# Uniform geometry of faithful weights The Connes cocycle can compare weights by more than its real-time values. A contractive continuation through a lower strip defines an order, and allowing all strip widths produces an extended metric on faithful normal semifinite weights. We prove the order and metric arguments, identify the trace-density model, correct a factor of two in the printed model, and prove completeness. The source section fixes a separable properly infinite von Neumann algebra. The arguments below use neither restriction: throughout, \(M\) is a von Neumann algebra and \(\mathcal W_0(M)\) is its set of faithful normal semifinite weights. The trace model separately assumes that \(M\) is semifinite. The Fourier-filter convention is the one fixed in OA-FLOW. The two spectral applications below have complete contour and Poisson arguments relative to explicit Fourier and analytic inputs; their prerequisites and the half-strip domination theorem are not proved here. ## Contractive strips form an order For \(\lambda>0\), put \[ \begin{aligned} S_\lambda &=\{z\in\mathbb C:\\ &\qquad-\lambda\leq\Im z\leq0\}. \end{aligned} \tag{UW.1} \] For \(\varphi,\psi\in\mathcal W_0(M)\), write \(\varphi\preceq_\lambda\psi\) when \[ u_t=[D\varphi:D\psi]_t \tag{UW.2} \] has a sigma-strongly continuous extension to \(S_\lambda\), holomorphic in its interior, such that \[ \|u_z\|\leq1\qquad(z\in S_\lambda). \tag{UW.3} \] Local boundedness and scalar holomorphy make the interior extension norm-holomorphic by the norming-predual argument in CX-03. Consequently, products of two such extensions are holomorphic. For each fixed \(\lambda\), the relation \(\preceq_\lambda\) is an order. Reflexivity follows from \([D\varphi:D\varphi]_t=1\). If \(\varphi_1\preceq_\lambda\varphi_2\) and \(\varphi_2\preceq_\lambda\varphi_3\), the ordered chain rule gives \[ \begin{aligned} [D\varphi_1:D\varphi_3]_t &= [D\varphi_1:D\varphi_2]_t\\ &\quad\cdot[D\varphi_2:D\varphi_3]_t. \end{aligned} \tag{UW.4} \] The product of the two strip extensions is contractive, proving transitivity. For antisymmetry, suppose both directions hold and denote the two lower-strip extensions by \(u_z\) and \(v_z\). On the real axis, \(v_t=u_t^*\). Let \(\omega\) be a normal state. The two scalar functions \[ \begin{gathered} \omega(v_z)\quad(\Im z\leq0),\\ \overline{\omega(u_{\bar z})}\quad(\Im z\geq0). \end{gathered} \tag{UW.5} \] have equal boundary values and glue across the real axis. The elementary Morera gluing argument from MA-08 makes the result holomorphic on \(-\lambda<\operatorname{Im}z<\lambda\). Its modulus is at most one, while its value at zero is one. The maximum-modulus principle therefore makes it constant. Hence \(\omega(v_t)=1\) for every normal state and every real \(t\). Normal states separate \(M\), so \(v_t=1\). Fixed-reference cocycle injectivity in CX-10 gives \(\varphi_1=\varphi_2\). Write \[ \begin{gathered} \varphi\preceq_\infty\psi\\ \Longleftrightarrow\\ \varphi\preceq_\lambda\psi\quad(\lambda>0). \end{gathered} \tag{UW.6} \] The extensions for different widths agree on overlaps by scalar analytic uniqueness. Thus (UW.6) is equivalently a contractive holomorphic extension of (UW.2) to the closed lower half-plane. ## The half-strip recovers pointwise order We use the following exact form of the weight-domination theorem, recorded as OA-MOD-UW-DEP-DOMINATION. If \(\alpha,\beta\in\mathcal W_0(M)\) and \(C>0\), then \[ \alpha(x)\leq C\,\beta(x) \quad(x\in M_+) \tag{UW.7} \] is equivalent to continuation of \([D\alpha:D\beta]_t\) through the strip \(-1/2\leq\operatorname{Im}z\leq0\), with lower-boundary norm at most \(C^{1/2}\). The real boundary consists of unitaries, so the bounded-strip maximum principle gives the corresponding bound throughout the strip. This is Takesaki II, VIII.3, Theorem 3.17; its complete analytic proof remains an explicit prerequisite rather than being hidden in the present argument. Taking \(C=1\) gives the precise bridge \[ \boxed{ \begin{gathered} \alpha\preceq_{1/2}\beta\\ \Longleftrightarrow\\ \alpha(x)\leq\beta(x)\quad(x\in M_+). \end{gathered}} \tag{UW.8} \] All values in (UW.7)--(UW.8) are extended nonnegative values. No subtraction of infinite weights is involved. ## Infinite strips and the balanced spectral subspace Let \(\Phi=\varphi_1\oplus\varphi_2\) be the balanced weight on \(N=M\mathbin{\overline\otimes}M_2\). The balanced-matrix formula SI-14 gives \[ \begin{aligned} \sigma_t^\Phi(1\otimes e_{12}) &=[D\varphi_1:D\varphi_2]_t\\ &\quad\otimes e_{12}. \end{aligned} \tag{UW.9} \] We use the OA-FLOW Fourier-filter definition: if \(f=\mathcal F b\), then \[ \begin{aligned} f(p)&=\int_{\mathbb R}e^{ipt}b(t)\,dt,\\ \gamma_f(x)&=\int_{\mathbb R}b(t)\gamma_t(x)\,dt. \end{aligned} \tag{UW.50} \] The operator integral is weak-star. The spectrum is the hull of the ideal of Fourier functions whose filters annihilate \(x\). For the orbit \(\gamma_t(x)=e^{i\nu t}x\), with \(x\ne0\), (UW.50) gives \(\gamma_f(x)=f(\nu)x\), so its spectrum is \(\{\nu\}\). Smooth compactly supported Fourier functions separate any other point from \(\nu\), proving both inclusions in this hull equality. The positive Fourier transform of the scalar function \(e^{i\nu t}\), regarded as a distribution, is instead supported at \(-\nu\). These are different uses of the transform. Since \(|e^{i\nu(t-is)}|=e^{\nu s}\), boundedness below the real axis selects \(\nu\leq0\) in the filter spectrum. Here is a proof of the required half-plane criterion, relative to the exact Fourier, integration and local ideal inputs of OA-FLOW's spectral calculus. Their proofs are not given here. The general flow theorem is treated in OA-FLOW; this is its local application with the convention made explicit. **A bounded lower extension excludes positive frequencies.** Suppose \(F(t)=\gamma_t(x)\) extends holomorphically below the real axis, with \(\|F(z)\|\leq C\), and has its weak-star continuous boundary values. For \(f\in C_c^\infty(\mathbb R)\) supported in \((c,\infty)\), \(c>0\), set \[ \begin{gathered} b_f(z)\\ =\frac1{2\pi}\int e^{-ipz}f(p)\,dp. \end{gathered} \tag{UW.51} \] This is entire. Two integrations by parts in \(p\), together with the undifferentiated estimate for \(|t|\leq1\), give \[ \begin{gathered} \int |b_f(t-is)|\,dt\\ \leq C_f(1+s)^2e^{-cs},\\ s\geq0. \end{gathered} \tag{UW.52} \] In particular \(b_f\in L^1(\mathbb R)\) and \(\mathcal F b_f=f\). For completeness, the latter inversion follows by inserting \(e^{-\eta t^2}\), applying Fubini and the Gaussian Fourier identity specified in OA-FLOW, and then letting \(\eta\downarrow0\). The resulting Gaussian approximate identity tends to \(f(p)\), while dominated convergence on the \(L^1\) function \(b_f\) removes the factor. Thus no distributional inversion or closed-set spectral synthesis is being assumed in this test-function argument. For a normal functional \(\omega\), shift the contour of \(b_f(z)\omega(F(z))\) down by \(s\). Its vertical integrals vanish: on any fixed horizontal strip the two integrations by parts give \(O(|\operatorname{Re}z|^{-2})\), and \(F\) is bounded. If the boundary is only weak-star continuous, first start at depth \(\eta>0\), then pass to \(\eta\downarrow0\) by the same integrable bound. Consequently \[ \begin{gathered} |\omega(\gamma_f x)|\\ \leq C\|\omega\|C_f(1+s)^2e^{-cs}\\ \longrightarrow0\quad(s\to\infty). \end{gathered} \tag{UW.53} \] Normal functionals separate the algebra, so every such filter kills \(x\). At each \(p>0\) choose such an \(f\) with \(f(p)\ne0\). It belongs to the annihilator ideal and excludes \(p\) from its hull. Hence \(\operatorname{Sp}_\gamma(x)\subseteq(-\infty,0]\). **Negative spectrum supplies the contractive extension.** Now let \(\gamma\) be a sigma-strong-star continuous automorphism flow on a von Neumann algebra and assume that spectrum inclusion. For \(s>0\), put \[ \begin{gathered} P_s(r)=\frac{s}{\pi(s^2+r^2)},\\ \int P_s(r)\,dr=1,\\ \mathcal F P_s(p)=e^{-s|p|}. \end{gathered} \tag{UW.54} \] The mass identity is the elementary arctangent integral. For \(p>0\), integrate \(e^{ipz}s/(\pi(z^2+s^2))\) on the rectangle with vertices \(-R,R,R+iR,-R+iR\), where \(R>s\). The two vertical integrals are \(O(R^{-1})\); the upper integral is \(O(R^{-1}e^{-pR})\). The only pole inside is \(is\), with coefficient \(e^{-ps}/(2\pi i)\). Subtract that simple-pole term and apply the rectangle Cauchy formula to obtain the real integral \(e^{-ps}\). For \(p<0\) use the lower rectangle, with clockwise orientation and pole \(-is\), giving \(e^{sp}\). This proves (UW.54) using the scalar rectangle Cauchy input, without importing an arbitrary-cycle residue theorem. Both \(\partial_sP_s\) and \(P_s'\) are in \(L^1\), locally continuously in \(s>0\), as their explicit rational formulas show. Define the weak-star Poisson integral \[ \begin{gathered} F(t-is)\\ =\int P_s(r)\gamma_{t-r}(x)\,dr. \end{gathered} \tag{UW.55} \] Positivity and unit mass give \(\|F(t-is)\|\leq\|x\|\). We verify holomorphy without assuming spectral synthesis at zero. For \(\delta>0\), use the dual Banach action \(\beta_t^\delta=e^{-i\delta t}\gamma_t\). It is isometric and has continuous predual orbits; it need not be an automorphism action. The frequency-origin identity in OA-FLOW gives \(\operatorname{Sp}_{\beta^\delta}(x) =\operatorname{Sp}_\gamma(x)-\delta\subseteq(-\infty,-\delta]\). This also follows directly from (UW.50) by translating the Fourier function. Let \(F_\delta\) be (UW.55) with \(\gamma\) replaced by \(\beta^\delta\). Writing its kernel as \(P_s(t-v)\) makes both derivatives norm derivatives: the difference quotients converge in \(L^1\), and the action is bounded. The kernel for \(\partial_sF_\delta+i\partial_tF_\delta\), written as a filter of \(\beta^\delta_t(x)\), is \[ \begin{gathered} K_s(v)\\ =\partial_sP_s(v)-iP_s'(v),\\ \mathcal F K_s(p)\\ =-(|p|+p)e^{-s|p|}. \end{gathered} \tag{UW.56} \] Its Fourier support is contained in \([0,\infty)\), disjoint from \(\operatorname{Sp}_{\beta^\delta}(x)\). OA-FLOW's filter-support inclusion (S16) and its empty-spectrum criterion therefore make this filter zero. Thus \(\partial_sF_\delta+i\partial_tF_\delta=0\), the Cauchy--Riemann equation for \(z=t-is\). Continuous norm derivatives prove norm holomorphy. Put \(E_\delta=F_\delta(t-is)-F(t-is)\). The integral and (UW.54), with Cauchy--Schwarz for the probability measure \(P_s(r)\,dr\), give \[ \begin{aligned} &\|E_\delta\|\\ &\leq\|x\|\delta|t|\\ &\quad+\|x\|\sqrt{2(1-e^{-\delta s})}. \end{aligned} \tag{UW.57} \] This tends to zero uniformly on every compact subset of the lower half-plane. The norm limit is holomorphic and retains its norm bound. In a faithful normal representation, the vector integral corresponding to (UW.55) exists: each continuous vector orbit on \(\mathbb R\) is separable and bounded. Put \(Y=F(t-is)-\gamma_{t_0}(x)\) and \(D(r)=\gamma_{t-r}(x)-\gamma_{t_0}(x)\). Jensen's inequality gives \[ \begin{gathered} \|Y\xi\|^2\\ \leq\int P_s(r)\|D(r)\xi\|^2\,dr. \end{gathered} \tag{UW.58} \] As \((t,s)\to(t_0,0)\), split the integral at a small fixed \(|r|\). Strong continuity controls the part near zero; the remaining Poisson mass tends to zero, with integrand at most \(4\|x\|^2\|\xi\|^2\). The same argument for \(x^*\) gives strong-star continuity. Uniform boundedness makes this sigma-strong-star continuity as well: each summable family of vector seminorms is handled by a finite head and its uniformly bounded tail. Thus (UW.55) has the required boundary regularity, without a separability assumption on the algebra or its representation. Apply both directions to \(\gamma=\sigma^\Phi\) and \(x=1\otimes e_{12}\). Every integrand in (UW.55) remains in the 12-corner by (UW.9), so the extension has the form \(u_z\otimes e_{12}\). The corner embedding is isometric and preserves the boundary topology. We obtain the corrected equivalence \[ \boxed{ \begin{gathered} \varphi_1\preceq_\infty\varphi_2\\ \Longleftrightarrow\\ \operatorname{Sp}_{\sigma^\Phi}(1\otimes e_{12})\\ \subseteq(-\infty,0]. \end{gathered}} \tag{UW.10} \] Taking adjoints instead gives the positive half-line for \(e_{21}\). Indeed the filter identity for \(x^*\) replaces \(f(p)\) by \(\overline{f(-p)}\), so its annihilator hull is reflected. The earlier positive half-line for \(e_{12}\) was incompatible with the unchanged OA-FLOW convention. This identifies a local error; it makes no claim about an error in the source book's conventions. **A complete scalar sign test.** Take \(M=\mathbb C\), \(\psi(r)=r\) and \(\varphi(r)=r/2\) for \(r\geq0\). These are faithful normal finite weights. The balanced density on \(M_2(\mathbb C)\) is \(\operatorname{diag}(1/2,1)\) relative to its usual trace. Hence \[ \begin{gathered} u_t=2^{-it},\\ |u_{t-is}|=2^{-s}\leq1,\\ \operatorname{Sp}_{\sigma^\Phi}(e_{12}) =\{-\log2\}. \end{gathered} \tag{UW.59} \] Thus \(\varphi\preceq_\infty\psi\), while the earlier positive half-line condition fails. The adjoint \(e_{21}\) has frequency \(\log2\). For equal weights the frequency is zero, retained by both closed half-lines. The zero vector has empty spectrum, as in OA-FLOW. ## The extended uniform distance Scaling a weight changes the cocycle by \[ \begin{aligned} [D(c\varphi):D(d\psi)]_t &=(c/d)^{it}\\ &\quad\cdot[D\varphi:D\psi]_t,\\ &\hspace{5em}c,d>0. \end{aligned} \tag{UW.11} \] Define \[ \begin{aligned} d(\varphi,\psi) &=\inf\bigl\{a>0:\\ &\quad e^{-a}\psi\preceq_\infty\varphi,\\ &\quad \varphi\preceq_\infty e^a\psi \bigr\}. \end{aligned} \tag{UW.12} \] This is an extended metric. Multiplying both weights in a comparison by the same positive scalar leaves its cocycle unchanged. Hence the two inequalities in (UW.12), after reciprocal rescaling, also give \[ e^{-a}\varphi\preceq_\infty\psi \preceq_\infty e^a\varphi, \tag{UW.13} \] which proves symmetry. If \(a\) and \(b\) are admissible for \((\varphi,\psi)\) and \((\psi,\chi)\), respectively, transitivity gives \[ \begin{gathered} e^{-(a+b)}\chi\preceq_\infty\varphi,\\ \varphi\preceq_\infty e^{a+b}\chi. \end{gathered} \tag{UW.14} \] Taking infima proves the triangle inequality. If \(d(\varphi,\psi)=0\), (UW.8) gives, for every \(a>0\), \[ e^{-a}\psi\leq\varphi\leq e^a\psi. \tag{UW.15} \] Letting \(a\downarrow0\) at each positive element, including an infinite value, gives \(\varphi=\psi\). If \(d(\varphi,\psi)0\), rescale time by \(2\lambda\). The function \[ \begin{aligned} w_t &=[D\varphi:D\psi]_{2\lambda t}\\ &=h^{2\lambda it}k^{-2\lambda it}\\ &=[D\tau_{h^{2\lambda}} :D\tau_{k^{2\lambda}}]_t. \end{aligned} \tag{UW.20} \] has a contractive half-strip extension exactly when the cocycle in (UW.19) has a contractive strip of width \(\lambda\). Equations (UW.8) and PT-08's order-preserving density correspondence therefore prove \[ \boxed{ \begin{gathered} \varphi\preceq_\lambda\psi\\ \Longleftrightarrow\\ h^{2\lambda}\leq k^{2\lambda}. \end{gathered}} \tag{UW.21} \] The printed XII.5.6 omits the factor \(2\) in (UW.21). Its formula cannot be reconciled with the half-strip identity (UW.8). A concrete obstruction occurs in \(M_2(\mathbb C)\). Let \[ \begin{aligned} h&=\begin{pmatrix}2&0\\0&1\end{pmatrix},\\ k&=\begin{pmatrix}3&1\\1&2\end{pmatrix}. \end{aligned} \tag{UW.22} \] Then \(k-h\) is the positive rank-one matrix with every entry equal to one, but \[ \begin{gathered} k^2-h^2= \begin{pmatrix}6&5\\5&4\end{pmatrix},\\ \det(k^2-h^2)=-1. \end{gathered} \tag{UW.23} \] Thus \(h\leq k\) while \(h^2\nleq k^2\). At \(\lambda=1\), the printed criterion would accept this pair, whereas the defining strip and (UW.8) correctly reject it. The infinite-width conclusion printed after that formula remains valid. If \(p(s)=1_{[0,s]}(h)\) and \(q(s)=1_{[0,s]}(k)\), then \[ \begin{gathered} \varphi\preceq_\infty\psi\\ \Longleftrightarrow h^r\leq k^r\;(r>0)\\ \Longleftrightarrow h^n\leq k^n\;(n\geq1)\\ \Longleftrightarrow p(s)\geq q(s)\;(s\geq0). \end{gathered} \tag{UW.24} \] Only the implication from integer powers to spectral projections needs care. If \(\xi\in q(s)H\) and \(\mu>s\), the inequality with exponent \(2n\) gives \[ \begin{gathered} \|\mu^{-n}h^n\xi\|^2 \leq\|\mu^{-n}k^n\xi\|^2\\ \leq(s/\mu)^{2n}\|\xi\|^2. \end{gathered} \tag{UW.25} \] The spectral theorem then puts \(\xi\) in \(p(s)H\), so \(q(s)\leq p(s)\). Conversely, this projection order gives \(1-p(s)\leq1-q(s)\). The layer-cake formula for quadratic forms yields, for every \(r>0\), \[ \begin{aligned} \langle h^r\xi,\xi\rangle &=\int_0^\infty r s^{r-1}\\ &\quad\cdot \langle(1-p(s))\xi,\xi\rangle\\ &\qquad ds\\ &\leq\langle k^r\xi,\xi\rangle. \end{aligned} \tag{UW.26} \] The equality and inequality allow the value \(+\infty\). This proves all of (UW.24), and (UW.21) converts the all-real-power condition back to \(\preceq_\infty\). ## Differentiable cocycles are bounded logarithmic perturbations Keep the trace setting and put \[ \begin{aligned} H&=\log h,\\ K&=\log k,\\ u_t&=[D\psi:D\varphi]_t\\ &=e^{itK}e^{-itH}. \end{aligned} \tag{UW.27} \] The map \(t\mapsto u_t\) is sigma-strong-star differentiable if and only if \[ \begin{gathered} K=H+a,\\ D(H)=D(K). \end{gathered} \tag{UW.28} \] for a bounded self-adjoint \(a\in M\). Suppose first that the derivative at zero exists. Differentiating \(u_t^*u_t=1\) shows that \(u'_0=ia\) for a bounded self-adjoint \(a\). For \(\xi\in D(H)\), the identity \[ e^{itK}\xi=u_t e^{itH}\xi \tag{UW.29} \] has a strong derivative at zero. Stone's exact derivative-domain theorem SG-04 gives \(\xi\in D(K)\) and \(K\xi=(H+a)\xi\). Applying the same argument to \(e^{itH}=u_t^*e^{itK}\) gives the reverse domain inclusion. This proves (UW.28). Conversely, a bounded self-adjoint perturbation preserves self-adjointness and the domain. The cocycle identity and the derivative at zero give the strong-star differential equation \[ \begin{gathered} u'_t=i\,u_t\sigma_t^\varphi(a),\\ u_0=1. \end{gathered} \tag{UW.30} \] and hence \[ u_t=1+i\int_0^t u_s\sigma_s^\varphi(a)\,ds. \tag{UW.31} \] The integral is sigma-strong-star. Iteration gives the time-ordered Dyson series; its \(n\)-th term has norm at most \(\|a\|^n|t|^n/n!\). It therefore converges in norm on compact time intervals and verifies (UW.30), proving the converse without differentiating an unbounded product formally. ## Uniform distance bounds the perturbation spectrum Let \(\varphi,\psi\in\mathcal W_0(M)\) and suppose \[ \alpha=d(\varphi,\psi)<\infty. \tag{UW.32} \] Write \(u_t=[D\psi:D\varphi]_t\), as in (UW.16). The entire extension in UW-04 is norm differentiable. Set \(a=-iu'_0\in M\); differentiation of \(u_t^*u_t=1\) at zero gives \(a=a^*\). Differentiating the cocycle identity \(u_{t+s}=u_t\sigma_t^\varphi(u_s)\) in \(s\) at zero gives (UW.30). In the trace-density setting of UW-06, its domain argument identifies \(a=K-H\) on \(D(H)=D(K)\). For arbitrary weights, the bounded cocycle derivative defines \(a\) without introducing trace densities. For every \(\varepsilon>0\), put \(b=\alpha+\varepsilon\). Equation (UW.17) gives \[ \|u_z\| \leq e^{(\alpha+\varepsilon)|\operatorname{Im}z|}. \tag{UW.33} \] The differential equation supplies the actual modular orbit \[ \sigma_t^\varphi(a) =-i\,u_t^*u'_t. \tag{UW.34} \] The functions \(u_t\) and \(u'_t\) are cocycle functions; they have not been identified as orbits of fixed algebra elements. Therefore the fixed-element spectral-product theorem does not justify a product-support step here. We instead construct the entire continuation of (UW.34) and prove its Fourier-filter support directly. Put \(u^\sharp(z)=u(\bar z)^*\). This is entire: if \(u(w)=\sum_n c_n(w-\bar z_0)^n\) near \(\bar z_0\), then \(u^\sharp(z)=\sum_n c_n^*(z-z_0)^n\) near \(z_0\). Its norm has the same bound as \(u\). The norm Cauchy estimate on the circle of radius one about \(z\) gives \[ \|u'_z\|\leq e^b e^{b|\operatorname{Im}z|}. \tag{UW.60} \] Consequently \[ \begin{gathered} G(z)=-i\,u^\sharp(z)u'_z,\\ G(t)=\sigma_t^\varphi(a),\\ \|G(z)\|\leq e^b e^{2b|\operatorname{Im}z|}. \end{gathered} \tag{UW.61} \] Both factors are norm holomorphic and multiplication is a bounded bilinear map, so \(G\) is entire. The reflection in \(u^\sharp\) is essential; \(z\mapsto u(z)^*\) alone would not be holomorphic. Take \(f\in C_c^\infty(\mathbb R)\) supported in \((c,\infty)\), where \(c>2b\), and use its inverse Fourier function \(b_f\) from (UW.51). The inversion and integrability proved in UW-03 make \(\sigma_f^\varphi(a)\) exactly the filter in (UW.50). For a normal functional \(\omega\), the rectangle contour shift gives \[ \begin{aligned} &\omega(\sigma_f^\varphi(a))\\ &=\int b_f(t-is)\\ &\quad\cdot\omega(G(t-is))\,dt. \end{aligned} \tag{UW.62} \] For fixed \(s\), the vertical sides vanish because \(b_f\) is \(O(|t|^{-2})\) uniformly on that strip and \(G\) is bounded there by (UW.61). All horizontal integrals are absolutely convergent. Equations (UW.52) and (UW.61) give \[ \begin{aligned} &|\omega(\sigma_f^\varphi(a))|\\ &\leq\|\omega\| e^b C_f(1+s)^2\\ &\quad\cdot e^{-(c-2b)s}\longrightarrow0. \end{aligned} \tag{UW.63} \] Thus every such filter annihilates \(a\). For \(f\) supported in \((-\infty,-c)\), shift upwards instead: \(b_f(t+is)\) has the same exponential decay, and (UW.61) is symmetric in the two half-planes. This proves annihilation for the negative tests too. At any point outside \([-2b,2b]\), choose a smooth compactly supported test function nonzero there and supported entirely on the corresponding side, with some \(c>2b\). It lies in the annihilator ideal, so the point is absent from the OA-FLOW hull. Hence \(\operatorname{Sp}_{\sigma^\varphi}(a)\subseteq[-2b,2b]\). Intersecting these closed intervals over all \(\varepsilon>0\) proves \[ \boxed{ \operatorname{Sp}_{\sigma^\varphi}(a) \subseteq[-2\alpha,2\alpha].} \tag{UW.35} \] The argument proves the claimed support at the filter definition, without a product of distributions or a spectral-synthesis assumption. Reflecting the Fourier sign leaves this symmetric interval unchanged. The case \(\alpha=0\) is included by the same intersection (and the metric gives \(\varphi=\psi\), hence \(a=0\)). The factor two in (UW.35) is present in the source and is distinct from the missing factor two corrected in (UW.21). ## Entire functions of small exponential type vary little Let \(E\) be a complex Banach space. Given \(\varepsilon,R>0\), there is \(\delta>0\) such that every entire \(f:\mathbb C\to E\) satisfying \[ \begin{gathered} \|f(z)\|\leq e^{\delta|\Im z|},\\ z\in\mathbb C. \end{gathered} \tag{UW.36} \] also satisfies \[ \begin{gathered} \|f(z)-f(0)\|\leq\varepsilon,\\ |z|0\). If \(d(\varphi_n,\varphi_m)<\delta\), write \[ W_{nm}(z)=[D\varphi_n:D\varphi_m]_z. \tag{UW.42} \] UW-08 makes \(W_{nm}(z)\) uniformly close to one for \(|z|\leq R\) once \(\delta\) is small. Analytic continuation of the chain rule gives \[ U_n(z)=W_{nm}(z)U_m(z). \tag{UW.43} \] The common compact bound on \(U_m\) now shows that \((U_n)\) is uniformly Cauchy on \(|z|\leq R\). Its locally uniform norm limit \(V\) is entire. On the real axis it is a strongly continuous unitary \(\sigma^\rho\)-cocycle: unitarity and the cocycle law pass through the norm limit. The reconstruction theorem CX-10 gives a unique \(\psi\in\mathcal W_0(M)\) with \[ V_t=[D\psi:D\rho]_t. \tag{UW.44} \] For \(\varepsilon>0\), choose \(n\) so late that \(d(\varphi_m,\varphi_n)<\varepsilon\) for every later \(m\). The functions \([D\varphi_m:D\varphi_n]_z\) have the common bound \(e^{\varepsilon|\operatorname{Im}z|}\). Their locally uniform limit is the analytic continuation of \([D\psi:D\varphi_n]_t\), by (UW.43)--(UW.44). The same bound survives the limit, so \[ d(\psi,\varphi_n)\leq\varepsilon. \tag{UW.45} \] Thus \(\varphi_n\to\psi\) in the uniform topology. Choosing the fixed tail reference \(\rho\) explicitly avoids assuming in advance that the whole sequence lies in one finite-distance component. For every \(x\in M_+\), the evaluation map \[ \varphi\longmapsto\varphi(x) \tag{UW.46} \] is continuous into \([0,\infty]\) with its order topology. Indeed, \(d(\varphi,\psi)