaystyle \frac{1}{2}}\end{array}}{m}_{\xi }\left(\rho \right).$$

Thus $D_{T_0}\left(T\right)$ is the same object as the full off-line count

$$N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right).$$

Lemma 6.124 

(compatibility with the finite-window defect). Suppose that $T>T_0$ is a $\xi$-admissible height. That is, T is not the imaginary part of a zero of $\xi$, and $R_{\eta }(T_0,T)$ is admissible in the sense of Definition 6.111. Then

$$D_{T_0}\left(T\right)=D_{T_0}^{\mathrm{fw}}\left(T\right)=N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right)-J_{\mathrm{line}}(T_0,T).$$

Proof. 

By Definition 6.123,

$$D_{T_0}\left(T\right)=N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right).$$

Also, by Definition 6.119,

$$D_{T_0}^{\mathrm{fw}}\left(T\right)=N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right).$$

Therefore

$$D_{T_0}\left(T\right)=D_{T_0}^{\mathrm{fw}}\left(T\right).$$

Furthermore, by Theorem 6.118,

$$N_{\mathrm{off},T_0}^{\mathrm{full}}\left(T\right)=N^{\mathrm{cl}}\left(T\right)-N^{\mathrm{cl}}\left(T_0\right)-J_{\mathrm{line}}(T_0,T).$$

Combining these identities gives the assertion. □

Lemma 6.125 

(staircase properties). The function

$$D_{T_0}:[T_0,\infty )\to {\mathbb{Z}}_{\ge 0}$$

is nonnegative-integer-valued, monotonically nondecreasing, right-continuous, and has only finitely many jumps on any bounded interval. Moreover, the jump size at $T>T_0$ is

$$\Delta D_{T_0}\left(T\right):=D_{T_0}\left(T\right)-D_{T_0}\left(T^{-}\right)=\sum _{\begin{array}{c}\xi \left(\rho \right)=0\\ \rho =\beta +iT\\ \beta \ne {\displaystyle \frac{1}{2}}\end{array}}{m}_{\xi }\left(\rho \right).$$

Here

$$D_{T_0}\left(T^{-}\right):=\underset{u\uparrow T}{lim}{D}_{T_0}\left(u\right).$$

Proof. 

By definition, $D_{T_0}\left(T\right)$ is the number of zeros counted with multiplicity, and hence

$$D_{T_0}\left(T\right)\in {\mathbb{Z}}_{\ge 0}.$$

Moreover, if $T_1\le T_2$, then

$$\{T_0<\gamma \le T_1\}\subset \{T_0<\gamma \le T_2\},$$

and therefore

$$D_{T_0}\left(T_1\right)\le D_{T_0}\left(T_2\right).$$

We show right-continuity. For fixed $T\ge T_0$, $\xi$ is an entire function, and its zero set is discrete. Therefore there exists $\epsilon >0$ such that the interval

$$(T,T+\epsilon ]$$

contains no imaginary part of a zero of $\xi$. Then, for $0<u\le \epsilon$,

$$D_{T_0}(T+u)=D_{T_0}\left(T\right),$$

and right-continuity follows.

Local finiteness follows for the same reason. For arbitrary $A<B$,

$$\{s\in \mathbb{C}:0\le Res\le 1,A\le Ims\le B\}$$

is compact, and since the zero set of $\xi$ is discrete, there are only finitely many zeros in this region. Nontrivial zeros lie in the critical strip

$$0<Res<1,$$

and hence there are only finitely many nontrivial zeros whose imaginary parts lie in $[A,B]$. Thus $D_{T_0}$ has only finitely many jumps on bounded intervals.

Finally, by the half-open convention

$$T_0<\gamma \le T,$$

the jump at T is exactly the contribution, counted with multiplicity, of the off-line zeros whose imaginary part is exactly T. Therefore

$$\Delta D_{T_0}\left(T\right)=\sum _{\begin{array}{c}\xi \left(\rho \right)=0\\ \rho =\beta +iT\\ \beta \ne {\displaystyle \frac{1}{2}}\end{array}}{m}_{\xi }\left(\rho \right).$$

□

Lemma 6.126 

(symmetric pairing of off-line zeros). If $\rho =\beta +i\gamma$, $\gamma >0$, $\beta \ne {\displaystyle \frac{1}{2}}$, is a zero of $\xi$, then

$$1-\overline{\rho }=1-\beta +i\gamma$$

is also a zero with the same multiplicity. Therefore

$$D_{T_0}\left(T\right)$$

is even for every $T\ge T_0$.

Proof. 

If $\xi \left(\rho \right)=0$, then the functional equation

$$\xi \left(s\right)=\xi (1-s)$$

gives $\xi (1-\rho )=0$, and the reality property

$$\overline{\xi \left(s\right)}=\xi \left(\overline{s}\right)$$

gives

$$\xi (1-\overline{\rho })=0.$$

If $\beta \ne 1/2$, then

$$\rho \ne 1-\overline{\rho }.$$

Moreover, the multiplicity of a zero is preserved under composition with a holomorphic function, and therefore the two zeros have the same multiplicity. Thus off-line zeros with positive imaginary part split into symmetric pairs at the same height. Therefore $D_{T_0}\left(T\right)$ is even. □

Definition 6.127 

(symmetric off-line pair count). By Lemma 6.126, $D_{T_0}\left(T\right)$ is even. Define the symmetric off-line pair count by

$$R_{T_0}^{\mathrm{off}}\left(T\right):={\displaystyle \frac{1}{2}}{D}_{T_0}\left(T\right).$$

This paper mainly uses the full count $D_{T_0}$, but $R_{T_0}^{\mathrm{off}}$ may equivalently be used when needed.

Definition 6.128 

(regular anchored Gram lattice). Fix $T_0>0$. A regular anchored Gram lattice above $T_0$ means a strictly increasing sequence

$$\mathcal{G}\left(T_0\right)={\left\{g_n^{\left(T_0\right)}\right\}}_{n\ge 0}$$

satisfying the following.

1.

$$g_0^{\left(T_0\right)}=T_0,g_n^{\left(T_0\right)}\to +\infty (n\to \infty ).$$

2.

For each $n\ge 1$,

$$g_n^{\left(T_0\right)}$$

is not the imaginary part of a zero of $\xi$.

3.

Each interval

$$I_n^{\left(T_0\right)}:=\left(g_n^{\left(T_0\right)},g_{n+1}^{\left(T_0\right)}\right]$$

has finite length and satisfies

$$(T_0,\infty )=\underset{n\ge 0}{⨆}{I}_n^{\left(T_0\right)}.$$

When needed, in a sufficiently large region where the Riemann–Siegel theta function $\vartheta \left(t\right)$ is monotone, one may take phase-anchored Gram points satisfying

$$\vartheta \left(g_n^{\left(T_0\right)}\right)=\vartheta \left(T_0\right)+n\pi$$

as the reference, and perturb only those points that coincide with imaginary parts of zeros by arbitrarily small amounts to obtain a regular lattice. The arguments below use only the three conditions above.

Lemma 6.129 

(existence of regular anchored lattices). For every $T_0>0$, a regular anchored Gram lattice exists.

Proof. 

The set of imaginary parts of zeros of $\xi$ is finite in every bounded interval of $[T_0,\infty )$. Therefore, for every $n\ge 1$, there exists a point in the interval

$$(T_0+n,T_0+n+1)$$

which is not the imaginary part of a zero of $\xi$. Choosing one such point, and if necessary choosing recursively so as to preserve the ordering, we obtain a strictly increasing sequence

$$T_0=g_0^{\left(T_0\right)}<g_1^{\left(T_0\right)}<g_2^{\left(T_0\right)}<\dots$$

such that

$$g_n^{\left(T_0\right)}\to +\infty$$

and $g_n^{\left(T_0\right)}$, for $n\ge 1$, is not the imaginary part of a zero. This gives

$$(T_0,\infty )=\underset{n\ge 0}{⨆}\left(g_n^{\left(T_0\right)},g_{n+1}^{\left(T_0\right)}\right].$$

Hence a regular anchored Gram lattice exists. □

Definition 6.130 

(anchored interval defects). Let $\mathcal{G}\left(T_0\right)={\left\{g_n^{\left(T_0\right)}\right\}}_{n\ge 0}$ be a regular anchored Gram lattice. For each $n\ge 0$, write

$$I_n^{\left(T_0\right)}:=\left(g_n^{\left(T_0\right)},g_{n+1}^{\left(T_0\right)}\right].$$

Define the off-line defect contained in this interval by

$$d_n^{\left(T_0\right)}:=\#\left\{(\rho ,k)\in {\mathcal{Z}}_{\mathrm{off}}^{+}\left(T_0\right):Im\rho \in I_n^{\left(T_0\right)}\right\}.$$

Equivalently,

$$d_n^{\left(T_0\right)}=D_{T_0}\left(g_{n+1}^{\left(T_0\right)}\right)-D_{T_0}\left(g_n^{\left(T_0\right)}\right).$$

Clearly,

$$d_n^{\left(T_0\right)}\in {\mathbb{Z}}_{\ge 0}.$$

When $T_0$ is fixed in context, abbreviate

$$I_n=I_n^{\left(T_0\right)},d_n=d_n^{\left(T_0\right)}.$$

Lemma 6.131 

(lattice decomposition at anchored heights). For every $M\ge 1$,

$$D_{T_0}\left(g_M^{\left(T_0\right)}\right)={\displaystyle \sum _{n=0}^{M-1}}{d}_n^{\left(T_0\right)}.$$

Proof. 

The intervals

$$I_0^{\left(T_0\right)},I_1^{\left(T_0\right)},\dots ,I_{M-1}^{\left(T_0\right)}$$

are disjoint and satisfy

$$(T_0,g_M^{\left(T_0\right)}]={\displaystyle \underset{n=0}{\overset{M-1}{⨆}}}{I}_n^{\left(T_0\right)}.$$

Each $d_n^{\left(T_0\right)}$ counts the off-line zeros in $I_n^{\left(T_0\right)}$, with multiplicity. Therefore, taking the sum gives the number of all off-line zeros in

$$(T_0,g_M^{\left(T_0\right)}]$$

counted with multiplicity, and this is equal to

$$D_{T_0}\left(g_M^{\left(T_0\right)}\right).$$

□

Definition 6.132 

(partial interval defect). Let $T\ge T_0$. Take the unique $M\ge 0$ such that $T\in I_M^{\left(T_0\right)}$, and define the partial interval defect by

$$d_M^{\left(T_0\right)}\left(T\right):=\#\left\{(\rho ,k)\in {\mathcal{Z}}_{\mathrm{off}}^{+}\left(T_0\right):g_M^{\left(T_0\right)}<Im\rho \le T\right\}.$$

Then

$$0\le d_M^{\left(T_0\right)}\left(T\right)\le d_M^{\left(T_0\right)}.$$

Lemma 6.133 

(lattice decomposition at arbitrary heights). Let $T\ge T_0$, and suppose that $T\in I_M^{\left(T_0\right)}$. Then

$$D_{T_0}\left(T\right)={\displaystyle \sum _{n=0}^{M-1}}{d}_n^{\left(T_0\right)}+d_M^{\left(T_0\right)}\left(T\right).$$

In particular, when $T=g_M^{\left(T_0\right)}$,

$$D_{T_0}\left(T\right)={\displaystyle \sum _{n=0}^{M-1}}{d}_n^{\left(T_0\right)}.$$

Proof. 

By the half-open interval decomposition

$$(T_0,T]=\left({\displaystyle \underset{n=0}{\overset{M-1}{⨆}}}{I}_n^{\left(T_0\right)}\right)\bigsqcup \left(g_M^{\left(T_0\right)},T\right],$$

the number of off-line zeros in

$$(T_0,T]$$

counted with multiplicity decomposes into the contribution of the complete intervals $I_0,\dots ,I_{M-1}$, and the contribution of the final partial interval

$$(g_M^{\left(T_0\right)},T].$$

The former is

$${\displaystyle \sum _{n=0}^{M-1}}{d}_n^{\left(T_0\right)},$$

and the latter is

$$d_M^{\left(T_0\right)}\left(T\right).$$

The assertion follows. □

Corollary 6.134 

(first positive defect index). If

$$D_{T_0}\left(T\right)>0$$

for some $T\ge T_0$, then

$$n_{*}:=min\{n\ge 0:d_n^{\left(T_0\right)}>0\}$$

is well-defined. Moreover,

$$d_n^{\left(T_0\right)}=0(0\le n<n_{*}),d_{n_{*}}^{\left(T_0\right)}>0.$$

Proof. 

By Lemma 6.133, if $D_{T_0}\left(T\right)>0$, then some finite interval contribution or partial interval contribution is positive. In that case, at least one $d_n^{\left(T_0\right)}$ is positive. By the well-ordering property of the natural numbers,

$$n_{*}=min\{n\ge 0:d_n^{\left(T_0\right)}>0\}$$

exists. By definition,

$$d_n^{\left(T_0\right)}=0(n<n_{*}),d_{n_{*}}^{\left(T_0\right)}>0.$$

□

Remark 6.135 

(role of the anchored staircase). The objects constructed in this section are the nonnegative-integer-valued right-continuous staircase function

$$D_{T_0}\left(T\right)$$

and its anchored lattice decomposition

$$D_{T_0}\left(g_M^{\left(T_0\right)}\right)={\displaystyle \sum _{n=0}^{M-1}}{d}_n^{\left(T_0\right)}.$$

This staircase does not supply an additional assumption for the determinant identity. It only records the off-line part of the zero count after the identity $F_K\equiv \xi$ has already been established. In the next section, using the spectral localization of the regularized determinant and the staircase-function structure of this section, we show that the off-line defect vanishes identically.

6.7. No-First-Hit THEOREM

This section is the first point at which zero localization is used. The input is the self-adjoint Hilbert–Schmidt realization of Section 6.3 together with the global determinant identity of Section 6.4,

$$F_K\left(s\right)\equiv \xi \left(s\right).$$

The finite-window bridge and the defect staircase of Section 6.5–6.6 are used only to record this localization in bounded height windows and anchored intervals. They are not used to prove $F_K\equiv \xi$.

The core of the argument is spectral localization: the zeros of the regularized determinant

$${det}_2\left(I+i(s-\frac{1}{2})K\right)$$

are localized on the critical line by the real eigenvalues of the self-adjoint operator K. The defect staircase of Section 6.6 then translates this spectral localization into the vanishing of a nonnegative-integer-valued staircase function. Thus the no-first-hit argument is a subsequent zero-counting consequence of the determinant closure, not a replacement for the finite-word contradiction mechanism of the earlier hybrid formulation.

Definition 6.136 

(regular admissible heights above $T_0$). Fix $T_0>0$, and assume that $T_0$ is not the imaginary part of a zero of $\xi$. Define

$$\mathcal{H}\left(T_0\right):=\{T>T_0:T\mathrm{is}\mathrm{not}\mathrm{the}\mathrm{imaginary}\mathrm{part}\mathrm{of}\mathrm{a}\mathrm{zero}\mathrm{of}\xi \}.$$

Lemma 6.137 

(regular heights give admissible windows). If $T\in \mathcal{H}\left(T_0\right)$, then for every $\eta >0$, $R_{\eta }(T_0,T)$ is $\xi$-admissible in the sense of Definition 6.111.

Proof. 

Nontrivial zeros lie in the critical strip

$$0<Res<1.$$

Therefore no zero lies on the vertical sides $-\eta$ and $1+\eta$. Moreover, since $T_0$ and T are not imaginary parts of zeros, no zero lies on the upper or lower horizontal side. Therefore

$$\xi \left(s\right)\ne 0(s\in \partial R_{\eta }(T_0,T)),$$

and $R_{\eta }(T_0,T)$ is $\xi$-admissible. □

Lemma 6.138 

(right-density of admissible heights). For every $T\ge T_0$ and every $\epsilon >0$,

$$\mathcal{H}\left(T_0\right)\cap (T,T+\epsilon )\ne ⌀.$$

In particular, for each $T\ge T_0$, one can take a sequence

$$T_j\in \mathcal{H}\left(T_0\right),T_j\downarrow T.$$

Proof. 

Nontrivial zeros lie in the critical strip

$$0<Res<1.$$

Therefore the nontrivial zeros whose imaginary parts lie in $[T,T+\epsilon ]$ are contained in the compact rectangle

$$\{s:0\le Res\le 1,T\le Ims\le T+\epsilon \}.$$

Since $\xi$ is an entire function and its zero set is discrete, there are only finitely many zeros in this compact set. Hence there are only finitely many values in the interval $(T,T+\epsilon )$ that occur as imaginary parts of zeros, and the complement is nonempty. Thus

$$\mathcal{H}\left(T_0\right)\cap (T,T+\epsilon )\ne ⌀.$$

Taking $\epsilon =1/j$ and choosing a point gives a sequence satisfying $T_j\in \mathcal{H}\left(T_0\right)$, $T<T_j<T+1/j$. If necessary, by taking a monotone subsequence, one can arrange that $T_j\downarrow T$. □

Lemma 6.139 

(spectral product for the determinant comparison function). Let $K=K^{*}\in {\mathfrak{S}}_2\left(H_{\alpha ,+}\right)$, and denote its nonzero eigenvalues, counted with algebraic multiplicity, by

$${\left\{{\lambda }_j\right\}}_{j\ge 1}\subset \mathbb{R}\setminus \left\{0\right\}.$$

Then

$$F_K\left(s\right)=e^{a_K+b_K(s-{\displaystyle \frac{1}{2}})}\prod _j\left(1+i(s-\frac{1}{2}){\lambda }_j\right)exp\left(-i(s-\frac{1}{2}){\lambda }_j\right),$$

and the product converges uniformly on compact sets.

Proof. 

By Theorem 6.54, K is a self-adjoint Hilbert–Schmidt operator. Therefore K is a compact normal operator, and by the spectral theorem it is diagonalized by a real eigenvalue sequence ${\lambda }_j$ and an orthonormal system of eigenvectors. Applying the product representation of Definition 6.57 to $A=i(s-\frac{1}{2})K$, one obtains

$${det}_2\left(I+i(s-\frac{1}{2})K\right)=\prod _j\left(1+i(s-\frac{1}{2}){\lambda }_j\right)exp\left(-i(s-\frac{1}{2}){\lambda }_j\right).$$

Since $\left\{{\lambda }_j\right\}\in {\ell }^2$, this ${det}_2$-product converges uniformly on compact sets. Multiplying by the exponential normalization factor gives the displayed formula. □

Theorem 6.140 

(spectral localization of zeros). All zeros of $F_K$ lie on the critical line. Namely,

$$F_K\left(\rho \right)=0⟹Re\rho ={\displaystyle \frac{1}{2}}.$$

More concretely, the zeros are of the form

$${\rho }_j={\displaystyle \frac{1}{2}}+{\displaystyle \frac{i}{{\lambda }_j}}$$

for nonzero eigenvalues ${\lambda }_j\in \mathbb{R}\setminus \left\{0\right\}$, and their multiplicities agree with the corresponding eigenvalue multiplicities.

Proof. 

In the representation of Lemma 6.139, the exponential factors

$$e^{a_K+b_K(s-{\displaystyle \frac{1}{2}})}\mathrm{and}exp\left(-i(s-\frac{1}{2}){\lambda }_j\right)$$

have no zeros. Therefore $F_K\left(s\right)=0$ is equivalent to the existence of a nonzero eigenvalue ${\lambda }_j$ such that

$$1+i(s-\frac{1}{2}){\lambda }_j=0.$$

Solving this gives

$$s={\displaystyle \frac{1}{2}}+{\displaystyle \frac{i}{{\lambda }_j}}.$$

By self-adjointness, ${\lambda }_j\in \mathbb{R}$, and hence

$$Res={\displaystyle \frac{1}{2}}.$$

If the same eigenvalue occurs with multiplicity, then the corresponding linear factor occurs with the same multiplicity, and therefore the zero multiplicity is equal to the eigenvalue multiplicity. □

Corollary 6.141 

(spectral localization of zeros of $\xi$). All nontrivial zeros of the completed zeta function ξ lie on the critical line. Namely,

$$\xi \left(\rho \right)=0⟹Re\rho ={\displaystyle \frac{1}{2}}.$$

Proof. 

By Theorem 6.106,

$$F_K\left(s\right)\equiv \xi \left(s\right).$$

Therefore the zeros of $\xi$ agree with the zeros of $F_K$, counted with multiplicity. By Theorem 6.140, all zeros of $F_K$ lie on the critical line. Hence all zeros of $\xi$ also lie on the critical line. □

Theorem 6.142 

(admissible-height vanishing of the tail defect). For every

$$T\in \mathcal{H}\left(T_0\right),$$

one has

$$D_{T_0}\left(T\right)=0.$$

Proof. 

By Corollary 6.141, $\xi$ has no off-line zeros. Therefore, for every $T\ge T_0$,

$$\{\rho =\beta +i\gamma :\xi \left(\rho \right)=0,T_0<\gamma \le T,\beta \ne \frac{1}{2}\}$$

is empty. In particular, if $T\in \mathcal{H}\left(T_0\right)$, then

$$D_{T_0}\left(T\right)=0.$$

□

Lemma 6.143 

(extension from admissible heights to all heights). For every

$$T\ge T_0,$$

one has

$$D_{T_0}\left(T\right)=0.$$

Proof. 

Take arbitrary $T\ge T_0$. By Lemma 6.138, one can take a sequence

$$T_j\in \mathcal{H}\left(T_0\right),T_j\downarrow T.$$

By Theorem 6.142, for each j,

$$D_{T_0}\left(T_j\right)=0.$$

On the other hand, by Lemma 6.125, $D_{T_0}$ is right-continuous. Therefore

$$D_{T_0}\left(T\right)=\underset{j\to \infty }{lim}{D}_{T_0}\left(T_j\right)=0.$$

□

Theorem 6.144 

(no-first-hit theorem). There is no first hit of the off-line defect in the upper tail. Namely,

$$D_{T_0}\left(T\right)=0(T\ge T_0).$$

Proof. 

This follows immediately from Lemma 6.143. □

Definition 6.145 

(RH above $T_0$). $\mathrm{RH}\mathrm{above}\left(T_0\right)$ means that, for every nontrivial zero

$$\rho =\beta +i\gamma ,$$

one has

$$\gamma >T_0⟹\beta ={\displaystyle \frac{1}{2}}.$$

Theorem 6.146 

(RH above $T_0$).

$$\mathrm{RH}\mathrm{above}\left(T_0\right)$$

holds.

Proof. 

By Theorem 6.144,

$$D_{T_0}\left(T\right)=0(T\ge T_0).$$

If there existed an off-line zero

$$\rho =\beta +i\gamma ,\beta \ne {\displaystyle \frac{1}{2}},$$

with $\gamma >T_0$, then taking $T=\gamma$ would give

$$D_{T_0}\left(\gamma \right)\ge m_{\xi }\left(\rho \right)>0.$$

This contradicts Theorem 6.144. Therefore all nontrivial zeros with $\gamma >T_0$ satisfy

$$Re\rho ={\displaystyle \frac{1}{2}}.$$

□

Corollary 6.147 

(vanishing of anchored interval defects). For every regular anchored Gram lattice

$$\mathcal{G}\left(T_0\right)={\left\{g_n^{\left(T_0\right)}\right\}}_{n\ge 0},$$

one has

$$d_n^{\left(T_0\right)}=0(n\ge 0).$$

Proof. 

By Theorem 6.144,

$$D_{T_0}\left(T\right)=0(T\ge T_0).$$

By Definition 6.130,

$$d_n^{\left(T_0\right)}=D_{T_0}\left(g_{n+1}^{\left(T_0\right)}\right)-D_{T_0}\left(g_n^{\left(T_0\right)}\right).$$

The right-hand side is

$$0-0=0.$$

Therefore

$$d_n^{\left(T_0\right)}=0(n\ge 0).$$

□

Theorem 6.148 

(Riemann Hypothesis). For every nontrivial zero

$$\rho =\beta +i\gamma ,$$

one has

$$\beta ={\displaystyle \frac{1}{2}}.$$

That is, the Riemann Hypothesis holds.

Proof. 

By Corollary 6.141, all nontrivial zeros of the completed zeta function $\xi$ lie on the critical line. Therefore, for every nontrivial zero

$$\rho =\beta +i\gamma ,$$

one has

$$Re\rho =\beta ={\displaystyle \frac{1}{2}}.$$

This is precisely the Riemann Hypothesis. □

Remark 6.149 

(logic of the spectral closure). The argument of this section connects the spectral localization on the regularized-determinant side with the nonnegative-integer-valued defect staircase of Section 6.6. From the reality of the eigenvalues of the self-adjoint Hilbert–Schmidt operator K, the zeros of $F_K$ are localized on the critical line. By the global uniqueness of Section 6.4,

$$F_K=\xi ,$$

the zeros of $\xi$ are identified with the same critical-line zeros. Therefore the off-line defect staircase is identically zero, and the Riemann Hypothesis follows.

7. Conclusions

This section records the logical combination of the preceding constructions. The main theorem has already been proved as Theorem 6.148; no new assumptions, external inputs, or additional comparison principles are introduced here.

Section 2 constructs the analytic operator setting. The weighted Hilbert space, the quadratic form $q_R$, its admissible core, the closed form, the associated self-adjoint operator $A_R$, and the positive shifted operator $L=A_R+I$ are fixed there. The compact embedding of the form domain and the compact-resolvent reference operator give a purely discrete spectral framework. The scalar Herglotz-type resolvent function provides the analytic resolvent data used later in the kernel and determinant constructions.

Section 3 constructs the coefficient-space arithmetic data. The formal Dirichlet algebra, the completed augmentation ideal, the exact prime indicator, the exact von Mangoldt lift

$${\Lambda }^{\mathrm{ex}},$$

the composite-cancellation operator, and the arithmetic derivation are defined in that section. The coefficient Hilbert space and the weighted diagonal arithmetic trace then provide an exact Hilbert-space evaluation of the prime-power contribution. These arithmetic objects are not used in Section 4; they are combined with the singular-boundary data only in the orthogonal-decomposition framework of Section 5.

Section 4 constructs the singular-boundary data inside the analytic Hilbert-space setup. The Gelfand triple

$${\mathcal{D}}_R\subset H_{\alpha ,+}\subset {\mathcal{D}}_R^{\prime }$$

is fixed, and point evaluations, singular boundary traces, distribution kernels, and boundary forms are separated by type. The boundary parameter space, zero-area singular locus, trace maps, support maps, and regular boundary trace-mass functionals are constructed. The boundary bilinear form and the cancellation identity remove the regular boundary trace-mass contribution while preserving the singular-boundary component. The one-sided singular-boundary subspace $K_R^{+}$, the projection ${\Pi }_R^{+}$, the Friedrichs-type realization, the singular-boundary transport group, the anti-self-adjoint generator, and the boundary-distribution kernel representation are thereby obtained. These objects supply the analytic boundary data used in Section 6.

Section 5 places the arithmetic and singular-boundary constructions in the common ambient Hilbert space X. The one-sided singular-boundary subspace $K_R^{+}\subset H_{\alpha ,+}$ is embedded as

$${\mathcal{K}}_R=J_{\mathrm{an}}{K}_R^{+}\subset X,$$

and the ambient projection is

$${\Pi }_R=J_{\mathrm{an}}{\Pi }_R^{+}{J}_{\mathrm{an}}^{*}.$$

Together with the arithmetic embedding, this gives the orthogonal decomposition

$$X={\mathcal{K}}_R\oplus J_{\mathrm{arith}}{\mathcal{H}}_{\mathrm{arith}}\oplus Ran{\Pi }_{\mathrm{res}}.$$

For localized finite-window comparison data, the arithmetic contribution is evaluated by the weighted diagonal arithmetic trace, and the residual component is removed by passing to the canonical representative

$$x^{♯}=({\Pi }_R+{\Pi }_{\mathrm{arith}}^X)x.$$

Thus

$${\Pi }_{\mathrm{res}}{x}^{♯}=0,{\Pi }_R{x}^{♯}\in {\mathcal{K}}_R,$$

and the remaining ${\mathcal{K}}_R$-projected component is represented in the singular-boundary subspace ${\mathcal{K}}_R$.

Section 6 converts this residual-free comparison interface into an analytic determinant identity. The singular boundary trace and the continuous extension of the localized comparison interface give the map

$${\mathcal{D}}_R\stackrel{{Tr}_{\partial ,R}}{\to }{\mathcal{D}}_{R,\mathrm{adm}}^{\prime }\stackrel{{LCI}_R}{\to }X\stackrel{{\Pi }_R}{\to }{\mathcal{K}}_R.$$

The functional equation first gives the boundary reflection

$${\Theta }_R:{\mathcal{D}}_R^{\prime }\to {\mathcal{D}}_R^{\prime }.$$

Definition 6.34, Lemma 6.35, and Lemma 6.37 establish its involutive, pairing-preserving, admissibility-preserving, and residual-free compatibility properties. Proposition 6.38 then descends this reflection to a bounded self-adjoint involution

$${\mathcal{S}}_R:{\mathcal{K}}_R\to {\mathcal{K}}_R.$$

The signed boundary-distribution comparison kernel is

$${\mathfrak{k}}_R(f,g)={〈{\mathcal{S}}_R{\Pi }_R{\mathcal{J}}_{R}f,{\Pi }_R{\mathcal{J}}_{R}g〉}_X.$$

Lemma 6.42 shows its Hermitian symmetry using only ${\mathcal{S}}_R^{*}={\mathcal{S}}_R$ and the Hilbert-space inner product. The smoothing estimates of Theorem 6.50 and Theorem 6.51, together with Proposition 6.52, yield the self-adjoint Hilbert–Schmidt realization

$$K=K^{*},K\in {\mathfrak{S}}_2$$

in Theorem 6.54. Remark 6.43 records that this construction uses the functional equation, the boundary-distribution framework, and the orthogonal projections, but not any zero-location information or RH-equivalent positivity assumption.

From K, Definition 6.57 and Definition 6.62 define

$$F_K\left(s\right)=e^{a_K+b_K(s-{\displaystyle \frac{1}{2}})}{det}_2\left(I+i(s-\frac{1}{2})K\right).$$

The constants $a_K,b_K$ fix only the central value and first logarithmic derivative, as recorded in Remark 6.63. Lemma 6.64 proves that $F_K$ is entire, and Theorem 6.68 collects the trace-ideal determinant output of Section 6.3.

The determinant identity is obtained in Section 6.4 through the central Cauchy–Laplace comparison. Definition 6.75 fixes the central kernel, Definition 6.76 fixes the raw finite-window cutoffs, Definition 6.77 fixes the central finite-jet map, and Definition 6.78 fixes the finite-dimensional principal-part space, universal principal-part map, and principal-part embedding into the pre-completion test-input space. Definition 6.79 then defines the regularized finite-window input by the algebraic subtraction of the embedded counterterm inside ${\mathcal{C}}_{\mathrm{cen}}^0$. Lemma 6.80 shows that this counterterm is the common local principal part for the Archimedean, arithmetic-trace, and singular-boundary contributions; Lemma 6.81 shows that this finite-jet subtraction is controlled by the defining seminorms. Definition 6.83 fixes the topology of ${\mathcal{C}}_{\mathrm{cen}}$, Definition 6.84 records the resulting central finite-part operation, and Remark 6.85 records that this regularization does not define the conclusion $F_K\equiv \xi$.

Lemma 6.88 proves the convergence

$${\Psi }_{w,M}^{\mathrm{fw}}⟶{\Psi }_w^{\mathrm{cen}}\mathrm{in}{\mathcal{C}}_{\mathrm{cen}},$$

locally uniformly in w, and Theorem 6.98 proves the continuity of the ${\mu }_L$- and ${\mu }_{\xi }$-pairings on ${\mathcal{C}}_{\mathrm{cen}}$. Lemma 6.100 gives the finite-window residual-free equality for the central cutoffs. Therefore Theorem 6.101 passes to the limit and obtains

$$\langle {\mu }_L,{\Psi }_w^{\mathrm{cen}}\rangle =\langle {\mu }_{\xi },{\Psi }_w^{\mathrm{cen}}\rangle .$$

The two sides of this equality are identified separately. Lemma 6.92 identifies the K-side central transform with

$${\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}\right),$$

while Lemma 6.90 identifies the zeta-side central transform with

$${\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}+w\right)-{\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}\right).$$

The zeta-side identification uses the standard Hadamard product of the completed zeta function and does not assume RH. Theorem 6.102 then gives

$${\displaystyle \frac{d}{dw}}log{F}_K\left({\displaystyle \frac{1}{2}}+w\right)={\displaystyle \frac{d}{dw}}log\xi \left({\displaystyle \frac{1}{2}}+w\right).$$

Lemma 6.104 uses the normalization $F_K(1/2)=\xi (1/2)\ne 0$ to obtain local analytic equality. The identity theorem, recorded in Theorem 6.105, gives the global identity

$$F_K\left(s\right)\equiv \xi \left(s\right)$$

in Theorem 6.106.

The remaining step is spectral localization. Since $K=K^{*}$ is compact, its nonzero eigenvalues are real. Hence a zero of

$${det}_2\left(I+i(s-\frac{1}{2})K\right)$$

has the form

$$1+i(s-\frac{1}{2}){\lambda }_j=0,{\lambda }_j\in \mathbb{R}\setminus \left\{0\right\},$$

or equivalently

$$s={\displaystyle \frac{1}{2}}+{\displaystyle \frac{i}{{\lambda }_j}}.$$

The exponential factor in $F_K$ has no zeros, so all zeros of $F_K$ lie on the critical line. Since $F_K=\xi$, the same zero configuration holds for the completed zeta function. Hence all nontrivial zeros of $\xi$ lie on the critical line, and the Riemann Hypothesis follows.

Remark 7.1 

(role of the finite-window material). The finite-window bridge and the anchored defect staircase of Section 6.5 and 6.6 do not add an independent hypothesis to the determinant argument and are not used to prove $F_K\equiv \xi$. They record, in finite-window form, the zero configuration obtained from the global identity $F_K=\xi$ and the spectral localization of the self-adjoint Hilbert–Schmidt operator K. The closure of the main proof is provided by the trace-ideal realization of Section 6.3, the central comparison and global uniqueness argument of Section 6.4, and the spectral localization in Section 6.7.