bibkey: fukushima2011dirichlet authors: Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda year: 2011 title: Dirichlet Forms and Symmetric Markov Processes, second revised and extended edition doi: 10.1515/9783110218091 url: https://doi.org/10.1515/9783110218091 claim: The symmetric-kernel construction and regular minimal-extension theorem apply to the actual continuous-plus-prime-graph theta energy in the pole measure. They supply its minimal closed realization, not the one-half variance lower bound or equality with the maximal domain. strata_touched: [] license: bibliographic-reference-only triage: anchor

The minimal mixed theta jump form

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The source is Fukushima–Oshima–Takeda, second revised and extended edition, Example 1.2.4, printed pp.14–15, conditions (j.1)–(j.3) and equations (1.2.22)–(1.2.25), together with Theorem 3.1.2, printed p.110. The first construction permits a measurable destination kernel $j(x,dy)$, including atoms, when $m(dx)j(x,dy)$ is symmetric, its off-$\epsilon$ rates are locally integrable and its local small-jump second moment is finite. More explicitly, (j.1) is $x\mapsto j(x,X\setminus U_{\epsilon }(x))\in L_{\mathrm{loc}}^1(m)$ for every $\epsilon >0$, (j.2) is symmetry of $m(dx)j(x,dy)$, and (j.3) is the finite $|x-y{|}^2$ moment on every compact $K\times K$. Its maximal finite-energy domain is closed when it is dense in $L^2(m)$. The regular minimal-extension theorem has separate algebra, cutoff and Markov hypotheses; it does not automatically equate that extension to the maximal domain.

Theorem 3.1.2 assumes a closable nonnegative Markovian symmetric form on $L^2(X;m)$, with $X$ locally compact separable metric and $m$ a full-support positive Radon measure. Its initial domain must be a uniformly dense subalgebra of compactly supported continuous functions, admitting a nonnegative cutoff equal to one on any compact $K$ and zero outside any relatively compact open $G$ containing $K$. The conclusion is a regular minimal closed extension with that initial domain as a special standard core. The theorem’s additional locality assertion is conditional on the initial form being local and is not invoked for this jump form.

Theorem 1.3.1, printed p.20, and Corollary 1.3.1, printed p.21, equation (1.3.10), give the unique associated self-adjoint generator $A$: $D(A)\subset D[\mathcal{E}]$ and $\mathcal{E}(u,v)=(-Au,v)$ for $u\in D(A)$ and $v\in D[\mathcal{E}]$. We use the nonnegative convention $T=-A$.

The exact energy, speed and graph atoms

Use the theta-weighted even Weil interface, with $\rho (x)=2\Phi (x)\cosh (x/2)$ and $d\nu =\rho (x)dx$. The original positive smooth even $\Phi$ makes $\nu$ a full-support Radon probability, equivalent to Lebesgue measure. Put $\psi (t)=e^{-t/2}/(1-e^{-2t})$ and $w_n=\Lambda (n)/\sqrt{n}$, retaining every prime power. The actual conductance measure off the diagonal is

$$\begin{array}{rl}{J}_{\Gamma }(dx,dy)& =\Phi (x)\Phi (y)\psi (|x-y|)dxdy,\\ J_p(dx,dy)& =\sum _{n\ge 2}{w}_n\Phi (x)dx\left[\Phi (x+\log n){\delta }_{x+\log n}(dy)+\Phi (x-\log n){\delta }_{x-\log n}(dy)\right].\end{array}$$

With $J=J_{\Gamma }+J_p$, the real preform is exactly

$$D(h)=\frac{1}{2}\iint |h(y)-h(x){|}^{2}J(dx,dy)=E_{\Gamma }(h)+E_{\mathrm{prime}}(h).$$

FOT (1.2.23) has no prefactor one-half. Its conductance measure must therefore be $J_{\mathrm{FOT}}=J/2$, with $m=\nu$. The corresponding measurable destination kernel, set to zero on the diagonal, is exactly

$$j_{\mathrm{FOT}}(x,dy)=\frac{1}{4\cosh (x/2)}\left[\Phi (y)\psi (|x-y|)dy+\sum _{n\ge 2}{w}_n\left(\Phi (x+\log n){\delta }_{x+\log n}(dy)+\Phi (x-\log n){\delta }_{x-\log n}(dy)\right)\right].$$

Thus $\nu (dx)j_{\mathrm{FOT}}(x,dy)=J(dx,dy)/2$ preserves the normalization. Translation exchanges the two prime graphs, so $J$ is symmetric. The continuous part and each integrated graph marginal charge no $\nu$-null set. This is not pointwise absolute continuity of $j(x,\cdot )$: that would exclude the actual atoms.

The integrability checks retain all long edges

Romik’s theta tail gives $\Phi (x)\le C{e}^{-2|x|}$. Thus, for $t\ge 0$,

$$C_{\Phi }(t):=\int \Phi (x)\Phi (x+t)dx\le C^2(t+\frac{1}{2})e^{-2t}.$$

The actual global second jump moment is finite:

$$M_2:=\iint |x-y{|}^{2}J(dx,dy)=2{\int }_0^{\infty }{t}^2\psi (t)C_{\Phi }(t)dt+2\sum _{n\ge 2}{w}_n(\log n{)}^2{C}_{\Phi }(\log n)<\infty .$$

At zero use $C_{\Phi }(t)\le \Vert \Phi {\Vert }_2^2$ and $\psi (t)\sim (2t{)}^{-1}$; at infinity use exponential decay. For the prime part, $\Lambda (n)\le \log n$ gives the summable majorant $C(\log n{)}^3(\log n+1/2)n^{-5/2}$, without PNT or RH. Off any fixed diagonal band the total $J$ mass is finite, and the row prime sums are uniformly finite on compact $x$ sets, since $\Phi (x\pm \log n)\le C_K{n}^{-2}$ there. The Gamma row rate at the diagonal is infinite; these hypotheses concern off-diagonal mass and square differences, not a finite total Gamma rate.

In particular every compact Lipschitz $h$ has $D(h)\le \mathrm{Lip}(h{)}^2{M}_2/2$. Use compact Lipschitz functions as the uniformly dense algebra for the minimal-extension theorem: this domain is closed under unit clipping and admits the required compact-set/open-set cutoffs. Compact smooth functions are not literally closed under that clipping.

They have the same form closure here. For compact Lipschitz $h$, ordinary mollification gives compact smooth $h_{\epsilon }$ with common support in a slightly larger compact set, uniform convergence and $\mathrm{Lip}(h_{\epsilon })\le \mathrm{Lip}(h)$. The difference increments converge pointwise and their squares are bounded by $4\mathrm{Lip}(h{)}^2|x-y{|}^2$. Dominated convergence using $M_2$ gives $D(h_{\epsilon }-h)\to 0$; the $L^2(\nu )$ error also tends to zero. This verifies the smooth-core interface rather than assuming clipping preserves smoothness.

Minimal closure, constants and the even restriction

Let ${\mathcal{F}}_{\max }=\{h\in L^2(\nu ):D(h)<\infty \}$ and

$${\mathcal{F}}_{\min }={\overline{C_c^{\infty }(\mathbb{R})}}^{\Vert \cdot {\Vert }_{D,1}},\Vert h{\Vert }_{D,1}^2=\Vert h{\Vert }_{L^2(\nu )}^2+D(h).$$

The preceding smooth inclusion makes the maximal domain dense in $L^2(\nu )$. The source construction supplies a closed maximal form and a regular minimal form with ${\mathcal{F}}_{\min }\subseteq {\mathcal{F}}_{\max }$. The initial compact Lipschitz form is closable as a restriction of that closed maximal form, so the minimal-extension theorem applies. Equality of the minimal and maximal domains is not used or asserted. For $0\le \chi \le 1$ even compact smooth and one on $[-1,1]$, put ${\chi }_R(x)=\chi (x/R)$. Then

$$\Vert {\chi }_R-1{\Vert }_{L^2(\nu )}\to 0,D({\chi }_R)\le \frac{\Vert {\chi }^{\prime }{\Vert }_{\infty }^2{M}_2}{2R^2}\to 0.$$

Thus $1\in {\mathcal{F}}_{\min }$ with zero energy, including all prime tails. Reflection preserves both $\nu$ and $J$. Averaging a function with its reflection is contractive in the form norm and preserves the smooth core, hence

$${\overline{C_{c,\mathrm{even}}^{\infty }}}^{\Vert \cdot {\Vert }_{D,1}}={\mathcal{F}}_{\min }\cap L_{\mathrm{even}}^2(\nu ).$$

Complexification gives the even complex closed realization in that even Hilbert space. It is not claimed densely defined on the full non-even Hilbert space. Denote its associated nonnegative self-adjoint operator by $T$. The cited representation gives

$$h\in D(T)⟺h\in {\mathcal{F}}_{\min ,\mathrm{even}}\text{ and }D(h,g)=⟨u,g{⟩}_{\nu }\text{ for some }u\in L_{\mathrm{even}}^2(\nu )\text{ and every }g\in {\mathcal{F}}_{\min ,\mathrm{even}},Th=u.$$

The energy and speed measure are the original ones. Here the pairing is linear in its first argument and $D(h,g)$ is the polarized complex form.

This is a source application and model-specific verification, without new Lean certification or an originality claim. The source supplies no one-half Poincare constant or spectral description. The same-test target $D(h)\ge {\mathrm{Var}}_{\nu }(h)/2$, hence RH and full Robin, remains unproved. The minimal realization suffices for the compact-test target; no stronger maximal-domain core equality is a prerequisite.

Weighted Fourier cutoff and a complete finite cosine family

The sharp-band center interface uses the same even minimal realization and full operator with the later high-block gap. It supplies actual operator-domain cosine columns and an analytic approximation of their complete Schur center. A prescribed rank at most 96 pays a 1/16 center remainder at $c=3/8$; retained matrix positivity, high residuals and cofinal control remain unresolved.

Use (JE), (BT), (JF) and (JR) from the same-form exterior analysis. This alternative to the translated-kernel mesh retains the original minimal realization, full measure and complete operator. Write $A=T$ for its even operator above and $P_{\epsilon }=1_{[0,1/2-\epsilon ]}(A)$. The original-series coefficient suppliers give $s\in H^3(\mathbb{R})$, bounded $s,s^{\prime }$, and ${\sum }_{n\ge 2}\Vert b_n^{\prime }{\Vert }_{\infty }<\infty$ for

$$Uh=\sqrt{\rho }h,s=\sqrt{\Phi /(2\cosh (x/2))},b_n(x)=w_{n}s(x)s(x+\log n).$$

These are paper-level model deductions from the cited inputs. They provide a prescribed finite map with uniform form error on each whole fixed-gap spectral subspace. They do not certify a matrix sign, useful numerical size, cofinal bound, originality or new Lean result.

Transformed form and a global derivative comparison

Write $\Vert h{\Vert }_{\mathcal{F}}^2=\Vert h{\Vert }_{\nu }^2+D(h)$ and transport this norm by $U$. Formula (JE) gives exactly

$$\tilde{D}(v)=\frac{1}{2}\Vert v{\Vert }_2^2+G_{\mathrm{flat}}(sv)+c_{\Gamma }\Vert sv{\Vert }_2^2-⟨Bv,v⟩,-8<c_{\Gamma }<0,B=\sum _{n\ge 2}(M_{b_n}{\tau }_n+(M_{b_n}{\tau }_n{)}^{\ast }).\text{\hspace{1em}(WF1)}$$

Every prime power and shifted adjoint remains present. Let $M_2=\int_0^\infty t^2\psi(t)dt =\sum_{k\ge0}2/(2k+1/2)^3$. Translation differences and $|a+b{|}^2\le 2|a{|}^2+2|b{|}^2$ give, for even $v\in H^1(\mathbb{R})$,

$$\begin{array}{rl}\Vert U^{-1}v{\Vert }_{\mathcal{F}}^2& \le c_0\Vert v{\Vert }_2^2+c_1\Vert v^{\prime }{\Vert }_2^2,\\ c_0& =\frac{3}{2}+\Vert B\Vert +2M_2\Vert s^{\prime }{\Vert }_{\infty }^2,c_1=2M_2\Vert s{\Vert }_{\infty }^2.\end{array}\text{\hspace{1em}(WF2)}$$

The negative $c_{\Gamma }$ term is dropped only for this upper bound. Compact smooth even approximants, with (WF2) applied to their differences, put $U^{-1}v$ in the actual minimal form domain. Neither a compact minimum density nor an identification with a maximal domain is used.

High-frequency coercivity and the scalar deficit

Use the unitary Fourier convention. The flat multiplier is

$$\begin{array}{rl}m(\xi )& =2\sum _{k\ge 0}\frac{{\xi }^2}{a_k(a_k^2+{\xi }^2)},a_k=2k+\frac{1}{2},\\ \frac{1}{2}\log |\xi |-1& \le m(\xi )\le 16(1+\log (1+|\xi |))(|\xi |\ge 1).\end{array}\text{\hspace{1em}(WF3)}$$

The symbol is nonnegative and increasing in $|\xi |$; its upper bound also holds below one. For $X=|\xi |\ge 1$ and $K=\lfloor (X-1/2)/2\rfloor$, retaining $a_k\le X$ gives $m(X)\ge\sum_{k=0}^K1/a_k \ge\tfrac12\log(4K+5)\ge\tfrac12\log(2X)$. The complementary split and inverse-cubic tail give $m(X)\le 4+\log (2X)+2/X+1/2\le 13/2+\log (2X)$, which implies the displayed upper bound; monotonicity covers $X<1$.

Define the complete symmetric row and its comparison potential by

$$R_B(x)=\sum _{n\ge 2}(b_n(x)+b_n(x-\log n)),W(x)=\frac{1}{2}+c_{\Gamma }s(x{)}^2-R_B(x).$$

The symmetric graph inequality yields $|⟨Bv,v⟩|\le \int R_B|v{|}^2$. The coefficient estimate underlying (BT) bounds the sum of both coefficient supremum bounds outside $[-R,R]$ by $b_R^{\ast }$. Thus $R_B(x)\le b_R^{\ast }$ there; this is a row estimate, rather than an inference from the operator norm. With $p_R$ from (JL), $W\ge 1/2-d_R$ outside that interval. For $0<\epsilon \le 1/2$ set

$$\beta =\frac{1}{2}-\epsilon /2,{\mu }_{\epsilon }=\underset{x}{\sup }\frac{(\beta -W(x){)}_{+}}{s(x{)}^2}<\infty .\text{\hspace{1em}(WF4)}$$

For example, $R\ge \frac{1}{2}\log ((8/3)\log (320/\epsilon ))$ makes $d_R\le \epsilon /2$. The numerator then vanishes outside that interval, while positivity and continuity of $s$ bound the quotient inside. This deficit uses the same theta row tail as (JL), without a quantitative prime-counting remainder; it does not remove the PNT input from the earlier model construction.

If $\hat{v}$ is supported in $|\xi |\ge N$, Young’s inequality and Cauchy–Schwarz on the Fourier tail of $s$ give

$$\Vert 1_{|\mathsf{D}|<N/2}sv{\Vert }_2\le {\eta }_N\Vert v{\Vert }_2,{\eta }_N\le \sqrt{\frac{8}{3\pi }}{N}^{-3/2}\Vert s^{\prime \prime }{\Vert }_2.\text{\hspace{1em}(WF5)}$$

Indeed, the Fourier convolution bound is $(2\pi {)}^{-1/2}\Vert 1_{|\omega |>N/2}\hat{s}{\Vert }_1$; use ${\omega }^{-2}$ in Cauchy–Schwarz. Consequently

$$\tilde{D}(v)\ge \int (m(N/2)s^2+W)|v{|}^2-m(N/2){\eta }_N^2\Vert v{\Vert }_2^2.$$

For $a=1/2-\epsilon$, the conditions $m(N/2)\ge {\mu }_{\epsilon }$ and $m(N/2){\eta }_N^2\le \epsilon /4$ give a high-frequency lower bound $a+\epsilon /4$. An explicit sufficient threshold is

$$N\ge N_0(\epsilon ):=\max \left\{2e^{2{\mu }_{\epsilon }+2},\sqrt{\frac{512}{3\pi \epsilon }}\Vert s^{\prime \prime }{\Vert }_2,2\right\}.\text{\hspace{1em}(WF6)}$$

Formula (WF3) supplies the first condition. For the second, use $m(N/2)\le 16N$ for $N\ge 2$ in (WF5). This threshold is sufficient; direct symbol and leakage bounds can permit smaller bandwidths.

The direct actual-theta supplier verifies both conditions at $\epsilon =1/4$, $N=64$, without claiming that this meets the coarse displayed $N_0$ formula. The joint full-row floor further gives $\tilde{D}\ge 0.4133682545007734\Vert v{\Vert }_2^2$ on the even high-frequency restriction, above $c=3/8$. Its positive variance gap retains the complete operator and mean term. The low-frequency block and its coupling still require estimation.

The full Fourier commutator and the whole projector

Fix a real even $p\in C_c^{\infty }(\mathbb{R})$ with $0\le p\le 1$, $p=1$ on $[-1,1]$ and $p=0$ outside $[-2,2]$. Put $p_N=p(\mathsf{D}/N)$, $Q_N=I-p_N$, $L_p=\Vert p^{\prime }{\Vert }_{\infty }$ and $C_p=\int |x||{\check{p}}_0(x)|dx$, where ${\check{p}}_0$ is the inverse transform normalized by $(2\pi {)}^{-1}$. On the even core (WF1) has operator $M_{s}m(\mathsf{D})M_s+M_{1/2+c_{\Gamma }{s}^2}-B$.

Translations commute with $p_N$, so the full prime commutator has bound $2C_p{N}^{-1}{\sum }_n\Vert b_n^{\prime }{\Vert }_{\infty }$, including shifted adjoints. The potential commutator has bound $8C_p{N}^{-1}\Vert (s^2{)}^{\prime }{\Vert }_{\infty }$. For the principal double Fourier kernel put $u=\xi -\zeta$, $v=\zeta -\eta$. A nonzero cutoff difference implies $\min \{|\xi |,|\eta |\}\le 2N$, and its Lipschitz bound and (WF3) give, for $N\ge 2$,

$$m(\zeta )|p(\eta /N)-p(\xi /N)|\le \frac{32L_p(1+\log N)}{N}(1+|u|)(1+\log (1+|u|))(1+|v|)(1+\log (1+|v|)).$$

For $F(\omega)=(1+|\omega|)(1+\log(1+|\omega|)) |\widehat s(\omega)|$, Cauchy–Schwarz gives $(2\pi {)}^{-1/2}\Vert F{\Vert }_1\le \sqrt{2}\Vert s{\Vert }_{H^3}$. One sufficient integral estimate uses $\log (1+t)\le \sqrt{t}$, $(1+t{)}^2\le 2(1+t^2)$ and $(1+\sqrt{t}{)}^2\le 2(1+t)$, giving a bound $2\pi +4<4\pi$. Two Young inequalities give a principal bound with coefficient $64$; the following retains the conservative coefficient $128$:

$$\begin{array}{rl}\Vert [\tilde{A},p_N]\Vert & \le {\kappa }_N:=K_p\frac{1+\log N}{N},\\ K_p& =128L_p\Vert s{\Vert }_{H^3}^2+C_p\left(8\Vert (s^2{)}^{\prime }{\Vert }_{\infty }+2\sum _n\Vert b_n^{\prime }{\Vert }_{\infty }\right).\end{array}\text{\hspace{1em}(WF7)}$$

All coefficient hypotheses are supplied by (WC1)–(WC5) in the linked original-series note. No derivative of the separate prime diagonal or regularity assumption on unknown spectral vectors enters this estimate.

At fixed $N$, $p_N:L^2\to H^1$, and (WF2) bounds its output in the actual minimal form norm by $(c_0+4N^2{c}_1{)}^{1/2}$ times the input norm. Core approximation therefore extends the bounded form commutator to that minimal domain. The operator representation of the form gives $p_{N}D(\tilde{A})\subset D(\tilde{A})$ and the displayed bounded operator commutator there. This establishes the Fourier domain passage without an operator-core or maximal-domain identification.

Let $P=U{P}_{\epsilon }{U}^{-1}$, $\mathcal{L}=\mathrm{ran}P$ and $A_{\mathcal{L}}=\tilde{A}{|}_{\mathcal{L}}$. Restrict the closed form to even functions supported in $|\xi |\ge N$ in Fourier space, and denote its associated operator by $C_N$. Even $H^1$ functions with bounded Fourier support in that set give density. By (WF6), $C_N\ge a+\epsilon /4$. Since $Q_N\mathcal{L}\subset D(C_N)$, restriction of the full operator identity yields

$$C_N{Q}_N{|}_{\mathcal{L}}-Q_N{|}_{\mathcal{L}}{A}_{\mathcal{L}}=1_{|\mathsf{D}|\ge N}[\tilde{A},Q_N]{|}_{\mathcal{L}}.$$

Reuse the separated-spectra semigroup argument of (LP) to obtain $\Vert Q_{N}P\Vert \le 4{\kappa }_N/\epsilon$. Because $A_{\mathcal{L}}$ preserves the whole subspace, the same form estimate as (JF) then gives

$$\begin{array}{rl}\Vert Q_{N}P{\Vert }_{L^2\to \tilde{\mathcal{F}}}^2& \le {\kappa }_N^2\left(\frac{16(1+a)}{{\epsilon }^2}+\frac{4}{\epsilon }\right)\le \frac{26{\kappa }_N^2}{{\epsilon }^2},\\ \Vert (I-U^{-1}{p}_{N}U)P_{\epsilon }{\Vert }_{L^2(\nu )\to \mathcal{F}}& \le \frac{6K_p}{\epsilon }\frac{1+\log N}{N}.\end{array}\text{\hspace{1em}(WF8)}$$

This is a bound for the entire fixed-gap subspace, with no scalar eigenvalue assigned to a spectral mixture.

Spatial multiplier and prescribed finite cosine generators

Let ${\chi }_R=1-{\eta }_R$ be the real even smooth cutoff of (JR). It is one on $[-R,R]$, zero outside $[-R-2,R+2]$, between zero and one, and Lipschitz with constant at most one. Put $J={\int }_0^{\infty }\psi (t)\min \{t^2,1\}dt$. The existing short- and long-jump estimates give $J\le 3/4+2e^{-1/2}/(1-e^{-2})<9/4$. For example, $e^2>7$ and $e>8/3$ give the strict upper bound $3/4+35/24=53/24<9/4$.

The product difference yields $G_{\mathrm{flat}}({\chi }_{R}f)\le 2G_{\mathrm{flat}}(f)+2J\Vert f{\Vert }_2^2$. Use this in the same (WF1), retaining complete $B$ and dropping the negative $c_{\Gamma }$ term only for the upper bound. Since $\Vert B\Vert \le 216/5$ and $\Vert s{\Vert }_{\infty }^2\le 9/5$,

$$\begin{array}{rl}\Vert {\chi }_{R}h{\Vert }_{\mathcal{F}}^2& \le 2D(h)+\left(\frac{1}{2}+3\Vert B\Vert +(16+2J)\Vert s{\Vert }_{\infty }^2\right)\Vert h{\Vert }_{\nu }^2\\ & \le 2D(h)+167\Vert h{\Vert }_{\nu }^2\le 169\Vert h{\Vert }_{\mathcal{F}}^2.\end{array}\text{\hspace{1em}(FF1)}$$

Core approximation extends this multiplier to the actual minimal domain. Thus $\Vert M_{{\chi }_R}{\Vert }_{\mathcal{F}\to \mathcal{F}}\le 13$, independently of $R$.

Fix $0<\epsilon \le 1/2$, $0<\tau \le 1$, and choose exactly

$$R=\max \left\{2,\frac{1}{2}\log \left(\frac{8}{3}\log \frac{728}{\epsilon \tau }\right)\right\}.\text{\hspace{1em}(FF2)}$$

Then (JF)–(JR) give $|(I-M_{\chi_R})P_\varepsilon|_{\nu\to\mathcal F} \le\sqrt7,\tau/8<\tau/3$. For $N\ge N_0(\epsilon )$ put

$$X_R=\Vert x{\chi }_R{\Vert }_2,Y_R=\Vert {\chi }_R+x{\chi }_R^{\prime }{\Vert }_2,K_{R,N}^2=(c_0+4N^2{c}_1)X_R^2+c_1{Y}_R^2.\text{\hspace{1em}(FF3)}$$

Partition $[0,2N]$ into $m=\lceil 2N/\Delta \rceil$ equal cells $I_j$, with midpoints ${\xi }_j$ and width at most $\Delta$. Define

$$\begin{array}{rl}{\phi }_j(x)& =\frac{{\chi }_R(x)}{\sqrt{\rho (x)}}\cos ({\xi }_{j}x),\\ c_j(h)& =\frac{2}{\sqrt{2\pi }}{\int }_{I_j}p(\xi /N)\widehat{Uh}(\xi )d\xi ,Vh=\sum _{j=1}^m{c}_j(h){\phi }_j.\end{array}\text{\hspace{1em}(FF4)}$$

The generators lie in the original compact smooth even core, because $\rho$ is positive smooth and even on their support. The Fourier cell integrals are bounded linear functionals, with $\Vert c_j\Vert \le \sqrt{|I_j|/\pi }$; point values of an arbitrary $L^2$ transform are not required. The map is complex linear with real generators. For complex even $Uh$, its Fourier transform is even, without conjugation, giving the factor two in (FF4).

For the even generator $g_{\xi }=U^{-1}({\chi }_R\cos (\xi x))$, put $a_x=x{\chi }_R$ and $b_x={\chi }_R+x{\chi }_R^{\prime }$. The weighted frequency derivative $-a_x\sin (\xi x)$ is even. The pointwise identity

$$(b_x\sin (\xi x)+\xi a_x\cos (\xi x){)}^2+(b_x\cos (\xi x)-\xi a_x\sin (\xi x){)}^2=b_x^2+{\xi }^2{a}_x^2$$

and the even comparison (WF2) give $|\partial_\xi g_\xi|{\mathcal F}^2 \le c_0X_R^2+c_1(Y_R^2+\xi^2X_R^2)\le K{R,N}^2$ for $|\xi |\le 2N$. The second square is used only in this algebraic identity; no odd-form comparison is applied. Midpoint displacement is at most $\Delta /2$, and positive frequencies carry half the squared Fourier norm. The Bochner triangle inequality and Cauchy–Schwarz therefore give

$$\Vert M_{{\chi }_R}{U}^{-1}{p}_{N}U-V{\Vert }_{\nu \to \mathcal{F}}\le \Delta K_{R,N}\sqrt{N/(2\pi )}.\text{\hspace{1em}(FF5)}$$

Combine (FF1), (WF8), (FF2), and (FF5) on the same original projector:

$$\Vert (I-V)P_{\epsilon }{\Vert }_{\nu \to \mathcal{F}}\le \frac{\sqrt{7}}{8}\tau +\frac{78K_p}{\epsilon }\frac{1+\log N}{N}+\Delta K_{R,N}\sqrt{N/(2\pi )}.\text{\hspace{1em}(FF6)}$$

Make the count-producing choices exact:

$$\delta =\min \left\{\frac{1}{4},\frac{\epsilon \tau }{234K_p}\right\},N=\max \{N_0(\epsilon ),2{\delta }^{-1}\log (2/\delta )\},\Delta =\frac{\tau \sqrt{2\pi /N}}{3K_{R,N}}.\text{\hspace{1em}(FF7)}$$

The second and third errors in (FF6) are each at most $\tau /3$, while the first is strictly smaller. For the bandwidth estimate set $L=\log (2/\delta )$ and use $\log L\le L-1$ and monotonicity of $(1+\log N)/N$ on $N\ge 1$. Thus the total error is $<\tau$, and

$$\mathrm{rank}V\le m\le 1+\frac{6K_{R,N}{N}^{3/2}}{\tau \sqrt{2\pi }}.\text{\hspace{1em}(FF8)}$$

For fixed $\epsilon$, all coefficient constants and $N_0(\epsilon )$ are independent of $\tau$. These exact choices give $N=O_{\epsilon }({\tau }^{-1}\log (e/\tau ))$, $R=O_{\epsilon }(\log \log (e^e/\tau ))$, and $K_{R,N}=O(N(R+2{)}^{3/2})$. Hence the prescribed generator count is

$$O_{\epsilon }\left({\tau }^{-7/2}\log (e/\tau {)}^{5/2}\log \log (e^e/\tau {)}^{3/2}\right)(\tau \downarrow 0).$$

This controls accuracy dependence, not actual feasible size or matrix conditioning. With $\tau =\epsilon /8$, PSD of the complete original form/full-variance matrix on these generators at $c_{\epsilon }=1/2-\epsilon /2$ feeds the existing complete-window transfer (MT). Its entries, signs and cofinal $\epsilon \downarrow 0$ certificates remain uncomputed. The translated-kernel/FIB construction retains its own margin and floor conditions. This cosine construction supplies no arithmetic advantage from relabeling frequency cells by FIB addresses.

Ordered-transition hypotheses must be checked separately

The Karlin–McGregor source application tests the unchanged minimal even semigroup in radial order. Three separated nonnegative core functions give a strictly negative ordered two-by-two bilinear minor for all sufficiently small positive times. The Gamma conductance alone supplies the crossing. Thus the inherited positivity preservation does not supply all-times order-two positivity for a spectral-ordering argument. This paper application retains every prime power and supplies no sharp Poincare or RH conclusion.

Even polynomials in the same minimal form norm

The following is a paper application of the existing theta tail, cutoff and Fourier-approximation estimates, without a new generic density theorem, originality claim or Lean certification. It concerns the unchanged $J$, $\nu$ and ${\mathcal{F}}_{\min ,\mathrm{even}}$ above. The Fourier/Laurent core at bounded support belongs to a different realization and is not used as a core theorem for this domain. The project’s finite polynomial Mellin windows likewise do not establish density in this form norm.

First let $h$ be smooth, with $|h(x)|+|h^{\prime }(x)|\le C{e}^{a|x|}$ for some fixed $C,a>0$. The original theta bound implies, for every fixed $0<b<\pi$, $\Phi (x)\le C_b\exp (-b{e}^{2|x|})$. In particular $h\in L^2(\nu )$. For Gamma jumps $0<t<1$, the square increment is bounded by $C_a{t}^2{e}^{2a|x|}$; boundedness of $\Phi (x+t)$ and integrability of $e^{2a|x|}\Phi (x)$ pay the diagonal singularity. For $t\ge 1$, use the two separate bounds $|h(x+t)-h(x){|}^2\le 2C^2(e^{2a|x|}+e^{2a|x+t|})$ and $\psi (t)\le C{e}^{-t/2}$. The resulting double integral is finite.

For the prime shifts put $t=\log n$ and $S=e^{2|x|}+e^{2|x+t|}$. Absorbing the fixed exponential growth into the two theta tails gives, with a fixed $b>0$,

$$\begin{array}{rl}\Phi (x)\Phi (x+t)|h(x+t)-h(x){|}^2& \le C_a{e}^{-bS}\\ & \le C_a{e}^{-bn}\exp \left(-\frac{b}{2}{e}^{2|x|}\right),S\ge 2n.\end{array}\text{\hspace{1em}(PC1)}$$

The final factor is integrable in $x$; multiplication by $\Lambda (n)/\sqrt{n}\le \log n/\sqrt{n}$ leaves a summable series. Thus (PC1) pays every prime power, rather than integrating a constant over an infinite line. The same majorants, with constants uniform in $R\ge 1$, apply to $(1-{\chi }_R)h$ and its derivative. Their increments tend pointwise to zero. Dominated convergence yields ${\chi }_{R}h\to h$ in the original form norm, hence $h\in {\mathcal{F}}_{\min }$. Even $h$ lies in its even restriction.

Next fix $h\in C_{c,\mathrm{even}}^{\infty }$. Choose an even smooth Fourier cutoff $\eta$ supported on $[-1,1]$, equal to one near zero, and put

$$h_A(x)=\frac{1}{2\pi }{\int }_{-A}^A\eta (\xi /A)\hat{h}(\xi )e^{i\xi x}d\xi .$$

Fourier inversion and integrability of $\hat{h}$ and $\xi \hat{h}$ give uniform convergence of $h_A$ and $h_A^{\prime }$ to $h$ and $h^{\prime }$. Each $h_A$ is even and entire, with bounded function and derivative on the real line, so the preceding cutoff argument puts it in the same minimal domain. The global second moment gives directly, also for these noncompact Lipschitz differences,

$$D(h_A-h)\le \frac{1}{2}{M}_2\Vert h_A^{\prime }-h^{\prime }{\Vert }_{\infty }^2⟶0.\text{\hspace{1em}(PC2)}$$

For fixed $A$, let $p_{A,N}$ be the Taylor polynomial of $h_A$ through degree $2N$. Odd coefficients vanish. Termwise integration of the finite Taylor sum, and then the exponential series, give

$$|p_{A,N}(x)|\le C_A{e}^{A|x|},|p_{A,N}^{\prime }(x)|\le A{C}_A{e}^{A|x|},$$

uniformly in $N$. Both polynomials and their derivatives converge pointwise to $h_A$ and $h_A^{\prime }$. The complete Gamma and prime majorants above therefore show $p_{A,N}\to h_A$ in the same form norm. Combining this with (PC2) and the established compact smooth even core shows that even polynomials are form-dense in ${\mathcal{F}}_{\min ,\mathrm{even}}$.

For $j,k\ge 1$ define the original finite matrices

$$H_{jk}^{(m)}=D(x^{2j},x^{2k})-\frac{1}{2}\left[\nu (x^{2j+2k})-\nu (x^{2j})\nu (x^{2k})\right],1\le j,k\le m.$$

Constants have zero energy and variance. Continuity in the original form norm consequently makes the full half-bound equivalent to $H^{(m)}⪰0$ for every $m$. The two-direction polynomial certificate only addresses $m=2$. No all-degree PSD result, effective exhaustion rate, uniformly conditioned polynomial basis or RH/Robin conclusion is supplied by this density argument.

Quantitative polynomial approximation by published weighted estimates

The weighted Favard and Brascamp–Lieb application uses (WF2), the original coefficient suppliers and the same full fixed-gap spectral map to obtain a conditional polynomial approximation in this minimal form norm. It projects the physical derivative in an admissible auxiliary weight and integrates that one projection; it does not treat a Fourier cutoff of $Uh$ as a physical-bandlimited input. For fixed $\epsilon$, its prescribed dimension has asymptotic cost $O_{\epsilon }({\tau }^{-3}\log (e/\tau {)}^5)$, with unevaluated external Favard constant. This is an application of published approximation and variance inequalities, not a new generic core theorem or a numerical rank certificate. Original matrix positivity and cofinal control remain unproved, so it supplies no RH or Robin conclusion.

The sharp pointwise gradient route fails on the original core

This is a conditional paper check of a sufficient semigroup mechanism, using the same minimal even form, original measure and all prime powers. It reuses the Markov semigroup’s invariance, the existing critical eigenvector and the original PNT rate argument. It supplies no new generic curvature theorem, numerical acquisition or Lean certification.

Write $L=-T$, $P_t=e^{-tT}$, and for a real even form input define its jump energy density

$$\mathcal{G}(h)(x)=\frac{1}{4\cosh (x/2)}\left[\int \Phi (y){\psi }_{\Gamma }(|x-y|)|h(y)-h(x){|}^{2}dy+\sum _{n\ge 2}\frac{\Lambda (n)}{\sqrt{n}}\sum _{ϵ=\pm 1}\Phi (x+ϵ\log n)|h(x+ϵ\log n)-h(x){|}^2\right].\text{\hspace{1em}(SG1)}$$

The diagonal integral uses squared increments, not a finite total jump rate. Tonelli gives $\int \mathcal{G}(h)d\nu =D(h)$; in particular this nonnegative function lies in $L^1(\nu )$. The original minimal jump representation supplies its almost-everywhere value on the form domain.

Consider the sharp pointwise semigroup estimate

$$\mathcal{G}(P_{t}h)\le e^{-t}{P}_t\mathcal{G}(h)\nu \text{-almost everywhere},t>0,\text{\hspace{1em}(SG2)}$$

for every real even compact smooth $h$. The semigroup on the right uses its invariant $L^1(\nu )$ extension. This is the explicit estimate being tested. It is a standard sufficient gradient route toward a one-half Poincare inequality; no diffusion chain rule is assumed and no equivalence to a curvature condition on an unspecified algebra is claimed for this jump operator.

First transport (SG2) through the existing minimal closure. The actual increment representation and Cauchy–Schwarz give

$$\Vert \mathcal{G}(f)-\mathcal{G}(g){\Vert }_{L^1(\nu )}\le \sqrt{D(f-g)}[\sqrt{D(f)}+\sqrt{D(g)}].\text{\hspace{1em}(SG3)}$$

The spectral semigroup contracts the form norm, and its invariant $L^1$ extension contracts the $L^1$ norm. Thus core approximation and the closed positive cone of $L^1$ extend (SG2) to every real even form input. These are standard continuity properties applied to the unchanged increment maps; (SG3) does not replace the original covariance.

Use the already established $v={\Phi }^{\prime \prime }/\Phi -1/4$, with $v\in D(T)$ and $Tv=v/2$. If (SG2) held, $P_{t}v=e^{-t/2}v$ would give

$$\mathcal{G}(v)\le P_t\mathcal{G}(v).\text{\hspace{1em}(SG4)}$$

Invariance makes the two $L^1$ integrals equal, so (SG4) is equality almost everywhere. Every invariant nonnegative $g\in L^1(\nu )$ here is constant: Jensen applied to $g\wedge M$ and invariance make each bounded truncation invariant, and the spectral theorem puts it in $\ker T=\mathbb{R}1$. Taking $M\to \infty$ gives the assertion for $g$. This uses the known constant kernel of the same operator, rather than positivity on the unknown critical quotient. Consequently (SG2) would force $\mathcal{G}(v)$ to be constant.

That conclusion fails in the actual prime geometry. Differentiating the normally convergent theta series twice and retaining its first term and differentiated tail gives

$$v(x)\sim 4{\pi }^2{e}^{4x}(x\to +\infty ).\text{\hspace{1em}(SG5)}$$

This does not differentiate an asymptotic remainder. Choose a real even nonzero $0\le \zeta \le 1$ in $C_c^{\infty }(-1,1)$ and put $M={\sup }_{|y|\le 1}|v(y)|$. The incoming compact-target rate is

$$a_{\zeta }(x)=\frac{1}{2\cosh (x/2)}\sum _{n\ge 2}\frac{\Lambda (n)}{\sqrt{n}}\Phi (x-\log n)\zeta (x-\log n).$$

The existing ordinary PNT rate argument applies to the same smooth compact weight $y^{-1/2}\Phi (\log y)\zeta (\log y)$, giving

$$a_{\zeta }(x)⟶A_{\zeta }:={\int }_{\mathbb{R}}{e}^{r/2}\Phi (r)\zeta (r)dr>0.\text{\hspace{1em}(SG6)}$$

It retains every prime power; the target interval restricts only a nonnegative lower minorant. For all sufficiently large $x$, positivity of the other energy terms yields

$$\mathcal{G}(v)(x)\ge \frac{1}{2}{a}_{\zeta }(x)(|v(x)|-M{)}^2⟶\infty .\text{\hspace{1em}(SG7)}$$

The original density is strictly positive, so this is also a nonconstant almost-everywhere function. It contradicts the required constant energy density. Hence the all-input sharp estimate (SG2) cannot hold even when first requested only on the original compact core. The argument uses no prime truncation, fixed-zero-height hypothesis, finite matrix signs or RH assumption.

This excludes that sufficient pointwise gradient route at rate $e^{-t}$. It excludes neither a weaker gradient estimate nor an integrated or directly estimated global half-bound. The accepted conditional $0.41$ lower bound remains reusable, while positivity on the complete critical quotient, the one-half bound, Robin and RH remain unresolved. No precise smaller optimal curvature constant is asserted.

Spatially weighted high inverse at every subcritical parameter

The spatial comparison in (WF5) is reused before taking its scalar minimum. This is a conditional paper application of (WF1)–(WF6) on the same minimal even form, using the classical variational inverse comparison. It retains the complete prime row, Gamma term and original ground. It supplies an inverse-coupling allowance, not a low-block sign, cofinal positivity or Lean certification.

Fix $0<\epsilon \le 1/2$, put $c=1/2-\epsilon$, $\delta =\epsilon /4$, and choose

$$N\ge N_w(\epsilon ):=\max \left\{2e^{4{\mu }_{\epsilon }+4},\sqrt{\frac{512}{3\pi \epsilon }}\Vert s^{\prime \prime }{\Vert }_2,2\right\}.\text{\hspace{1em}(WH1)}$$

Here ${\mu }_{\epsilon }$ is exactly (WF4), computed from the same complete $W$ and $s$. Let $P=1_{|\mathsf{D}|<N}$, $Q=I-P$, $v_0=\sqrt{\rho }$, and $T_c=\tilde{A}-cI+c|v_0⟩⟨v_0|$. Denote its actual high closed form and associated operator by $C$. Set

$$m_N=m(N/2),a_N=m_N-{\mu }_{\epsilon },V_{\epsilon ,N}(x)=\delta +a_{N}s(x{)}^2.\text{\hspace{1em}(WH2)}$$

The existing symbol estimate gives $m_N\ge 2{\mu }_{\epsilon }+1$, so $a_N\ge {\mu }_{\epsilon }+1>0$. The leakage estimate (WF5) and $m_N\le 16N$ give $m_N{\eta }_N^2\le \epsilon /4$ under (WH1). Definition (WF4) supplies the pointwise relation $W+{\mu }_{\epsilon }{s}^2\ge 1/2-\epsilon /2$. Retain its unused spatial term in the same high-form calculation: for every $q$ in the actual high form domain,

$$\begin{array}{rl}C[q]& \ge \int (m_N{s}^2+W-c)|q{|}^{2}dx-m_N{\eta }_N^2\Vert q{\Vert }_2^2+c|⟨v_0,q⟩{|}^2\\ & \ge \int V_{\epsilon ,N}(x)|q(x){|}^{2}dx.\end{array}\text{\hspace{1em}(WH3)}$$

The nonnegative ground term is dropped only in the last lower bound. This uses the closed high restriction of the original form, rather than a new maximal domain. The sharp low projection maps $L^2$ into $H^1$ and the original operator domain as in the complete-column interface. Consequently its complementary projection preserves the minimal form domain. The original core approximation and (WF1), (WF5) extend the same inequality to that high form domain.

Since $V_{\epsilon ,N}\ge \delta >0$ and $s$ is bounded, $w_{\epsilon ,N}=V_{\epsilon ,N}^{-1}$ is bounded and positive. For every $r\in Q{L}_{\mathrm{even}}^2(dx)$, variational inversion of (WH3) gives

$$⟨r,C^{-1}r⟩\le \int w_{\epsilon ,N}(x)|r(x){|}^{2}dx.\text{\hspace{1em}(WH4)}$$

Indeed the left side is the supremum of $2\Re ⟨r,q⟩-C[q]$ over the high form domain. After (WH3), enlarging that supremum to all even $L^2$ and completing the pointwise square gives the right side. No commutation of $Q$ with the multiplier, or equality of inverse and compression, is asserted.

The accepted original cap $\Vert s{\Vert }_{\infty }^2\le 9/5$ used in (FF1) yields the explicit saving

$$\begin{array}{rl}{\delta }^{-1}\Vert r{\Vert }_2^2-\int w_{\epsilon ,N}|r{|}^2& =\frac{a_N}{\delta }\int \frac{s^2|r{|}^2}{\delta +a_N{s}^2}dx\\ & \ge b_{\epsilon ,N}\Vert sr{\Vert }_2^2,b_{\epsilon ,N}:=\frac{a_N}{\delta (\delta +9a_N/5)}>0.\end{array}\text{\hspace{1em}(WH5)}$$

For every nonzero $r$ this saving is strictly positive because $s>0$. There is no claimed uniform fractional saving for arbitrary sources: $s(x)\to 0$ at spatial infinity. The same-residual consumer uses (WH4)–(WH5) before any scalar residual norm is taken. The low Schur and complementary-low signs on the same bandwidth sequence remain unproved; no saved fixed-band matrix is transported to (WH1).

Keep the high constraint and the exact ground term

At the same (WH1) parameter and bandwidth, let $H=L_{\mathrm{even}}^2(dx)$, write $M_V,M_w$ for multiplication by (WH2) and its inverse, and put $V_{\max }=\delta +9a_N/5$. The existing cap gives $V_{\max }^{-1}I⪯M_w⪯{\delta }^{-1}I$. On $PH$ and $QH$ set

$$A=P{M}_{w}P{|}_{PH},D=Q{M}_{V}Q{|}_{QH},R=Q(M_w-M_{w}P{A}^{-1}P{M}_w)Q{|}_{QH}.\text{\hspace{1em}(WH6)}$$

Both compressed positive multipliers have bounded inverses; $P$ is infinite-dimensional. Reuse the standard positive block inverse identity: $R=D^{-1}$. The bounded positive shorting reference is Antezana–Corach–Stojanoff, Theorem 2.2(4), p.5, with operator $M_w$ and closed subspace $QH$; (SC2) supplies the existing positive block-elimination interface. For this application, the vector $q=M_w(r-P{A}^{-1}P{M}_{w}r)$ satisfies $Pq=0$ and $Q{M}_{V}q=r$ for every $r\in QH$. No commutation of $P$ with $M_w$ is assumed.

Set $g=Q{v}_0$ and $\beta =c/(1+c⟨g,Rg⟩)$. Before dropping the ground term, (WH3) gives $C[q]\ge\langle q,Dq\rangle+ c|\langle g,q\rangle|^2$ on the original high form domain. Variational inversion and the standard rank-one inverse identity yield

$$C^{-1}⪯(D+c|g⟩⟨g|{)}^{-1}=R-\beta |Rg⟩⟨Rg|.\text{\hspace{1em}(WH7)}$$

The rank-one identity reuses the general-ring Woodbury proof Matrix.invOf_add_mul_mul. Take its central factor to be $I$ and absorb $c$ into the update factor, including $c=0$. Its algebra applies to bounded endomorphism entries; finite block indices impose no finite-dimensional assumption on $QH$. Invertibility here comes from $D⪰\delta I$ and $c\ge 0$, not from that formula. The inspected shorting PDF SHA-256 is 449b7d53831f8f2ddec744ad297e43f5bbbbff3804df19833e48a1181cb90066. Only the variational supremum is enlarged to $QH$; the actual high operator and domain remain those of (WH3). Define, for $r\in QH$,

$$\begin{array}{rl}J(r)& =\int w|r{|}^{2}dx-⟨P(wr),A^{-1}P(wr)⟩-\beta |⟨Rg,r⟩{|}^2,\\ ⟨r,C^{-1}r⟩& \le J(r),\int w|r{|}^{2}dx-J(r)\ge \delta \Vert P(wr){\Vert }_2^2.\end{array}\text{\hspace{1em}(WH8)}$$

Here $A⪯{\delta }^{-1}I$ gives the last inequality. The projection and ground corrections are jointly valid because the latter uses the same constrained $R$, not the unconstrained $M_w$. Each correction can vanish; at $c=0$ the ground correction vanishes. The same-residual application keeps all cross entries and the exact ground trial cost. This is a conditional paper application of existing block and rank-one inverse tools. It supplies no evaluated entries, low or complementary-low signs, cofinal certificate, endpoint half-bound, RH/Robin proof or Lean certification.