bibkey: clason2021regularization authors: Christian Clason year: 2021 title: Regularization of Inverse Problems doi: null url: https://arxiv.org/abs/2001.00617v2 claim: The linear Tikhonov normal equation applies to the actual compact theta-translation map in the negative-edge metric. Fitted-source convergence is distinct from coefficient convergence and from a uniform projection error rate. strata_touched: [] license: citation-only triage: anchor

Regularization of Inverse Problems

Source contract

The inspected arXiv v2 lecture notes identify the version on the title page. Section 6, printed pp.58–60, Lemma 6.3 and equation (6.1), give the normal equation for arbitrary Hilbert-space data $y$ and positive $\alpha$:

$$(L^{\ast }L+\alpha I)x_{\alpha }=L^{\ast }y.$$

The section uses the filter $1/(\lambda +\alpha )$ and residual $\alpha /(\lambda +\alpha )$. Theorem 5.6, printed p.48, asserts coefficient convergence only for data in the pseudoinverse domain; Corollaries 6.1–6.2, printed p.59, require source conditions for rates. Neither supplies an unconditional coefficient limit or a rate for the project’s theta inputs. The inspected PDF SHA-256 is 696b4f580464ec0e1da6a1863b91d9142aa58b8b841b8ea437b5944152cd649e. The generic normal equation, filters and resolvent identity are reused.

The following is a conditional paper application under the actual theta form, critical-family and negative-edge metric premises. Use its $c_{\ast }=11/2500$ on the centered even space ${\mathcal{H}}_0$, $B=C_{-}^{\ast }{C}_{-}$ and closed critical source $N$. The source supplies no arithmetic half-bound, original-operator spectral estimate or Lean certification. This application preserves the original $\nu$ and both edge maps. It constructs a regularized coefficient problem, without claiming a certified finite numerical solution.

Actual theta translations generate the critical source

Take r=1/16, I=[-r,r] with Lebesgue measure, and define

$$w_t(x)=\frac{\Phi (x+t)+\Phi (x-t)}{2\Phi (x)}-\cosh (t/2).\text{\hspace{1em}(R1)}$$

For real t in I, w_t is even and nu-centered: shift the two integrals against cosh(x/2). On the complex box |Re z|,|Im z|<1/8, the inherited theta series and positive real first-term lower bound give a common squared-ratio majorant on either tail,

$$C{e}^{5|x|}\exp \{-\pi [2e^{-1/4}\cos (1/4)-1]e^{2|x|}\}.$$

The bracket is positive (e^-1/4>=3/4 and cos(1/4)>=31/32). On compact x intervals Phi has a positive minimum. Thus z->w_z is an H_0-valued holomorphic function on a neighborhood of I, and

$$w_t=\sum _{k\ge 1}\frac{t^{2k}}{(2k)!}{v}_k,\overline{\mathrm{span}\{w_t:t\in I\}}=N.\text{\hspace{1em}(R2)}$$

The first inclusion uses norm convergence of the Taylor series; the reverse uses Hilbert-valued difference quotients at zero. This does not use all real translations or a characterization by the complete zero set.

Define F:L2(I)->H_0 by Fq=integral_I w_t q(t)dt. It is Hilbert–Schmidt, its range closure is N, and every actual Fq lies in N and hence in the accepted form domain. Its adjoint is Fg(t)=<w_t,g> with the usual inner product antilinear in the first variable. The positive compact coefficient Gram K=FBF has the actual kernel

$$\kappa (s,t)=⟨C_{-}{w}_s,C_{-}{w}_t⟩=\frac{1}{4}\iint a(x,y)\overline{\Delta w_s(x,y)}\Delta w_t(x,y)d\nu (x)d\nu (y).\text{\hspace{1em}(R3)}$$

An explicit regularized common-source correction

For epsilon>0 and h in the actual form domain, h_0=h-nu(h)1, solve the bounded-window Fredholm normal equation

$$(K+\epsilon I)q_{\epsilon }=F^{\ast }B{h}_0,n_{\epsilon }=F{q}_{\epsilon }.\text{\hspace{1em}(R4)}$$

The coefficient inverse has norm<=1/epsilon and involves no exact P_N. It is the standard Tikhonov fit for L=B^(1/2)F and datum g=B^(1/2)h_0. Its fitted signal, not necessarily its coefficient vector, converges to P_M g, M=B^(1/2)N (closed because B>=c_*I on H_0). Thus n_epsilon converges in H_0 and in both critical edge norms to the same n_h=G^-1 P_N Bh_0 from G4. Coefficients q_epsilon need not converge: g’s projection may lie only in the closure of Ran L. Each n_epsilon belongs to N, so q(h-n_epsilon)=q(h) exactly.

For A=L L*, with spectral measure E_A, its remaining paired-edge error is exactly

$$\Vert C_{\pm }(n_h-n_{\epsilon }){\Vert }^2={\int }_{(0,\Vert A\Vert ]}{\left(\frac{\epsilon }{\lambda +\epsilon }\right)}^{2}d⟨E_A(\lambda )g,g⟩.\text{\hspace{1em}(R5)}$$

The zero eigencomponent is excluded. Since M is infinite dimensional while A is compact, eigenvalues on M tend to zero. The fitted-filter error has operator norm 1 on M for every epsilon>0. Strong convergence therefore provides no uniform rate over all unit inputs, and no actual source condition or small-spectral-value mass bound has been proved for the required input class. The source’s Theorem 5.6 applies only on the pseudoinverse domain; it is not used to infer bounded convergence of these coefficient vectors.

A finite-epsilon approximation bound with the same source

Suppose F_m maps L2(I) into a finite span of actual w_t, ||F-F_m||<=delta, and ||F||<=M_F. Define K_m=F_m* B F_m and n_epsilon,m=F_m(K_m+epsilon I)^-1 F_mBh_0. The usual resolvent identity for A and A_m=B^(1/2)F_mF_mB^(1/2) gives

$$\Vert C_{\pm }(n_{\epsilon }-n_{\epsilon ,m})\Vert \le \frac{\delta (2M_F+\delta )}{2\sqrt{2}\epsilon }\Vert h_0\Vert .\text{\hspace{1em}(R6)}$$

Both differences lie in N, so this is a common paired edge norm bound; the source norm bound is R6 divided by sqrt(c_*). With m equal cells of I and midpoint w_t samples, one may take delta<=L_w(2r)^(3/2)/(sqrt(12)m), where L_w is an independently validated upper bound for sup_I||w’_t||. The directed small-window bounds supply numerical M_F and L_w for this interface. No finite Fredholm matrix has been certified here. R6 controls discretization for fixed epsilon; it does not pay the regularization error R5.

Finite epsilon does not erase the old exact raw Schur obstruction

Let P_epsilon=F(K+epsilon I)^-1 F*B on H_0. The classical Woodbury identity gives

$$(I-P_{\epsilon }{)}^{-1}=I+{\epsilon }^{-1}F{F}^{\ast }B.\text{\hspace{1em}(R7)}$$

Extend $P_{\epsilon }$ by zero on constants, equivalently $P_{\epsilon }h=P_{\epsilon }(h-\nu (h)1)$; the inverse in (R7) extends by identity on constants. This agrees with the ambient formula because $B1=0$. Both operators preserve the actual form domain because their correction has range in $N$ and is bounded into its form norm. Suppose an ordinary measurable kernel $S$ has a finite, strictly positive weighted absolute Schur certificate of product at most one and exactly reconstructs $C_{-}(I-P_{\epsilon })h$ from $C_{+}(I-P_{\epsilon })h$ for every original core input. Schur supplies boundedness of $S$, so core density first extends that identity to the actual form domain. Equation (R7) then gives exact reconstruction of the original raw $C_{\pm }$ there. The already accepted theta-Schur obstruction excludes this certificate. This is reuse of that obstruction, not another fiber proof. Approximate reconstruction, absolute products>1 tending to1 and better actual norm estimates remain distinct possibilities.

The concrete new interface is R1–R4: it replaces an unspecified P_N primitive by a small-window integral kernel and a regularized normal equation, with the separately controlled discretization error R6. The outstanding estimate is the actual input’s small-spectral-value error R5, or a directly certified complementary residual B(h-n)-w with w in N-perp, followed by a jointly reconstructed and norm-bounded projected transfer. No half-bound, RH/Robin proof, complete zero-set characterization or executable full-space approximation algorithm is claimed.

Numerical inputs for the fixed-parameter discretization bound

The actual-theta translation norm computation uses the existing derivative supplier and the full original probability measure, including both spatial tails. Its outward interval bounds give

$$L_w<9/2,\Vert F\Vert \le \Vert F{\Vert }_{\mathrm{HS}}<3/50,\Vert F-F_m\Vert \le \frac{3}{50m}.$$

Substitution into R6 yields, for every positive epsilon and m equal midpoint cells, the same-source bound

$$\Vert C_{\pm }(n_{\epsilon }-n_{\epsilon ,m})\Vert \le \frac{9}{5000\sqrt{2}\epsilon }\left(\frac{2}{m}+\frac{1}{m^2}\right)\Vert h_0\Vert .$$

At epsilon=1/100 and m=256, its coefficient is strictly below 1/1000. This pays the fixed-parameter discretization term only. The actual-input regularization error R5, kernel-entry quadrature, a uniform complementary residual on the required input class and the projected reconstruction/norm estimate remain unresolved. No new original-energy lower bound or Lean certification is supplied.

One fixed quadratic-input complementary residual

The direct actual-operator enclosure uses seven finite critical directions and eight individual certified real Xi-zero witnesses for h=x^2. Under the inherited actual-model and numerical-supplier premises it supplies ||B(h-n)-w||<9/8000, with n in N and w in N-perp. The accepted inverse constant c*=11/2500 converts this to ||n_h-n||<=45/176 and paired edge errors <=9/(160 sqrt(11)). The 512 retained point rows are reused; derivative transport, all theta series and full spatial tails are paid. This is a bounded-B input calculation, without asserting original form-domain membership of x^2. Its upper enclosure does not reach 1e-4 and is not a lower bound on attainable residuals. It supplies neither R5 on the required input class nor the projected transfer/norm comparison, original half-bound, Robin or RH, and has no new Lean certification.

The real translation boundary retains fixed sharp-low mass

This is a conditional paper application of the original theta tail, (R1)–(R5), and the accepted actual-model premises above. It keeps the small-window generated space $N$ and the original measure. The theta asymptotic, analytic identity principle, weak translation convergence and Fourier projection tools are reused; no new generic regularization theorem, numerical producer, Lean certification or originality claim is supplied.

Put $v_0(x)=\sqrt{2\Phi (x)\cosh (x/2)}$, $Uh=v_{0}h$ and $N_c=UN$. The complex theta series is normally convergent on $|\Im z|<\pi /4$. For the translation parameter define

$$t_c=\frac{\log 2}{2},\mathcal{D}=\{z:|\Im z|<\pi /4,2e^{-2|\Re z|}\cos (2\Im z)>1\}.$$

This domain is connected, contains $I$ and has real section $(-t_c,t_c)$. For each compact $E\subset \mathcal{D}$, the normally convergent series and the positive real first-term denominator give on either spatial tail

$$|U{w}_z(x){|}^2\le C_E{e}^{5|x|}{e}^{-\pi k_E{e}^{2|x|}}+C_E{v}_0(x{)}^2,k_E=\underset{z\in E}{\min }(2e^{-2|\Re z|}\cos (2\Im z)-1)>0.$$

On compact spatial intervals use the positive minimum of $\Phi$. This common integrable majorant makes $z\mapsto w_z$ Hilbert-valued holomorphic. Its projection onto $N^{\perp }$ vanishes on $I$ by (R2), so the analytic identity principle makes it vanish on $\mathcal{D}$. Centering and evenness extend as well. Consequently $w_t\in N$ for every real $|t|<t_c$, without enlarging the original $N$.

For $0<t<t_c$ set

$$\begin{array}{c}{\eta }_t=2e^{-2t}-1,R_t=\frac{1}{2}\log (1/{\eta }_t),\\ G(y)=e^{5y/2-(\pi /2)e^{2y}},A_t=\pi e^{-9t/2}{\eta }_t^{-5/4}.\end{array}$$

The actual two-tail profile, as $t\uparrow t_c$, is

$${\Vert A_t^{-1}U{w}_t-G(\cdot -R_t)-G(-\cdot -R_t)\Vert }_2⟶0.\text{\hspace{1em}(E1)}$$

To pay both tails, fix $A>t_c+1$ and $t\ge t_c/2$. On $x\ge A$, the dominant numerator is $\Phi (x-t)$. The source tail comparison gives

$$\frac{v_0(x)\Phi (x-t)}{2\Phi (x)}\le C{e}^{5x/2}{e}^{-(\pi /2){\eta }_t{e}^{2x}}.$$

After $x=R_t+y$ and division by $A_t$, this is bounded by $CG(y)$ on $y\ge A-R_t$. For each fixed $y$, the original first-term relative asymptotic and $2\cosh (x/2)\sim e^{x/2}$ give convergence to $G(y)$. Extend the term by zero below $A-R_t$ and apply $L^2$ dominated convergence. On this right tail the other numerator has an ${\eta }_t$-independent integrable envelope, because $2e^{2t}-1$ is bounded away from zero. Its divided norm tends to zero, as does that of $\cosh (t/2)v_0$. The compact interval $[-A,A]$ contributes a bounded numerator divided by $A_t\to \infty$. Evenness supplies the left tail, where $\Phi (x+t)$ is dominant. The wrong-half profile tails vanish since $G\in L^2$; neither shifted numerator is asserted globally bounded.

The two full profiles have overlap tending to zero. Thus

$$\begin{array}{c}{n}_t:=\frac{U{w}_t}{\Vert U{w}_t{\Vert }_2}\in N_c,\Vert U{w}_t{\Vert }_2\sim \sqrt{2}{A}_t\Vert G{\Vert }_2,\\ {\Vert n_t-\frac{G(\cdot -R_t)+G(-\cdot -R_t)}{\sqrt{2}\Vert G{\Vert }_2}\Vert }_2⟶0,n_t\rightharpoonup 0.\end{array}\text{\hspace{1em}(E2)}$$

At real $|t|\ge t_c$, use $w_t=w_{|t|}$ and put ${\eta }_{|t|}=2e^{-2|t|}-1\le 0$. The same leading ratio has an $e^{5x/2}$ prefactor and exponent $-(\pi /2){\eta }_{|t|}{e}^{2x}$, so $w_t\notin L^2(\nu )$. The bounded centering subtraction cannot cancel it. This is a real translation-integrability boundary, not a zero ordinate or an eigenvalue.

For any fixed bandwidth $\Lambda >0$ let $P_{\Lambda }=1_{|\mathsf{D}|<\Lambda }$ on the even physical space. Compute the projection of $G$ in the full physical space, since $G$ itself is not even. Plancherel and Riemann–Lebesgue give

$$\Vert P_{\Lambda }{n}_t{\Vert }_2^2⟶{\gamma }_{\Lambda }:=\frac{\Vert 1_{|\mathsf{D}|<\Lambda }G{\Vert }_2^2}{\Vert G{\Vert }_2^2}>0.\text{\hspace{1em}(E3)}$$

The two projected translated profiles have equal norms and an oscillatory cross integral with an $L^1$ frequency density. Positivity uses $G\in L^1$, $G>0$ and its Fourier transform nonzero near zero. Also ${\gamma }_{\Lambda }\uparrow 1$ as $\Lambda \to \infty$. Hence the fixed-band restriction of the actual critical space is noncompact; this is a restricted-class interface to (R5).

A compact fit has a fixed error on the whole sharp-low sphere

Use the family (E2), the actual negative-edge metric (G2) and its already supplied positive coercivity. No inverse-constant computation is repeated. In physical coordinates,

$$\tilde{B}=UB{U}^{-1}=M_d-\mathcal{K},d(x)=\frac{1}{2}\int a(x,y)d\nu (y),\mathcal{K}(x,y)=\frac{1}{2}{v}_0(x)v_0(y)a(x,y),$$

with the same $a(x,y)$ as (G2); assign its null diagonal value zero. Since $0\le a\le 1$ and $\nu$ is a probability, $\mathcal{K}$ is Hilbert–Schmidt. For each fixed $y$, $a(x,y)\to 1$ as $|x|\to \infty$, so $d(x)\to 1/2$. Compactness makes $\mathcal{K}{n}_t\to 0$, and (E2) makes the mass on every fixed compact interval tend to zero. Therefore

$$\Vert (\tilde{B}-\frac{1}{2}I)n_t{\Vert }_2⟶0.\text{\hspace{1em}(E4)}$$

Remove the original ground state exactly. With first-slot-linear pairings, put $p_0=P_{\Lambda }{v}_0\ne 0$ and

$$b_t=P_{\Lambda }{n}_t-\frac{⟨P_{\Lambda }{n}_t,p_0⟩}{\Vert p_0{\Vert }^2}{p}_0,p_t=b_t/\Vert b_t\Vert .$$

Here $p_0\ne 0$ because $v_0$ is positive, belongs to $L^1$ and has Fourier transform nonzero at zero. Weak escape makes the removed coefficient tend to zero. For $t$ sufficiently close to $t_c$ the normalization is nonzero, and $p_t$ is an even sharp-low unit input, $⟨p_t,v_0⟩=0$, $p_t\rightharpoonup 0$, $\Vert b_t\Vert \to \sqrt{{\gamma }_{\Lambda }}$ and $⟨p_t,n_t⟩\to \sqrt{{\gamma }_{\Lambda }}$. The existing sharp-center (SC1) places every physical finite-band input in the original minimal operator domain; in $L^2(\nu )$ the corresponding input is $U^{-1}{p}_t$.

Let $\mathcal{R}p=U{n}_{U^{-1}p}$ be the exact common critical correction from (G4). It is bounded on the centered ambient space and has range in $N_c$. Its defining metric projection gives

$$⟨\mathcal{R}p,\tilde{B}z⟩=⟨p,\tilde{B}z⟩(z\in N_c).$$

With $z=n_t$, boundedness and (E4) give $⟨\mathcal{R}{p}_t,n_t⟩\to \sqrt{{\gamma }_{\Lambda }}$. For every fixed $\epsilon >0$ the original (R4) correction ${\mathcal{R}}_{\epsilon }=U{P}_{\epsilon }{U}^{-1}$ is compact, so ${\mathcal{R}}_{\epsilon }{p}_t\to 0$. Consequently

$$\begin{array}{c}\underset{t\uparrow t_c}{\mathrm{liminf}}\Vert (\mathcal{R}-{\mathcal{R}}_{\epsilon })p_t{\Vert }_2\ge \sqrt{{\gamma }_{\Lambda }},\\ \underset{\begin{array}{c}p\in P_{\Lambda }{L}_{\mathrm{even}}^2,\Vert p{\Vert }_2=1\\ ⟨p,v_0⟩=0\end{array}}{\sup }\Vert (\mathcal{R}-{\mathcal{R}}_{\epsilon })p{\Vert }_2\ge \sqrt{{\gamma }_{\Lambda }}.\end{array}\text{\hspace{1em}(E5)}$$

The paired-edge statement retains the same source. Set $z_t=(\mathcal{R}-{\mathcal{R}}_{\epsilon })p_t\in N_c$. Then $\langle z_t,\widetilde Bn_t\rangle\to \sqrt{\gamma_\Lambda}/2$ and $⟨n_t,\tilde{B}{n}_t⟩\to 1/2$. Cauchy–Schwarz in the $B$ metric, with mixed critical nullity, gives

$$\underset{t\uparrow t_c}{\mathrm{liminf}}\Vert C_{\pm }{U}^{-1}{z}_t\Vert \ge \sqrt{{\gamma }_{\Lambda }/2}.\text{\hspace{1em}(E6)}$$

This applies to the whole fixed infinite-dimensional sharp-low centered sphere for each positive regularization parameter. The source norm argument also tests any fixed compact source approximation into the ambient space. Extending the paired-edge conclusion requires the approximation’s range in $N_c$ with the inherited form-domain membership, as satisfied by ${\mathcal{R}}_{\epsilon }$. It does not contradict strong convergence on each individual input, or give this lower bound on a finite frame or a spatially restricted source class. Uniform accuracy on the stated sphere needs a noncompact approximation mechanism or a source restriction excluding this escaped family. This interface determines a boundary of the bounded-window fit; it supplies no divergence result for the actual projected inverse, no original all-input half-bound, and no signs on a common cofinal sequence. Full Robin and RH remain unresolved.

A bounded synthesis from the actual endpoint translations

This conditional paper interface retains the original $N$, $\nu$, unitary $U$, domain and common correction from (R1)–(R6), (E1)–(E6) and (G1)–(G4). The original theta tail supplies the additional quantitative step below. Gamma, convolution, Hilbert–Schmidt and compact-class convergence facts are reused. No new generic theorem, numerical producer, Lean certification or originality claim is made.

For sufficiently large $R_0$ and $R\ge R_0$ put

$$t(R)=\frac{1}{2}\log \frac{2}{1+e^{-2R}},n_R=n_{t(R)},E_R(x)=\frac{G(x-R)+G(-x-R)}{\sqrt{2}\Vert G{\Vert }_2}.$$

The original two-term theta tail improves (E2) to

$$\Vert n_R-E_R{\Vert }_2\le C{e}^{-R}(R\ge R_0).\text{\hspace{1em}(NC1)}$$

To verify the rate, fix $A>t_c+1$. Write on the positive tail $\Phi (x)=4{\pi }^2{e}^{9x/2-\pi e^{2x}}\ell (x)$, where the supplied two-term expansion gives $\ell (x)=1+O(e^{-2x})$. Uniformly for $t_c/2\le t<t_c$ and $x\ge A$, the dominant physical ratio equals

$$\frac{v_0(x)\Phi (x-t)}{2\Phi (x)}=\pi e^{-9t/2}{e}^{5x/2-(\pi /2){\eta }_t{e}^{2x}}\frac{\sqrt{1+e^{-x}}\ell (x-t)}{\sqrt{\ell (x)}}.$$

The last factor is $1+O(e^{-x})$. After $x=R+y$ and division by $A_{t(R)}$, its error is at most $C{e}^{-R}G(y)e^{-y}$, whose $L^2(dy)$ norm is finite. The compact-region, centering and wrong-shift terms in (E1) have divided norm $O(e^{-5R/2})$; the wrong-half ideal profile has that bound as well. Evenness pays the other spatial tail. The overlap integrand of the two ideal profiles is

$$e^{-5R}\exp \{-\frac{\pi }{2}(e^{2(x-R)}+e^{-2(x+R)})\}.$$

Its integral is $O((1+R)e^{-5R})$: on $x\ge 0$ drop the second positive exponential and split $x-R$ at zero, then reflect. Normalization therefore preserves the $O(e^{-R})$ error. This uses the actual original-series remainder, without differentiating an asymptotic or inferring a rate from (E2) alone.

For compactly supported coefficients in $L^2((R_0,\infty ),dR)$ define $S_{n}h={\int }_{R_0}^{\infty }{n}_{R}h(R)dR$, and define $S_E$ with the $E_R$ columns. Extending $h$ by zero, Young’s convolution bound and reflection make $S_E$ bounded. Equation (NC1) gives

$$\Vert S_n-S_E{\Vert }_{\mathrm{HS}}^2\le {\int }_{R_0}^{\infty }{C}^2{e}^{-2R}dR=\frac{1}{2}{C}^2{e}^{-2R_0}.\text{\hspace{1em}(NC2)}$$

Thus $S_n$ extends boundedly to all such $L^2$ coefficients. Finite coefficient truncations have range in the closed $N_c$, so its full range is in $N_c$. The infinite integral means this bounded extension; pointwise absolute integrability for every coefficient is not assumed. No lower frame bound or completeness for all of $N_c$ is asserted.

Use the unitary angular-frequency convention $\hat{f}(\xi )=(2\pi {)}^{-1/2}\int f(x)e^{-i\xi x}dx$. The Euler substitution $v=(\pi /2)e^{2y}$ gives

$$\hat{G}(\xi )=\frac{1}{2\sqrt{2\pi }}(\pi /2{)}^{-(5/4-i\xi /2)}\Gamma (5/4-i\xi /2).\text{\hspace{1em}(NC3)}$$

The classical Gamma nonvanishing theorem gives a positive minimum of $|\hat{G}|$ on each fixed compact band. Define the bounded coefficient map $W_{\Lambda }$ on sharp-low inputs by $\widehat{W_{\Lambda }p}=\hat{p}/(\sqrt{2\pi }\hat{G})$. Then $G\ast (W_{\Lambda }p)=p$. The coefficient map need not preserve evenness, and its norm is not asserted uniformly bounded as $\Lambda$ grows. These are applications of the existing Gamma and Fourier tools, not a new Wiener theorem.

The actual low residual is Hilbert–Schmidt

Let $J_{+}$ restrict the coefficient line to $(R_0,\infty )$. Define on the whole even sharp-low space

$${\mathcal{L}}_{\Lambda }p=\sqrt{2}\Vert G{\Vert }_2{S}_n{J}_{+}{W}_{\Lambda }p,K_{\Lambda }p=p-{\mathcal{L}}_{\Lambda }p.\text{\hspace{1em}(NC4)}$$

The principal map is bounded and has range in the original $N_c$. For $h=W_{\Lambda }p$, compare its ideal version on $x>0$ with $p(x)={\int }_{\mathbb{R}}G(x-R)h(R)dR$. The residual has a missing coefficient kernel $G(x-R)$ for $R<R_0$, and a reflected-tail kernel $G(-x-R)$ for $R>R_0$. Their full squared kernel integrals are

$$\begin{array}{rl}{\int }_{x>0,R<R_0}|G(x-R){|}^{2}dxdR& ={\int }_{y>-R_0}(y+R_0)|G(y){|}^{2}dy<\infty ,\\ {\int }_{x>0,R>R_0}|G(-x-R){|}^{2}dxdR& ={\int }_{y<-R_0}(-y-R_0)|G(y){|}^{2}dy<\infty .\end{array}$$

Evenness transports this residual to the negative half-line. The actual-minus-ideal synthesis is Hilbert–Schmidt by (NC2). Composing these maps with bounded $W_{\Lambda }$ proves that $K_{\Lambda }$ is Hilbert–Schmidt on the entire even sharp-low space. This supplies the original-theta interface; a generic compactness theorem alone does not supply these kernels or their range in $N_c$.

For the actual centered class set ${\mathcal{E}}_{\Lambda }=P_{\Lambda }{L}_{\mathrm{even}}^2\cap v_0^{\perp }$. Each ${\mathcal{L}}_{\Lambda }p$ is exactly ground-orthogonal, since each $n_R$ is. For $p\in {\mathcal{E}}_{\Lambda }$, $K_{\Lambda }p$ is centered and lies in the original form domain: (SC1) admits $p$, and the accepted critical-domain premise admits ${\mathcal{L}}_{\Lambda }p\in N_c$ after transport by $U^{-1}$.

A corrected fit converges uniformly at each fixed band

The actual correction $\mathcal{R}$ fixes $N_c$. Clason’s normal equation accepts arbitrary Hilbert data; its $B$-metric contraction and the existing $cI\le B\le I/2$ on the centered space give ${\sup }_{\epsilon >0}\Vert {\mathcal{R}}_{\epsilon }\Vert \le (2c{)}^{-1/2}$. The known strong convergence consequently extends from the dense form domain to all centered ambient inputs. Reuse that convergence on the compact image of the $K_{\Lambda }$ unit ball, rather than on the entire original unit ball.

Define the corrected common-source fit on ${\mathcal{E}}_{\Lambda }$ by

$$\begin{array}{rl}{\mathcal{R}}_{\Lambda ,\epsilon }p& ={\mathcal{L}}_{\Lambda }p+{\mathcal{R}}_{\epsilon }{K}_{\Lambda }p,\\ \mathcal{R}p-{\mathcal{R}}_{\Lambda ,\epsilon }p& =(\mathcal{R}-{\mathcal{R}}_{\epsilon })K_{\Lambda }p.\end{array}\text{\hspace{1em}(NC5)}$$

For each fixed $\Lambda >0$, the right side tends to zero in operator norm as $\epsilon \downarrow 0$. It also tends to zero in Hilbert–Schmidt norm by the standard strong-times-Hilbert–Schmidt convergence fact. Both terms of the fit lie in the same $N_c$. The existing common critical-edge bound therefore gives

$$\underset{\begin{array}{c}p\in {\mathcal{E}}_{\Lambda }\\ \Vert p{\Vert }_2=1\end{array}}{\sup }\Vert C_{\pm }{U}^{-1}(\mathcal{R}p-{\mathcal{R}}_{\Lambda ,\epsilon }p)\Vert \le \frac{1}{\sqrt{2}}\Vert \mathcal{R}-{\mathcal{R}}_{\Lambda ,\epsilon }{\Vert }_{{\mathcal{E}}_{\Lambda }\to L^2}⟶0.\text{\hspace{1em}(NC6)}$$

The principal map is noncompact, consistently with (E5). Replacing its synthesis by any fixed finite $R$ interval makes it compact and cannot preserve this full-sphere uniform conclusion. The construction is an infinite-source representation, without an all-input finite acquisition algorithm or an effective regularization rate. Constants depend on the band; no growing-band or common cofinal error estimate is supplied. The full residual comparison, original all-input half-bound, actual joint cofinal signs, full Robin and RH remain unresolved. This is conditional paper analysis with no new Lean certification or originality claim.

Actual endpoint jets in the original form

Under the same original-theta, whole-critical-space minimal-domain, mixed-nullity and coercive negative-edge premises as (NC1)–(NC6), the original series supplies spatial and parameter jets of the normalized endpoint columns. This is a conditional paper application of that series and the existing global form comparison (WF2), without a new generic regularization theorem, numerical producer, Lean certification or originality claim.

Retain $t(R)$, $n_R$, $E_R$ and sufficiently large $R_0>0$. For $j,\ell \in \{0,1\}$ there are finite original-series constants $C_{j,\ell }$ such that

$$\Vert {\partial }_x^j{\partial }_R^{\ell }(n_R-E_R){\Vert }_2\le C_{j,\ell }{e}^{-R},R\ge R_0.\text{\hspace{1em}(J1)}$$

To pay the derivatives, fix $A>t_c+1$ and write $\Phi (x)=4{\pi }^2{e}^{9x/2-\pi e^{2x}}{\ell }_{\theta }(x)$ on $x\ge A-t_c$. The first theta term gives ${\ell }_{\theta }(x)=1-(3/(2\pi ))e^{-2x}$ plus the terms with $n\ge 2$. Apply the supplied derivative polynomials to these terms before removing the first exponential. Their normally convergent series gives, for $0\le m\le 2$,

$$|{\ell }_{\theta }^{(m)}(x)-1_{m=0}|\le D_m{e}^{-2x},\underset{x\ge A-t_c}{\inf }{\ell }_{\theta }(x)>0.$$

The polynomial factors $e^{2mx}$ in the remaining terms are absorbed by $e^{-3\pi e^{2x}}$. This differentiates the original series, rather than an asymptotic remainder.

Put $A_R=\pi e^{-9t(R)/2}{e}^{5R/2}$, $d_R=A_R^{-1}U{w}_{t(R)}$ and $p_R=G(\cdot -R)+G(-\cdot -R)$. On the positive tail the dominant term of $d_R$ is exactly $G(x-R)b(x,t(R))$, where

$$b(x,t)=\frac{\sqrt{1+e^{-x}}{\ell }_{\theta }(x-t)}{\sqrt{{\ell }_{\theta }(x)}},t^{\prime }(R)=\frac{e^{-2R}}{1+e^{-2R}}.$$

For $j=0,1$ the same series bounds give $|{\partial }_x^j(b-1)|\le D{e}^{-x}$ and $|{\partial }_x^j{\partial }_{R}b(x,t(R))|\le D{e}^{-2R-2x}$. After $y=x-R$, the errors are bounded by $e^{-R}$ times finite linear combinations of $e^{-y}G(y)$, $e^{-y}{G}^{\prime }(y)$ and $e^{-y}{G}^{\prime \prime }(y)$, all in $L^2$. The compact-region, centering, wrong-shift and wrong-half ideal jets are $O(e^{-5R/2})$: $A_R^{\prime }/A_R$ is bounded, and the wrong shift retains a uniform positive tail exponent, including its differentiated series. Reflection pays the other spatial tail. These facts prove (J1) first with $d_R-p_R$ in place of $n_R-E_R$.

The ideal overlap and its $R$ derivative are $O((1+R)e^{-5R})$, by differentiating the overlap integrand in (NC1) and using the same two-half-line split. Set $a_0=\sqrt{2}\Vert G{\Vert }_2$ and $a_R=\Vert d_R{\Vert }_2$. The preceding jet bounds therefore give $a_R\ge a_0/2$ after increasing $R_0$, $|a_R-a_0|=O(e^{-R})$ and $|a_R^{\prime }|=O(e^{-R})$, using $a_R^{\prime }=\mathrm{Re}⟨d_R,d_R^{\prime }⟩/a_R$. Spatial derivatives commute with this scalar normalization. Substitute $n_R=d_R/a_R$ and $E_R=p_R/a_0$ to obtain (J1), including the mixed spatial–parameter derivative.

Use $\Vert v{\Vert }_{{\mathcal{F}}_c}=\Vert U^{-1}v{\Vert }_{\mathcal{F}}$ for the original minimal form norm in physical coordinates. The existing (WF2) comparison supplies

$$\Vert v{\Vert }_{{\mathcal{F}}_c}^2\le c_0\Vert v{\Vert }_2^2+c_1\Vert v^{\prime }{\Vert }_2^2,c_0=\frac{3}{2}+\Vert B_{\mathrm{prime}}\Vert +2M_2\Vert s^{\prime }{\Vert }_{\infty }^2,c_1=2M_2\Vert s{\Vert }_{\infty }^2.$$

Here $B_{\mathrm{prime}}$ is the complete prime operator in (WF1), distinct from the negative-edge metric $B$ in (R3)–(R5). Every prime power remains. The $C_{j,\ell }$ are unevaluated original-series constants, rather than numerical certificates. Ordinary spatial smoothness does not assert a source condition for the actual small-window Gram.

Truncate and sample only the actual compact residual

For $x>0$ define a kernel on the whole coefficient line by

$$k_R(x)=1_{R<R_0}G(x-R)-1_{R>R_0}G(-x-R)-a_0{1}_{R>R_0}(n_R-E_R)(x).$$

Let $Hh$ be its integral against $h(R)$ on $x>0$, reflected evenly. The missing and reflected squared kernel integrals are those in (NC4), with $G$ replaced by $G^{(j)}$ for $j=0,1$; (J1) pays the actual-error kernel. Even reflection preserves $H^1$, so $H$ is Hilbert–Schmidt into $H^1$ and hence into the original ${\mathcal{F}}_c$ by (WF2). The separate reflected columns need not have zero derivative trace at zero; they are not declared $H^2$.

For $p\in {\mathcal{E}}_{\Lambda }$, the same $W_{\Lambda }$ gives

$$K_{\Lambda }p=H{W}_{\Lambda }p=H_0{W}_{\Lambda }p,H_0=Q_{0}H,Q_0=I-|v_0⟩⟨v_0|.\text{\hspace{1em}(J2)}$$

The second equality uses the exact centering of $K_{\Lambda }p$. It does not make $Hh$ centered for arbitrary coefficients. Since the original constant has zero energy, $Q_0$ is contractive in ${\mathcal{F}}_c$.

For $T>R_0$, restrict the coefficient kernel to $[-T,T]$, giving $H_T$ and $H_{0,T}=Q_0{H}_T$. The same actual kernels imply

$$\begin{array}{rl}\Vert H_0-H_{0,T}{\Vert }_{\mathrm{HS}({\mathrm{L}}^2\to {\mathcal{F}}_{\mathrm{c}})}& \le d(T),\\ d(T)& ={\left[2\sum _{j=0}^1{c}_j{\int }_{y>T}(y-T)|G^{(j)}(y){|}^{2}dy\right]}^{1/2}\\ & +{\left[2\sum _{j=0}^1{c}_j{\int }_{y<-T}(-y-T)|G^{(j)}(y){|}^{2}dy\right]}^{1/2}\\ & +\Vert G{\Vert }_2{e}^{-T}(c_0{C}_{0,0}^2+c_1{C}_{1,0}^2{)}^{1/2}.\end{array}\text{\hspace{1em}(J3)}$$

The first integral comes from $R<-T$, the second from $R>T$, and the last term truncates the actual-minus-ideal columns. Their respective orders are super-exponential, $O(e^{-5T/2})$ and $O(e^{-T})$. This truncates the Hilbert–Schmidt residual, not the noncompact principal $S_n$.

Split $[-T,T]$ exactly at $R_0$. On its two open pieces the kernel columns are continuously differentiable in $R$ with values in ${\mathcal{F}}_c$. Equation (J1) supplies the uniform bounds

$$\begin{array}{rl}{M}_{-}& =[2(c_0\Vert G^{\prime }{\Vert }_2^2+c_1\Vert G^{\prime \prime }{\Vert }_2^2){]}^{1/2},\\ M_{+}& =M_{-}+a_0{e}^{-R_0}(c_0{C}_{0,1}^2+c_1{C}_{1,1}^2{)}^{1/2},M=\max (M_{-},M_{+}).\end{array}$$

For cells $I_i$ in these pieces, lengths ${\Delta }_i$ and midpoints $R_i$, put $r_i=Q_0{k}_{R_i}^{\mathrm{even}}$ and define

$$\begin{array}{rl}{D}_{\Lambda ,T,\mathcal{I}}p& =\sum _i{r}_i{\int }_{I_i}(W_{\Lambda }p)(R)dR,\\ \Vert K_{\Lambda }-D_{\Lambda ,T,\mathcal{I}}{\Vert }_{{\mathcal{E}}_{\Lambda }\to {\mathcal{F}}_c}& \le Q_{\Lambda }[d(T)+M{\Delta }_{\max }\sqrt{T/6}],Q_{\Lambda }=\Vert W_{\Lambda }\Vert .\end{array}\text{\hspace{1em}(J4)}$$

The columns are centered and in the original minimal form domain by (WF2). Reuse the midpoint mean-square column estimate: its Hilbert–Schmidt sampling error is at most $({\sum }_i{M}_i^2{\Delta }_i^3/12{)}^{1/2}$, bounded by the displayed mesh term because the total coefficient length is $2T$. The split at $R_0$ avoids an across-jump derivative estimate. All complex coefficients retain one actual $W_{\Lambda }$.

Equation (J4) is a joint parameter inequality in the original jet constants, band inverse norm, tail cutoff and mesh. It gives no band-independent constant. Column evaluation and integration certification remain acquisitions to perform; no sampled values or numerical producer are supplied here. If finite coefficient-functional representations are required, reuse the repository’s finite Fourier-window supplier. The infinite principal synthesis remains present.

Pay the finite remainder with the existing common-source certificate

The new kernel columns in (J4) can be used by the existing actual primal/dual residual certificate, conditionally on acquiring their simultaneous residual Gram. This reuses (G4)’s metric inverse and the common coefficient estimate; it is not a new generic Gram or projection theorem.

Write $\mathcal{B}=UB{U}^{-1}$ for the negative-edge metric on the centered physical space, with $b_{0}I\le \mathcal{B}\le I/2$. Retain the same $\mathcal{R}$ and $N_c$. Normalize the cells by

$$e_i=\sqrt{{\Delta }_i}{r}_i,z_i(p)={\Delta }_i^{-1/2}{\int }_{I_i}(W_{\Lambda }p)(R)dR,D_{\Lambda ,T,\mathcal{I}}p=\sum _i{e}_i{z}_i(p).$$

The normalized cell indicators are orthonormal in coefficient $L^2$, so $\Vert z(p){\Vert }_{{\ell }^2}\le Q_{\Lambda }\Vert p{\Vert }_2$. There is no dimension factor or independent column optimization. For each of these same $e_i$, choose an actual critical ${\eta }_i\in N_c$ and an exact dual ${\omega }_i\in N_c^{\perp }$, and put

$$b_i=\mathcal{B}(e_i-{\eta }_i)-{\omega }_i.$$

A positive Hermitian upper Gram $G_b$ must certify $\Vert {\sum }_i{z}_i{b}_i{\Vert }_2^2\le z^{\ast }{G}_{b}z$ for all complex coefficients. Individual samples and the previously acquired quadratic-input rows do not certify this different family. Each dual needs a legitimate orthogonality witness; no completeness of a real-zero family or RH is assumed.

The existing inverse $G_N=P_{N_c}\mathcal{B}{|}_{N_c}$ satisfies $G_N(\mathcal{R}{e}_i-{\eta }_i)=P_{N_c}{b}_i$ and $\Vert G_N^{-1}\Vert \le b_0^{-1}$. Retain the infinite principal map and set

$$\begin{array}{rl}{\mathcal{F}}_{\Lambda ,T,\mathcal{I}}p& ={\mathcal{L}}_{\Lambda }p+\sum _i{\eta }_i{z}_i(p),\\ \underset{\begin{array}{c}p\in {\mathcal{E}}_{\Lambda }\\ \Vert p{\Vert }_2=1\end{array}}{\sup }\Vert \mathcal{R}p-{\mathcal{F}}_{\Lambda ,T,\mathcal{I}}p{\Vert }_2& \le (2b_0{)}^{-1/2}{\delta }_{\Lambda }+\frac{Q_{\Lambda }}{b_0}\sqrt{\Vert G_b\Vert },\end{array}\text{\hspace{1em}(J5)}$$

Here ${\delta }_{\Lambda }$ is any valid (J4) upper allowance. Use the existing $\Vert \mathcal{R}\Vert \le (2b_0{)}^{-1/2}$ on $K_{\Lambda }-D_{\Lambda ,T,\mathcal{I}}$, then the same metric inverse on its finite columns. The difference is critical, so both original edge errors are bounded by the (J5) right side divided by $\sqrt{2}$. They use one source family, coefficient map and simultaneous Gram. Only the compact residual is finitely acquired.

For a growing band, sufficient conditions for the explicit (J5) upper allowance to vanish are that ${\delta }_{\Lambda }\to 0$ and $Q_{\Lambda }\sqrt{\Vert G_b\Vert }\to 0$ on the required common sequence. These are not asserted necessary conditions for actual approximation. No numerical $G_b$, selected ${\eta }_i,{\omega }_i$, effective regularization rate or finite all-input algorithm is supplied. Equation (J5) does not say that an arbitrary finite dual family can attain those residuals. The old quadratic-input experiment is not rerun, and its values are not reassigned to these columns.

Under the inherited mixed-nullity premise, $p-K_{\Lambda }p={\mathcal{L}}_{\Lambda }p$ is critical, so the exact original half-slack $\mathfrak{q}(v)=D(U^{-1}v)-\Vert Q_{0}v{\Vert }_2^2/2$ satisfies $\mathfrak{q}(p)=\mathfrak{q}(K_{\Lambda }p)$. The usual bounded-form estimate gives, for $p\in {\mathcal{E}}_{\Lambda }$,

$$|\mathfrak{q}(p)-\mathfrak{q}(D_{\Lambda ,T,\mathcal{I}}p)|\le {\delta }_{\Lambda }(2\Vert K_{\Lambda }{\Vert }_{{\mathcal{E}}_{\Lambda }\to {\mathcal{F}}_c}+{\delta }_{\Lambda })\Vert p{\Vert }_2^2.$$

This is an error bound, not positivity: a finite-rank approximation has an infinite-dimensional low kernel, and its absolute error alone cannot certify the entire low-space sign.

Equations (J1)–(J5) supply original-form residual, tail, mesh and conditional acquisition interfaces. They give no small-eigenvalue mass bound for the actual $\mathcal B^{1/2}UFF^*U^{-1} \mathcal B^{1/2}$, hence no effective rate for the old $(\mathcal{R}-{\mathcal{R}}_{\epsilon })K_{\Lambda }$. Actual residual certification, source and archimedean costs, low and complementary-low signs on one common cofinal sequence, the full half-bound, Robin, RH and Lean certification remain unresolved. Unevaluated series constants and paper reviews do not supply numerical certificates or a runtime guarantee.

The actual normalized endpoint columns have a complex parameter strip

Retain the original theta, small-window critical space, whole-critical-space minimal-domain and mixed-nullity premises, and negative-edge metric from (E1)–(E6), (NC1)–(NC6) and (J1)–(J5). The new interface is the actual normalized theta column in a complex endpoint parameter; ideal-profile analyticity alone does not supply it. This is conditional paper analysis, without a numerical constant, Lean certificate or originality claim. The theta series, derivative polynomials, analytic identity principle and global form comparison (WF2) are reused.

Fix $0<{\vartheta }_0<{\vartheta }_1<\pi /4$ and take sufficiently large $R_{\ast }$ and $R_0$. On $\Re z>R_{\ast }$, $|\Im z|<{\vartheta }_1$, define

$$\begin{array}{c}t(z)=\frac{1}{2}\mathrm{Log}\frac{2}{1+e^{-2z}},A(z)=\pi e^{-9t(z)/2}{e}^{5z/2},\\ d_z=A(z{)}^{-1}U{w}_{t(z)},p_z=G(\cdot -z)+G(-\cdot -z),a_0=\sqrt{2}\Vert G{\Vert }_2.\end{array}$$

The principal logarithm is legitimate since $\Re (1+e^{-2z})>0$. For large $R_{\ast }$, $\Re t(z)>0$, $|\Im t(z)|<\pi /4$ and

$$\Re (2e^{-2t(z)})-1=e^{-2\Re z}\cos (2\Im z)>0.$$

Also $\Re (2e^{2t(z)})-1=\Re [4/(1+e^{-2z})]-1$ is uniformly positive. Thus $t(z)$ lies in the original domain $\mathcal{D}$ from (E1). No enlargement of that integrability domain or of $N_c$ is used.

The normally convergent original theta series, its spatial derivative series and the positive real denominator $\Phi (x)$ make $d_z$ $H^1$-valued holomorphic. Uniformly on $|\Im z|\le {\vartheta }_0$,

$$\Vert d_z-p_z{\Vert }_{H^1}\le C_{{\vartheta }_0}{e}^{-\Re z}.\text{\hspace{1em}(A1)}$$

To pay this bound from the actual series, keep the positive-tail factor $b(x,t)$ used in (J1). The dominant term is exactly $G(x-z)b(x,t(z))$. The series at $x-t(z)$ has a uniformly positive exponential real part and gives the same $|{\ell }_{\theta }-1|+|{\ell }_{\theta }^{\prime }|=O(e^{-2x})$ tail bound. The factor $\sqrt{1+e^{-x}}$ remains independent of $z$. After $y=x-\Re z$, the error majorants are $e^{-\Re z}$ times linear combinations of $e^{-y}G(y-i\Im z)$ and its first spatial derivative. Their $L^2$ bounds are uniform because $\cos (2{\vartheta }_0)>0$. The compact-region, centering, wrong-shift and wrong-half terms and their first spatial derivatives have order $O(e^{-5\Re z/2})$. Reflection pays the other spatial tail. These are differentiated original-series estimates, rather than a complex extension of a real asymptotic remainder. They hold on every fixed closed strip of width less than $\pi /4$, after increasing $R_{\ast }$.

The ordinary complex $L^2$ norm is not a holomorphic normalization. Use the continuous complex bilinear scalar

$$q(z)={\int }_{\mathbb{R}}{d}_z(x{)}^{2}dx.$$

It is holomorphic and agrees with $\Vert d_R{\Vert }_2^2$ for real $R$, because those actual columns are real. The single-profile integral $\int G(x-z{)}^{2}dx$ is holomorphic on $|\Im z|<\pi /4$ and constant there by its real translation identity and the identity theorem. The ideal cross integrand is

$$G(x-z)G(-x-z)=e^{-5z}\exp \{-\pi e^{-2z}\cosh (2x)\}.$$

Its integral has modulus at most $C_{{\vartheta }_0}(1+\Re z)e^{-5\Re z}$: use $\cos (2{\vartheta }_0)>0$ and the two-half-line split from (NC1). Consequently $\int p_z^2=a_0^2+ O((1+\Re z)e^{-5\Re z})$, and (A1) with the uniformly bounded profile norms gives

$$|q(z)-a_0^2|\le C_{{\vartheta }_0}{e}^{-\Re z}.$$

Apply the same estimate on a slightly larger closed strip still narrower than $\pi /4$, and increase $R_{\ast }$ so that $|q(z)-a_0^2|<a_0^2/2$ throughout the holomorphic domain. This disk avoids zero and the negative real axis. Its principal holomorphic square root $a(z)$ therefore exists and satisfies $a(R)=\Vert d_R{\Vert }_2$ on real $R$. Set

$$n_z=d_z/a(z),E_z=p_z/a_0.$$

These columns match the actual $n_R$ and $E_R$ from (J1) and obey

$$\Vert n_z-E_z{\Vert }_{H^1}\le C_{{\vartheta }_0}{e}^{-\Re z},\underset{\begin{array}{c}\Re z>R_{\ast }\\ |\Im z|\le {\vartheta }_0\end{array}}{\sup }\Vert n_z{\Vert }_{H^1}<\infty .\text{\hspace{1em}(A2)}$$

Membership in $N_c$ follows from the original (E1) domain and survives scalar normalization. The continuous embedding (WF2) transports this holomorphy and these bounds to the original minimal form ${\mathcal{F}}_c$; the whole-critical-domain premise is unchanged. This constructs an actual zero-free normalization bridge, not an assumption that the ordinary norm has an analytic extension.

Apply existing analytic approximation to the actual compact residual

For $x>0$ extend the two separate (J2) kernel formulas by

$$k_z^{-}(x)=G(x-z),k_z^{+}(x)=-G(-x-z)-a_0(n_z-E_z)(x).$$

Even-reflect them and apply $Q_0$ to obtain the centered form columns $r_z^{-}$ and $r_z^{+}$. Choose a fixed $d>0$ with $d<{\vartheta }_0$ and $R_0-d>R_{\ast }$. Each formula is holomorphic on the radius-$d$ disks around all its real mesh centers, with a uniform ${\mathcal{F}}_c$ bound $M_{\vartheta }$. The left formula uses the strip bounds for translated $G$ at arbitrary real center; the right uses (A1)–(A2). Even reflection is bounded in $H^1$, and (WF2) and $Q_0$ preserve the form bound. The right formula extends slightly below $R_0$, and the left formula beyond it. The real coefficient integration still splits exactly at $R_0$. No analytic gluing at this jump or $H^2$ regularity is assumed.

Partition $[-T,T]$, $T>R_0$, into cells $I_i$ split at $R_0$, with lengths at most $d$. On each cell use the Taylor polynomial $P_i$ of degrees $0,\dots ,m-1$, $m\ge 1$, of the appropriate $r_z^{\pm }$ at its midpoint. Apply the standard Banach-valued Cauchy remainder on its radius-$d$ disk: the pointwise form error is at most $2M_{\vartheta }{2}^{-m}$. This is reuse of the analytic approximation theorem on the new actual columns, not a new generic interpolation result.

Define

$$D_{\Lambda ,T,\mathcal{I}}^{(m)}p=\sum _i{\int }_{I_i}{P}_i(R)(W_{\Lambda }p)(R)dR.$$

The unchanged tail allowance $d(T)$ from (J3) and the Hilbert–Schmidt column estimate give

$$\Vert K_{\Lambda }-D_{\Lambda ,T,\mathcal{I}}^{(m)}{\Vert }_{{\mathcal{E}}_{\Lambda }\to {\mathcal{F}}_c}\le Q_{\Lambda }[d(T)+2M_{\vartheta }\sqrt{2T}{2}^{-m}].\text{\hspace{1em}(A3)}$$

The parameter polynomial approximates only the compact residual; the principal ${\mathcal{L}}_{\Lambda }$ retains its infinite synthesis. Every cell uses the same actual $W_{\Lambda }$, not separately optimized coefficients.

Reuse the real orthonormal Legendre polynomials ${\psi }_{i,k}$ on each cell, extended by zero to the coefficient line. For $0\le k<m$ put

$$e_{i,k}={\int }_{I_i}{P}_i(R){\psi }_{i,k}(R)dR,z_{i,k}(p)={\int }_{I_i}{\psi }_{i,k}(R)(W_{\Lambda }p)(R)dR.$$

Then $D^{(m)}p={\sum }_{i,k}{e}_{i,k}{z}_{i,k}(p)$ and Bessel gives $\Vert z(p){\Vert }_{{\ell }^2}\le Q_{\Lambda }\Vert p{\Vert }_2$, with no dimension factor. All these columns lie in the original centered form domain. For this same family, (J5) applies with (A3) as ${\delta }_{\Lambda }$, provided an actual simultaneous complex residual Gram certifies its chosen primal and dual witnesses. Neither the old quadratic-input rows nor a separately optimal column choice certifies these new residuals.

If $d(T)\le C_d{e}^{-T}$ above a fixed threshold $T_{\ast }>R_0$, sufficient parameters for the (A3) upper allowance to be at most $\tau >0$ are

$$\begin{array}{rl}T& \ge \max \{T_{\ast },\log (2C_d{Q}_{\Lambda }/\tau )\},\\ m& \ge \max \{1,\lceil {\log }_2(4M_{\vartheta }{Q}_{\Lambda }\sqrt{2T}/\tau )\rceil \}.\end{array}\text{\hspace{1em}(A4)}$$

There exists a partition split at $R_0$ with at most $2T/d+2$ cells, hence at most $m(2T/d+2)$ columns. The usual Gamma/Stirling growth for (NC3) makes this count polynomial in $\Lambda$ and $\log (1/\tau )$ when $0<\tau \le 1$ and the original constants are fixed and certified. This sufficient column-count estimate is not an optimal acquisition-cost bound or a runtime guarantee; derivative evaluation, coefficient integrals and simultaneous Gram certification still require work.