# Compact smooth-test topology and its Fourier neighborhoods *Original proof exposition by GPT-6.1 Sol (OpenAI), Ultra, October 2026. CC0 1.0.* We prove four exact descriptions of compact smooth-test neighborhoods, their exhaustion-wise Fourier-envelope description, and their description by a single strict PSH weighted integral. The proof includes the complete logarithmic-contour homotopy and every successive choice needed for its infinite envelope. Its weight construction retains the actual curvature exponent and the whole real-frequency extent of each stabilized imaginary strip. Let \(X\subset\mathbb R^n\) be open and write \(\mathcal D(X)=C_c^\infty(X)\), with \(D=-i\partial\). Choose a compact exhaustion \[ K_0=\varnothing,\qquad K_j\subset\operatorname{int}K_{j+1}, \qquad \bigcup_{j\ge1}K_j=X. \tag{TP1} \] Empty initial terms are harmless and can be discarded. The space \(\mathcal D_{K_j}\) of smooth functions supported in \(K_j\) has the seminorms \(p_{j,N}(v)=\max_{|\alpha|\le N}\sup_x|D^\alpha v(x)|\). The locally convex inductive topology on \(\mathcal D(X)\) is generated by all seminorms whose restrictions to these spaces are continuous. Cofinality of the exhaustion gives the same topology as using every compact support set. ## TP1. The four exact neighborhood descriptions **Theorem TP1.** For a convex subset \(V\subset\mathcal D(X)\), these four statements are equivalent: 1. \(V\) is a neighborhood of zero in that inductive topology. 2. There are integers \(N_j\ge0\) and numbers \(\epsilon_j>0\) such that \(V\) contains every finite sum \(\sum_jv_j\), with \(v_j\in\mathcal D_{K_j}\) and \(p_{j,N_j}(v_j)<\epsilon_j\). 3. There are smooth coefficient functions \(\rho_\alpha\in C^\infty(X)\), indexed by multiindices, whose supports form a locally finite family, such that \(V\) contains every \(v\) satisfying \[ Q_\rho(v):=\sum_\alpha\sup_x|\rho_\alpha(x)D^\alpha v(x)|<1. \tag{TP2} \] 4. There are integers \(L_j\ge0\) and positive \(\delta_j\), \(j\ge0\), such that \(V\) contains every compact smooth \(v\) satisfying \[ |D^\alpha v(x)|\le\delta_j \quad(x\in X\setminus K_j,\ |\alpha|\le L_j,\ j\ge0). \tag{TP3} \] In particular the \(j\)-condition is on the entire exterior, not on one shell. We first justify the fixed-stage convex criterion rather than treating it as an unexplained definition. If \(V\) is a neighborhood, each stage intersection is a neighborhood by continuity of inclusion. Conversely suppose every stage intersection is a neighborhood. Then \(0\in V\). Set \[ W=\{v:\lambda v\in V\text{ for every complex }|\lambda|\le1\}. \tag{TP4} \] It is convex and balanced. On any fixed stage a sufficiently small ball of \(p_{j,N}\) is contained in \(V\), and all its complex scalar contractions are in that same ball. Thus \(W\) contains that stage ball. It is absorbing, since every compact smooth function lies in some stage. Its Minkowski gauge is a seminorm continuous on every stage and hence in the inductive topology just described. Its open unit ball lies in \(W\subset V\): gauge less than one implies membership in \(tW\) for some \(t<1\), and balancedness gives \(tW\subset W\). This proves the criterion for arbitrary convex \(V\), without assuming \(V\) was balanced. **(1) implies (2).** In each stage choose \(N_j,\epsilon_j\) so that \[ \{v\in\mathcal D_{K_j}:p_{j,N_j}(v)<2^j\epsilon_j\}\subset V. \tag{TP5} \] For a finite admissible sum \(v=\sum_{j\in J}v_j\), its summands \(2^jv_j\) are in \(V\). Since \(\sum_{j\in J}2^{-j}<1\) and \(0\in V\), write \(v\) as their convex combination together with the remaining coefficient on zero. This proves inclusion. **(2) implies (3).** Choose \(0\le\theta_j\le1\), smooth and supported in \(\operatorname{int}K_j\), with \(\theta_j=1\) near \(K_{j-1}\); take \(\theta_0=0\). Such cutoffs follow by covering \(K_{j-1}\) with finitely many small balls compact inside \(\operatorname{int}K_j\) and combining their usual smooth steps. Define \[ \psi_j=\theta_j-\theta_{j-1}. \tag{TP6} \] The signs of \(\psi_j\) need not be positive. Its support lies in \(K_j\) and vanishes near \(K_{j-2}\) for \(j\ge3\). The family and all its derivatives are locally finite. On any compact support, \(\sum_j\psi_j=1\) and only finitely many terms are nonzero. Thus \(v=\sum_j\psi_jv\) is a finite decomposition. For each needed derivative \(\beta\), \(|\beta|\le N_j\), of each active \(\psi_j\), choose a nonnegative smooth compact majorant \(b_{j,\beta}\ge|D^\beta\psi_j|\), supported in the same locally finite annular region enlarged slightly. One construction is a constant times a smooth cutoff equal to one on the compact derivative support; the constant is its finite supremum. For the few initial terms use cutoffs in \(K_2\). For later \(j\), these enlarged supports can avoid \(K_{j-3}\), preserving local finiteness. Put \[ \rho_\alpha(x)= \sum_{j:N_j\ge|\alpha|} \frac1{\epsilon_j} \sum_{|\alpha+\beta|\le N_j} \binom{\alpha+\beta}{\alpha}\,b_{j,\beta}(x). \tag{TP7} \] This is a locally finite sum of smooth nonnegative functions. On any compact set only finitely many \(j\)'s contribute, and among those only finitely many \(\alpha\)'s occur, so the coefficient supports are a locally finite family as required. For \(v_j=\psi_jv\), the product rule for \(D=-i\partial\) gives, for \(|\gamma|\le N_j\), \[ \frac{|D^\gamma v_j(x)|}{\epsilon_j} \le\sum_{\alpha+\beta=\gamma} \binom{\gamma}{\alpha} \frac{b_{j,\beta}(x)}{\epsilon_j}|D^\alpha v(x)| \le\sum_\alpha|\rho_\alpha(x)D^\alpha v(x)|. \tag{TP8} \] Taking a supremum bounds this by \(Q_\rho(v)\). If TP2 is less than one, every \(v_j\) therefore satisfies the condition in (2), giving \(v\in V\). The majorants make the coefficients actually smooth; replacing them by the nonsmooth absolute values of derivatives would not meet statement (3). **(3) implies (4).** On \(K_{j+1}\) only finitely many coefficient supports meet. Let \(I_j\) be their multiindices and choose \[ L_j\ge\max_{\alpha\in I_j}|\alpha|,\qquad \delta_j>0\quad\hbox{so small that}\quad \delta_j\sup_{K_{j+1}}|\rho_\alpha| \le2^{-n-1-|\alpha|}\quad(\alpha\in I_j). \tag{TP9} \] If \(I_j\) is empty, use \(L_j=0,\delta_j=1\). TP3 then bounds each product on the shell \(K_{j+1}\setminus K_j\) by the right side of TP9. These shells, together with \(K_1\) for \(j=0\), cover \(X\). For every \(\alpha\) its global supremum has the same bound, because the coefficient vanishes on shells where it is not in \(I_j\). Hence \[ Q_\rho(v)\le \sum_{\alpha\in\mathbb N^n}2^{-n-1-|\alpha|} =\tfrac12<1. \tag{TP10} \] The series identity is the product of \(n\) geometric series \(\sum_{l\ge0}2^{-l}=2\). This leaves a strict margin even though the exterior bounds themselves are non-strict. The bound is imposed on the whole exterior as TP3 specifies; the shells are used only to prove its consequence. **(4) implies (1).** Fix a compact support \(S\Subset X\). For all sufficiently large \(j\), \(S\) is in the interior of \(K_j\); the corresponding exterior restrictions are automatic because every derivative of \(v\) vanishes outside its support. Only finitely many \(j\)'s remain. Choose the maximum of their derivative orders and half the minimum of their positive \(\delta_j\)'s. The resulting small fixed-support seminorm ball satisfies every restriction in TP3. Thus \(V\) has a zero-neighborhood on every support stage. The convex criterion proved after TP4 gives (1). This proves all four implications and the exact first source theorem. \(\square\) ## TP2. Exact entire decay on a fixed smooth support Write \(F_v(z)=\int v(x)e^{-ix\cdot z}\,dx\). If \(v\) is supported in a compact \(S\), integration by parts in the real variables gives \(z^\alpha F_v=F_{D^\alpha v}\), and hence, for every integer \(L\ge0\), \[ |F_v(z)|\le C_{n,L}(1+|z|)^{-L}e^{H_S(\operatorname{Im}z)} \sum_{|\alpha|\le L}\|D^\alpha v\|_{L^1}. \tag{TP11} \] For \(|z|\ge1\), choose a coordinate with \(|z_l|\ge|z|/\sqrt n\) and use its \(L\)-th derivative; for \(|z|<1\), use the zeroth derivative. The comparison of \(|z|^L\) with \((1+|z|)^L\) gives the displayed fixed constant. The integrable norms are bounded by finitely many derivative suprema times the volume of a fixed bounded set containing \(S\). This proves the whole-complex estimate, rather than only real-frequency decay. Fourier inversion here needs no additional contour theorem. Insert \(e^{-\epsilon|\xi|^2}\) into the real Fourier integral. Fubini and the elementary Gaussian integral identify it with convolution of \(v\) and \((4\pi\epsilon)^{-n/2}e^{-|x|^2/(4\epsilon)}\). The latter tends to \(v(x)\): split the integral into a small neighborhood, where uniform continuity applies, and its Gaussian tail. TP11 on the real plane gives an integrable majorant, so dominated convergence removes the multiplier. The same argument applies to every derivative. ## TP3. The complete logarithmically shifted contour For a real unit vector \(\eta\), \(R\ge0\), and real \(\xi\), put \[ \ell(\xi)=\log(2+|\xi|^2),\qquad z_s(\xi)=\xi+i sR\eta\,\ell(\xi)\quad(0\le s\le1). \tag{TP12} \] The complex Jacobian of this real parametrization is the rank-one determinant \[ J_s(\xi)=\det(I+i sR\eta(\nabla\ell)^{\mathsf T}) =1+i sR\,\eta\cdot\nabla\ell,\qquad |J_s|\le1+R. \tag{TP13} \] The determinant identity follows by multilinearity: selecting two added rank-one columns gives zero; selecting one gives its diagonal contribution. Also \(|\nabla\ell|=2|\xi|/(2+|\xi|^2)\le1\). For any entire scalar \(H\), the chain rule gives the exact homotopy identity \[ \partial_s[H(z_s)J_s] =\sum_{l=1}^n\partial_{\xi_l} [iR\eta_l\ell(\xi)H(z_s)]. \tag{TP14} \] Indeed its right side is \(iR(\eta\cdot\nabla\ell)H+ iR\ell(\eta\cdot\partial_zH)(1+i sR\eta\cdot\nabla\ell)\), which is its left side. Thus only the real fundamental theorem of calculus on a cube is required for the contour homotopy. Take \(H(z)=z^\alpha F_v(z)e^{ix\cdot z}\), for fixed real \(x\) and a compact smooth \(v\) supported in \(S\). On the boundary of \([-A,A]^n\), TP11 gives, uniformly in \(s\) and unit \(\eta\), \[ |H(z_s)|\le C_L(1+A)^{|\alpha|-L+2R(r_S+|x|)} \quad(A\ge1),\qquad r_S=\sup_{y\in S}|y|. \tag{TP15} \] Here \(|z_s|\ge|\xi|\ge A\), while its imaginary part is \(sR\eta\log(2+|\xi|^2)\); the exponential growth is bounded by a constant times \(A^{2R(r_S+|x|)}\). Choose \(L>|\alpha|+2R(r_S+|x|)+n+2\). The boundary flux from TP14 has at most an extra \(\log(2+nA^2)\) and face area \(C_nA^{n-1}\), so its integral tends to zero. TP11 also gives an integrable bound for the full cube integrals, uniformly in \(s\). Integrate TP14 over the cube and over \(s\), then pass to \(A=\infty\). Together with real inversion this proves \[ D^\alpha v(x)=(2\pi)^{-n} \int_{\mathbb R^n}z_1(\xi)^\alpha F_v(z_1(\xi)) e^{ix\cdot z_1(\xi)}J_1(\xi)\,d\xi. \tag{TP16} \] This is an absolutely convergent identity for each fixed \(x,R,\eta,\alpha\). We have verified the homotopy boundary, the differential and its Jacobian, rather than assuming that a variable-height contour has the same measure as the real plane. ## TP4. The exhaustion-wise Fourier-envelope theorem Assume now that \(X\) is convex and the exhaustion \(K_j\) is also convex and nonempty for \(j\ge1\). **Theorem TP4.** A convex \(V\subset\mathcal D(X)\) is a zero-neighborhood if and only if there are positive numbers \(d_j,M_j\), \(j\ge1\), such that \(V\) contains every \(v\in\mathcal D(X)\) with \[ |F_v(z)|\le B(z):= \sum_{j=1}^\infty d_j(1+|z|)^{-M_j} e^{H_{K_j}(\operatorname{Im}z)} \quad(z\in\mathbb C^n). \tag{TP17} \] We may choose the \(M_j\) increasing positive integers and the series locally uniformly convergent. First suppose TP17 implies membership in \(V\). On each stage \(\mathcal D_{K_j}\), TP11 with \(L\ge M_j\) makes its single \(j\)-term dominate \(F_v\) if a fixed finite derivative seminorm of \(v\) is sufficiently small. Thus \(V\) contains a stage neighborhood. Its convexity and TP4's fixed-stage criterion prove that it is a zero-neighborhood. Conversely obtain \(L_j,\delta_j\) of the exterior criterion TP3. The envelope will imply all those bounds. To avoid referring to the support function of the empty set, handle \(j=0\) and \(j=1\) by real inversion alone. For either order \(L_j\), if \(M_k\ge L_j+n+1\), the \(k\)-term contributes at most \[ C_n d_k,\qquad C_n=(2\pi)^{-n}\int_{\mathbb R^n}(1+|\xi|)^{-n-1}d\xi<\infty \tag{TP18} \] to every derivative up to that order, everywhere in real space. We will make these sums smaller than their respective \(\delta_j\)'s. For \(j\ge2\), let \(c_j=\operatorname{dist}(K_{j-1},\mathbb R^n\setminus\operatorname{int}K_j)>0\). If \(x\notin K_j\), let \(p\) be its nearest point in the compact convex \(K_{j-1}\), and put \(\eta=(x-p)/|x-p|\). The elementary nearest-point inequality, obtained by differentiating \(|x-(p+t(y-p))|^2\) at \(t=0+\), gives \((x-p)\cdot(y-p)\le0\) for \(y\in K_{j-1}\). Thus \[ x\cdot\eta-H_{K_{j-1}}(\eta)=|x-p|\ge c_j. \tag{TP19} \] This supplies a uniform separation gap for the entire exterior. Suppose all \(d_k,M_k\) with \(k0\) so small that \[ C_{jk}d_k\le2^{-k-2}\delta_j\ (0\le j\le k),\qquad d_k\le2^{-k}e^{-k(1+r_{K_k})}, \tag{TP23} \] using \(C_{0k}=C_{1k}=C_n\). There are only finitely many requirements at this step, and increasing \(M_k\) only helps them. This gives positive coefficients. The last restriction implies local uniform convergence of TP17: on \(|\operatorname{Im}z|\le A\), all \(k\ge A\) terms are bounded by \(2^{-k}e^{-k}\). For \(j\ge2\), the earlier contribution is below \(\delta_j/2\), and the later sum is bounded by \(\delta_j\sum_{k\ge j}2^{-k-2}<\delta_j/2\). For \(j=0,1\), all terms are controlled by TP18 and TP23. Absolute convergence and Tonelli justify summing these nonnegative derivative bounds. The result is TP3 for every required derivative and every point of each entire exterior. Hence TP17 implies \(v\in V\), proving the exact Fourier-envelope theorem. \(\square\) ## TP5. A single strict PSH weighted neighborhood **Theorem TP5.** For open convex \(X\) and convex \(V\subset\mathcal D(X)\), \(V\) is a zero-neighborhood if and only if it contains \[ \mathcal U_\phi= \left\{v:\int_{\mathbb C^n}|F_v(z)|^2e^{-2\phi(z)}\,dV(z)<1\right\}, \tag{TP24} \] for a finite real function \(\phi\) with all three properties \[ \begin{aligned} e^{-\phi(\xi+i\eta)} &\le C_K(1+|\xi+i\eta|)^{N_K}e^{-H_K(\eta)} && (K\Subset X\text{ compact convex}),\\ |\nabla\phi(\xi+i\eta)| &\le C_b+\log(1+|\eta|) &&\text{almost everywhere; }\phi\text{ locally Lipschitz},\\ \mathcal L_\phi(w) &\ge c(1+|\eta|^2)^{-3/4}|w|^2 &&\text{distributionally, }c>0 . \end{aligned} \tag{TP25} \] The last property includes plurisubharmonicity. Its source exponent is \(-3/4\), as verified on the actual statement and seed-proof pixels. For the easy implication, TP11 and the first part of TP25 show that this weighted transform norm is finite and continuous on each fixed smooth-support stage: choose a compact convex \(K\Subset X\) containing that support and \(L>N_K+n\). Its squared integrand is bounded by a finite derivative seminorm squared times \((1+|z|)^{-2(L-N_K)}\), integrable in real dimension \(2n\). The triangle inequality in \(L^2\) makes the resulting expression a seminorm. It is therefore continuous in the inductive topology. Its open unit ball is a zero-neighborhood, as is every \(V\) containing it. For the converse use the increasing integer powers and coefficients in TP17. Enlarge each successive \(t_j\ge1\), strictly increasing to infinity, as specified below, and set \[ a_j=\frac{M_{j+1}+1}{2},\qquad M_t(\eta)=(t^2+|\eta|^2)^{1/2},\qquad p_t(\eta)=\frac{M_t(\eta)}{\sqrt t}-\sqrt{M_t(\eta)}, \qquad \psi_j(z)=H_{K_j}(\eta)-a_j\log(t_j^2+|z|^2)+p_{t_j}(\eta). \tag{TP26} \] Every \(a_j\) is nondecreasing. We next prove, rather than presume, the gradient and curvature needed for these varying seeds. ## TP6. Full seed bounds and their uniform active region The support function \(H_K(\eta)=\sup_{x\in K}x\cdot\eta\) is convex, globally Lipschitz with constant \(r_K\), and PSH as a function of \(\xi+i\eta\). For the last claim, convolution preserves convexity and gives a nonnegative real Hessian in \(\eta\); the complex Hessian is one quarter of it. Uniform convergence of these regularizations on compact sets passes the line submean inequality and the distributional matrix inequality to the original function. The ordinary complex Hessian of \(\log(t^2+|z|^2)\) is \[ \frac{I}{t^2+|z|^2} -\frac{\overline z\,z^{\mathsf T}}{(t^2+|z|^2)^2}, \qquad 0\le\mathcal L_{\log(t^2+|z|^2)}\le (t^2+|z|^2)^{-1}I. \tag{TP27} \] For \(p_t\), let \(s=t^2+|\eta|^2\), \(q=t^2/s\in(0,1]\). Its transverse real Hessian eigenvalue, divided by \(s^{-3/4}\), is \(t^{-1/2}s^{1/4}-\frac12\ge\frac12\). Its radial eigenvalue, with the same normalization, is \[ q^{3/4}+\tfrac14-\tfrac34q\ge\tfrac14, \quad\text{because }q^{3/4}\ge q. \tag{TP28} \] These follow by differentiating \(t^{-1/2}s^{1/2}-s^{1/4}\) twice. In dimension one only the radial eigenvalue is needed; at \(\eta=0\) the limiting formulas agree. Dividing the real Hessian by four gives \[ \mathcal L_{p_t}\ge\frac1{16}(t^2+|\eta|^2)^{-3/4}I,\qquad |\nabla p_t|\le t^{-1/2},\qquad p_t\ge0. \tag{TP29} \] The gradient formula is \(\eta[t^{-1/2}/M_t-\frac12M_t^{-3/2}]\), whose bracket is nonnegative and whose norm is at most \(t^{-1/2}\). Positivity of \(p_t\) follows from \(M_t\ge t\). Choose \(\sqrt{t_j}\ge32a_j=16(M_{j+1}+1)\). The negative logarithmic Hessian in TP27 is then at most \(\frac1{32}(t_j^2+|\eta|^2)^{-3/4}I\). Thus \[ \mathcal L_{\psi_j}\ge \frac1{32}(t_j^2+|\eta|^2)^{-3/4}I. \tag{TP30} \] For \(|\eta|>t_j\) this is at least \(c_*(1+|\eta|^2)^{-3/4}I\), where \(c_*=2^{-3/4}/32\) is independent of \(j\). The first seed has a global positive bound \(c_1=t_1^{-3/2}/32\) in the same form. Its global Lipschitz bound is \[ |\nabla\psi_j|\le r_{K_j}+a_j/t_j+t_j^{-1/2}. \tag{TP31} \] The logarithm's real gradient has norm \(2|z|/(t_j^2+|z|^2)\le1/t_j\). Increase \(t_j\) so that the right side of TP31 is at most \(\log t_j\). This is possible with \(a_j,r_{K_j}\) fixed. On \(|\eta|>t_j\) that gives the uniform bound \(\log(1+|\eta|)\). Set \(C_b\) equal to the finite global Lipschitz bound of the first seed; it will serve for all finite maxima. Since \(K_j\Subset\operatorname{int}K_{j+1}\), choose \(\rho_j>0\) with \(K_j+\rho_j\overline B\subset K_{j+1}\), and also require \(t_j^{-1/2}\le\rho_j\). Then \[ H_{K_j}(\eta)+t_j^{-1/2}|\eta|\le H_{K_{j+1}}(\eta),\qquad p_{t_j}(\eta)\le t_j^{-1/2}|\eta|+\sqrt{t_j}. \tag{TP32} \] The second inequality uses \(M_t\le t+|\eta|\) and discards the nonpositive term \(-\sqrt{M_t}\). ## TP7. Compatible maxima, finite stabilization and all three properties Let \(C_{\mathrm{mean}}=(n!/\pi^n)^{1/2}\). Because \(t_j\ge1\), \((t_j^2+|z|^2)^{1/2}\ge(1+|z|)/\sqrt2\). Equations TP26 and TP32 therefore permit a vertical constant \(G_j\) with \[ C_{\mathrm{mean}}e^{C_b}(2+|z|) e^{\psi_j(z)-G_j} \le d_{j+1}(1+|z|)^{-M_{j+1}} e^{H_{K_{j+1}}(\eta)} . \tag{TP33} \] For example it suffices initially to require \(G_j\ge C_b+\log C_{\mathrm{mean}}+(a_j+1)\log2+ \sqrt{t_j}-\log d_{j+1}\). The single extra power \(M_{j+1}+1\) accounts for the factor \(2+|z|\); \(C_b\) appears as a multiplicative exponential constant, not as an extra polynomial exponent. Fix \(G_1\) this way. For every later \(j\), choose increasing \(B_j\ge\max(j,t_j+1,B_{j-1}+1)\), and increase \(G_j\) further so that \[ \psi_j-G_j<\psi_1-G_1 \quad\hbox{on the whole strip }|\eta|\le B_j,\qquad \phi_j=\max_{1\le l\le j}(\psi_l-G_l). \tag{TP34} \] The comparison is uniform in all \(\xi\). Indeed \(a_j\ge a_1,t_j\ge t_1\), so \(-a_j\log(t_j^2+|z|^2)+a_1\log(t_1^2+|z|^2)\le0\). The support-function and radial-correction differences are bounded above on the prescribed strip. A finite vertical increase therefore suffices. Inductively \(\phi_j\) has the common gradient bound TP25 and common curvature \(c=\min(c_1,c_*)>0\). On \(|\eta|t_j\) both the previous maximum and the new branch have those bounds. These two open regions cover the space. The full variable-Hessian maximum argument and weak-gradient argument are proved in [L149 PW4](../../AN02-L149.html#pw4-a-maximum-preserves-the-variable-hessian-lower-bound): on a small ball subtract its constant lower bound times \(|z|^2\), take a PSH maximum, and recover the varying lower bound by a partition of nonnegative tests and shrinking the oscillation. The common local Lipschitz bounds pass to the maximum and yield the same weak gradient bound. Thus no differentiability of a support function or of a maximum is assumed. For any fixed strip, every sufficiently late branch is inactive there by TP34. Consequently \[ \phi(z)=\lim_j\phi_j(z) \quad\text{is finite and locally equals a fixed finite maximum}. \tag{TP35} \] It is locally Lipschitz and satisfies the last two parts of TP25. Also \(\phi\ge\psi_j-G_j\); since \(p_{t_j}\ge0\), this gives \[ e^{-\phi(z)} \le C_j(1+|z|)^{M_{j+1}+1}e^{-H_{K_j}(\eta)} . \tag{TP36} \] For any compact convex \(K\Subset X\), choose \(j\) with \(K\subset K_j\). Then \(H_{K_j}\ge H_K\), so TP36 proves the first part of TP25. All three required properties are established with one weight. Finally let the integral in TP24 be less than one. The complete holomorphic unit-ball mean inequality follows by applying the line-circle submean inequality successively and averaging radially, or the positive Laplacian submean proof in L131 NP2–NP3, to \(|F_v|^2\). The real \(2n\)-ball volume is \(\pi^n/n!\). The gradient bound compares \(\phi(z+w)\) with \(\phi(z)\), for \(|w|<1\), by at most \(C_b+\log(2+|\operatorname{Im}z|)\). Thus \[ \begin{aligned} |F_v(z)| &\le C_{\mathrm{mean}}e^{\phi(z)+C_b} (2+|\operatorname{Im}z|) \left(\int|F_v|^2e^{-2\phi}\,dV\right)^{1/2}\\ &\le C_{\mathrm{mean}}e^{\phi(z)+C_b}(2+|z|). \end{aligned} \tag{TP37} \] At each point \(e^\phi=\max_j e^{\psi_j-G_j}\); bounding this maximum by the sum of its nonnegative terms and using TP33 bounds TP37 by the envelope TP17 (even with its unused positive first term omitted). Therefore \(v\in V\). This proves the converse and the full weighted-neighborhood theorem. \(\square\) If \(X\) is empty, \(\mathcal D(X)=\{0\}\), and all neighborhood statements mean that \(V\) contains zero. For TP25 use \(\phi=p_1(\eta)\); its growth condition has no nonempty compact set to test, and TP29 gives its gradient and strict curvature. The weighted unit ball consists of zero. This covers the empty-domain endpoint without assigning a support function to \(K_0=\varnothing\). ## Source scope and remaining course obligations The exact classical targets are Hörmander, *The Analysis of Linear Partial Differential Operators II*, §15.4, Theorems 15.4.1–15.4.3, printed pp. 296–300. The approved local source statement and proof pixels were read; this is original proof exposition, with the logarithmic homotopy, successive envelope choices and smooth coefficient majorants expanded in full. No protected source text or source-page image is included. Actual source pixels give curvature \(c(1+|\operatorname{Im}z|^2)^{-3/4}|w|^2\) in 15.4.3, rather than the historical inventory's incorrect weaker exponent \(-3/2\). The historical inventory remains unchanged and the current exact mapping is corrected. Completing these three formal targets does not complete the source's embedded representation exercises, its notes, Chapter 16 or any remaining assigned work in Chapters 10–13.