Proof. The shrinker identities imply \(\operatorname{div}_S X^\top=2-|X^\perp|^2/2\). Weighted integration of \(\operatorname{div}_G(u^2X^\top)\) therefore gives \[\tfrac12\int_S |X|^2u^2\,d\gamma =2\int_Su^2\,d\gamma+2\int_SuX^\top\cdot\nabla u\,d\gamma.\] Cauchy’s inequality proves \(\||X|u\|_G\le C(\|\nabla u\|_G+\|u\|_G)\). The same calculation applied to the norm squared of \(\nabla u\) gives \(\||X|\nabla u\|_G\le C(\|\nabla^2u\|_G+\|\nabla u\|_G)\). Applying the first inequality to \((1+|X|^2)^{1/2}u\) then controls \(|X|^2u\). These calculations initially use compact supports, and pass to the completion.

For completeness the domain can also be read from an ordinary Schrödinger operator. The unitary map \(Uu=\rho_G^{1/2}u\) satisfies \[ -ULU^{-1}=-\Delta_S+\frac{|X|^2}{16} +\frac{|X^\perp|^2}{16}-1-|\mathrm{II}|^2. \tag{8}\] Indeed, for \(\phi=|X|^2/4\) one has \(\nabla\phi=X^\top/2\) and \(\Delta\phi=1-|X^\perp|^2/4\). The last three terms in (8) are bounded. If \(q=|X|^2/16\), direct integration by parts yields \[\|(-\Delta+q)w\|_2^2 =\|\Delta w\|_2^2+\|qw\|_2^2 +2\int q|\nabla w|^2-\int(\Delta q)|w|^2.\] Here \(\Delta q\) is bounded. The Bochner identity and bounded Ricci curvature control \(\nabla^2w\) by \(\Delta w\) and lower derivatives. It follows that the closed form realization has domain \[\{w:\nabla^2w,\ |X|\nabla w,\ |X|^2w,w\in L^2(S)\},\] with equivalent graph norm. Cutoffs and local elliptic regularity show that compactly supported smooth functions are a core. Conjugating back, and using the moment estimates just proved, identifies this domain with the derivative-defined \(H^2_G\) and proves (7). Finally \(q\to\infty\) on every end. The \(\|qw\|_2\) bound makes the \(L^2\) tails uniformly small, and Rellich compactness on compact subsets gives compactness of the domain inclusion. ◻

Proof. Choose \(k=\dim K\) compact smooth functions \(\chi_i\) whose pairings with a basis of \(K\) form a nonsingular matrix. For this initial list, projection of the first equation of (9) onto \(K\) determines \(\lambda\). Inversion of \(L\) on \(K^\perp\) determines the component of \(u\) in \(K^\perp\), and \(Pu\) determines its component in \(K\).

Let \(G=(L|_{K^\perp})^{-1}\) and \(\mathcal V=C_c^\infty(S)\cap K^\perp\). This space is dense in \(K^\perp\): approximate by compact functions and remove their finitely many kernel pairings using the initial \(\chi_i\). The symmetric form \[Q(f,g)=\langle Gf,g\rangle_G,\qquad f,g\in\mathcal V,\] is nondegenerate. If \(Q(f,g)=0\) for all \(g\in\mathcal V\), density gives \(Gf=0\) and hence \(f=0\). The sources \(q_j=Lu_j\) lie in \(\mathcal V\), by self-adjointness.

Every finite subspace \(V\subset\mathcal V\) has a finite nondegenerate enlargement for \(Q\). To see this, split off a nondegenerate part of \(V\) and let \(r_1,\ldots,r_s\) span its radical. Nondegeneracy on \(\mathcal V\) supplies vectors \(w_i\) with \(Q(r_i,w_j)=\delta_{ij}\). Subtract their projections to the already nondegenerate part. On \(\operatorname{span}(r_i,w_i)\) the matrix is \(\left(\begin{smallmatrix}0&I\\I&C\end{smallmatrix}\right)\) and is invertible. Apply this construction to the span of the \(q_j\) and add the resulting compact functions as observation duals.

For the enlarged homogeneous block, kernel projection first forces the original \(k\) multipliers to vanish. Its other multipliers vanish by the nondegeneracy of \(Q\) on the added space; the original observations then kill the remaining kernel component of \(u\). The same decomposition solves the inhomogeneous equations and proves bijectivity. Finally \(Lu_j=q_j\in\operatorname{range}P^*\), so \(u_j\) is the normalized solution associated with the observation value \(Pu_j\) and a suitable multiplier vector. No positivity of \(Q\) is required. ◻

Proof. Freeze \(a,c\) and first omit the small mixed coefficient. For real \(\xi\) and integer \(j\), accretivity and then the imaginary part give \[ 1+|i\xi+a\xi^2+cj^2| \ge c\bigl(1+|\xi|+E\xi^2+j^2+\sqrt E|j\xi|\bigr). \tag{18}\] Indeed the real part controls \(E\xi^2+j^2\), their imaginary parts have size at most a constant times that quantity, and \(2\sqrt E|j\xi|\le E\xi^2+j^2\). The mixed term is absorbed by choosing \(\varepsilon_*\) small. Fourier transformation in \(t\) and Fourier series in \(\theta\) give the interior estimate. The bounded normalized derivatives allow freezing on intervals of fixed sufficiently small length. Commutators with a cutoff contain \(a\chi'v_t\) and \((\chi'-a\chi'')v\), so interpolation absorbs the derivative term and leaves precisely the lower norm in (14).

For the boundary estimate add a fixed positive zeroth-order shift \(K\). The characteristic roots satisfy \[a\lambda^2-\lambda-cj^2-K=0.\] There is one root in each open half-plane, and, with labels \(+\) and \(-\), \[ \begin{split} \operatorname{Re}\lambda_+\asymp|\lambda_+| &\asymp E^{-1}(1+\sqrt E|j|),\\ -\operatorname{Re}\lambda_-\asymp|\lambda_-| &\asymp\frac{1+j^2}{1+\sqrt E|j|}. \end{split} \tag{19}\] To verify the half-plane assertion, an imaginary root would make the real part of the equation equal to \(-\operatorname{Re}a\,\xi^2-\operatorname{Re}c\,j^2-K<0\). Continue from positive real coefficients. For \(\sqrt E|j|\) small the roots have the two asserted scales by the quadratic formula; for this quantity large they are asymptotic to \(\pm j\sqrt{c/a}\), whose real parts have a uniform angle margin. The intervening scaled compact set, together with the preceding exclusion of imaginary roots, gives uniform constants. These arguments also cover the bounded transition between the two scales.

At an entrance the correction to a whole-line particular solution is \(\varphi_j e^{\lambda_-t}\). At an exit, written as \(t=0\) with \(t<0\) inside, the derivative lift is \(\psi_j\lambda_+^{-1}e^{\lambda_+t}\). Integrating their five squared norms in (12) gives, respectively, (16) and (17), up to uniform constants. This constructs the boundary parametrices. Freezing and absorption as above give the variable-coefficient boundary estimate; removing \(K\) only restores the lower norm in (14).

The simpler traces follow by product integration, in particular from \[\partial_t\|v_\theta\|_{H^h}^2 =-2\operatorname{Re}\langle v_t,v_{\theta\theta}\rangle_{H^h}, \qquad \partial_t\|v_t\|_{H^h}^2 =2\operatorname{Re}\langle v_{tt},v_t\rangle_{H^h}.\] For the stronger Dirichlet trace apply the same argument in each mode also to \(\sqrt E|j|^3|v_j|^2\); its derivative is bounded by a constant times \(|j^2v_j|\,|\sqrt E jv_{t,j}|\). For the Neumann trace set \(w=v_{t,j}\) and apply the one-dimensional trace inequality on length \(\ell=E(1+Ej^2)^{-1/2}\). Multiplication by \(E\sqrt{1+Ej^2}\) bounds its right hand side by \[C\bigl((1+Ej^2)\|v_{t,j}\|_2^2 +E^2\|v_{tt,j}\|_2^2\bigr).\] Summing proves the assertions. Thus the additional angular regularity in a convenient smooth-data lift is imposed on prescribed data, not lost from the unknown. A derivative trace supplied by another solution already belongs to the natural space (17). ◻

Proof. Put \(R_z=\mu R_t\) and \(D=R^2(1+R_z^2)+R_\theta^2\). Direct inversion of \(g=\operatorname{diag}(1,R^2)+dR\otimes dR\) gives \[g^{zz}=\frac{R^2+R_\theta^2}{D},\qquad g^{z\theta}=-\frac{R_zR_\theta}{D},\qquad g^{\theta\theta}=\frac{1+R_z^2}{D}.\] These are analytic functions of the displayed first-derivative variables while \(D\) is a unit. In the curvature numerator above, \(R_{zz}=\mu^2(R_{tt}-R_t)\) and \(R_{z\theta}=\mu R_{t\theta}\); the remaining terms have only the same first variables and \(R_{\theta\theta}\). The support term \((R-R_t)/2\) is affine. Thus every highest second derivative remains in its \(L^2_tH^h_\theta\) slot, while its coefficient is in \(L^\infty_tH^h_\theta\) by (15). The product estimate \[\|AB\|_{L^2_tH^h_\theta} \le C_h\|A\|_{L^\infty_tH^h_\theta} \|B\|_{L^2_tH^h_\theta}\] and analytic inversion of the unit denominator prove the map assertion. In its derivative, a coefficient variation is estimated in the first trace norm and multiplies a background second derivative in the bulk norm. This proves the stated small mean-value bound, strip by strip; division by \(\rho\) gives the weighted version. Fixed translations preserve the factors \(\mu^2,\mu\), and angular commutations use the same algebra and module estimate. ◻

Proof. High Fourier numbers satisfy the coercive estimate after any fixed shift \(\alpha\), so only finitely many angular modes need separate treatment. In such a mode the equation is \[v_t-2\mu^2(v_{tt}-v_t)-\sigma_jv=f.\] Write \(a=2\mu^2\), \(q=v_t\), \(k=(1+a)/a\), and \[H(t,s)=\exp\!\left(-\int_t^s k(\zeta)\,d\zeta\right),\qquad (\mathsf A_T w)(t)=\int_t^T a(s)^{-1}H(t,s)w(s)\,ds.\] Solving the first-order equation for \(q\) backward gives the exact formula \[q(t)=H(t,T)q(T)+\mathsf A_T(\sigma_jv+f)(t).\] The kernel has bounded mass, width \(O(\mu(t)^2)\), and an adjacent-strip tail bounded by \(C\exp(-c|s-t|/\mu(t)^2)\). Its finite mass is \[\begin{align*} \mathsf A_T1 &=1-H(t,T)-\int_t^T\frac{a(s)}{1+a(s)}k(s)H(t,s)\,ds\\ &=1-H(t,T)+O(a(t)). \end{align*}\] In particular the infinite average has mass \(1+O(a(t))\), whereas the finite one has a terminal layer. On its last strip \(\|H(\cdot,T)\|_{L^2}\le C\sqrt{a(T)}=C\mu(T)\). Translation across the kernel and the fundamental theorem of calculus therefore give, for \(\rho=e^{\alpha t}\), \[ \begin{split} \|(\mathsf A_T-I)v\|_{Y_{\rho,h}} \le{}&C\sup_{t\ge t_0}\mu(t)\,\|v_t\|_{Y_{\rho,h}} +C\sup_{t\ge t_0}\mu(t)^2\,\|v\|_{Y_{\rho,h}}\\ &+C\mu(T)\rho(T)^{-1}\|v(T)\|_{H^h}. \end{split} \tag{24}\] One may equivalently retain \(-H(t,T)v(T)\) as part of the terminal lift; replacing \(v(t)\) by \(v(T)\) in that layer costs the first term above. For the finitely many modes under consideration, the one-dimensional \(H^1\) trace on a terminal interval of length \(\min(1,\ell_*)\) bounds the last term by \(C\sup\mu(\|v\|_{Y_{\rho,h}}+\|v_t\|_{Y_{\rho,h}})\). At an infinite exit the layer is absent. The estimate thus retains a small factor at every permitted finite exit, with no additional angular trace requirement. Keep this average on \(f\). The remaining first-order operator \(\partial_t-\sigma_j\) is inverted by integration from the entrance if \(\sigma_j<\alpha\), and by integration from the right if \(\sigma_j>\alpha\). Its weighted kernel has mass at most \(|\alpha-\sigma_j|^{-1}\). Taking \(t_0\) large absorbs the displayed averaging error. The equation then controls \(v_t\) and \(\mu^2v_{tt}\). This constructs the separated inverse, including finite exits. Variation of constants shows that crossing \(\sigma_j\) adds one solution for each vector of its angular eigenspace, with that vector times \(e^{\sigma_jt}\) as leading term. Condition (22) makes the perturbation norm on a far tail arbitrarily small at each fixed weight; the Neumann series and the same variation-of-constants formulas then give the asserted perturbation and crossing statements.

For the first assertion of (23), we first need small-coefficient invertibility, without a coefficient decay hypothesis. Put \(v=R-\sqrt2\), \(a=2\mu^2\), and let \(K\) denote the exact curvature numerator preceding (11). Stationarity gives \[q:=v_t=zR_z=R+2K\longrightarrow0,\qquad a v_{tt}=2(R_{zz}+R_z/z)\longrightarrow0,\qquad \sqrt a\,v_{t\theta}=\sqrt2R_{z\theta}\longrightarrow0.\] Indeed spatial convergence gives \(K\to-1/\sqrt2\); angular differentiation gives the same assertions at every fixed angular order. Together with \(v,v_{\theta\theta}\to0\), these identities prove that the actual graph has small tail norm in \(X_0:=X_{1,h}\), with all five slots of (12). No unweighted bound for \(v_{tt}\) has been used. The homogeneous term of the model fast derivative inverse is proportional to \(\exp(t+e^{2t}/4)\) and is excluded by \(q=zR_z=o(e^t)\).

Fix \(0<\eta<1\), set \(\alpha=-2+\eta\), and write \(X_\beta=X_{e^{\beta(t-t_0)},h}\) and similarly \(Y_\beta\). For both \(\beta=0\) and \(\beta=\alpha\), the model construction above has the same entrance projection \(S=P_{\{|j|\ge2\}}\) and gives \[\|u\|_{X_\beta}\le C_{\eta,h} \bigl(\|\mathcal L_{\mathrm{cyl}}u\|_{Y_\beta} +\|Su(t_0)\|_{\mathcal T^D_{a(t_0),h}}\bigr).\] The other slow channels use their backward integrals at infinity; every fast outward channel is excluded. For the finitely many low modes the integral kernel is bounded by the inverse spectral gap. For \(|j|\ge2\), conjugation by \(e^{\beta(t-t_0)}\) gives interior energy \[\int\!\left[a|w_t|^2+ \{j^2-2+\beta-a(1+\beta+\beta^2)\}|w|^2\right].\] Here \(a'=-2a\) and the leading mass is at least \(\eta\). Homogeneous outgoing derivative data give terminal coefficient \((1-a(T))/2>0\). The strip estimate supplies the remaining slots, and localization as in (21) gives a uniformly local constant independent of the terminal section. Finite problems and exhaustion therefore construct this inverse on each space, with the natural entrance trace (16).

Lemma 7 supplies the nonlinear coefficient estimate in these same strip norms, with the first-derivative traces and the strict principal margins just established.

Factor the actual equation as \((\mathcal L_{\mathrm{cyl}}+Q)v=0\) by integrating its derivative along \(sv\), \(0\le s\le1\). The radial coefficient variables \(R,R_\theta,\mu R_t\), the first traces (15), and the coefficient–module rule of Lemma 7 give \(\|Q\|_{X_\beta\to Y_\beta}\le\delta(t_0)\to0\) for these two weights. Background second derivatives stay in their bulk slots; no log derivative of a coefficient is needed here. If \(H_\beta\) lifts the stable entrance value and \(K_\beta\) is the zero-entrance model inverse, solve \[u_\beta=H_\beta Sv(t_0)-K_\beta Q u_\beta\] by a Neumann series, choosing \(C_{\eta,h}\delta(t_0)<1/2\). The constructed \(u_\alpha\) lies in \(X_0\), as does the actual \(v\). Uniqueness in \(X_0\) with the same stable data gives \(v=u_0=u_\alpha\). This proves the domain upgrade and the first bound in (23); it does not assume that bound when estimating \(Qv\).

To obtain log regularity, use the nonlinear equation itself. Replace its explicit \(\mu(t)\) by \(e^{-\tau}\mu(t)\) on the fixed tail and prescribe stable entrance data \(Sv(t_0+\tau)\). The coefficient–module formulas make this a smooth family of maps \(X_\alpha\to Y_\alpha\), whose derivative in the unknown is a small perturbation of the same invertible model. The implicit function theorem constructs its smooth local solution family. The shifted actual graph is small in \(X_0\) and has these data. The same mean-value estimate gives uniqueness of small nonlinear solutions in \(X_0\), identifying it with that family without assuming translation differentiability in \(X_\alpha\). Differentiating the explicit coefficients preserves the factors \(\mu^2\) and \(\mu\) on second derivatives. Iteration, at a sufficiently large fixed angular order, gives \(\partial_t^k v\in X_\alpha\) for each fixed \(k\); a small extra loss may be allowed in \(\eta\). Consequently the normalized coefficient differences and their required log derivatives decay like \(O(e^{-(2-\eta)t})\). Only at this point do we use the decaying-coefficient assertion for the sharper Jacobi asymptotics.

We also explain why the Gaussian condition permits use of polynomial weights. Local elliptic estimates on spatial balls of radius \((1+|X|)^{-1}\) and the Gaussian \(L^2\) bound give, for a homogeneous exterior solution, a pointwise bound by a polynomial times \(\exp(|X|^2/8+o(|X|^2))\). Choose \(\alpha_0\) strictly between \(1/8\) and \(1/4\). Direct differentiation gives \[L e^{\alpha_0|X|^2} =\bigl((4\alpha_0^2-\alpha_0)|X|^2+O(1)\bigr) e^{\alpha_0|X|^2}.\] Likewise \(L(1+|X|^2)^{M/2}\) is negative on a far end for sufficiently large fixed \(M\), and dominates a given polynomial source. Compare the real and imaginary parts with a fixed multiple of this power plus \(\varepsilon e^{\alpha_0|X|^2}\). The latter dominates the remote boundary; after exhausting the end, let \(\varepsilon\downarrow0\). The comparison principle is justified by division by the positive supersolution, which makes the zeroth-order coefficient nonpositive. Thus Gaussian solutions of polynomial-source equations have polynomial growth. For a nonsmooth source in an exterior range space, first subtract a cutoff of the exterior particular solution; the same argument applies to the homogeneous remainder.

Starting with that polynomial bound, descend through weights above \(2\), then through \(2\) and \(1\). The resulting coefficients are \(c\) and \(\ell\) in (23). Coefficient errors multiplying \(cz^2\) have size \(O(z^\eta)\), explaining the remainder stated there. When \(c=0\), subtract an exact rotation field, whose leading term is \(z(\ell\cdot e_r)\). No remaining slow rate lies between \(0\) and \(-2+\eta\), giving the sharper remainder. Conversely prescribe each slow coefficient in the variation-of-constants construction on the tail and multiply the resulting exact end solution by an inward cutoff. It has polynomial growth and compact source.

Finally log regularity does not require holomorphic spatial charts. Pull the operator back by a small real translation of \(t\), cut off near the fixed entrance. Its coefficients are differentiable in the translation parameter in the same mapping norms. Invert the differentiated equation and use uniqueness to identify its derivative with \(\partial_t v\). Iteration, increasing the fixed angular order if needed, proves the asserted fixed-order log estimates. ◻

Proof. First take real data, zero entrance value, and a finite interval with zero terminal time derivative. Put \[U=\|\mu^{-1}v_t\|_2,\quad V^2=\|\mu v_{t\theta}\|_2^2+\|\mu v_{\theta\theta}\|_2^2, \quad F_0=\|f/\mu\|_2,\quad \mu_0=e^{-t_0}, \quad \ell=dv_t+qv_\theta+ev.\] Multiplication of \(\mathcal Cv=f\) by \(-v_{\theta\theta}\), followed by one integration in \(t\) and one in \(\theta\) for the \(a\) term, gives \[\begin{align*} &\tfrac12\|v_\theta(T)\|_2^2 +\int\mu^2 (a v_{t\theta}^2+2b v_{t\theta}v_{\theta\theta} +c v_{\theta\theta}^2) \\ &\quad=-\int f v_{\theta\theta} -\int\mu^2a_\theta v_t v_{t\theta} +\int(\mu^2a)_t v_t v_{\theta\theta} -\int\mu^2\ell v_{\theta\theta}. \tag{29}\end{align*}\] The boundary term suppressed in this identity is \(-[\int\mu^2a v_t v_{\theta\theta}]_{t_0}^T\). It vanishes because \(v_{\theta\theta}(t_0)=0\) and \(v_t(T)=0\).

The second multiplier is \(\mu^{-2}v_t\). Integrating the \(a\) term in time and the mixed term in angle gives \[\begin{align*} U^2+\tfrac12\int_{S^1}a(t_0)v_t(t_0)^2 ={}&\int(f/\mu)(v_t/\mu)+\int c v_{\theta\theta}v_t +\int\ell v_t\\ &-\int(\tfrac12a_t+b_\theta)v_t^2. \tag{30}\end{align*}\] In particular the entrance term is favorable and is generally nonzero. The cross term \(\int c v_{\theta\theta}v_t\) is bounded directly by \(CUV\). Integrating it in time would unnecessarily introduce an unweighted \(c_t v_\theta^2\) term; no integrability of \(c_t\) is assumed.

Let \(H_0=\|\mu(qv_\theta+ev)\|_2\). The two identities imply \[c_*V^2\le F_0V+C\mu_0^2UV+H_0V, \qquad U^2\le F_0U+CUV+C\mu_0^2U^2+H_0U.\] Product integration, using the zero entrance value, gives \[\sup_t\|v_\theta(t)\|_2^2\le2UV, \qquad \sup_t\|v(t)\|_2\le(\int_{t_0}^\infty\mu^2)^{1/2}U =\mu_0 U/\sqrt2.\] Consequently \(H_0\le C\mu_0\sqrt{UV}+C\mu_0^2U\). Add a sufficiently small fixed multiple of the second energy inequality to the first, and then choose \(\mu_0\) sufficiently small. Young’s inequality absorbs the cross terms and yields \(U+V\le CF_0\) with a constant independent of \(T\). The equation recovers the missing derivative: \[a\mu v_{tt}=\mu^{-1}v_t-2b\mu v_{t\theta} -c\mu v_{\theta\theta}-\mu\ell-f/\mu.\] Thus \(W=\|\mu v_{tt}\|_2\le CF_0\). Since \(v_t(T)=0\), product integration also gives \(\sup_t\|v_t(t)\|_2^2\le2UW\). These are all the zeroth-angular-order norms in (26).

Commute with \(\partial_\theta^j\) and keep the same principal operator on the left. After division of the new source by \(\mu\), the new highest terms are \[\mu(\partial_\theta^k A^{rs}) D_{rs}\partial_\theta^{j-k}v,\qquad 1\le k\le j.\] Each involves a strictly lower angular commutation of the already controlled second derivatives. Induction in \(j\) therefore proves (28) at every fixed \(h\), without a time derivative of \(f\). Nonzero entrance data are removed by an extension on the first fixed unit collar. A Fourier extension \(\varphi_j e^{-(1+|j|)(t-t_0)}\), with a fixed cutoff, has commuted \(H^2\) norm controlled by \(\|\varphi\|_{H^{h+3/2}}\); its forcing in (27) has the same bound with a constant depending on \(t_0\).

For each finite \(T\) this is a scalar elliptic problem with entrance Dirichlet and transverse outgoing derivative boundary conditions. Its index is zero, as follows by deforming the positive principal matrix to the identity and the lower terms to zero on the finite domain. The estimate excludes a kernel, so the problem is surjective. Exhaustion, weak compactness in the weighted bulk spaces, and local elliptic compactness give an infinite-end solution with the same bound.

For uniqueness on the infinite end it is necessary to justify removal of terminal terms for an arbitrary element of (26). Indeed \[\int_{t_0}^\infty \left|\frac{d}{dt}\|v_t(t)\|_{H^h}^2\right|dt \le2\|\mu^{-1}v_t\|_{L^2H^h}\|\mu v_{tt}\|_{L^2H^h}<\infty.\] The weighted \(L^2\) condition forces the limiting derivative trace to be zero. Also, after angular integration, \[\int\mu^2a v_t v_{\theta\theta} =-\int\mu^2a v_{t\theta}v_\theta -\int\mu^2a_\theta v_tv_\theta.\] Choose terminal sections tending to infinity where \(\|\mu v_{t\theta}\|_{H^h}\to0\). The other factors have bounded traces, so this boundary term vanishes along those sections. The terminal angular energy in (29) remains nonnegative. The same two-energy argument now proves the estimate and uniqueness on the infinite domain. Finally all second derivatives, including \(\mu v_{tt}\), have been controlled; a small complex change is therefore a small bounded perturbation of the full isomorphism and is inverted by a Neumann series. ◻

Proof. Fix a far entrance and write \(z=e^t\). The normal component of link change is \(z a_n(t,\theta)\), where \(a_n\) is a real smooth unit with bounded inverse and bounded fixed log and angular derivatives. The scalar normalization of the conical equation gives another such unit \(c_n\) for which, on the tail, \[L(z a_n v)=z c_n\mathcal Cv.\] The functions \(a_n,c_n\) include the normal-speed and drift factors; they multiply the source as well as the unknown. This identity follows by differentiating the prescribed link graph equation, whose coefficient of \(v_t\) is a nonzero unit before its drift normalization.

Local elliptic regularity makes the trace \(v_0=J/(z a_n)\) at the entrance smooth. Solve \(\mathcal Cv=0\) with this trace by Lemma 9. The normal lift \(\widetilde J=z a_n v\) is in the Gaussian \(H^2\) domain on the tail. Indeed smooth conical coordinates give \(dA\asymp z^2dt\,d\theta\) and \[|\nabla\widetilde J|\le C(|v|+|Dv|),\qquad |\nabla^2\widetilde J| \le Cz^{-1}(|D^2v|+|Dv|+|v|).\] The highest Gaussian integral is bounded by \[\int e^{-z^2/4}|D^2v|^2\,dt\,d\theta \le C\int z^{-2}|D^2v|^2\,dt\,d\theta,\] and the other terms follow from the traces and weighted bulk terms in (26). A fixed collar extension at the entrance preserves this conclusion.

The difference \(J-\widetilde J\) is therefore a Gaussian exterior \(L\)-solution with zero Dirichlet trace at the entrance. Local elliptic estimates give the pointwise bound used in the proof of Lemma 8, namely a polynomial times \(\exp(|X|^2/8+o(|X|^2))\). For \(1/8<\alpha_0<1/4\) the positive function \(e^{\alpha_0|X|^2}\) satisfies \(Le^{\alpha_0|X|^2}<0\) on a sufficiently far tail. On each finite tail, divide the equation by this function and apply the maximum principle to both signs of the real and imaginary parts. The entrance value is zero, and for each \(\varepsilon>0\) the remote value is bounded by \(\varepsilon e^{\alpha_0|X|^2}\) once the remote section is sufficiently far out. Exhaustion and then \(\varepsilon\downarrow0\) give \(J=\widetilde J\). Thus the conical estimate applies after, rather than before, identification of its domain.

For \(f\in Y^{\mathrm{con}}_h\), its physical source \(z c_n f\) belongs to \(L^2_G\), because \[\int z^4|f|^2e^{-z^2/4}\,dt\,d\theta \le C\int z^2|f|^2\,dt\,d\theta.\] Solve the conical particular problem with zero entrance, cut its lift off on a fixed collar, and subtract it. The remaining source is compact and the preceding argument applies. The maximum principle is used only for the real operator \(L\); analytic perturbations in complex parameters will be taken later on fixed real-coordinate function spaces. ◻

Proof. Pull the core and end fiber maps once by the fixed real coordinate identifications, so that their ambient values are functions of the same base point \(y\). At \((\vartheta,s)=(0,0)\) all these values equal \(X(y)\). Orient their scalar variables so that the normal component of each scalar derivative is positive. On each fixed compact overlap interpolate the ambient fiber maps pointwise with a smooth partition of unity in \(y\). At the reference, the base derivatives of the interpolated map are \(D_yX\), and its scalar derivative still has positive normal component. The full derivative in \((y,s)\) is therefore invertible. Compactness preserves this transversality for small parameters. Choose the partition to select the core map on the observation support and the relevant end map outside the overlap collars. Because its cutoffs depend only on the real base point, the interpolation is analytic in the finite variables and invokes no parameter-dependent spatial composition. The far tail charts retain their already stated scaled domains and units. ◻

Proof. After dividing the conical output by \(\mu\), its equation is \[\mu^{-1}v_t-\mu\bigl(A^{ij}(t,\theta,v,Dv)D_{ij}v +B(t,\theta,v,Dv)\bigr).\] The norm (26) gives uniformly bounded first-derivative algebra traces, which multiply \(\mu D^2v\) in \(L^2_tH^h_\theta\). The lower terms have the integrable factor \(\mu\). Analytic composition on a small trace neighborhood and the same algebra–module estimate as in Lemma 7 therefore prove the conical map assertion. For example the trace of \(v_t\) follows from product integration of \(\mu^{-1}v_t\) with \(\mu v_{tt}\) on neighboring strips.

For completeness consider a compact flow box \((s,y)\) of one nonvanishing commuting field, straightened to \(\partial_s\). Include \(\partial_s^j u\) in \(H^2\) for \(0\le j\le h\), and use overlapping flow boxes for the finite field family. In a differentiated coefficient \(A(u,Du)\), the sole potentially highest factor \(\partial_s^h Du\) has \(L^\infty_yL^2_s\) control by its \(y\) derivative and product integration. The other factor \(D^2u\) has \(L^2_yL^\infty_s\) control by one additional \(s\) derivative. Their product is in \(L^2_{s,y}\), since \[\|fg\|_{L^2_{s,y}} \le\|f\|_{L^\infty_yL^2_s}\, \|g\|_{L^2_yL^\infty_s}.\] When all commutations fall on \(D^2u\), the coefficient has its ordinary uniform first-derivative bound. Intermediate distributions obey the one-dimensional Sobolev product inequality in \(s\), followed by this same inequality in \(y\). Iterating the analytic coefficient expansion covers any number of first-derivative factors. Smooth changes of boxes and cutoffs only create lower commutations. This proves the required product map on the compact transitions, including at a place where only one of the retained fields is nonzero. ◻

Proof. The type-preserving inverse and the reference family. For each end, Lemma 8 constructs Jacobi solutions with its opening and two tilt coefficients prescribed independently. Extend them inward by cutoffs. They lie in \(H^2_G\) and have compact Jacobi sources. Apply Lemma 5 to these functions. The map from the enlarged observation values to the \(\kappa\) opening coefficients of the homogeneous normalized linear solution is surjective. Split the observation space into its kernel \(V_b\) and a fixed \(\kappa\)-dimensional complement \(V_x\) on which this map is an isomorphism.

The type-preserving domain consists of the compact graph variables, cutoff end rotations together with cylindrical remainders of weight \(-2+\eta\), and the conical link variables of (26). Its range has the corresponding cylindrical source weight, conical source norm, and compact norms. Impose only the \(V_b\) components of the observations, while retaining the full multiplier vector. We verify that the linearization is an isomorphism in these spaces. For a general exterior source, first solve a cylindrical particular problem with zero slow coefficients at infinity and zero stable entrance data in Lemma 8, and a conical particular problem with zero entrance value. Cut their normal lifts off inside fixed far end collars where the retained fields \(\mathcal Z\) have reduced to angular commutations. Choose these collars outside the common compact observation and multiplier-source support, so the cutoff lifts vanish on that support and have zero observations. Their Gaussian norms and those of their lifted sources are controlled: this follows from their polynomial cylindrical bounds and from the lift estimates in Lemma 10.

The remaining source is compact. Solve the Gaussian normalized block with the required \(V_b\) observations and arbitrary \(V_x\) observations, then vary the latter uniquely until all opening coefficients vanish. The finite-dimensional inverse used here is the opening map just constructed. Lemma 8 gives the resulting rotations and decaying cylindrical remainders. Enlarge the compact source set to include the common multiplier-source support, and choose the later trace collars outside this set. Lemma 10, applied at such an entrance, identifies the conical link component with the solution in \(X^{\mathrm{con}}_h\). These estimates give a bound in the full type-preserving norm directly for the stated range spaces. On the core, the commuted compact elliptic estimate bounds the \(C_h^2\) norm by the Gaussian \(H^2_G\) norm, the commuted compact source norm, and the multiplier norm; the source columns are fixed smooth compact functions. The end inverses accept their full sources; after the cutoffs the Gaussian source is compactly supported in \(L^2_G\). Near a farther fixed entrance its Gaussian solution is homogeneous and therefore smooth, so the entrance trace estimates apply there even for the original rough source. A homogeneous element of this domain lifts to \(H^2_G\); if its \(V_b\) observations and its openings vanish, the \(V_x\) observations also vanish. The Gaussian block then kills the element and its multipliers. This proves the asserted isomorphism.

In prescribed graph coordinates the stationary equation is affine in the highest second derivatives, with coefficients analytic in position and first derivatives. The denominator units are bounded away from zero near the reference. The trace and product rules above make it an analytic map between precisely these domain and range spaces. On a conical end, for example, the divided range expression is \(\mu^{-1}v_t-\mu(A^{ij}D_{ij}v+B)\): bounded analytic coefficients of the first-derivative traces multiply \(\mu D^2v\) in \(L^2\), and lower terms have the integrable factor \(\mu\). The cylinder uses \(R,R_\theta,\mu R_t\) as coefficient variables and the second-derivative slots in (12); compact transitions use (31). End rotations are prescribed ambient rotations with a fixed cutoff on the core. They are analytic in their finite parameters without requiring analytic spatial charts. The analytic implicit function theorem gives \(S_b\) and its multiplier vector \(\lambda_b\) on one parameter ball. Write its full observation value as \(q_0(b)\) and set \[ \mathsf P(b,x)=q_0(b)+V_xx. \tag{35}\] This is a local analytic coordinate system for the observation values. Write the preceding solution on the fixed atlas as \(X_b(y)=\Gamma_{\vartheta(b)}(y,u_b(y))\), and center its scalar chart by \[ \Psi_b(y,s)=\Gamma_{\vartheta(b)}(y,u_b(y)+s),\qquad \Psi_b(y,0)=X_b(y). \tag{36}\] This is the exact chart for subsequent prescribed graphs. The low-space family makes it analytic in the fixed-coordinate Banach spaces; the local bootstrap below supplies every fixed spatial derivative. On the observation support the core fiber map remains fixed and each \(\chi_i\) is evaluated at the same real \(y\). The variational source columns there therefore acquire only the pointwise analytic density and normal-speed units. No spatial coordinate is composed with a \(b\)-dependent map. For any smaller fixed cylindrical loss \(\eta_0>0\), repeat this same type-preserving inverse and implicit construction with weight \(-2+\eta_0\). Its solution lies in the original weaker domain and has the same \(V_b\) observations. Small uniqueness in that weaker domain identifies it with \(S_b\). After shrinking the parameter ball for this fixed loss and angular order, the same family is therefore analytic in the stronger weight. The same real-translation argument supplies its fixed log derivatives there. This justifies every fixed choice of the cylindrical remainder loss used below.

Uniform regularity of the real family. The preceding construction supplies the small conical norm, but the smooth geometry required below also uses the stationary equation. Write the full link equation on a fixed tail as \[\mathcal C^{\mathrm{nl}}(v)=v_t-\mu^2 \{A^{ij}(t,\theta,v,Dv)D_{ij}v+B(t,\theta,v,Dv)\}=0.\] In the preceding construction choose one extra fixed entrance collar, and analyze the tail starting at a later \(t_0\). On a fixed larger entrance collar, local parameter-dependent elliptic bootstrap for this already constructed low-norm family gives every fixed spatial Sobolev norm uniformly on a closed inner complex \(b\)-ball. The coefficients and compact sources are smooth there and their ellipticity is uniform. The usual commuted local estimates prove the bounds successively; testing against smooth functions and using the low-space holomorphy then gives holomorphy in each of these locally bounded higher Sobolev norms. In particular the entrance data used below are smooth in their real translation parameters into \(H^{h+3/2}\), uniformly on that ball. This is a fixed-collar statement.

For small real translations \((\tau,\sigma)\) of the log and periodic angular coordinates, prescribe on the fixed tail \[\begin{align*} \mathcal C^{\mathrm{nl}}_{\tau,\sigma}(w) =w_t-e^{-2\tau}\mu(t)^2 \{&A^{ij}(t+\tau,\theta+\sigma,w,Dw)D_{ij}w\\ &+B(t+\tau,\theta+\sigma,w,Dw)\},\\ w(t_0,\theta)&=v_b(t_0+\tau,\theta+\sigma). \end{align*}\] All explicit reference functions and the angular frame are shifted in this formula. The bounded reference derivatives and the conical coefficient–module rule make it a \(C^\infty\) family of maps from one fixed low \(X^{\mathrm{con}}_h\), \(h\ge4\), to \(Y^{\mathrm{con}}_h\), holomorphic in \(b\). Its derivative in \(w\) is a uniformly small perturbation of the conical entrance isomorphism. The implicit function theorem therefore gives one joint solution family, smooth in \((\tau,\sigma)\) and holomorphic in \(b\).

The actual translated graph remains in the same small domain, even though strong continuity of translation in the supremum part of that norm has not been assumed. On the enlarged tail, \[\|\mu^{-1}v_{b,t}(\,\cdot+\tau)\|_2 \le e^{|\tau|}\|\mu^{-1}v_{b,t}\|_2,\qquad \|\mu D^2v_b(\,\cdot+\tau)\|_2 \le e^{|\tau|}\|\mu D^2v_b\|_2;\] the supremum bounds are unchanged up to restriction, and angular translation is an isometry. Small nonlinear uniqueness identifies the actual translate with the implicit family. Differentiating that one family gives \(\partial_t^k\partial_\theta^jv_b\in X^{\mathrm{con}}_h\) for every fixed \(j,k\). Constants may depend on \(j,k\), but these derivatives are taken on one translation neighborhood and one closed inner \(b\)-ball.

The resulting first traces bound every fixed second log derivative pointwise. The equation now yields \(\|\partial_t v_b(t)\|_{H^h}\le C_h e^{-2t}\), and its differentiated versions give the same factor for every fixed log, angular, and parameter derivative of \(\partial_t v_b\). Integration to infinity proves (32), with \[v_{b,\infty}=v_b(t_0)+\int_{t_0}^{\infty}v_{b,t}(t)\,dt.\] The integrable uniform estimates justify differentiation in \(b\), so the limiting link is analytic. For the cylinder, the translation argument in Lemma 8 also permits the angular translation \(\sigma\): its stable projection onto \(|j|\ge2\) commutes with angular rotations. Shift the explicit coefficients, angular frame, and stable entrance data together, and use the same low-space uniqueness comparison with the rotated family. One joint translation family then gives all its fixed mixed derivative bounds on a closed small parameter ball.

For real \(b\), write a conical parametrization as \(X_b=r\omega_b(t,\theta)\). The preceding estimates give uniformly smooth rescaled annuli, \(|\mathrm{II}_b|\le C/r\), their fixed physical derivative bounds, and \(|X_b^\perp|\le C/r\) because \(\omega_{b,t}=O(r^{-2})\). The normalized metrics and their fixed derivatives vary by \(O(|b|)\). For the separated end data in the final assertion, the embedded limiting links remain embedded and separated on a small closed ball. Their tubular radii on the unit sphere are uniformly positive. Scaling gives a tube of radius \(cr\) on each far conical annulus; the annular asymptotics and separated end directions prevent other ends from entering that tube. The rotated cylindrical tails have a fixed positive tube, and compact embeddedness gives one on the core and the fixed transition collars. Taking the minimum proves the asserted global positive tubular radius. A conical parameter variation is \(r\partial_b\omega_b\) and a cylindrical tilt also has size \(O(r)\); their physical first derivatives are bounded. This proves the final parameter-variation assertion. In particular, on a region of radius \(R\), normal conversion between nearby real family members costs at most \(C(1+R)|b'-b|\) in height and slope while that quantity is within the tube.

The augmented Gaussian inverse about the family. We establish this inverse before solving any opening coefficient. Use the prescribed rotations and radial conical charts to pull all differential expressions to the fixed real atlas of \(S\). Put \(r=|X|\) there, and let \[\mathcal D_G=\{w:\nabla^2w,\ (1+r)\nabla w,\ (1+r^2)w\in L^2(S)\}\] with its sum norm. This is the half-density domain in Lemma 4. For real \(b\), let \(F_b:S\to S_b\) be the prescribed parametrization and \(J_b\) its area Jacobian. Pull back the scalar component in the unit normal frame of \(S_b\); if a prescribed graph variable is used to compute that component, first multiply by its known graph-speed factor. Multiply the resulting physical normal scalar by \(J_b^{1/2}\exp(-(1+\delta)|F_b|^2/8)\). This is the usual Gaussian half-density identification for real \(b,\delta\).

For complex \(b\), take the analytic bilinear expressions for \(F_b\cdot F_b\), \(J_b^{1/2}\), and the normalized normal frame on this same real atlas, and define the conjugated differential expression by their chain rules. The branches are fixed from \(b=0\). This defines an analytic operator \(\widetilde{\mathcal A}_{b,\delta}:\mathcal D_G\oplus\mathbb C^m \to L^2(S)\oplus\mathbb C^m\); it does not define a positive measure on a complex surface or evaluate a spatial function at a complex point. The uniform geometry just proved puts its coefficient difference from \(\widetilde{\mathcal A}_{0,0}\) on the three domain terms \[o(1)|\nabla^2w|+ o(1)(1+r)|\nabla w|+ o(1)(1+r^2)|w|.\] For the complex coefficient bound, use the preceding joint translated family as a holomorphic Banach-valued map at every fixed translated order. The required first-trace maps are bounded; Cauchy’s estimate on a closed inner complex ball therefore gives \(O(|b|)\) differences for all normalized coefficient derivatives used here. These are the analytic estimates behind the real geometric interpretation. Indeed normalized metric differences and their physical derivatives are \(O(|b|)\); differentiating the half-density contributes at most one factor of \(r\) in first-order coefficients and two in zeroth-order coefficients. The measure-Jacobian derivatives have the bounded-geometry scales. Changing \(\delta\) contributes \(O(|\delta|)((1+r)|\nabla w|+(1+r^2)|w|)\). The derivative of the base multiplier term \(\sum_i\lambda_{b,i}\chi_i(u)\) is retained and is a small compactly supported operator, as are the observation and source-column changes. Consequently \[ \|(\widetilde{\mathcal A}_{b,\delta} -\widetilde{\mathcal A}_{0,0})(w,\lambda)\| \le \varepsilon(b,\delta)(\|w\|_{\mathcal D_G}+|\lambda|), \qquad \varepsilon(b,\delta)\longrightarrow0. \tag{37}\] The Gaussian block at \((0,0)\) is invertible. A Neumann series therefore gives a uniform inverse for small parameters, holomorphic in complex \(b\), on these fixed spaces.

For real \(b,\delta\), its domain is also the derivative-defined Gaussian \(H^2\) domain on \(S_b\) and controls the two moment terms. To see this directly, the compactly supported defect \(\Phi_b=\mathbf H_b+X^\perp/2\) gives the following divergence for the weight \(e^{-(1+\delta)|X|^2/4}\): \[\operatorname{div}_{G,b,\delta}X^\top =2-\frac{1+\delta}{2}|X|^2+ \frac{\delta}{2}|X^\perp|^2+\Phi_b\cdot X^\perp.\] The last two terms are uniformly bounded. The moment integrations in Lemma 4 therefore hold uniformly for this density. Together with (37) and its half-density domain, they identify the domain and give its \(|X|\nabla u\) and \(|X|^2u\) bounds. This establishes the real shifted estimate recorded after this proof, with the derivative of the nonzero base multiplier included.

The mixed inverse for a fixed opening degree. Fix \(h\ge4\) and a cylindrical rate \(\gamma>2\). Choose the tail entrances far enough for this pair of parameters, absorbing the intervening finite collars into the compact norms. On the fixed real atlas let \(\mathcal X_{\gamma,h}\) consist of the compact norm (31), the cylindrical norms \(X_{e^{\gamma(t-t_0)},h}\), and the original conical norms \(X^{\mathrm{con}}_h\), with the linear graph-unit identifications obtained by differentiating (36) on fixed collars. Let \(\mathcal Y_{\gamma,h}\) have the corresponding compact, cylindrical, and conical source norms. The full augmented derivative about \(S_b\), written in these units and including the base compact multiplier derivative, satisfies \[ \mathcal A_b: \mathcal X_{\gamma,h}\oplus\mathbb C^m \longrightarrow\mathcal Y_{\gamma,h}\oplus\mathbb C^m \quad\hbox{is an isomorphism for small $b$, with analytic inverse.} \tag{38}\] The parameter neighborhood and its inverse bound may depend on \(\gamma,h\). All spaces in this assertion are fixed real-coordinate spaces; the conical part is the bounded-link domain, not a growing-link space.

Here is the proof. At \(b=0\), solve a zero-entrance exterior particular problem on each cylinder using Lemma 8 at \(\gamma>2\), where every slow channel is an entrance channel. Solve the original zero-entrance problem of Lemma 9 on each cone. Cut their normal lifts off on fixed far collars where \(\mathcal Z\) contains only the retained angular commutations, and denote the result by \(u_{\rm ext}\). Let \(f_{\rm phys}\) be the prescribed source in physical normal units, with the graph and row factors applied, and put \(r_c=f_{\rm phys}-Lu_{\rm ext}\). The exterior estimates and the fixed cutoff product estimates give \[\|u_{\rm ext}\|_{\mathcal X_{\gamma,h}} +\|u_{\rm ext}\|_{H^2_G} +\|r_c\|_{C_h^0(K)} +\|r_c\|_{L^2_G} +|Pu_{\rm ext}| \le C_{\gamma,h}\|f\|_{\mathcal Y_{\gamma,h}},\] where \(K\) is one fixed compact set containing the remainder support and the common compact observation and multiplier-source support. The Gaussian observation datum is the prescribed observation datum minus \(Pu_{\rm ext}\). The Gaussian lift bounds on cylinders follow by summing their polynomial strip bounds against the Gaussian, and the conical lift and source bounds are those proved in Lemma 10. Solve this remaining compact source and the remaining observations by the Gaussian augmented block. Its solution \((J_G,\lambda_G)\) is bounded in \(H^2_G\oplus\mathbb C^m\). On fixed nested compacts containing \(K\), the commuted elliptic estimate applied to \(LJ_G=r_c-P^*\lambda_G\) gives \[C_h^2(J_G;K_{\mathrm{core}}) \le C\bigl(\|J_G\|_{H^2_G}+\|r_c\|_{C_h^0(K)}+|\lambda_G|\bigr).\] The smooth compact source columns account for the last term, which the Gaussian inverse already controls. On nested fixed collars outside \(K\), the commuted local elliptic estimates bound the cylindrical \(\mathcal T^D\) entrance traces and conical \(H^{h+3/2}\) entrance traces by \(C(\|J_G\|_{H^2_G}+\|r_c\|_{C_h^0(K)})\). The polynomial Gaussian barrier and the separated estimate at \(\gamma>2\) now bound the entire cylindrical component by those traces; equivalently its expansion is (23). On a cone, Lemma 10 identifies \(J_G\) with the conical entrance solution, whose full norm is bounded by its trace. These inequalities give the claimed real inverse bound for the full source spaces. In particular the cutoff source may be only compactly supported \(L^2_G\); homogeneity near the farther entrance is what gives the smooth traces. Conversely an element of \(\mathcal X_{\gamma,h}\) has polynomial cylindrical local Sobolev bounds, and its conical normal lift lies in \(H^2_G\) by Lemma 10’s lift calculation. Its Gaussian norm is finite, so the Gaussian block proves injectivity.

For complex \(b\) close to zero, keep exactly the same link displacement as domain variable on each conical tail. The full drift normalization is already part of \(\mathcal C^{\mathrm{nl}}\), so its derivative at \(v_b\) is directly a small analytic bounded perturbation from the fixed \(X^{\mathrm{con}}_h\) to \(Y^{\mathrm{con}}_h\). No global multiplication of the conical domain by a graph-speed factor is made. A further scalar row factor, if used, acts only on \(Y^{\mathrm{con}}_h\) and is an analytic bounded \(L^\infty_tH^h_\theta\) multiplier. The factors \(z a_n\) were used for the real Gaussian lift above, and on fixed collars their conversions are bounded in the compact norms. On each cylinder the prescribed rotation removes the tilt, and the small decaying remainder gives the same small normalized coefficient–module perturbation on the fixed weight \(\gamma\). The constant can depend on that weight. The cylindrical row and graph-speed ratios used to identify the \(b\) and \(0\) equations, and the scalar identifications on fixed collars, are analytic bounded multiplication maps in their strip or compact modules; all spatial coordinates remain the fixed real ones. On the compact support, the coefficients, base multiplier derivative, source columns, and observation maps are analytic bounded maps in their fixed commuted norms. It follows that \[b\longmapsto\mathcal A_b\in\mathcal B( \mathcal X_{\gamma,h}\oplus\mathbb C^m, \mathcal Y_{\gamma,h}\oplus\mathbb C^m)\] is holomorphic in the full operator norm, including both finite-dimensional rows and columns, and \(\|\mathcal A_b-\mathcal A_0\|\le C_{\gamma,h}|b|\) on a smaller ball. A Neumann series proves (38), including its analytic complex parameter dependence. This step supplies a polynomial norm for the coefficients; Gaussian-space analyticity alone would not supply that norm.

At degree one, choose \(\gamma=2+\epsilon\) with \(\epsilon>0\) small and apply this inverse to the prescribed \(V_x\) observations. On a cylindrical tail write its equation as \(\mathcal L_{\mathrm{cyl}}u_1=-Q_bu_1\), where the normalized gain (22) for the rotated family is holomorphic in \(b\) in its full operator norm. Choose its loss \(\eta_0>0\) so that \(\epsilon+\eta_0<1\), using the corresponding stronger realization of \(S_b\) on its smaller parameter ball. Then \(Q_bu_1\) belongs holomorphically to \(Y_{e^{(\epsilon+\eta_0)(t-t_0)},h}\). The fixed model variation-of-constants splitting is a bounded linear map from this source and the entrance trace of \(u_1\) to the rate-\(2\) coefficient, the two rate-\(1\) coefficients, and a remainder of rate \(\epsilon+\eta_0\). Every fast outward mode is excluded by the polynomial domain. The trace map is bounded on \(\mathcal X_{\gamma,h}\), so this splitting proves analytic dependence of the opening and tilt coefficients without assuming them in advance. The two small losses can be chosen below any specified remainder loss in (33). At \(b=0\) the resulting opening matrix is the isomorphism on \(V_x\), so it stays invertible. With the normalization \(c_j=\beta_j/\sqrt2\) it is \(M(b)\) in (33). Removing the first-order tilt gives the displayed remainder.

Now suppose all opening coefficients of degree below \(n\) have been constructed in prescribed charts. Their cylindrical source is controlled directly by the weighted coefficient–module rule, before any pointwise regularity bootstrap. On a strip put \[\mathcal J_1v=(v,v_\theta,\mu v_t),\qquad \mathcal J_2v=(\mu^2(v_{tt}-v_t),\mu v_{t\theta},v_{\theta\theta}).\] For \(v\in X_{\rho,h}\) the first tuple has algebra trace norm at most \(C\rho\|v\|_{X_{\rho,h}}\), and the second has bulk norm at most the same quantity. Each formal coefficient of the radial equation is a finite sum of products of first tuples and at most one second tuple, with bounded analytic background coefficients. The product rule therefore puts a monomial with input weights \(\rho_1,\ldots,\rho_j\) in the source weight \(\rho_1\cdots\rho_j\), with norm bounded by the product of the input norms. The lower terms use one first-derivative factor in bulk and obey the same bound. Choose a fixed \(\gamma>2\) above the finitely many resulting power exponents, allowing a small extra power for logarithms. On a conical end, keep every coefficient in the link variable. The conical source at degree \(\alpha_0\) is a finite sum of terms \[ -\frac1{j!}D^j\mathcal C^{\mathrm{nl}}(v_b) [v_{\alpha_1},\ldots,v_{\alpha_j}], \qquad j\ge2,\quad \alpha_1+\cdots+\alpha_j=\alpha_0,\quad |\alpha_i|\ge1. \tag{39}\] The analytic map \(\mathcal C^{\mathrm{nl}}:X^{\mathrm{con}}_h\to Y^{\mathrm{con}}_h\) has bounded multilinear derivatives on a small ball. Every term in (39) therefore belongs to \(Y^{\mathrm{con}}_h\), with a bound by the product of the lower \(X^{\mathrm{con}}_h\) norms, and is analytic in \(b\). The compact source has the analogous finite Taylor bound. Equation (38), with the prescribed coefficient of (35), now gives the unique coefficient of degree \(n\). In particular its conical link coefficient again belongs to \(X^{\mathrm{con}}_h\). The chosen cylindrical weight and the small complex \(b\)-ball may change with \(n\). This is precisely the coefficientwise analyticity required by \(\mathbb R\{b\}[[x]]\).

The following local argument closes the derivative part of this induction at each degree. Cylindrical bounds follow from the commuted separated estimates, increasing the weight and angular order by a fixed amount when needed. On a conical unit log strip at height \(z\), the \(X^{\mathrm{con}}_h\) norm initially bounds the local second derivatives by a fixed multiple of \(z\), and bounds the first traces. After division by \(\mu^2\), a coefficient equation has uniformly elliptic principal matrix and a first-order drift of size \(O(z^2)\); all fixed derivatives of its other coefficients are bounded by the family regularity already proved. Commuted local elliptic estimates on nested fixed log patches, with interpolation to absorb the first-order term, therefore have constants bounded by a fixed power of \(z\) at every fixed order. This follows inductively because each commutation differentiates a coefficient a finite number of times, and the only growing coefficient and its derivatives are \(O(z^2)\). The source at degree \(n\) is a finite product of lower-degree derivatives. Starting from the preceding local \(H^2\) bound, induction on degree and on the local elliptic order gives polynomial local Sobolev bounds for every fixed derivative. Sobolev embedding gives the corresponding pointwise bounds. These bounds are uniform on a closed smaller degree-dependent complex \(b\)-ball; fixed coordinate differentiation and locally bounded holomorphy give the same statement for parameter derivatives.

The conversion to normal jets is coefficientwise. On a real scaled annulus choose a local projection branch and use its normal chart to write the equality between the prescribed graph parametrization and a normal graph with an unknown real tangential displacement. At \(x=0\) this displacement is zero. At a fixed \(x\) degree, expand that equality at the identity using only the finitely many spatial derivatives of the lower jets and of \(S_b\) which occur at that degree. The pointwise matrix formed by the tangent vectors and the normal is invertible. On a conical annulus, scale the ambient and normal variables by its radius; the matrix and its inverse are then uniformly bounded in the two link coordinates. On a cylindrical tail the axial and angular tangent scales are different in log coordinates: the axial tangent has size comparable to \(z\) and the angular tangent has size comparable to one. The prescribed cylinder charts therefore give an inverse with at most a fixed polynomial factor in \(z\); on compact charts it is uniformly bounded. Solving this matrix equation gives the tangential coefficient and the normal coefficient at that degree. The local polynomial bounds just proved give polynomial bounds for these coefficients and all fixed derivatives. Differentiating the finite real identities in \(b\) gives the same fixed-order polynomial bounds for their parameter derivatives. For real parameters the ordinary smooth implicit function theorem and Taylor’s theorem on the chosen projection branch identify these finite coefficients and the corresponding remainder with the actual normal conversion. More quantitatively, let \(v_M^{\rm pres}(b,x)\) be the finite prescribed graph polynomial through degree \(M\), let \(v_M^{\rm nor}(b,x)\) be its coefficientwise normal polynomial, and write \(\mathfrak n_b\) for the actual normal-height conversion for real parameters. Wherever its scalar values lie in the chart domain, the first polynomial represents the exact immersion \(y\mapsto\Psi_b(y,v_M^{\rm pres}(b,x)(y))\); its fixed-overlap identifications agree exactly by (36). For the separated family, work on an inner patch whose inverse projection stays in a controlled larger collar on the chosen branch. Suppose that throughout this larger collar, for every parameter on the segment \(\{tx:0\le t\le1\}\), the physical graph displacement stays within a fixed fraction of the local tubular radius and its first derivative is correspondingly small in the reference metric. Then the projection derivatives through any chosen finite order have at most a fixed polynomial cost in \(r=|X|\). Taylor’s theorem in the finite variable \(x\), applied to the smooth real projection equation, therefore gives for every fixed spatial order \(j\) \[ \left|\nabla_{S_b}^j\bigl(\mathfrak n_b(v_M^{\rm pres}(b,x)) -v_M^{\rm nor}(b,x)\bigr)\right| \le C_{M,j}|x|^{M+1}(1+r)^{D_{M,j}}. \tag{40}\] The constants are uniform on a closed smaller real \(b\)-ball and on such inner patches with the stated collar and tubular controls. The complex parameter assertions are made for the prescribed-coordinate jets, where they were proved by the mixed inverse; the action is computed in those coordinates.

The formal action. Each fixed action coefficient is a polynomially bounded combination of the finite graph jets and their first derivatives times the Gaussian. On a sufficiently small degree-dependent complex \(b\)-ball, the prescribed tilted and conical immersions satisfy \(\operatorname{Re}(F_b\cdot F_b)\ge c r^2-C\). Gaussian domination permits coefficientwise integration and differentiation in \(b\). In the formal first variation on exhausted domains, the boundary terms are polynomially bounded times a Gaussian and tend to zero. The plus normalization in (10) gives \(d\mathcal F=\lambda\cdot d\mathsf P\), proving (34). The same bounds control every fixed truncation. The augmented coefficient equations are unique in the mixed spaces. To see uniqueness among all formal solutions with polynomial growth, take the first degree at which two such solutions differ. Their lower sources and observation coefficients agree, so the difference solves the homogeneous full augmented derivative about \(S_b\). At this degree its prescribed graph difference converts linearly to a polynomially growing physical normal scalar on the fixed real atlas. Local elliptic estimates for the homogeneous equation give polynomial bounds for its first two derivatives as well. After the analytic half-density conjugation it belongs to \(\mathcal D_G\), also for small complex \(b\). The full augmented Gaussian inverse proved before the jets kills this difference and its multiplier. Induction proves uniqueness in the stated polynomial class. For real parameters the finite normal-coordinate identities above identify the same geometric jets; their action coefficients are those computed in the prescribed charts. ◻

Proof. The inverse was established before the opening-jet construction in (37). For real parameters its half-density identification is unitary in the displayed positive measure, and the moment calculation there identifies its domain with \(H^2_{G,b,\delta}\) and gives the two additional moment bounds. Conjugating that inverse back proves (41). All parameter perturbations are fixed before any later localization radius is allowed to increase. ◻

Proof. The coefficient equations and their polynomial bounds are those of Proposition 13, including its coefficientwise conversion to normal graph variables. The stated absolute smallness keeps every intermediate finite polynomial at \((b,tx)\), \(0\le t\le1\), in the same normal graph neighborhood. On such a region, Taylor’s formula for the pointwise geometric graph formula leaves \(|x|^{M+1}\) times a fixed polynomial in \(1+|Y|\). That formula is analytic in height and first derivatives and affine in the highest second derivatives, so the local coefficient–module estimates apply to the same derivatives as in the formal equations. The compact observation and source maps after the real graph conversion need only be smooth: their bounded derivatives through the fixed order \(M+1\) give the corresponding finite Taylor remainders on their fixed support. Gaussian integration proves the equation estimate, and the compact Taylor formula proves the observation estimate.

Expand the normal-coordinate graph area density, including its Gaussian, in the finite opening variables. Every fixed coefficient is a polynomially bounded function times a Gaussian, by the family estimates. The finite formal change of variables between the prescribed and normal coordinates identifies their complete coefficient integrals: on an exhaustion, the change of local density contributes the corresponding boundary divergence, whose polynomial Gaussian boundary terms vanish at infinity. The resulting complete normal-coordinate integrals are the coefficients of \(f_M\). The local Taylor comparison uses these normal-coordinate densities and their own tails. On a real small normal graph its Taylor remainder is again \(|x|^{M+1}\) times a fixed polynomial and a slightly weaker Gaussian. This proves the local area assertion and the coefficient-tail bound. This assertion concerns the density coefficients on the complete \(S_b\); it does not treat an uncut polynomial graph as a complete immersed surface.

Choose \(D_M\) to bound the growth of the finite height coefficients and all their derivatives through order two, and any higher fixed orders used in the application. Take \(\sigma_MD_M<1/2\) and cut off at \(|Y|\asymp |x|^{-\sigma_M}\). Every nonconstant term and these derivatives are then \(O(|x|^{1/2})\). The cutoff commutators are supported at that polynomial radius and have Gaussian norm smaller than every fixed power of \(|x|\). The interior Taylor estimates just proved give (44) and (45). Coefficientwise first variation in (34) gives (46); an \(x\) derivative can lower the remainder degree by one. No differentiation of the cutoff with respect to \(x\) is used, and the cut-off graph is not asserted to be analytic in \(x\). ◻

Proof. Write \(r_M=\mathcal N_b(v,\lambda_M)\). Before linearization, the exact subtraction is \[\begin{align*} &\mathcal Q_b(u)-\mathcal Q_b(v) +(\mathcal C_b(u)-\mathcal C_b(v))\lambda_b+C_b\ell\\ &\quad=r_U-r_M-(\mathcal C_b(u)-C_b)\ell -(\mathcal C_b(u)-\mathcal C_b(v))(\lambda_M-\lambda_b). \end{align*}\] Subtract \(A_b(u-v)\) from the first line. In the quasilinear principal part retain the coefficients evaluated on \(u\) on the second derivatives of \(u-v\). Their difference from the coefficients of \(A_b\) is small by the height and slope tolerance. The remaining coefficient differences multiply the second derivatives of \(v\), and their bound is small because \(v\) is \(C^2\)-small. The lower terms are controlled in the weighted slots of (42). This arrangement uses no bound for the second derivatives of \(u\).

On the compact source support, \(\mathcal C_b(u)-C_b\) is a small bounded map, so its product with the unknown \(\ell\) is absorbed in (43). The last term in the identity is a small map on \(u-v\), since \(\lambda_M-\lambda_b=O(|x|)\). This argument does not need an a priori bound for \(\lambda_U\). The observation equation, with \(\chi_R=1\) on its support, gives \[|DP_bw|\le |e_P|+C_M|x|^{M+1} +\varepsilon\|w\|_{\mathcal H^2_a}.\] Multiplication by \(\chi_R\) creates only height and first-derivative commutators on its annulus. The coefficients of the normal graph equation and its drift have at most the polynomial growth measured by \(\mathfrak a_R\). Apply (43), use Lemma 15 for \(r_M\), and absorb all the small terms. This proves (47).

For the action assertion, interpolate the real graphs \(v+\tau w\), \(0\le\tau\le1\), on \(|Y|<R\); the variation \(w\) is compactly supported there. In the normal tubular neighborhood, the derivative of their area density obeys \[|D\mathcal L_{v+\tau w}[w]| \le C(1+|Y|)^d e^{-c|Y|^2/4} (|w|+|\nabla w|).\] The uniform bound for \(Y^\perp\) on the real family and the small height and slope give the retained exponent \(c\) and the polynomial prefactor. Gaussian Cauchy–Schwarz with \(a<2c\) bounds its integral by \(C\|w\|_{\mathcal H^2_a}\). This direct density estimate does not use stationarity of either graph. On \(|Y|\le\vartheta R\) the interpolated endpoint is \(u\); on the transition annulus all three small graph densities have Gaussian tails. Compare the other endpoint with the integrated finite coefficient densities by Lemma 15, and add their full-surface tail and the omitted action \(\tau_{\vartheta R}\) of \(U\). The result is (48). ◻

Proof. Scale \(z^2\) by \(|a|\) and the absolute leading coefficient so that the leading pole is \(p_*\), with \(|p_*|=1\). For the lower bypass write \(u=h-iY(h)\), \(h'>0\), and choose real \(c,d\) by \(\operatorname{Re}(u/p_*)=ch+dY\); thus \(c^2+d^2=1\). Put \(Q=h^2+Y^2\). The exact identity \[\frac{|u-p_*|^2}{|u|^2} =1-2H,\qquad H=\frac{ch+dY-\tfrac12}{Q},\] shows that the second condition in (51) is \(H'>0\). Direct differentiation gives \[ Q^2H'=N=cA+dJ+h+YY',\quad A=Y^2-h^2-2hYY',\quad J=(h^2-Y^2)Y'-2hY. \tag{52}\] The first condition is simply \(dh>0\).

Near the adverse direction let \(e=4\delta\), \(0<e\le0.1\), and take the positive branch \[|d|\le e/2,\qquad c=\sqrt{1-d^2}\ge\sqrt{1-e^2/4}.\] Join the origin to \((h_0,Y_0)=(1/4,\,0.6/e)\) by the ray \(Y=mh\), \(m=2.4/e\). On this ray \[N=(1+m^2)h\{1-(c+dm)h\}.\] For \(c\le1\) and \(|d|\le e/2\), its worst endpoint projection is \(1/4+0.3=0.55<1\). From \((h_0,Y_0)\) follow the larger positive root of \[ K(h^2+Y^2)=h+eY-\tfrac12,\qquad K=\frac{0.35}{0.0625+0.36/e^2}. \tag{53}\] Here \(0.970<K/e^2<0.973\), and \[Y'=\frac{1-2Kh}{2KY-e}>0 \quad\hbox{for }\quad 1/4\le h\le0.01/e^2.\] The denominator is positive at the starting point and remains positive on this branch. If \(h\le1\), then \(Y\ge0.6/e\ge6>h\). If \(h\ge1\), the circle identity implies \[KY^2\ge h-\tfrac12-Kh^2\ge0.49h,\qquad \frac Yh\ge\sqrt{\frac{0.49}{Kh}}>7.\] Consequently \(J<0\). The circle has \(N(1,e)=0\); eliminating \(A\) from (52) gives the decisive exact identity \[ N(c,d)=(1-c)(h+YY')+(d-ce)J>0. \tag{54}\] Both terms are nonnegative and the second is positive, since \(ce>d\). Thus a level circle for the auxiliary parameters gives strict increase for every actual pole parameter under consideration.

If the absolute nonzero leading coefficients are \(b_j>0\), the required horizontal endpoint is \(h=b_jD\). The choice \[16\eta^2\max_j b_j<0.01\] places every such endpoint in the circle interval. This proves the claimed \(D^{-1/2}\) angular reach. Constants such as the imaginary height are allowed to depend on the now fixed \(D\).

We give the blending argument, since endpoints alone do not suffice for subsequent action comparisons. On the circle define \[Q_c=cY^2+(1-2ch)YY'.\] Equation (53) gives \[Q_1=\frac{Y\{eY+2Kh(h-1)\}}{2KY-e}>0,\qquad Q_c=cQ_1+(1-c)YY'>0.\] For \(h\le1\), use \(eY\ge0.6\) and \(2Kh(h-1)\ge-K/2\); for \(h\ge1\) positivity is immediate. At \(d=0\), replacing \(Y\) by \(tY\), \(t\ge1\), changes the numerator to \(h(1-ch)+t^2Q_c\), which is positive at \(t=1\) and increases. On the favorable overlap \(d\le-\delta/2\), the extra contribution \(dJ_t\) is positive. Hence arbitrary fixed magnification of \(Y\) preserves admissibility. On a fixed compact interval write a magnified profile as \(Y=Mf(h)\). Interpolate \(f\) to a positive increasing ray profile, keeping both \(f\) and \(f'\) bounded below. The favorable contribution is \[|d|M^3f^2f'+O(M),\] while the remaining terms are \(O(M^2)\). A sufficiently large fixed \(M\) makes the entire interpolation admissible. Initial pieces may already be rays. Away from the adverse direction, a ray with \(\operatorname{Re}du>0\) and \(\operatorname{Re}(u/p_*)<0\) works: for \(p_*=e^{i\varphi}\), \(0<\varphi<\pi\), take \(u=h(1-im)\) with \(\cos\varphi-m\sin\varphi<0\). This joins the favorable overlap. Reflection supplies the other family. At every join the strict conditions are linear in the tangent vector at the point, so smoothing corners preserves them. Near \(u=0\) the logarithmic condition reduces to increasing modulus, and the start can be made real.

For a real leading adverse pole, set \(V=Y^2\). Formula (52) becomes \[ N(V)=h(1-h)+V+(\tfrac12-h)V'. \tag{55}\] On \(0<h\le3/4\), scaling \(V\) to \(tV\), \(0\le t\le1\), gives \(N(tV)=(1-t)h(1-h)+tN(V)>0\). This is an actual common-real-contour homotopy. Beyond the pole, convex interpolation of two admissible \(V\)-profiles with a common horizontal endpoint preserves the inequality and avoids the pole provided both profiles are positive at \(h=1\). Any zero-height portion lies strictly before the pole, where the positive real-ray margin permits smoothing the square-root parametrization in both variables.

For clarity, the required small-radius turn can be written explicitly. In \(q=r^2\), take \[q(\vartheta)=\rho e^{(-\sigma_{\rm sp}+i)\vartheta}, \qquad 0\le\vartheta\le2\pi/3,\qquad \sigma_{\rm sp}=1/4.\] Its modulus decreases and \[\frac{d}{d\vartheta}\operatorname{Re}q =-|q|(\sigma_{\rm sp}\cos\vartheta+\sin\vartheta)<0.\] Under \(u=1-q/2\), the exact other sign is \[\frac{d}{d\vartheta}\log\left|\frac u{1-u}\right| =\operatorname{Re}\frac{\sigma_{\rm sp}-i}{1-q/2}>0\] for small fixed \(\rho\). The terminal real part is \(-|q|/2\). Scale \(\rho\) to obtain the homothetic shrinking turns. A small preliminary phase correction has the form \(q=q_0e^{-t+i\psi(t)}\); when \(\psi,\psi'\) are small, \[\operatorname{Re}\frac{d\log r}{dt}=-\tfrac12,\qquad \operatorname{Re}\bigl(Z^2q'\bigr) =-|Z|^2|q|\{\cos\alpha+\psi'\sin\alpha\}<0, \quad\alpha=2\arg Z+\psi.\] This permits alignment of the final translating coordinate. The affine interpolation above joins these turns to the original bypasses after shortening at a common endpoint beyond the pole.

All remaining contour pieces form a compact collection of normalized intervals for fixed \(D\). Its strict inequalities survive small changes in the leading coefficients and higher-order parameter terms. Smaller-order openings have \(\beta z^2=o(1)\) and may use the real path. Choosing the complex parameter ball to be a sufficiently high fixed power of \(|s|\) preserves every margin. ◻

Proof. Write \(a=\zeta_t\), \(b=R_t\) and \[ D=c_2a-c_1b =c_2\zeta_0'-c_1R_0'+(c_2c_1'-c_1c_2')u. \tag{59}\] In particular \(D\) contains neither \(u_t\) nor \(u_\theta\). The tangent vectors give \[\begin{align*} g_{tt}&=B^2a^2+b^2=:A, &g_{t\theta}&=(B^2ac_1+bc_2)u_\theta, &g_{\theta\theta}&=R^2+(B^2c_1^2+c_2^2)u_\theta^2,\\ J:=\det g&=R^2A+B^2D^2u_\theta^2, &g^{tt}&=\frac{R^2+(B^2c_1^2+c_2^2)u_\theta^2}{J},\\ g^{t\theta}&=-\frac{(B^2ac_1+bc_2)u_\theta}{J}, &g^{\theta\theta}&=\frac A J. \tag{60}\end{align*}\] The cancellation in the determinant is exact. In particular, a term \(B^2\zeta_\theta^2\) cannot be estimated independently of the off-diagonal square in the determinant. For \(E=B^{-2}\), \(d=a^2+Eb^2\), \(L=c_1a+Ec_2b\) and \(T=R^2d+D^2u_\theta^2\), the same identities become \[ g^{tt}=\frac E d+\frac{L^2u_\theta^2}{dT},\qquad g^{t\theta}=-\frac{Lu_\theta}{T},\qquad g^{\theta\theta}=\frac dT,\qquad g^{tt}-\frac{(g^{t\theta})^2}{g^{\theta\theta}}=\frac E d. \tag{61}\] This is the exact Schur complement identity.

An unnormalized normal, sufficient for taking normal components, is \[ N=\left(-b/B,\ a e_r+ \frac{b\zeta_\theta-aR_\theta}{R}e_\theta\right). \tag{62}\] It satisfies \(N\cdot X_t=N\cdot X_\theta=0\) and \(N\cdot X_u=D\). Consequently the stationary equation is precisely \[ \mathcal Q(u)=g^{ij}\frac{N\cdot X_{ij}}D +\frac{N\cdot X}{2D}=0. \tag{63}\] No connection term remains in \(N\cdot X_{ij}\). Direct differentiation yields \[\begin{align*} \frac{N\cdot X_{tt}}D&=u_{tt}+C_{tt}, &C_{tt}&=\frac{a(R_0''+c_2''u+2c_2'u_t) -b(\zeta_0''+c_1''u+2c_1'u_t)}D,\\ \frac{N\cdot X_{t\theta}}D&=u_{t\theta}+C_{t\theta}u_\theta, &C_{t\theta}&=\frac{ac_2'-bc_1'}D-\frac bR,\\ \frac{N\cdot X_{\theta\theta}}D &=u_{\theta\theta}-\frac{aR}D-\frac{2c_2u_\theta^2}R, &\frac{N\cdot X}{D}&=\frac{Ra-\zeta b}D. \tag{64}\end{align*}\] Thus the complete operator is affine in the second derivatives. \(C_{tt}\) is at most quadratic in \(u_t\) and \(C_{t\theta}\) at most affine in \(u_t\), with coefficients analytic in \(u\). Moreover \[\begin{align*} Ra-\zeta b={}&R_0\zeta_0'-\zeta_0R_0' +(c_2\zeta_0'+R_0c_1'-c_1R_0'-\zeta_0c_2')u\\ &+(R_0c_1-\zeta_0c_2)u_t+(c_2c_1'-c_1c_2')u^2. \tag{65}\end{align*}\] The terms \(u u_t\) cancel. The normalized support function is affine in \(u_t\). At circular data, irrespective of its speed, \[ g^{tt}=A^{-1}=O(B^{-2}),\qquad g^{t\theta}=0,\qquad g^{\theta\theta}=R^{-2}. \tag{66}\] In particular the circular angular curvature term is \(-a/(DR)\), which is affine in \(u_t\); it has no unsuppressed quadratic speed term.

Here is a full coefficient estimate. Use the Banach algebra \(\mathcal A=L^\infty(I;H^h(\mathbb S^1))\) and its module \(\mathcal Y_h\). By (56), \[ \begin{gathered} \|(u,u_\theta)\|_{\mathcal A}\le W,\qquad \|u_t\|_{\mathcal A}\le BW,\\ \|(u_t,u_{\theta\theta})\|_{\mathcal Y_h}\le W,\qquad \|u_{t\theta}\|_{\mathcal Y_h}\le BW,\qquad \|u_{tt}\|_{\mathcal Y_h}\le B^2W. \end{gathered} \tag{67}\] The functions \(R,D,d,T\) are analytic units in this algebra when \(BW\) is small. Equations (61)–(64) partition the entire operator into the following classes:

1. The circular radial coefficient is \(E\) times an analytic function of \((u,u_t)\). Its change multiplies \(u_{tt}\) with bound \(C E(W+BW)B^2W\le CBW^2\). Its Taylor terms on bounded reference derivatives satisfy this bound too.

2. Every extra radial coefficient has two factors \(u_\theta\), by (61). Its product with \(u_{tt}\) is bounded by \(CB^2W^3\).

3. The mixed coefficient has a factor \(u_\theta\) and analytic dependence on \((u,u_t,u_\theta)\). Its product with \(u_{t\theta}\) is \(O(BW^2)\); any additional speed factor costs at most \(BW\).

4. The angular coefficient equals \(R^{-2}\) plus a term divisible by \(u_\theta^2\). Its change multiplying \(u_{\theta\theta}\) is \(O(W^2)+O(W^3)\), with analytic speed factors uniformly bounded.

5. The remaining terms are exactly the lower terms in (64) and (65). Pure speed squares in \(C_{tt}\) retain the factor \(E\); all other pure speed dependence at \(u_\theta=0\) is affine. Terms with angular tilt retain its one or two explicit factors. A product containing only first derivatives is estimated with one factor in \(\mathcal Y_h\), so even \(u_t^2\) costs \(BW^2\), rather than \(B^2W^2\).

This list exhausts the three inverse-metric contractions and the support term. Every further Taylor factor is small in \(\mathcal A\), being bounded by \(C(W+BW)\). The convergent analytic series therefore sum to (58). Differencing a product gives the asserted Lipschitz estimate. The multiplication inequality \(\|fg\|_{H^h}\le C_h\|f\|_{H^h}\|g\|_{H^h}\) proves these statements with all \(h\) angular commutations, keeping exactly one second derivative in \(L^2\); none is put into a trace.

For the last assertion let \(U=\langle u\rangle_\theta\) and \(\mathcal N(u)=\mathcal Q(u)-\mathcal Q(0)-D\mathcal Q(0)u\). The linearized coefficients are independent of \(\theta\), whence \[ \mathcal Q(U)=\mathcal N(U)-\langle\mathcal N(u)\rangle_\theta \quad\hbox{if }\mathcal Q(u)=0. \tag{68}\] The circular curve is exactly \((B(\zeta_0+c_1U),R_0+c_2U)\). If instead one expands about \(U\), its second derivatives need not have uniform traces: split them into reference derivatives and perturbation derivatives and keep the latter in the same \(L^2\) slot. The coefficient Taylor remainder then costs at most \(B^2W^3\). This proves the claim also for a varying circular mean. ◻

Proof. The physical slopes in the polar orthonormal frame are \(p=-Zg_t/r\), \(q=Zg_\theta/r\). With \(H=1+p^2+q^2\), the curvature matrix has entries \[ A=\frac{1+q^2}{H},\qquad D_\theta=\frac{1+p^2}{H},\qquad C=-\frac{pq}{H}. \tag{73}\] The stationary equation, multiplied by \(r^2/Z\), is exactly \[ A(g_{tt}+g_t)+D_\theta(-g_t+g_{\theta\theta}) -2C(g_{t\theta}+g_\theta) =-\tfrac12r^2(g_t+g). \tag{74}\] Complex tangent factors on a prescribed contour are included by the chain rule in (69). Its row is the fixed circular unit, including the \(J_\xi\) term; the graph-dependent \(H_1\) normalization remains inside \(\mathcal F_t\) and its linearization. The preceding normalized inequalities retain the required strict margins.

Set \(p_0=-Z(g_0)_t/r\), \(e=p_0^{-2}\), \(s=(p-p_0)/p_0\) and \(q=p_0 v_1\). Then \(|e|\asymp E\), \(s= v_t/(g_0)_t\) and \(v_1=-v_\theta/(g_0)_t\), and the exact coefficient formulas read \[ A=\frac{e+v_1^2}{e+(1+s)^2+v_1^2},\quad D_\theta=\frac{e+(1+s)^2}{e+(1+s)^2+v_1^2},\quad C=-\frac{(1+s)v_1}{e+(1+s)^2+v_1^2}. \tag{75}\] For an explicit check of the drift division, write instead \(P=g_t/r^2\), \(Q=g_\theta/r^2\), \(e_1=(Zr)^{-2}\) and \[H_1=(1-r^2/2)P^2-(1+r^2/2)Q^2-r^2e_1/2.\] Equation (74) becomes \[ g_t-a_1g_{tt}-b_1g_{\theta\theta} +2c_1^*g_{t\theta}+2c_1^*g_\theta-d_1g=0, \tag{76}\] where \[a_1=\frac{e_1+Q^2}{H_1},\qquad b_1=\frac{e_1+P^2}{H_1},\qquad c_1^*=\frac{PQ}{H_1},\qquad d_1=\frac{r^2(e_1+P^2+Q^2)}{2H_1}.\] Here \(P\) is a bounded unit at the reference. Put \(\mathfrak p=E^{-1/2}W/|r|^2\) and \(\mathfrak q=E^{-1/2}W_o/|r|^2\). In the angular algebra these exact normalized coefficients satisfy \[ \frac{\|a_1-a_{1,0}\|}{E} +\frac{\|b_1-b_{1,0}\|}{E} +\frac{\|d_1-d_{1,0}\|}{|r|^2E} \le C(\mathfrak p+\mathfrak q^2),\qquad \frac{\|c_1^*\|}{\sqrt E}\le C\mathfrak q. \tag{77}\] In particular at zero tilt the speed variation of \(b_1\) retains a factor \(E\), and that of \(d_1\) a factor \(r^2E\); their remaining variations have two tilt factors. Drift division introduces no unweighted radial-speed term.

Consequently \(A-A_0=O(E s)+O(v_1^2)\), \(D_\theta-1=-v_1^2/(e+(1+s)^2+v_1^2)=O(v_1^2)\), and \(C=O(v_1)\), as analytic algebra estimates. Every pure radial-speed variation in \(A\) retains the factor \(E\); angular changes of \(A\) have two tilt factors. The condition that the mixed coefficient be small relative to \(\sqrt E\) is precisely the second smallness condition in (70).

For the full map, assign the traces of \(v,v_\theta\) size \(W\), the trace of \(v_t\) size \(E^{-1/2}W\), and the bulk sizes of \(v_{tt},v_{t\theta},v_{\theta\theta}\) respectively \(E^{-1}W,E^{-1/2}W,W\). A pure speed change in \(A\) times \(v_{tt}\) is bounded by \(E|r|^{-2}(E^{-1/2}W)(E^{-1}W)=A_rW^2\). A mixed tilt times \(v_{t\theta}\) costs \(A_rW_oW\). The two-tilt part of \(A\) times \(v_{tt}\) costs \(C_rW_o^2W\). The angular coefficient costs less. These are all second derivative terms in (74). For a coefficient Taylor remainder on a background derivative, use its size \(O(|r|^2)\). The pure speed part has the factor \(E\), and use one \(v_t\) in bulk rather than trace. For instance \(E|r|^{-4}v_t^2 O(|r|^2)\) has bound \(E^{1/2}|r|^{-2}W^2\le A_rW^2\). Tilt squares have bound \(|r|^{-2}W_o^2\le A_rW_o^2\). For the normalized zeroth-order term \(d_1g_0\), use the factor \(r^2E\) in (77) and \(g_0=O(1)\), giving these same bounds; it is not necessary to assign \(g_0\) the derivative size \(O(r^2)\). All lower terms are already displayed in (74); the support term before division is affine in \(g,g_t\). The analytic \(H_1\) division already included in \(\mathcal F_t\) preserves these bounds by its unit expansion. The fixed row in (69) and its bounded chain-rule coefficients preserve them as well. Higher powers are absorbed using (70).

Finally rotation of \(\theta\) commutes with the equation. At a circular input, its linearization has circular coefficients. A mean output therefore has no term linear in the oscillation, and an oscillatory output has no term containing only mean inputs. Applying the preceding multilinear estimates with these absent terms gives (72). Analytic composition in \(H^h\) and the single highest-derivative bulk slot prove every angular commutation and every difference estimate asserted above. ◻

Proof. We first prove one exterior estimate on the real interval in (iii). This avoids imposing boundary data at changes of graph gauge. For a scalar change \(u=A\widetilde u\), conjugate the source by \(A^{-1}\). The principal coefficients of (13) are unchanged, while the lower coefficients become \[\begin{align*} \widetilde d_1&=d_1-2aA_s/A-2bA_\theta/A,\\ \widetilde d_2&=d_2-2bA_s/A-2cA_\theta/A,\\ \widetilde d_0&=d_0+(1+d_1)A_s/A+d_2A_\theta/A -aA_{ss}/A-2bA_{s\theta}/A-cA_{\theta\theta}/A. \end{align*}\] The \(a\) contribution to \(\widetilde d_1\) is \(O(E_B)\); the \(b\) contribution is \(O(\varepsilon_*\sqrt{E_B})\) and is an allowed small strip operator perturbation. The remaining additions are bounded lower terms. In particular \(A_s/A\) must have an ordinary bounded derivative norm, since it occurs without an \(E_B\) factor in the mass. The transition is taken in the prescribed reference frames; the coefficient–module bounds control the rougher actual backgrounds. For the one-dimensional real coordinate changes in (iii), the chain rule is (78) with positive real \(H\). Their bounded factors preserve the radial, mixed, and angular units.

The fixed-radius radial-to-height transition illustrates why these requirements hold for the later circular overlap. Write \(z=Z_0\varphi(r)\) with \(|Z_0|\asymp|Z|\asymp B\) and \(\varphi^2=1-r^2/2\), and put \(H_r=(\log(r_0/r))_s\). Then \[(\log z)_s=\frac{r^2}{2\varphi^2}H_r,\qquad u_{\rm rel} =-\frac{z_s}{Zr_s}u_{\rm radius} =\frac{Z_0}{Z}\frac r{2\varphi}u_{\rm radius}.\] Here \(u_{\rm rel}\) is relative height, so the factor follows by equating the two variations’ normal components. The last factor and its inverse have bounded fixed derivatives when \(|r|\asymp r_0>0\) and \(\varphi\) is a unit. They need not do so at \(|r|\asymp B^{-1}\), so the transition is made on that fixed-radius overlap. In the common real parameter, the circular radial and angular diffusions, before the small axial-curvature correction to the drift, are \[\frac{4}{Z_0^2r^2H_r(1+4\varphi^2/(Z_0^2r^2))}, \qquad \frac{H_r}{\varphi^2}.\] These have the polar units \(E_B\asymp(B|r|)^{-2}\) and \(1\). Equivalently, in the mixed gauge of Lemma 18, the angular-curvature and support terms have slow drift \(-\varphi^2/(r^2H_r)\), independent of the prescribed transverse direction. The formulas follow from \(r_s=-rH_r\), \(z_s=Z_0r^2H_r/(2\varphi)\) and the metric in (60). The remaining axial drift is \(O(E_B)\) on this overlap. On a long height chart (78) and the strict margins in (ii) are checked with its own scale \((B|r|)^{-2}\), including the tangent derivatives. No uniformly bounded conversion to physical normal height along the whole cap is needed here.

Trivialize on the fixed overlap bands as in (iii), keeping the prescribed exit variable. Put \(u=\rho_B w\) on the entire interval and \(d=(\log\rho_B)_s\). On the long real cylinder the model shifted mass at \(d=\alpha_{\mathrm{cyl}}\) is \[d-2-2\mu^2(d^2-d).\] It has a strict positive buffer for a far entrance depending on this fixed rate; at \(d=2.2\) the limiting buffer is \(0.2\). The small full operator-norm perturbation in (ii) preserves this estimate by absorption; decay of that perturbation is not needed on the interval \(z\le cB\). Raise the rate on a fixed band while this margin still holds, before entering a region whose lower terms require \(d_D\), and similarly enter the height rate. On the other bands the algebraic mass is \[(1+\widetilde d_1)d-a(d^2+d_s)+\widetilde d_0.\] The chosen rate exceeds the bounded reaction costs, and \(E_Bd\ll1\) controls both the quadratic term and the smooth interpolation term. The coefficient–module perturbations are absorbed with the strict local margins. Perform the integration in the proof of (21) once on this whole interval, including the same additional exponentially decreasing localization centered at an arbitrary strip. There are only the true entrance and terminal boundary terms. The exit variable is unchanged there, so its conjugated condition is \(w_s+d w=\rho_B(T_B)^{-1}g\) and has the favorable terminal sign already computed. The geometric strip sum is uniform in the interval length. Local patch estimates recover every bulk term across the smooth overlap bands. We obtain \[ \|u\|_{X_B(\mathrm{exterior})} \le C\left(C_h^2(u;K_{\mathrm{collar}}) +\|L_Bu\|_{Y_B(\mathrm{exterior})} +\|g/\rho_B(T_B)\|_{\mathcal T^N_{E(T_B),h}}\right). \tag{80}\] The overlap constants depend only on the prescribed fixed data and \(D\). Lemma 9 supplies the conical part of this estimate. The compact multiplier sources are absent on all sufficiently far collars.

Interior elliptic estimates on nested fixed collars, including the commutations in (31), now yield \[ \|(u,\lambda)\|_{X_B} \le C\left(\|\mathcal A_B(u,\lambda)\|_{Y_B} +\|u\|_{L^2(K')}+|\lambda|\right) \tag{81}\] for a fixed compact \(K'\). In particular the estimate has been proved in the weighted norm, before any power of \(B\) is spent.

For a fully stopped problem write \(T\) for the smallest logarithmic stopping coordinate on the conical ends. If there are no conical ends, omit \(T\). Suppose there were no thresholds \(B_0,T_0\) and constant \(C\) giving (79) whenever \(B\ge B_0\) and \(T\ge T_0\). We could then choose a failing sequence with \(B\to\infty\) and every conical stop tending to infinity. Normalize its unknowns to have \(X_B\) norm one and its output to tend to zero. Interior compactness and finite dimensionality of the multipliers give a local limit \((u_\infty, \lambda_\infty)\). On every fixed cylindrical interval the weight is eventually \(e^{\alpha_{\mathrm{cyl}}(t-t_0)}\). Consequently \(u_\infty\) has at most polynomial growth there. On conical ends its ambient normal height has at most linear growth, by (26). The derivative estimates give \(u_\infty\in H^2_G\). It solves the homogeneous limiting normalized block and hence vanishes, together with its multipliers. Local elliptic estimates upgrade this convergence sufficiently to make the compact terms in (81) vanish. The latter inequality contradicts the normalization. This is the exclusion of escape along a long end: it uses the fixed strict buffer in (80), and would not follow by normalizing an unweighted estimate with a factor \(B^q\). The contradiction gives one bound on the entire quadrant \(B\ge B_0\), \(T\ge T_0\). Thus the subsequent conical exhaustion may fix any \(B\ge B_0\) before sending the conical stops to infinity.

Surjectivity is a separate argument. Stop all ends, including the conical ends, at remote finite sections. At each fixed \(B\) and each fixed conical stopping radius, the compact scalar boundary problem is realized from its homogeneous mixed-boundary \(H^2\) domain to \(L^2\). The full boundary-data map is Fredholm by the complementing-boundary theorem (Nazarov and Plamenevsky 1994, chap. 1, Section 1, subsection 3, Theorem 1.1), localized in surface charts. Entrance Dirichlet and transverse derivative conditions occur on disjoint boundary components \(\Gamma_D,\Gamma_N\). To pass to the stated homogeneous realization, let \(\mathcal B\) be the boundary trace map into \(\mathcal T=H^{3/2}(\Gamma_D)\oplus H^{1/2}(\Gamma_N)\) and choose its bounded right inverse \(R\) from fixed smooth collars. The decomposition \(u=v+Rg\), \(v\in\ker\mathcal B\), identifies the full map with \((v,g)\mapsto(L_Bv+L_BRg,g)\). A bounded triangular change in the range reduces it to \(\operatorname{diag}(L_B|_{\ker\mathcal B},I_{\mathcal T})\), so the homogeneous realization is Fredholm with the same index. Hypothesis (iv) identifies its index with that of a real coercive problem, hence zero. Continuity of the index and invariance under finite-rank perturbations are the Banach-space facts in (Kato 1995, IV, Sections 5.2–5.3, Theorems 5.17 and 5.26). Adding equally many multiplier variables and observation equations first gives \(\operatorname{ind}\operatorname{diag}(L_B,0_{\mathbb C^m}) =\operatorname{ind}L_B\). The off-diagonal observation and source maps are finite rank, so the augmented block still has index zero.

Apply the already proved weighted a priori estimate to the actual stopped operator. Its constants are independent of the conical stopping radii by Lemma 9 and the compact-limit argument above. For sufficiently remote stops and large \(B\) it excludes a kernel: elliptic boundary regularity (Nazarov and Plamenevsky 1994, chap. 1, Section 1, subsection 3, Theorem 1.5) first makes every kernel element smooth, so the commuted estimate applies. The stopped augmented operator is therefore bijective on this \(H^2\)–\(L^2\) realization. For smooth sources its solution is smooth; the actual commuted estimates then bound it in \(X_B\) by the source in \(Y_B\). Approximation by smooth sources extends this inverse to the stated angularly commuted spaces, with the same uniform bound. Thus no identification of those spaces with an isotropic higher Sobolev scale is needed. Let the conical stops tend to infinity, keeping \(B\) fixed. The uniform conical estimates, weak compactness in their weighted bulk spaces, and local elliptic compactness pass these solutions to the infinite-conical domain. The prescribed compact observations and finite cylindrical or height exits pass to the limit as well. This proves surjectivity of the actual operator with the asserted bound.

Only compact stopped problems enter the artificial index homotopy. No uniform inverse along that homotopy, no geometric realization of its intermediate operators, and no deformation of an infinite conical boundary problem are required. Fixed-parameter elliptic Fredholm theory supplies the index; the weighted estimates for the actual geometric operators supply uniformity in \(B\) and in the conical exhaustion.

Finally the real-cylinder weight is at most \(CB^{\alpha_{\mathrm{cyl}}}\), and bounded bands cost a fixed factor depending on \(D\). By the stated length hypothesis and the bounded real pullbacks, a cap has \(s\)-length at most \(C_{\mathrm{par}}\log B+C_{\mathrm{par}}\), with \(C_{\mathrm{par}}\) fixed before \(B\). Thus \(\sup\rho_B\le C_{D,Y_0,\alpha_{\mathrm{cyl}}} B^{\alpha_{\mathrm{cyl}}+C_{\mathrm{par}}d_{\mathrm{cap}}}\). Conversions of the prescribed height units and any fixed number of their derivatives add only fixed powers. This proves the last assertion without altering the uniform weighted estimate. ◻

Proof. Put \(Z=B\), \(d=c_1Z^2\), \(D=\partial_t+d\), and \(v=e^{d(t-T)}w\), with \(d\) constant on the overlap. Choose \(c_1>0\) after the reference margins so that \(d\sup E\) is small. All bulk norms below mean \(L^2_tH^h_\theta\) on the fixed overlap, or on a fixed enlarged patch. The shifted reference energy has mass \(d-O(Ed^2)\ge cd\); the symbol and boundary argument of (21) control \[N_d(w):=d\|w\|+\|Dw\|+\|w_{\theta\theta}\| +\|ED^2w\|+\|\sqrt E D w_\theta\|.\] Mixed and imaginary terms use the small derivative energies exactly as in that argument. For clarity the first traces also have constants independent of \(Z\). With \(E\) replaced by its comparable patch value, product integration gives \[\begin{align*} E\|Dw\|_{L^\infty H^h}^2 &\le C\bigl((E+Ed)\|Dw\|^2+\|Dw\|\,\|ED^2w\|\bigr),\\ \|w_\theta\|_{L^\infty H^h}^2 &\le C\|w_{\theta\theta}\| \bigl((1+d)\|w\|+\|Dw\|\bigr). \end{align*}\] The ordinary product identity supplies the \(w\) trace. Hence \((w,w_\theta,\sqrt E Dw)\) is bounded by \(CN_d(w)\) in \(\mathcal A=L^\infty_tH^h_\theta\). The term \(d\|w\|\) is retained in this estimate; no second derivative has a trace.

Explicitly, for a perturbation \(\delta aD^2+2\delta bD\partial_\theta+\delta c\partial_\theta^2 +pD+q\partial_\theta+r\), a sufficient coefficient–module norm is \[\mathfrak M=\|\delta a/E\|_{\mathcal A} +\|\delta b/\sqrt E\|_{\mathcal A}+\|\delta c\|_{\mathcal A} +E^{-1/2}\|p\|+\|q\|+\|r\|.\] The last three norms are bulk norms: the preceding traces give the operator bound \(C\mathfrak M N_d(w)\). Bounded lower coefficients may also be estimated directly on their bulk slots.

Here is the shifted perturbation check for the geometric difference operators used below. In the mixed prescribed gauges of Lemma 18, suppose the two background perturbations have norm at most \(W\) in (56), and set \(\varepsilon=ZW\ll1\). Relative to the circular reference, (61) and (67) give, uniformly along the averaged segment, \[\|\delta g^{tt}\|_{\mathcal A}\le CE(\varepsilon+\varepsilon^2), \quad\|\delta g^{t\theta}\|_{\mathcal A}\le C\sqrt E\varepsilon, \quad\|\delta g^{\theta\theta}\|_{\mathcal A}\le C(W+W^2).\] Analytic drift division preserves the corresponding normalized bounds. Applying these coefficients to \(D^2w,Dw_\theta,w_{\theta\theta}\) costs at most \(C(\varepsilon+\varepsilon^2)N_d(w)\). Keeping \(D\) intact includes the terms with \(d^2w\) and \(dw_\theta\).

The averaged Jacobian also differentiates the coefficients multiplying a background second derivative. Keep that derivative in its original unshifted \(L^2\) slot. From (61)–(67), the resulting coefficients of \(Dw,w_\theta,w\) have respective \(L^2_tH^h_\theta\) bounds \(CW\), \(C(\varepsilon+\varepsilon^2)\), and \(CW\). For example differentiating the circular radial coefficient costs \(CE\), and \(\|u_{tt}\|\le CZ^2W\); differentiating its two-tilt correction costs \(CW\), giving \(CZ^2W^2\) in the angular slot. Use \(\|Dw\|_{\mathcal A}\le CZN_d(w)\) and the other first traces to bound all these module products by \(C(\varepsilon+\varepsilon^2)N_d(w)\). The lower terms are precisely (64) and (65): pure speed squares retain \(E\), and the circular angular and support terms are affine in speed. They obey the same bound. Thus the full difference operator satisfies \[\|e^{-d(t-T)}(L_*-L_{\rm ref})e^{d(t-T)}w\|_{L^2H^h} \le C(\varepsilon+\varepsilon^2)N_d(w).\] This is a module estimate, not pointwise control of every lower coefficient. In a pure radial chart the variables in Lemma 7 give the same argument, with any small circular opening included in the reference. The reference margins and this shifted smallness are checked before applying the lemma: smallness of the opening alone does not assert \(ZW\ll1\) for the correction relative to that reference.

Keep the actual exit condition throughout: \[ v_t(T)=g\quad\Longleftrightarrow\quad Dw(T)=g. \tag{83}\] For a reference fast root the lift denominator is \((\lambda_+-d)+d=\lambda_+\), so the natural norm (17) retains its gain. For homogeneous data the principal terminal coefficient is \(d\operatorname{Re}a+(1-2d\operatorname{Re}a)/2=1/2\); \(d_1=O(E)\) adds only \(O(E)\). The full perturbation is inverted on this same Robin domain by a Neumann series once its displayed relative norm is sufficiently small. No trace of a rough lower coefficient is introduced. A scalar conversion \(v=A(t)\widetilde v\) changes both the interior operator and the exit to \((\partial_t+d+A_t/A)\widetilde w=g/A\). Bounded invertible \(A\) with the prescribed derivative bounds gives equivalent shifted domains. Returning to ordinary derivatives costs only fixed powers of \(Z\); commuted versions use the corresponding differentiated coefficient–module bounds.

One may alternatively take the farther actual trace as Dirichlet data, or cut off the solution near the far side and impose zero data beyond the cutoff. The resulting source is supported a fixed distance to the right of the band being estimated. The factor \(e^{d(t-T)}\) suppresses its effect there by \(e^{-d\,\mathrm{gap}}\). The shifted estimates have only polynomial losses, which proves (82). All operations occur in prescribed charts. On the inner parameter ball in the statement, Cauchy’s estimate for a derivative of order \(j\) costs at most \(C_jB^{qj}\), which preserves the exponential. A ball’s radius by itself gives no estimate at points arbitrarily close to its boundary. Smooth contour parameters instead require differentiation of the equation and its prescribed real-domain pullback. When those differentiated coefficients and data have the stated polynomial bounds and their sources remain on the farther side, the same shifted estimate applies to each differentiated equation. ◻

Proof. Use the exact family \(S_b\) as base, with its multiplier and the first multiplier jet in the comparison pair. Remove the first-order transverse tilt by analytic rotations of the individual ends, cut off on fixed compact collars. On the real incoming part add the linear opening jet and the nonlinear part of \(R_c\), starting the latter beyond a fixed end entrance. Before changing the spatial contour, cut off the small noncircular remainders on a bounded logarithmic band and use \(R_c\) alone on the complex part. All cutoffs are prescribed functions on the real parameter domain. They are never evaluated at an unknown complex spatial argument.

Here is the size of the residual. On the compact and conical pieces the constant and linear terms have been solved, so it is \(O(B^{-4})\), including the observation error. On a cylindrical piece the no-axial-derivative circular equation vanishes exactly on \(R_c\). The base error is \(O(z^{-2+\eta})\); the nonleading part of the first opening jet is \(O(B^{-2}z^\eta)\). Their interactions beyond first order have size at most \(CB^{-4}(1+|z|)^{2+\eta'}\). The axial curvature terms carry \(z^{-2}\); expansion where \(|\beta z^2|\) is small gives the same bound. Choose the fixed losses so that \(\eta'<0.06\). On the final bounded logarithmic bands, direct differentiation of \(R_c\) and of the prescribed cutoff gives \(O(B^{-2+\eta'})\). Thus the output residual is bounded by \[C_D B^{-4}(1+|z|)^{2.06}.\] Small compact corrections enforce the observations exactly. The exit derivative is that of the comparison profile, so its initial derivative residual is zero.

Apply Proposition 20 with rate \(2.2\), keeping the strict buffer above \(2.06\), and with the bounded-band weights specified above. Its hypotheses hold: before the bounded outer part the coefficients are small perturbations of the cylindrical family; on that part Lemma 17 gives the diffusion signs, and the reference remains transverse with nonsingular drift.

We verify the index hypothesis separately, using a homotopy of operators on fully stopped domains. Straightening an adverse geometric bypass would cross the pole of the circular profile; the following argument does not do so.

Fix \(B\) and finite conical stopping sections, and pull every chart back to its prescribed real parameter domain. The scalar graph variations and scalar normal sources describe the same complex normal line of the reference immersion. Identifying a graph variable with its normal component multiplies the unknown and the source by the same nonzero graph-speed factor. This is conjugation of the operator and changes only lower-order terms. Its principal tensor is therefore a single geometric tensor \(A\). The local drift divisions give smooth nonzero row multipliers \(m_i\) for which \(\operatorname{Re}(m_iA)\) is positive definite, with the sign chosen for minus the second-order part. This is the strict radial and angular accretivity and small mixed coefficient on the cylindrical and height charts. It follows from closeness to the real reference on the core and its collars, and from Lemma 9 on conical pieces. The spatial coordinate changes on the pulled-back domains are real, so transform the tensors by real congruence and preserve positivity.

Choose a nonnegative real partition of unity \(\chi_i\) subordinate to these charts and define \(m=\sum_i\chi_i m_i\). Then, for every nonzero real covector \(\xi\), \[\operatorname{Re}\bigl(mA(\xi,\xi)\bigr) =\sum_i\chi_i\operatorname{Re}\bigl(m_iA(\xi,\xi)\bigr)>0.\] Consequently \(m\) is nonzero everywhere. Multiplication of the entire source row, including the compact multiplier columns, by \(m\) is a range isomorphism and produces one globally accretive principal tensor. This avoids imposing an unsupported common phase on the different local normalizing factors.

The graph-unit identification may write an actual exit condition as \((\partial_t+q)u=0\). Choose a smooth function \(F\) supported in a collar of that exit, with \(F_t=-q\) on its boundary, and put \(u=e^Fv\), multiplying the source by \(e^{-F}\) as well. The boundary condition becomes exactly \(v_t=0\), while the principal tensor is unchanged. Disjoint collars allow this at every exit; entrance Dirichlet data stay zero. Fix all these isomorphisms before the homotopy. For the compact index calculation take the ordinary \(H^2\) domain with these homogeneous boundary conditions and range \(L^2\), together with the finite multiplier and observation spaces. These spaces remain fixed along the homotopy. Positive smooth weights at a fixed stop give equivalent norms at this order; they do not identify the higher angularly commuted spaces with an isotropic higher Sobolev scale.

Choose a real Riemannian metric which is a product in the exit collars, so that \(\partial_t\) there is a nonzero multiple of its normal derivative. Convexly interpolate the globally normalized principal tensor to its positive inverse metric, and interpolate the lower coefficients to those of \(-\Delta+1\). Every intermediate principal tensor has positive real part. The scalar boundary symbol has one decaying normal root for each nonzero real tangential covector. This root cannot be zero, since the tangential quadratic form has positive real part. Thus the fixed transverse derivative condition stays complementing; so does Dirichlet data. The endpoint \(-\Delta+1\) with these mixed Dirichlet/Neumann conditions is invertible by its coercive variational form and boundary regularity, and has index zero. Keep the transformed compact observations and multiplier columns fixed along this homotopy. Their equal-dimensional augmentation has index zero too.

This proves the required index assertion on the actual fully stopped domains. Boundary regularity makes its kernel smooth, so the weighted estimate applies at the actual geometric operator and gives bijectivity on the compact \(H^2\)–\(L^2\) realization. For smooth data the solution obeys the full angularly commuted target estimates; smooth approximation then gives the inverse from \(Y_B\) to \(X_B\), with the same bound. Exhaustion of the conical ends then proceeds exactly as in Proposition 20; the artificial compact homotopy is not used on the infinite ends.

The resulting linear correction satisfies (87). On every outer strip its nonlinear Lipschitz factor is at most \[C_D(BW+B^2W^2)=O_D(B^{-0.8}),\] and on earlier strips the local axial scale is at most \(C_DB\), so the factor is no larger. Fixed compact and conical pieces have the ordinary small perturbation factors. For \(B\) large the correction map therefore preserves a fixed multiple of (87) and contracts there. This proves existence and uniqueness in that neighborhood.

This application needs the uniform weighted inverse. A merely polynomial bound for the original correction would not imply the small factor just displayed. Polynomial losses are used only after the exact perturbative solution has been constructed.

The construction is uniform over the compact collections of normalized contours for fixed \(D\) and fixed lower radius. The outgoing lift of (85) has \(X_B\) size at most \(Cc_{\rm exit}B^{-4}\). Choose a fixed \(c_{\rm hol}>c_{\rm exit}\) and \(C_{\rm pre}\) to include that larger trace ball; the same contraction then works on an open neighborhood of the stated closed data class for large \(B\). The full stops use \(g=g_{\rm app}\). Curtailment and smoothing use the comparison derivative of the same local reference at their moving exit; at the common pre-pole endpoint both problems have the same path, scalar exit unit, and comparison derivative. The real stopped family below checks the consecutive-stop data separately. At the initial fixed-radius height transition, 34 verifies the natural trace size \(O(B^{-2.8})\), below the available outer strip size \(B^{-4}\rho_B(T)\asymp B^{-1.8}\), and places the solve in the same preliminary ball. Its separate refined inverse covers the later shrinking paths and their trace interpolations. For each fixed contour, choose a bounded complex-linear outgoing trace lift on the identified spaces, supported outside the observation core. After subtracting this lift, the unknown has homogeneous exits, and its equation is jointly analytic in the unknown, \(p\), and the independent remainder \(\delta g\). Uniform contraction on the larger trace ball gives the asserted joint Banach-space holomorphy. The actual full-stop and curtailment data are comparison derivatives of the analytic reference profiles at prescribed real path parameters; their remainders, and affine interpolations with coefficients independent of \(p\), are holomorphic in \(p\). The consecutive real stops have the same property. Proposition 34 checks it for the fixed-radius comparison interpolation and for the later actual shorter traces in the refined solver. Cauchy estimates on \(\mathcal B_s\) cost \(C_D|s|^{-Lj}\) at each fixed derivative order \(j\). Differentiating a smooth contour homotopy uses the same linearized inverse and the stipulated differentiated coefficient and data bounds. This proves the final derivative assertion.

There is no requirement that \(b=O(B^{-2})\): typically \(b=O(s)=O(B^{-2/k})\). It belongs to a fixed small analytic neighborhood of \(S_b\), which is an exact base family. Only the openings have size \(O(B^{-2})\). ◻

Proof. Write the pulled-back area density as \(\mathcal L(X,DX)=e^{-X\cdot X/4}\sqrt{\det g}\). For a domain with moving boundary, differentiation and one integration by parts give \[\delta G =\int_\Omega \mathcal Q(X)\cdot\delta X +\int_{\partial\Omega}\Pi_\nu\cdot\delta X +\int_{\partial\Omega}\mathcal L\,v_{\partial\Omega}, \qquad \Pi_\nu=\mathcal L_{DX}\nu .\] The equation and the compact constraints turn the interior term into \(\lambda\cdot\delta\mathsf P(p)\). Tangential changes contribute only to the boundary: algebraic reparametrization covariance gives \(\mathcal Q(X)\cdot DX=0\). These identities continue bilinearly to the prescribed complex charts. The observation charts are fixed on their compact support, so no moving-coordinate term is hidden in \(\delta\mathsf P\). At fixed \(p\) the interior term is zero. Prescribed Neumann data determine a solve; they do not set \(\Pi_\nu\) equal to zero, and the displayed boundary terms are retained.

For those preliminary families, the area and momentum prefactors, endpoint velocities and solution variations have polynomial bounds by Proposition 23. Along these fixed scaled contours, \(\operatorname{Re}(X\cdot X)=\operatorname{Re}z^2+o(B^2)\); bounded transverse radii and the small relative graph corrections account for the error. This proves (88). At the full stop \(H=DB^2\), absorb the polynomial factors and the \(o(B^2)\) loss into the strict margin from \(1/4\) to \(1/5\). Variation of \(s\) at fixed \(p\) changes only the domain, contour and exit data. Applying the identity to its two real derivatives gives (89). Finite \(p\)-derivatives follow by Cauchy estimates on the slightly larger ball. The same reasoning treats changes of exit prescriptions at fixed observations. The refined family uses the same algebraic identity, with its separate shape and Gaussian-height estimates in 36; its trace class alone is not used to infer (88). ◻

Proof. The length \(H\) is a parameter of this boundary problem, not \(|s|^{-k/2}\). Since \(\beta(b,x)\) is linear in \(x\) with bounded coefficients, choosing \(d_0\) small keeps \(|\beta|H^2\) uniformly small. For real parameters the chosen circular branch has positive radius. On the complex polydisc it remains in a fixed small neighborhood of that positive real comparison, so its denominator units and the strict normalized diffusion bounds on the real path persist throughout the stop. Use the same base, first opening jet, and prescribed outer cutoff as in 23. On the outer cutoff band the base noncircular remainder is \(O(H^{-2+\eta})\), and the nonleading first jet is \(O(H^{-2+\eta})\). With \(\eta<0.06\), their cutoff residual is bounded by \(CH^{-4}(1+z)^{2.06}\) there. Before that band the solved constant and linear terms give the same bound; on the compact and conical pieces it is \(O(H^{-4})\). The weighted inverse and contraction from the preliminary proof therefore apply with \(H\) in place of \(B\). This argument includes \(x=0\). In that case the stopped base can differ from \(S_b\) because the noncircular tail was cut off before the exit.

The data and the operator are analytic in \((b,x)\) on the displayed polydisc, and uniform contraction gives the asserted holomorphy. For consecutive stops move the real endpoint and always use the comparison derivative in its current pure graph coordinate. Any interpolation needed to identify the two outer cutoffs changes a smooth derivative by \(O(H^{-2+\eta})\). Its natural Neumann norm is \(O(H^{-3+\eta})\), since \(\sqrt E\asymp H^{-1}\) there. This is below the permitted trace size \(H^{-4}\rho_H(T)\asymp H^{-1.8}\), so the whole homotopy consists of exact normalized solves. Along it \(\Re(X\cdot X)\ge cH^2\) at the exit, and all position, momentum, and variation factors grow at most polynomially. Endpoint first variation proves (91); changing \(c\) absorbs the polynomial factors and the fixed ratio \(\Lambda\). On the conical ends the small complex link graphs retain \(\Re(X\cdot X)\ge c|Y|^2-C\), and their first traces give polynomial area bounds. The resulting integrable Gaussian dominates the holomorphic density uniformly on a smaller parameter polydisc. Differentiation under that integral gives holomorphy of \(F_H\) and its uniform action bound.

For the final assertion take real parameters. On \(z\le3H^\nu\) the opening displacement is \(O(H^{-2}z^2)\), an end tilt contributes \(O(H^{-2}z)\), and the correction is \(O(H^{-4}(1+z)^{2.2})\) in the strip and first-trace norms. These quantities and their physical first derivatives tend to zero for \(\nu<1\). On a conical end the first opening jet with compact exterior source belongs to the conical entrance domain by the inverse identification in the family proof. Its link displacement is \(O(H^{-2})\), and the correction has link norm \(O(H^{-4})\). The conical first-derivative traces therefore give the same smallness on \(|Y|\le3H^\nu\). This larger region remains inside every cylindrical stop for large \(H\), since \(\nu<1\); the conical ends are infinite. Choose the real projection branches there. Their inverse projections over \(|Y|\le2H^\nu\) stay in the larger region, because the graph displacements and slopes tend to zero while the radial margin is \(H^\nu\). The uniform real tubular neighborhood of \(S_b\) then puts the graph descriptions in a common normal graph on \(|Y|\le2H^\nu\). On the remaining real cylindrical pieces \(|X|\asymp z\) and the area density has polynomial bounds from the first traces; on the conical pieces \(|X|\) is comparable to their radial coordinate and their area density is polynomial. Integrating the Gaussian on those tails proves the claimed action bound. ◻

Proof. Take the real stopped problems of Lemma 25 with \(R_j=R_0\Lambda^j\), \(\Lambda>1\), and set \[F_j(s,h)=F_{R_j}(p_0(s)+s^\ell h).\] There are full complex \(s\)-disks \[|s|\le r_j=dR_j^{-2/k}\] on which this construction is holomorphic in \((s,h)\), uniformly for \(h\) in a fixed complex ball. Indeed \(\ell\) is chosen at least \(k\), so \(x=O(s^k)=O(R_j^{-2})\), and \(b\) remains in the fixed neighborhood. Choose \(d\) small enough for the stopped polydisc. The independent length formulation includes the center \(s=0\); it does not substitute an infinite value for \(B\) there.

For consecutive radii the common disk is the smaller one. Equation (91) gives \[\|F_j-F_{j-1}\|_{|s|\le r_j}\le C e^{-cR_j^2};\] the fixed ratio of successive radii has merely changed \(c\). Polynomial factors are absorbed by decreasing \(c\). Set \(D_j=F_j-F_{j-1}\) for \(j\ge1\), and \(D_0=F_0\). Cauchy’s estimate yields \[\|[s^n]D_j\| \le C e^{-cR_j^2}d^{-n}R_j^{2n/k}.\] For \(t_j=R_j^2\), reserve \(e^{-ct_j/2}\) for the convergent geometric-scale sum, and maximize the remaining expression: \[\sup_{t>0}t^{n/k}e^{-ct/2} =\left(\frac{2n}{cke}\right)^{n/k}.\] It follows that \[ \sum_{j\ge1}e^{-cR_j^2}R_j^{2n/k} \le C A^n\Gamma(1+n/k). \tag{94}\] Thus \(\widetilde f_n=[s^n]F_0+\sum_{j\ge1}[s^n]D_j\) exists as a holomorphic function of \(h\) and satisfies the coefficient bound in (92). At this point it is the coefficient series of the stopped telescope; its identification is proved below.

For the all-orders remainder, fix \(0<\vartheta<1\) and take \(J=J(s)\) maximal with \(|s|\le\vartheta r_J\). Then \(R_J^2\asymp |s|^{-k}\). Writing \(T_{N-1}\) for the Taylor polynomial through degree \(N-1\), the exact identity is \[ F_J-\sum_{n<N}\widetilde f_ns^n =\sum_{j=0}^J(D_j-T_{N-1}D_j) -\sum_{j>J}T_{N-1}D_j . \tag{95}\] For \(j\le J\), the Taylor remainder is at most \[\frac{C e^{-cR_j^2}}{1-\vartheta} |s|^N r_j^{-N},\] with the ordinary analytic bound for \(D_0\). For \(j>J\), put \(x_j=|s|/r_j>\vartheta\). For every \(N\ge1\), \[\sum_{n<N}x_j^n\le N\vartheta^{-N}x_j^N.\] Use (94) on both sums in (95), absorbing \(N\vartheta^{-N}\) into \(A^N\). This proves the stated remainder for every \(N\), including orders beyond optimal truncation.

We now identify the coefficient series using only real graph comparisons. Fix an order \(n\) and choose \(M\) with \(k(M+1)>n\). Choose \(0<\nu_M<1\) so small that the degree-\(M\) polynomial height and its first two derivatives are \(o(1)\) on \(|Y|\le2H^{\nu_M}\) whenever \(|x|\le CH^{-2}\). This is possible by its fixed polynomial growth; for example, if that growth has degree \(D_M\), take \(\nu_MD_M<1/2\). On real parameters Lemma 25 puts \(U_H\) in the same small normal graph there. Its normalized residual and observation error vanish: the real change of graph variable transforms the scalar source row with the graph speed, while the compact observations and the multiplier \(\lambda_H\) retain their fixed variational normalization. Fix the real Gaussian exponent \(a_G=1\). The small graph neighborhood can be chosen with retained density exponent \(c_G>1/2\), so \(a_G<2c_G\). In the reference comparison lemmas use their weight \(a=a_G\). Applying [ref:real-comparison,ref:finite-taylor] with outer radius \(R=2H^{\nu_M}\) and \(\vartheta=1/2\) gives \[\|\chi_R(u_H-v^{(M)})\|_{\mathcal H^2_{a_G}} +|\lambda_H-\lambda_M| \le C_M|x|^{M+1}+H^{d_M}e^{-c_MH^{2\nu_M}}.\] Here the annular commutator is controlled by the real first-trace bounds and the \(C^2\)-small Taylor graph. The real action-tail bound of Lemma 25 and (48) then yield \[ |F_H(p)-f_M(p)| \le C_M|x|^{M+1}+H^{d_M}e^{-c_MH^{2\nu_M}}. \tag{96}\] The Taylor graph in this comparison is used only on the inner domain. The full \(f_M\) is the integral of its finite polynomial coefficient densities, whose tails were included in the estimate. Thus the argument neither requires a global uncut polynomial immersion nor changes real normal graphs by a complex spatial reparametrization.

Apply (96) with \(H=R_{J(s)}\) on \(s>0\) and \(h\) in a fixed smaller real ball. The balance \(H^2\asymp s^{-k}\) and \(|x|=O(s^k)\) make the right side \(O(s^{k(M+1)})\) plus a function smaller than every power of \(s\), uniformly on this real ball. For the fixed \(M\), the function \(f_M(p_0(s)+s^\ell h)\) is an ordinary analytic function of \((s,h)\) on a sufficiently small neighborhood. The formal composition of \(\mathcal F\) has the same coefficients through order \(n\): every omitted \(x\) monomial has formal \(s\)-valuation at least \(k(M+1)\). This valuation statement does not assign a numerical bound to the omitted formal tail.

The already proved telescoping remainder, with \(N=n+1\), is uniform on the smaller complex \(h\)-ball, even though \(J(s)\) jumps. Compare it with the ordinary Taylor expansion of \(f_M(p_0(s)+s^\ell h)\) and (96). Inductively subtract the lower identified powers, divide by \(s^n\), and let \(s\downarrow0\). This identifies \(\widetilde f_n(h)\) with the coefficient \(f_n(h)\) of the formal stationary composition on the real ball. Both sides are holomorphic in \(h\), so the identity extends to the complex ball. Since \(n\) was arbitrary, the stopped coefficient series is exactly the formal stationary action. The weaker cutoff exponential in (96) is used only for this identification; the \(e^{-cR_j^2}\) endpoint telescope supplies all Gevrey bounds.

The sector array and \(F_J\) can be curtailed to a common inner real-path region at radius \(c_DB\). Their boundary problems are connected by the homotopies already constructed, at fixed observations. Endpoint first variation gives \[\|G_\pm-F_J\|\le C_D e^{-b_D/|s|^k},\qquad b_D>0.\] For every \(N\), \[e^{-b_D/|s|^k} \le \left(\frac{N}{b_Dke}\right)^{N/k}|s|^N \le C A_D^N\Gamma(1+N/k)|s|^N .\] This transfers the all-orders remainder. Finally, Cauchy’s estimate on nested fixed \(h\)-balls gives these bounds for derivatives in the scaled variable \(h\). Derivatives in the original parameter \(p\) additionally require fixed division; their proof is given after Lemma 27, in (98). The gamma weight is equivalent, up to exponential factors in \(N\), to \((N!)^{1/k}\). ◻

Proof. Sums, products and a fixed analytic operation preserve exponential telescoping by their bounded derivatives on slightly smaller fixed neighborhoods. For division the exact remainder identity is \[R_N(s^{-q}F)=s^{-q}R_{N+q}(F).\] The ratio of gamma weights with the fixed shift \(q\) has at most polynomial growth in \(N\), which is absorbed in a larger exponential base.

The disk approximants need a correction: formal vanishing of the limiting coefficients does not make \(F_j/s^q\) holomorphic. Use instead \[ \widetilde F_j=\frac{F_j-T_{q-1}F_j}{s^q}. \tag{97}\] On a fixed smaller disk their differences are bounded by \[C r_j^{-q}e^{-cR_j^2}\le C e^{-c'R_j^2}.\] Each removed low coefficient has limit zero and is therefore an exponentially small tail of the coefficient telescoping sum. At \(J(s)\), its value after division by \(s^q\) has only a fixed polynomial loss, so remains exponentially small. Thus (97) approximates the actual quotient with the required accuracy. Uniform formal vanishing on the whole parameter ball is essential in this argument.

For implicit solution suppose the normalized equation is \(H(s,z)=0\), with leading equation \(H_0(z)=0\), root \(z_0\), and invertible \(A=D_zH_0(z_0)\). Take the corrected disk approximants \(H_j\). Their centered telescoping gives, for any fixed \(0<\vartheta<1\), \[\|H_j(s,\cdot)-H_0\|_{|s|\le\vartheta r_j} \le C r_j+C e^{-c'R_j^2}.\] In fact the constant-coefficient tail is exponentially small, while the nonconstant parts are bounded by \(|s|\) times the convergent sum \(\sum_{\nu\ge1}e^{-cR_\nu^2}r_\nu^{-1}\), together with the fixed initial analytic term. Cauchy’s estimate on a smaller \(z\)-ball gives the same convergence for the first \(z\)-derivative. Shrink that fixed ball and the disks so that \[\Psi_{j,s}(z)=z-A^{-1}H_j(s,z)\] is a contraction with a common constant \(q_*<1\) and maps that ball into itself. Its roots \(z_j(s)\) are holomorphic, and subtraction of their equations gives \[\|z_j-z_{j-1}\| \le\frac{\|A^{-1}\|}{1-q_*}\, \|H_j-H_{j-1}\|.\] They therefore telescope exponentially on full disks. Lemma 26 supplies their coefficient and all-orders remainder bounds. The same contraction estimate compares the actual sector root with the stopped approximant. This proves implicit closure with remainders. If every formal coefficient is zero, choose \(N=\lfloor c_0|s|^{-k}\rfloor\) in the remainder bound, where \(c_0>0\) is small. Stirling’s bound gives \(Ce^{-c/|s|^k}\). ◻

Proof. Let \(\mathcal A\subset\mathbb R[[p]]\) be the full differential polynomial algebra generated by the \(f_i\) and all coordinate functions. Let \[I=\{f\in\mathcal A:f(\widehat p(s))=0\},\qquad K_s=\mathbb R((s)).\] Choose \(g_1,\ldots,g_r\in I\) with maximal gradient rank over \(K_s\). Although the closure \(\mathcal A\) is infinite, each selected expression is finite and \(r\le\dim p\). Split \(p=(u,v)\) so \(A=g_u\) is invertible along the arc, and put \(\Delta=\det A\) and \(m=\operatorname{ord}_s\Delta(\widehat p)<\infty\). For each free coordinate set \[D_j=\Delta\partial_{v_j} -\bigl(\operatorname{adj}(A)g_{v_j}\bigr)\cdot\partial_u, \qquad E_j=\Delta^{-1}D_j.\] The operators \(D_j\) preserve \(\mathcal A\), and \(D_jg_i=0\) identically. For every \(f\in I\), maximality implies \(df(\widehat p)\in\operatorname{span}_{K_s}\{dg_i(\widehat p)\}\). Hence \(D_jf(\widehat p)=0\), or \(D_jI\subset I\). Ordinary partial differentiation need not preserve \(I\); it is this cofactor tangency that does.

The denominators of iterated intrinsic derivatives admit an explicit bound. For \(n\ge1\) and \(h\in\mathcal A\), \[ E_{j_1}\cdots E_{j_n}h =\frac{P_n}{\Delta^{2n-1}},\qquad P_n\in\mathcal A. \tag{99}\] The first denominator is \(\Delta\), and the induction is \[E_j(P/\Delta^q) =\frac{\Delta D_jP-qP D_j\Delta}{\Delta^{q+2}}.\] If \(h\in I\), all numerators remain in \(I\). Thus every iterated intrinsic derivative of every original constraint vanishes along the arc.

Translate to \(\widehat p(s)\) and apply formal implicit inversion over \(K_s\) to obtain a graph in \(\eta=v-\widehat v(s)\). On that graph \(E_j\) is differentiation in \(\eta_j\). Apply (99) also to the coordinate functions \(u_i\). Each coefficient of total \(\eta\)-degree \(n\) has \(s\)-valuation at least \(-(2n-1)m\). If \(\operatorname{ord}_s\eta\ge N>2m\), its degree \(n\) contribution therefore has valuation at least \[nN-(2n-1)m=m+n(N-2m)\longrightarrow\infty.\] The substitution is adically convergent and the dependent displacement has order at least \(N-m\). All original constraints have zero intrinsic Taylor coefficients, so remain zero after substitution. Both free and dependent displacements have positive valuation; the same bound justifies rearranging their formal evaluations. This is not an evaluation of a divergent numerical series. If \(r=0\), take \(\Delta=1\) and all coordinates free; the same argument says all relevant Taylor coefficients vanish. If there are no free coordinates, no substitution is necessary. ◻

Proof. First fix the inner evaluation ball of (86), with exponent \(L_0\), and the finite derivative orders occurring in the selected \(g_i\). Its polynomial distance from the outer solve ball fixes all Cauchy bounds and possible polynomial losses before any further rescaling. Let \(m\) be the valuation of the selected minor. If the second derivatives needed below are bounded, choose \[\ell\ge\max\{L_0+1,m+1\}.\] If they have a previously fixed loss \(|s|^{-M_0}\), choose instead \(\ell>\max\{L_0,m+M_0\}\). In either case enlarge \(\ell\) to the fixed threshold required in Lemma 26, in particular \(\ell\ge k\), and then set \(N=\ell+m+1\). Increase \(N\) and the polynomial approximation order if needed to preserve all finite derivatives and leading minor coefficients. Take a real polynomial center \(p_0=(u_0,v_0)\) agreeing with \(\widehat p\) modulo \(s^N\), and freeze \(v=v_0\). In either array put \(J_0(s)=g_u(s,p_0(s))\). The finite-order expansion and the nonzero formal minor give \[g(s,p_0)=O(s^N),\quad \det J_0=s^m(d_0+O(s)),\quad d_0\ne0,\quad \|J_0^{-1}\|=O(|s|^{-m}).\] Allowing a larger fixed order for the center makes these statements unchanged when a fixed derivative loss is present.

Set \(u=u_0+s^\ell h\), and consider the actual equation \[ H(s,h)=s^{-\ell}J_0(s)^{-1} g(s,u_0+s^\ell h,v_0)=0 . \tag{101}\] Taylor’s formula in \(u\) is the exact identity \[H=h+s^{-\ell}J_0^{-1}g(s,p_0) +s^\ell J_0^{-1}\int_0^1(1-t) g_{uu}(s,u_0+ts^\ell h,v_0)[h,h]\,dt.\] Thus \(H(s,0)=O(s)\) and \[H_h=I+O(|s|^{\ell-m-M_0})\] on a fixed \(h\)-ball, with \(M_0=0\) in the bounded case. It is a uniformly regular implicit problem, and contraction gives \(h=O(s)\). Since \(\ell>L_0\), the whole fixed \(h\)-ball used in this equation lies strictly inside the previously chosen inner evaluation ball for small \(s\). The physical displacement of its root is the still smaller \(O(s^{\ell+1})\). Free coordinates have been approximated more accurately than the allowed dependent displacement; this pays the amplification by \(s^{-m}\).

We check the divisions needed for Gevrey closure uniformly in \(h\). Write \[H=\bigl(s^{-m}\det J_0\bigr)^{-1} s^{-(\ell+m)} \operatorname{adj}(J_0) g(s,u_0+s^\ell h,v_0).\] The first factor is the inverse of a unit. The numerator of the second quotient has zero formal coefficients below order \(\ell+m\), identically in \(h\): the constant term has order \(N>\ell+m\), the linear term is \(\det J_0\,s^\ell h\), and the quadratic term starts at order \(2\ell\ge\ell+m\). Hence Lemma 27, with its low-Taylor subtractions for the disk approximants, applies to these actual quotients. Their leading equation is \(H_0(h)=h\). The regular implicit roots have the Gevrey expansion given by formal inversion. Lemma 28 identifies it with the substituted formal arc and proves that every original constraint has zero formal expansion. This gives the first estimate in (100).

For the antiholomorphic estimate differentiate the exact selected equations. Holomorphy in \(p\) and the polynomial nature of \(v_0\) give \[ \bar\partial_s u =-g_u(s,p(s))^{-1}\bar\partial_sg(s,p(s)),\qquad \bar\partial_s f(s,p(s)) =\bar\partial_sf+f_u\,\bar\partial_su . \tag{102}\] Here \(H_h=J_0^{-1}g_u\) is uniformly invertible, so the actual \(g_u^{-1}\) has only the fixed loss \(O(|s|^{-m})\). Every selected \(g_i\) uses finitely many parameter derivatives and polynomial operations, so (89) bounds its fixed-parameter antiholomorphic derivative with only further fixed powers. The chain rule therefore loses only powers of \(|s|^{-1}\). For fixed \(D\) these are absorbed between the constants \(D/5\) and \(D/6\), proving the second bound. The constants and the ball radius \(c_D|s|^{L_0}\) may depend on \(D\); no exponentially small ball is used. ◻

Proof. Put \(A=(D/8)e^{-i\delta/2}\). Choose \(\epsilon>0\) so small that \[\epsilon<\delta/2,\qquad D\sin\epsilon/8<c_2(D)/2.\] For each fixed \(D\) both conditions can be met, regardless of how small \(c_2(D)\) is. The sector between the rays \[\alpha=\delta/2-\pi/2+\epsilon,\qquad \beta=\delta/2+\pi/2-\epsilon\] has opening \(\pi-2\epsilon<\pi\), lies within the available sector, and contains the evaluation ray strictly. On its two sides \(\operatorname{Re}(Aw) =(D/8)|w|\sin\epsilon\), so \(H=e^{Aw}e\) is bounded. It also has at most order-one exponential growth and is bounded on the finite inner circular boundary. Its antiholomorphic source satisfies \[|\bar\partial H|\le C_D(1+|w|)^M e^{-D|w|/24}.\]

Extend this source, not \(H\), by zero to the plane, calling the extension \(Q\). It belongs to \(L^1\cap L^\infty\). The Cauchy transform \[V(w)=\frac1\pi\int_{\mathbb C} \frac{Q(\zeta)}{w-\zeta}\,dA(\zeta)\] is bounded. Indeed the contribution from \(|w-\zeta|<1\) is at most \(2\|Q\|_\infty\), and the complement is at most \(\pi^{-1}\|Q\|_1\). Distributionally \(\bar\partial V=Q\); hence \(H-V\) is holomorphic in the exterior sector, has bounded boundary values including the inner arc, and has order at most one.

For completeness the required Phragmén–Lindelöf step follows from the maximum principle. Choose \[1<\rho<\frac{\pi}{\pi-2\epsilon}.\] With sector center \(\theta_0=\delta/2\), the real part of \((e^{-i\theta_0}w)^\rho\) is positive on the closed sector. Multiply \(H-V\) by \(\exp\{-\tau(e^{-i\theta_0}w)^\rho\}\), \(\tau>0\). Its outer-circle values tend to zero as the radius tends to infinity, since \(\rho>1\); its side and inner-arc values are bounded by the original boundary bound. The maximum principle on truncated sectors followed by the limits at infinity and then \(\tau\downarrow0\) shows \(H-V\) bounded. Since \(V\) is bounded, so is \(H\). At \(\arg w=\pi/2\), \(\operatorname{Re}(Aw)=D|w|\sin(\delta/2)/8\), which proves (103). ◻

Proof. Use the same polynomial center and the same free coordinates for the two regular implicit solves. The finite expressions defining \(g_\pm\), the selected inverse minor and all rescalings have only polynomial losses. Apply (90) on their common inner evaluation ball and then subtract the two uniformly contracting normalized equations. It follows that \[ |p_+(s)-p_-(s)| \le C_D|s|^{-M}e^{-(0.7-o(1))A_{\mathrm{lead}}}. \tag{106}\] Their midpoint \(p=(p_++p_-)/2\) lies in the same ball, with room provided by the fixed-\(h\) inclusion proved above. When the data and evaluation parameter are real, choose conjugate contours and prescriptions; uniqueness gives \(p_-=\overline{p_+}\), so the midpoint is real. The estimate itself uses no conjugacy.

The Hessians of the actions have polynomial bounds on the common ball. Taylor expansion from each \(p_\pm\) to \(p\), with (104), gives \[|G_\pm(s,p)-\mathcal F(0)| \le C_K e^{-K/|a|} +C_K e^{-K/|a|}|p-p_\pm| +C_D|s|^{-M}|p-p_\pm|^2.\] Since \(A_{\mathrm{lead}}\le C_A/|a|\) for a fixed \(C_A\), choose \(K>1.4C_A\), so \(K/|a|>1.4A_{\mathrm{lead}}\). The last term has exponent \((1.4-o(1))A_{\mathrm{lead}}\); the fixed powers are absorbed in the margin between \(1.4\) and \(1.25\). Subtracting the two estimates proves (105). The common formal parameter series has real coefficients because the selected formal equations and the polynomial free path are real and formal implicit inversion is unique.

If all nonzero \(c_j\) are positive, changing the sign of real \(s\) does not solve the problem when \(k\) is even. Instead choose the continuous argument with \(\arg a=\pi\), so \(\arg s=\pi/k\). Use the two argument intervals \((-\delta_{\rm use},\pi+\delta_{\rm use})\) and \((\pi-\delta_{\rm use},2\pi+\delta_{\rm use})\) for \(a\), taking \(\arg s=(\arg a)/k\) continuously on both. Their overlap therefore gives the same value of \(s\) near \(\arg s=\pi/k\), not values differing by \(2\pi/k\). Equivalently, all the preceding constructions are made on a fixed local covering chart in the \(s\)-plane; \(a\) only labels its leading scale. The effective opening coefficients are now negative real to leading order. Higher terms need not be real. Strict contour margins persist under these \(o(1)\) changes, and the common-curtailment estimate has only \(1+o(1)\) changes in its Gaussian heights. The flatness-upgrade variable is rotated accordingly. Thus (106) and the same midpoint Taylor estimate hold with complex \(p\). No identification of the two arrays by conjugation, and no reality of their actual subleading parameters, has been assumed.

Finally fix \(D\), all contour margins, and a decay constant in (104) with room to spare. A sufficiently small fixed change of evaluation angle stays inside the sectors and preserves the positive multiplier gain in Lemma 30. The endpoint Gaussian real parts change continuously by a small relative amount. The preceding \(1.4\) quadratic margin therefore still gives exponent \(1.2\). This is the open range of angles available for the later separation of distinct Laurent principal parts. ◻

Proof. For the mean put \(A=a^{-1}\) and \[H(s,t)=\exp\left(-\int_s^t A(u)\,du\right),\qquad F=f_m-q_0v_m .\] Solving the equation for \(v_{m,s}\) backward from its actual exit gives the exact formula \[ v_{m,s}(s)=H(s,T)\psi_m+ \int_s^T H(s,t)A(t)F(t)\,dt,\qquad \int_s^T H(s,t)A(t)\,dt=1-H(s,T). \tag{127}\] In particular the finite kernel does not have mass one at the terminal section. Its missing terminal layer has \(L^2\) size \(O(\sqrt{E(T)})\) when it multiplies bounded data. We keep that layer in (127).

Accretivity gives \(|H(s,t)|\leq\exp(-c\int_s^t E^{-1})\) and \(|A(t)|\leq C/E(t)\). The kernel \(H(s,t)A(t)\) has bounded row and column masses. For the column estimate one uses the increasing \(E\) on the incoming pieces; its variation on the modified bands is bounded. Equivalently, bounded logarithmic derivatives and small \(E_*\) give the same estimate on consecutive strips. Integrating (127) once yields \[\begin{split} v_m(s)={}&\varphi_m+\psi_m\int_0^sH(u,T)\,du +\int_0^s m(t)F(t)\,dt+\int_s^T K_+(s,t)F(t)\,dt,\\ m(t)={}&A(t)\int_0^tH(u,t)\,du,\qquad K_+(s,t)=A(t)\int_0^sH(u,t)\,du . \end{split}\] Here \(|m(t)|\leq C\) and \(|K_+(s,t)|\leq C E(s)E(t)^{-1} \exp(-c\int_s^t E^{-1})\); the latter kernel has row mass \(O(E(s))\). The past integral has the ordinary Volterra Grönwall bound \(\exp(C\int|q_0|)\), which is uniform, and the future \(q_0v_m\) term is absorbed by taking \(E_*\) small. If \(L\) bounds the logarithmic rate of the mean weight, then \[\frac{\omega(t)}{\omega(s)} \leq C_{\rm mod}e^{L(t-s)} \leq C_{\rm mod}\exp\left(LE_*\int_s^t E^{-1}\right).\] Thus \(LE_*<c/2\) preserves a fixed part of the fast exponent in the weighted future and exit kernels. Schur’s inequality for the differentiated formula, followed by the equation for \(a v_{m,ss}\) and the first-trace estimate of 6, proves (125). The two source sums in (124) are respectively its cumulative past part and its fast future tail. The terminal lift has derivative bulk size \(O(\sqrt{E(T)}|\psi_m|)\) and value size \(O(E(T)|\psi_m|)\), so its contribution is precisely the stated natural Neumann term, uniformly after weighting.

For the nonconstant modes choose \(d=(\log\rho_o)_s=-9/10\) on the long incoming pieces away from the reserved collars and \(d=0\) on the bounded modified bands. Ramp \(d\) smoothly between these values only on the fixed \(J=1\) collars, where the unit incoming estimate for \(c\) holds; in particular \(d=0\) before entering each merely accretive band and until after leaving it. Their bounded geometry makes this weight comparable to \((|r|/r_0)^{9/10}\). Writing \(v_o=\rho_o w\) gives \[\rho_o^{-1}\mathcal L_0(\rho_o w) =-aw_{ss}+(1-2ad)w_s-cw_{\theta\theta} +\{d-a(d^2+d_s)+q_0\}w.\] For mean-zero functions the first angular eigenvalue is one. On the long incoming pieces the real part of the mass therefore has the strict buffer \(1-9/10\) after the small \(r_0\) and \(E_*\) errors. On every modified band \(d=0\), and \(\Re c\geq c_*\) supplies a fixed buffer because \(q_0=O(r^2)\). On each ramp collar, \(J=1\) and \(c=1+O(|r|^2(1+|\log r|)+E)\) while \(-9/10\le d\le0\); the bounded \(E d_s\) term is small, so the same fixed incoming buffer persists there. Integration against \(\bar w\) absorbs the normalized coefficient derivatives and imaginary drift terms into \(E|w_s|^2\), \(|w_\theta|^2\) and this positive mass. The actual homogeneous exit is \(w_s+dw=0\); its radial and drift boundary contributions add to \[d\Re a+\tfrac12\Re(1-2ad)=\tfrac12.\] Thus the negative incoming weight causes no terminal loss. Localizing the energy by a smooth weight comparable to \(e^{-\epsilon|s-s_\nu|}\), with \(\epsilon\) smaller than the unused mass buffer, sums the source strips geometrically. The local estimate in 6 then supplies all bulk and first-trace slots. Natural boundary lifts prove (126). Finally the finite Dirichlet–Neumann problem has index zero by its complementing boundary parametrices and an accretive homotopy at fixed \(T\); the estimates prove injectivity and hence solvability. This index calculation is on the ordinary compact \(H^2\) domain and \(L^2\) range; the commuted estimates and density then give the stated fixed angular order. ◻

Proof. For the first assertion take the first turn to end at \(|r|=r_1>0\), with \(r_1\) fixed below \(r_0\). Its logarithmic length is bounded and \(E\asymp B^{-2}\). On its terminal band the preliminary comparison is circular. In relative-height units it is \[g_R=(Z_0/Z)\phi,\qquad g_c-g_R=(1-Z_0/Z)\phi+Z^{-2}d_0.\] The last difference and every fixed logarithmic derivative are \(O(B^{-1.8})\) on this fixed-radius interval. In particular the known smooth comparison derivative changes by \(O(B^{-1.8})\) in \(H^{h+1}\), and its natural lift has size \(O(B^{-2.8})\) by (129). This assertion concerns the smooth comparison profiles, not an unknown solution trace. After the fixed graph-unit conversion, division by the preliminary weight \(\rho_B(T)\asymp B^{2.2}\) gives \(O(B^{-5})\), inside (85) throughout this interpolation. For each fixed path, \(Z\) is the bounded mean trace of the preliminary holomorphic solution at the fixed incoming section. The chosen branch of \(Z_0\) and the formulas for \(g_R\) and \(g_c\) are therefore holomorphic in \(p\) on the outer ball. Their comparison derivatives, evaluated at the prescribed real path parameter at the exit, are holomorphic in the identified trace space, as is their affine interpolation. This verifies the parameter-data condition in 23.

Apply 33 on this bounded interval with the constant total budget \(W_{\rm fix}=B^{-1.8}\). The incoming oscillation is \(O(W_{\rm fix})\), its mean is zero, the reference residual is \(O(B^{-4})\), and the preceding boundary lift is \(O(B^{-2.8})\). Here \(E^{-1/2}|r|^{-2}\asymp B\), so the full nonlinear remainder and Lipschitz factor are \[O(BW_{\rm fix}^2+B^2W_{\rm fix}^3),\qquad O(BW_{\rm fix}+B^2W_{\rm fix}^2)=O(B^{-0.8}).\] There is consequently a unique fixed-interval solution in \(\sup_I X_E^h(u;I)\leq C_{\rm fix}B^{-1.8}\) for a fixed \(C_{\rm fix}\). Project its equation onto the mean and use the zero incoming mean. The same inverse and the projected products give the stronger bound \[\sup_I X_E^h(u_m;I) \leq C\{B^{-4}+B^{-2.8} +BW_{\rm fix}^2+B^2W_{\rm fix}^3\} =O(B^{-2.6}).\] At fixed \(r\), \(W_o\asymp B^{-7/4}\) and \(W_m\asymp B^{-2.1}\). Thus its oscillation and mean lie in the refined ball for large \(B\). It has the same entrance and exit data as the canonical refined exterior, so uniqueness in (139) identifies those two exterior solves. The normalized correction (145) preserves this membership.

For clarity, the preliminary ball here is the growth-weighted \(X_B\) ball of radius \(C_{\rm pre}B^{-4}\) about the comparison pair of 23, after its prescribed boundary lift. The weight on this fixed outer band is comparable to \(B^{2.2}\), so \(B^{-1.8}/B^{2.2}=B^{-4}\). The core and conical parts equal those of \(V_{\rm pre}\) before the new normalized correction. There \(V_{\rm pre}\) differs from the shared preliminary comparison pair by \(O(B^{-4})\) by 23; (145) controls the additional discrepancy and the multiplier. Choose \(C_{\rm pre}\) larger than these fixed bounds before taking \(B\) large. The preliminary contraction remains a contraction on this fixed enlarged ball because its outer Lipschitz factor is \(O(C_{\rm pre}B^{-0.8})\). The new normalized solution and the preliminary solution solve the same augmented equation, observations and exit problem inside that ball. Its uniqueness identifies them. This proves the first assertion and connects the fixed-radius end of the array homotopy to the refined family.

For the second assertion, the natural trace theorem applied on an enlarged strip of each longer canonical solution gives \[\|\psi_{L,m}\|_{\mathcal T^N}\leq C W_m(T),\qquad \|\psi_{L,o}\|_{\mathcal T^N}\leq C W_o(T).\] Its normalized correction adds only \(B^M e^{-c_{\rm sh}B^2}\) in these norms. The earlier choice of \(C_{\rm tr}\) includes these two canonical trace families for large \(B\). Thus (133) holds for their actual traces and for their linear interpolations. For fixed paths and shorter sections, the longer zero-remainder-exit solutions are holomorphic in \(p\) by the preceding contractions. After the fixed real pullbacks, restriction and the natural trace map are bounded complex-linear maps into the identified shorter trace space. Thus \(\psi_-(p)\), \(\psi_+(p)\), and their affine interpolation are holomorphic there. Variation of the contour or shorter section uses the separately proved smooth pulled-domain derivatives.

Let \(g_{\rm short}^{\rm can}\) be the shorter canonical exterior with entrance \(g_{\rm pre}\) and datum \(\psi_L\). The restriction \(g_L\) satisfies the same exterior equation because the compact multiplier sources vanish on this height region. It has the same actual exit derivative, but its entrance differs by \[e_0=(g_L-g_{\rm pre})|_{s=0},\qquad \|e_0\|_{\mathcal T^D}\leq CB^M e^{-c_{\rm sh}B^2},\] by (145) and the trace estimate. Both exteriors lie in twice the refined ball. Applying (141) to their mean-value difference, with zero exit discrepancy and this nonzero entrance discrepancy, gives \[ \|g_L-g_{\rm short}^{\rm can}\|_{\mathcal X} \leq CB^{M'}e^{-c_{\rm sh}B^2}. \tag{146}\] Dividing by \(W_m(0)\) and \(W_o(0)\) has only increased the fixed power. No zero-entrance shielding has been applied to this difference.

On the core the longer restriction differs from \(V_{\rm pre}\) by its correction (145), including \(\lambda_L-\lambda_{\rm pre}\). On the shorter exterior it differs from \(g_{\rm short}^{\rm can}\) by (146). On the joining band the remaining comparison between \(g_{\rm short}^{\rm can}\) and \(g_{\rm pre}\) has zero entrance and is exactly the canonical comparison to which (143) applies. Combining these three estimates in the prescribed charts and in \(X_B\) costs only another fixed power, and hence \[\|(V_L|_{\rm short},\lambda_L) -(V_{\rm app,short},\lambda_{\rm pre})\|_{X_B} \leq CB^{M''}e^{-c_{\rm sh}B^2} <e^{-c_{\rm gl}B^2}.\] The restriction is therefore in the shorter augmented ball. It has the same PDE, observation values and outgoing derivative as the normalized shorter solution; uniqueness in (144) identifies them.

Finally put \(\psi_\xi=(1-\xi)\psi_-+\xi\psi_+\) for \(0\leq\xi\leq1\), simultaneously on all adverse ends. The natural trace balls in (133) are convex. The exterior contraction and augmented correction therefore give a smooth normalized family with these data. Differentiating the exterior contraction in \(\xi\) and using the lift of \(\psi_+-\psi_-\) gives a uniform bound for \(\partial_\xi u\) in the refined norm; its entrance derivative is zero. Shielding this canonical derivative on the fixed joining band and then differentiating the augmented correction gives (145) with a larger fixed power. At a fixed section \(y=Y\), conversion to the scaled height gives \[ \|\partial_\xi h(Y,\cdot)\|_{H^h} \leq C\{B^{-\varepsilon}+Y^{-\sigma} +B^{-13/20}Y^{9/10}\} +B^M e^{-c_{\rm sh}B^2}. \tag{147}\] Indeed \(h=(Z_*-Zg)Z_*/2\) and \(|ZZ_*|\asymp B^2\), so this is exactly the refined derivative bound multiplied by \(B^2\). The endpoint identification just proved makes this the required family between the two restrictions. ◻

Proof. Near any chosen \(\eta\), pull neighboring paths to its fixed real parameter interval by maps \(\Psi_\eta\) which are the identity before the last bounded bands. Their derivatives in the path coordinate and in \(\eta\) are bounded in logarithmic units. This is possible for the homothetic bypasses and their preceding shortening bands. First let \(g_\eta\) denote the canonical exterior before normalization, and put \(u_\eta=(g_\eta-g_c)\circ\Psi_\eta\) and use dots for derivatives at a fixed pulled coordinate. The exact remainder problem has the form \[\mathcal L_\eta u_\eta+\mathcal N_\eta(u_\eta)=-R_\eta,\qquad u_\eta(0)=u_{\rm in},\qquad (u_\eta)_s(T)=0.\] The incoming value is independent of \(\eta\). The pulled exit condition is still homogeneous because it is a coordinate derivative of the remainder. The locally parameterized normalized branches are smooth by the uniform contractions for these smooth coefficient maps. They agree on overlaps by uniqueness in (144), so they describe one smooth shortening family. Differentiation therefore gives \[ \begin{split} \{\mathcal L_\eta+D\mathcal N_\eta(u_\eta)\}\dot u &=-\dot R_\eta-(\dot{\mathcal L}_\eta)u_\eta -(\partial_\eta\mathcal N_\eta)(u_\eta),\\ \dot u(0)&=0,\qquad \dot u_s(T)=0 . \end{split} \tag{149}\] No second-derivative boundary trace appears.

The explicit coefficient variations are supported on the last bounded bands. Differentiating the full pulled drift and then dividing it as above gives, schematically, the source slots \[O(1)u_s,\quad O(E)u_{ss},\quad O(1)u_{\theta\theta}, \quad O(\sqrt E)u_{s\theta},\quad O(1)(u,u_\theta).\] Their coefficients are bounded by a fixed polynomial in \(Y\), with no independent power of \(B\). These are precisely the normalized bulk slots; a background Hessian stays in its bulk factor. The circular coefficient variations preserve the mean and nonconstant projections. Moreover \(\dot R_\eta=O(B^{-2}|Br|^{-2})\) up to such a polynomial, and the differentiated nonlinear coefficients obey the projected products in (134).

Apply the two parts of 33 to (149) and absorb \(D\mathcal N_\eta\) with (138). The explicit mean sources occupy only a bounded number of terminal strips, so their cumulative term in (124) contributes no factor \(\log B\). The oscillatory estimate uses its strict angular buffer. It follows that \[ X_E^h(\dot u_m;I)\leq P(Y)W_m(I),\qquad X_E^h(\dot u_o;I)\leq P(Y)W_o(I). \tag{150}\]

At the exit \(|r|\asymp Y/B\) and \(|\dot r|\leq C|r|\). Since \(Z\) is fixed, \[|Z\,\partial_\eta(g_c\circ\Psi_\eta)| \leq C\{B|r|^2+B^{-1}(1+|r|^2|\log r|)\} \leq B^{-1}P(Y).\] The first inequality uses \(r\partial_r\phi=O(r^2)\) and \(r\partial_rd_0=O(1+r^2|\log r|)\). The first traces in (150) give \[|Z\dot u|\leq CBP(Y)(W_m+W_o) \leq B^{-1}P(Y),\] because, up to fixed comparison constants, \(BW_m\leq CB^{-1}(B^{-\varepsilon}+Y^{-\sigma})\) and \(BW_o\leq CB^{1-\alpha-\gamma}Y^\gamma =CB^{-33/20}Y^{9/10}\). A derivative trace used for a slope can cost an additional factor \(E^{-1/2}\asymp Y\), which is also included in \(P\). Together with \(|\dot r|\leq CB^{-1}Y\), these are the desired physical scales for the canonical exterior.

It remains to check the global correction. On the fixed overlap before the support of the pulled coefficient variations, the canonical derivative \(\partial_\eta(g_\eta-g_{\rm pre})\) has zero entrance value and satisfies the homogeneous linearized equation. Its outer size from (150) is at most a fixed power of \(B\), since \(Y\leq cB\). Applying 21 to this zero-entrance derivative gives \(B^M e^{-c_{\rm sh}B^2}\) on the joining band. Thus the derivative of the spliced residual has that size. This step uses the canonical local derivative, not the nonzero-entrance restriction difference in (146).

Differentiate the augmented correction equation at fixed observations and with homogeneous pulled derivative data for the correction. The unknown includes \(\dot\ell\), the multiplier derivative. Its inverse and the coefficient derivatives cost only fixed powers in the global norms, while the differentiated residual and every term containing the correction have the exponent in (145). Hence, after increasing \(M\), \[\|(\dot k,\dot\ell)\|_{X_B} +\|\partial_\eta X_{k,{\rm e}}\|_{H^h} \leq CB^M e^{-c_{\rm sh}B^2}\leq B^{-1}.\] The last inequality is uniform for \(Y\in[Y_0,cB]\). Adding this correction proves (148). ◻

Proof. At \(r=2y/Z_*\), the oscillatory height error is bounded by \[CB^2W_o\leq CB^{2-\alpha-\gamma}|y|^\gamma =CB^{-13/20}|y|^{9/10},\] and the mean error by \(C(B^{-\varepsilon}+|y|^{-\sigma})\). Insert (112) into (115). The constant terms cancel because of the definition of \(Z_*\), giving \[ h-(y^2/2-\log y) =O(|y|^{-\sigma})+o_B(1) \tag{154}\] for fixed \(y\). The derivative traces may cost a further factor \(|y|\), which is harmless for the relative slope in (153). Fixed higher spatial regularity follows by differentiating the equation and the strip estimates; on each fixed exterior annulus the scaled coefficients are uniformly elliptic. The interpolation estimates in 34 give the same conclusion for the interpolated problems.

One can also identify the limiting equation before taking a limit. With \(D_yh=(h_y,h_\theta/y)\), substitution of (113) into (119) gives \[ \left(I-\frac{D_yh\otimes D_yh}{1+D_yh\cdot D_yh}\right) :D_y^2h =1+\frac{2(yh_y-h)}{Z_*^2}, \tag{155}\] where \(D_y^2h\) is the polar Euclidean Hessian. Oscillations vanish by the preceding bound. Letting \(B\to\infty\) in (155) gives \(h''/(1+(h')^2)+h'/y=1\), namely (152). This uses only regions \(|y|\geq Y_0\).

For the endpoint estimate retain the full \(\phi\) in (115). At a complex exit of size \(Y\asymp|Br|\), its negative \(\Re(Z^2r^2)\) and the identity \(\phi^2=1-r^2/2\) give \[ \Re(z^2-Z_*^2)\geq cY^2-C(1+\log Y). \tag{156}\] Here the logarithmic correction is in \(\log(Br)\), not \(\log B\): (113) subtracts exactly the latter constant. The residual errors in (140) are bounded in scaled height by \(C(1+Y^\gamma)\), uniformly in \(B\), and so can be absorbed in the right side of (156) for large \(Y_0\). The quadratic errors from replacing \(Z\) by \(Z_*\) are smaller still. On the initial fixed-radius pieces any polynomial in \(B\) is absorbed by the extra \(e^{-cB^2}\).

By the first part of 34, the canonical normalized family at fixed radius is the same array family after the preliminary deformation and comparison-data interpolation. That initial deformation costs a fixed power of \(B\) times \(e^{-\Re Z_0^2/4-cB^2}\). The relations \(Z/Z_0=1+O(B^{-1.8})\) and (113) give \(Z_*^2-Z_0^2=o(B^2)\), so this cost is negligible in the scale \(B^{-2}e^{-\Re Z_*^2/4}\). The earlier synchronization contributes (108). The fixed choice of \(D\) makes this \(o(B^{-2}e^{-\Re Z_{*,j}^2/4})\) for every adverse \(j\); at each fixed \(Y_0\) it is absorbed in (151) for large \(B\).

Along the subsequent shortening, (148) bounds the physical boundary displacement by \(B^{-1}P(Y)\). The boundary circumference is \(O(B^{-1}Y)\), and slopes, conormals and their continued area branches have at most polynomial size in \(Y\). Indeed the first traces give \(z_r=O(Y+1+Y^\gamma)\) and \(z_\theta/r=O(B^{-13/20}Y^{-1/10})\) at the exit, while the relative slope smallness in 19 keeps the continued denominators in the same branches as the circular profile. Thus both the displacement and momentum terms in first variation are bounded by \[CB^{-2}P(Y)\exp(-\Re Z_*^2/4-cY^2)\] per unit logarithmic size, for a fixed polynomial \(P\). The observation terms vanish because \(p\) is fixed. Integrating over \(Y_0\leq Y\leq CB\) and decreasing \(c\) absorbs \(P\) and proves (151). The nonadverse endpoints stay at \(\Re z^2=DB^2\). Their derivative prescriptions and paths are fixed, but their boundary heights can vary through the normalized correction; their fixed-exit momentum terms remain in the first variation and are bounded by \(C_D B^M e^{-DB^2/5}\) using its polynomial derivative bounds. The same choice of \(D\) absorbs them in the asserted scale. This proves all claims, with \(B\to\infty\) at fixed \(Y_0\) before \(Y_0\) is increased. ◻

Proof. Put \(\tau=y^2/2\) and \(q=1-l/y\). The exact equation is \[ q_\tau+q=2q^2-q^3+\frac{1-2q}{2\tau}. \tag{158}\] Choose \(\alpha_0>0\) and \(2\pi/3<\vartheta<\pi\) with \(\vartheta+\alpha_0<\pi\), and put \(v=e^{i\vartheta}\). On the translated cone \[\Omega_+(R)= \{a e^{-i\alpha_0}+bv:a>R,\ b>0\}\] the rays \(\tau+tv\), \(t\geq0\), remain in the cone and \(|\tau+tv|\geq c(|\tau|+t)\). As \(-\Re v>0\), the map \[ (\mathcal Tq)(\tau)=-v\int_0^\infty e^{tv} \left(2q(\tau+tv)^2-q(\tau+tv)^3+ \frac{1-2q(\tau+tv)}{2(\tau+tv)}\right)dt \tag{159}\] is a contraction on a fixed sufficiently large ball in \(\sup|\tau q(\tau)|\) when \(R\) is large. Indeed its inhomogeneous part is \(O(|\tau|^{-1})\), and its derivative in \(q\) is \(O(R^{-1})\) there. Its holomorphic fixed point satisfies (158), by differentiating under the integral and integrating once in \(t\). It covers every closed angular subsector of \((-\alpha_0,\vartheta)\) at sufficiently large radius. Reflection gives \(q_-\) below. Uniqueness in these classes is also immediate: the difference of two small solutions satisfies \(\delta'=(-1+O(\tau^{-1}))\delta\), and a nonzero such difference grows exponentially on the ray toward upper-left infinity. The same argument holds in the lower sector.

Here is an explicit factorial estimate for their common formal series. Set \(x=y^{-2}=(2\tau)^{-1}\) and \(D=x\partial_x\). Equation (158) is formally equivalent to \[ q=x+x\frac{2Dq-q^2}{(1-q)^2} =x+x\sum_{k\geq1}(2D-k+1)q^k. \tag{160}\] If \(q=\sum_{n\geq1}b_nx^n\), then \(b_1=1\) and \[ b_{n+1}=\sum_{k=1}^n(2n-k+1) \sum_{\substack{i_1+\cdots+i_k=n\\i_j\geq1}} \prod_{j=1}^k b_{i_j}. \tag{161}\] Every summand is positive. The \(k=1\) term and induction give the lower estimate in \[ 2^{n-1}(n-1)!\leq b_n\leq3^{n-1}(n-1)!. \tag{162}\] For the upper induction, \(\prod(i_j-1)!\leq(n-k)!\) and there are \(\binom{n-1}{k-1}\) positive compositions. Hence \[b_{n+1}\leq 2\,3^{n-1}n!\sum_{j=0}^{n-1}\frac{3^{-j}}{j!} \leq 2e^{1/3}3^{n-1}n!<3^n n!,\] which proves (162).

We next verify the corresponding remainder estimate, rather than infer it from coefficient growth. Write \(a_j=b_j/2^j\), \(P_N(\tau)=\sum_{j<N}a_j\tau^{-j}\), and \[D_N=P_N'+P_N-2P_N^2+P_N^3- \frac{1-2P_N}{2\tau}.\] As a polynomial in \(\zeta=1/\tau\), \(D_N\) vanishes to order \(N\). On \(|\zeta|=(LN)^{-1}\), with \(L\) fixed and large, (162) gives \(|P_N|+|\zeta^2\partial_\zeta P_N|\leq C/N\). The maximum principle applied after division by \(\zeta^N\) therefore gives \[|D_N(\tau)|\leq (C/N)(LN)^N|\tau|^{-N} \leq C A^N N!|\tau|^{-N} \quad (|\tau|\geq LN).\] The exact error \(e_N=q_+-P_N\) obeys \[e_N'+(1-\mathcal B)e_N=-D_N, \quad \mathcal B=2(q_++P_N)-(q_+^2+q_+P_N+P_N^2)-1/\tau.\] For \(|\tau|\geq L'N\), choose \(L'\) so large that along the ray in (159) one has \(\mathcal B=O(|\tau+tv|^{-1})\). The integrating kernel is bounded by \(e^{-ct}(1+t/|\tau|)^C\). Also \(|\tau|^N/|\tau+tv|^N\leq \exp(Nt/(c|\tau|))\). The latter is absorbed by the exponential when \(L'\) is large. Variation from infinity proves \[ |q_\pm(\tau)-P_N(\tau)| \leq C A^N N!|\tau|^{-N},\qquad |\tau|\geq L'N, \tag{163}\] uniformly on every closed smaller sector. The boundary term at infinity vanishes because the error is \(O(|\tau|^{-1})\) there.

Suppose the lateral solutions agreed. They would then give a single holomorphic \(q\) on an exterior sector of opening greater than \(\pi\), with (163). Choose \(\pi/2<\omega<\vartheta\) and let \(\Gamma_R\) run inward from infinity on angle \(-\omega\), counterclockwise along the right circle \(|\tau|=R\) to angle \(\omega\), then outward to infinity. For \(|\arg u|<\omega-\pi/2\) set \[ H(u)=\frac1{2\pi i}\int_{\Gamma_R} e^{\tau u}q(\tau)\,d\tau. \tag{164}\] The integral converges, is independent of sufficiently large \(R\), and is holomorphic near every positive \(u\). Move \(R\) to \(KN\), with \(K\) larger than the threshold in (163). Each monomial integral, closed through the left half-plane, is its residue \(u^{j-1}/(j-1)!\). The remainder on the arc and rays is at most \[C A^N N!(KN)^{1-N}e^{KNu} \leq CKN(A/K)^Ne^{KNu}\] for positive \(u\). First take \(K\) large, then \(u>0\) small. This tends to zero. Consequently (164) equals the Borel series \[ \widehat q(u)=\sum_{n\geq1} \frac{a_n}{(n-1)!}u^{n-1} \tag{165}\] near the positive origin and continues it analytically along the entire positive axis.

This is impossible by the positive-coefficient singularity principle known as Pringsheim’s theorem (Flajolet and Sedgewick 2009, Theorem IV.6 and Note IV.13), whose short argument we give here. The coefficients in (165) are positive, and (162) places its convergence radius \(R_H\) in \([2/3,1]\). If it extended holomorphically through \(R_H\), choose a positive \(r<R_H\) sufficiently close that its Taylor disk extends to a positive \(b>R_H\). All Taylor coefficients at \(r\) are nonnegative, being termwise derivatives of (165). Exchanging the resulting nonnegative double sums at \(b\) would show that the original series converges at \(b\), contradicting the definition of \(R_H\). Thus the two lateral solutions are distinct.

Finally their slope difference satisfies the exact linear equation \[ (l_+-l_-)'= \left(l_++l_--\frac1y- \frac{l_+^2+l_+l_-+l_-^2}{y}\right)(l_+-l_-). \tag{166}\] The common expansion is \(l=y-y^{-1}-2y^{-3}+O(y^{-5})\), so the coefficient is \(-y+3/y+O(y^{-3})\). Its integrable remainder gives (157); the constant is nonzero because a nonzero solution of this scalar linear equation never vanishes.

For a finite bypass compare its slope with the canonical lateral slope at the exit. Their difference there is at most polynomial in \(Y\). Along a path of length \(O(Y)\) with \(|y|\asymp Y\), the coefficient in their exact difference equation is \(-y+o(Y)\), by (153). Integrating back to \(y=Y\) therefore gives the factor \[\exp\left(-Y^2/2+\Re y_{\rm e}^{2}/2+o(Y^2)\right).\] By (118) this is bounded by \(\exp(-Y^2/2-c'Y^2)\). Polynomial factors are absorbed by decreasing \(c'\). This proves the final assertion and applies to every subsequential exterior limit. ◻

Proof of Proposition 32. Lemma 36, including the preliminary synchronization (108) and the retained far boundary momenta, changes the original actions to the ones used in (169), at a cost \(O(e^{-cY_0^2})\) in the sum of absolute end scales. At fixed \(Y_0\), (116) and the normalized homotopy give the sum of the limiting comparisons, with errors \(o_B(1)\) in each scale. Lemma 37 and (169) identify the same constant on every end as \(Y_0\to\infty\). There are only finitely many ends, so these errors sum with respect to their absolute scales. The order of limits is therefore the one stated.

There is no end-dependent orientation sign. On every end \(z\) increases outward along its chosen ray, the circular radius decreases toward the collapsing profile, and the branch of area starts positive. The change to (116) therefore always reverses the radial integration limits in the same way. Reversing a normal, or rotating an end’s axis in ambient space, changes none of these area conventions. With one common ordering of upper and lower bypasses, the same \(\epsilon_{\rm lab}c_0\) occurs throughout. This proves the proposition. The separate phase comparison of the \(Z_{*,j}\) will determine which terms can dominate the sum; no assertion of cancellation or noncancellation for arbitrary complex phases is built into (117). ◻

Proof. Choose a fixed collar containing the entrances to all ends and lying outside \(K_0\), where \(S_b\) is an exact shrinker. Include this collar and the observation core in a compact reference region \(K\). Since the conclusion concerns \(R\to\infty\), increase the lower radius threshold to \(R_{\eta,K}\) in Proposition 44; its cutoff \(\chi_R\) is then one on both regions. The compact estimate and (206) give exponentially small compact error and multiplier. Interpolation with the assumed derivatives makes the height and slope exponentially small on the entrance collar.

We first obtain pointwise control of the forcing on a curtailed ball. Fix \(\epsilon>0\), and let \(q\) have radius at most \((1-\epsilon)R\). On a uniform small geometric patch of radius \(\rho\) around \(q\), contained in \(B_{(1-\epsilon/2)R}\) for large \(R\), (211) implies \[ N_q:=\|\Phi\|_{L^2(\text{patch})} \le C\exp\left(-\frac{1-(1-\epsilon/2)^2}{8}R^2\right). \tag{212}\] A bound for \(A\) and \(\nabla A\) gives \(\|\nabla\Phi\|_\infty\le L_R\le C(1+R)^k\): differentiating \(X^\perp\) introduces at most one position factor. Put \(A_q=|\Phi(q)|\). A disk of radius comparable to \(\min\{\rho,A_q/(2L_R)\}\) has \(|\Phi|\ge A_q/2\). The uniform two-dimensional area lower bound on these graphical patches gives \[ A_q\le C\bigl(L_R^{1/2}N_q^{1/2}+\rho^{-1}N_q\bigr). \tag{213}\] When \(L_R=0\) only the second term is needed. Polynomial factors can be absorbed in a smaller positive quadratic exponent. Thus \(|\Phi|\le Ce^{-cR^2}\) on the curtailed ball. The exponent \(1/8\) in (212) is the Gaussian norm exponent; replacing it by the energy exponent \(1/4\) would be incorrect.

Let \(u\) be the actual normal height over an exterior portion of \(S_b\). Its equation, divided by the normal component of the graphical displacement, has the form \[ \begin{gathered} \mathcal L_u u=f,\qquad \mathcal L_u=a^{ij}\nabla_{ij} +\bigl(-\tfrac12Y^\top+\mathfrak b\bigr)\cdot\nabla+c,\\ |\mathfrak b|\le o(1)(1+|Y|)+C,\qquad |c|\le C,\qquad |f|\le C|\Phi|. \end{gathered} \tag{214}\] The second-order part is uniformly elliptic and close to the base metric. The bounded zeroth-order coefficient deserves verification. With the convention \(D\nu=-A\), the unnormalized graph normal is \(\nu-(I-uA)^{-1}\nabla u\). Therefore the support-function term, after this division, is exactly \[ \tfrac12\left(Y\cdot\nu+u -Y^\top\cdot(I-uA)^{-1}\nabla u\right). \tag{215}\] Keep its entire transport expression in the first-order part. In particular, do not differentiate its coefficient in \(u\) and put \(|Y|\,|\nabla u|\) into a purported bounded potential. In the curvature expression, keep the actual coefficients on \(\nabla^2u\) and expand the remaining smooth terms about \(u=0\). Their coefficients depend on the bounded base geometry and the small first graph variables, and are bounded. The base equation vanishes on this annulus. This proves (214) with no compact multiplier forcing.

Write \(r=|Y|\). Uniformly on the far ends, \(|Y^\perp|\le C\), so \(|\nabla r|^2=1+O(r^{-2})\) and \(\nabla^2r=O(r^{-1})\). After choosing the weak tolerance small, a sufficiently large fixed \(p\) and a sufficiently small fixed \(\alpha_0>0\) give \[ \mathcal L_u r^p\le-c_1r^p, \qquad \mathcal L_u e^{\alpha_0r^2} \le-c_2r^2e^{\alpha_0r^2} \tag{216}\] past a fixed entrance. For the first inequality the leading coefficient is \(-p/2+c+o(p)+O(p^2/r^2)\); for the second it is \((4\alpha_0^2(1+o(1))-\alpha_0+o(\alpha_0))r^2+O(1)\). Take, for example, \(\alpha_0<1/4\) with a strict margin, then decrease the graph tolerance and enlarge the entrance. The comparison principle applies despite the possible positivity of \(c\): dividing a tested function by the positive supersolution \(r^p\) gives a strictly negative zeroth-order coefficient.

On an annulus whose outer radius is \(R_1=(1-\epsilon)R\), inner data and \(f\) are exponentially small, while outer data are bounded by the weak height tolerance. The function \[V(Y)=C e^{-cR^2}r^p +C\varepsilon_{\rm weak} e^{\alpha_0(r^2-R_1^2)}\] can, by increasing its first constant, be arranged to dominate \(|u|\) on the two boundaries and to satisfy \(\mathcal L_uV\le-|f|\). Applying comparison to \(V-u\) and \(V+u\) gives \(|u|\le V\). A further fixed fractional curtailment makes the second summand exponentially small. The first stays exponentially small after multiplication by its fixed polynomial. Interpolation with the available derivative bounds gives the same conclusion for \(\nabla u\). Allocate the two curtailments and the small collar margins inside the prescribed \(\zeta\). ◻

Proof. The geometric bounds used here follow uniformly from the finite parameter family and its end asymptotics. Shrinking factors in \([\sqrt{1-U},1]\) preserve bounded geometry and a positive tubular radius. Although the raw homothetic parametrization has large tangential velocity at a distant point, its normal velocity is bounded uniformly, since \(|Y^\perp|\) is bounded on \(S_b\). Thus, after tangential reparametrization, comparison motion costs only a bounded additive spatial buffer on \([0,U]\).

The tubular bound here includes extrinsic separation. On the compact core and its collars, proper embeddedness and small smooth variation in the closed parameter family give a common tubular neighborhood. On each cylindrical tail the uniformly decaying graph over its rotated round cylinder retains a fixed tubular radius. On a conical tail, rescaled compact annuli are small smooth graphs over annuli of an embedded cone; the embedded link and its small variations have uniformly separated nonadjacent pieces. Scaling back gives a tubular radius bounded below by a fixed multiple of the distance from the origin. The separated end links and directions exclude close pairs from distinct tails. The remaining transition region is compact. These observations give one \(\iota>0\) for the complete family, preserved up to a fixed factor by the shrinking motion on \([0,U]\). Thus curvature bounds alone are not being used to exclude a second reference sheet from an entrance cylinder.

We first describe the compactness class of possible comparisons. If \(b_i\) stays in the closed parameter neighborhood and \(\lambda(b_i,0)\to0\), a subsequence has \(b_i\to b_\infty\). At bounded observation centers the limit \(S_{b_\infty}\) is a globally stationary shrinker, because its only possible defect was the compact source and its multiplier now vanishes. At centers going to infinity down a cylindrical end, translation gives a round cylinder, whose radius at time \(t\) is \(\sqrt{2(1-t)}\). At centers going to infinity down a conical end, translation gives a plane, stationary under the limiting motion. The smooth end estimates and separation of the finitely many ends give these limits on every fixed ball, in all fixed orders; another end cannot enter a bounded translated patch. The bounded-center models are complete shrinkers with bounded geometry, and all these models have smooth uniformly controlled motions on \([0,U]\). No assertion that \(b_i\to0\) has been used.

Fix a small INS output slope \(\upsilon<1/100\), obtain its entrance constants \(\varepsilon,\delta\in(0,1)\), and choose \(r<\iota/100\) so that every radius-\(4r\) reference patch is a graph of slope less than \(\varepsilon/4\). Set \(s=\delta r\). Choose \(\rho<s/100\), then choose \(h>0\) so that \[ h<s^2/8,\qquad Vh<\rho/8,\qquad 4h<\rho^2/32, \tag{218}\] and reference tangent variation during \(h\) is small. Here \(V\) bounds reference normal speed. All choices precede the small initial-error threshold, which in particular is less than \(\rho/8\). The number \(J\le\lceil U/h\rceil+1\) of slabs is fixed.

For \(y\) on the reference at the start of a slab, center the entrance cylinder at the actual point \(x=y+u(y)\nu(y)\), with horizontal plane parallel to \(T_yS_b\). On the larger reference patch, horizontal projection of the initial normal graph is a small \(C^1\) perturbation of the identity. Its restriction to a larger horizontal disk is injective and, by its boundary margin, covers the entire radius-\(r\) disk about \(x\). Its height stays inside the vertical cylinder. Uniform reach excludes a different reference patch; the complete initial graph and the outer data buffer exclude any other actual portion. Thus the whole entrance intersection satisfies INS. Apply it at every such actual center and retain half of each radius-\(s\) output cylinder.

Reserve a fixed additive data buffer, including an extra collar containing these retained cylinders. If \(p\) in this buffered region satisfies \(\operatorname{dist}(p,S_b(t))\ge\rho\) at \(t\le h\), then its distance from the initial reference is at least \(\rho-Vh\). Its ball \(B_{\rho/2}(p)\) is initially disjoint from the entire actual surface. Avoidance with a shrinking sphere, whose squared radius is \(\rho^2/4-4t\), excludes \(p\) from the actual flow. Every actual point is therefore in the fresh \(\rho\)-tube. If \(q\) is its nearest reference point, follow the reference normal parametrization back to \(y\), and choose \(x\) as above. Then \[ |p-x|\le |p-q|+|q-y|+|y-x| <\rho+Vh+\rho/8<s/4. \tag{219}\] Every actual point consequently lies in an interior retained output.

Normal projection has exactly one preimage on each reference fiber. To prove existence, fix \(q\) and its cylinder as above. Write its whole output graph as \(z=f(v)\) on the full horizontal disk \(B_s\). Since \(|q-x|\le Vh+\rho/8<\rho/4\), the disk used next is strictly inside \(B_s\). On the disk \(B_{s/2}(q_H)\), the horizontal coordinate \(G(v)\) of the normal projection of \((v,f(v))\) differs from \(v\) by at most \(\rho\), by confinement and reach. Its boundary therefore avoids \(q_H\), and the homotopy \(v+\sigma(G(v)-v)\) has degree one about \(q_H\). The projected image contains \(q\), proving existence. All the disks and their graphical lifts lie in the buffered region. For uniqueness, two points on that fiber are in the same retained cylinder by (219). If their normal parameters differ by \(a\), their vertical separation is at least \(\sqrt{1-\omega^2}|a|\) and their horizontal separation is at most \(\omega|a|\), where \(\omega\ll1\) is the reference normal’s tilt. The actual graph instead bounds the vertical separation by \(\upsilon\omega|a|\), forcing \(a=0\). The same small slopes make normal projection a local diffeomorphism. Full-disk coverage thus gives existence, and the slope bound gives uniqueness. The local graphs agree on overlaps and cover artificial patch edges. Notice that no inequality \(\upsilon s<\rho\) was assumed: avoidance supplies the sharper normal height independently of the INS height bound.

We verify the vanishing error modulus also at elapsed times tending to zero. Suppose a sequence with initial graph error, compact multiplier, and inverse data radius tending to zero has graph error at least \(\eta>0\) at times \(t_i\). Let \(K\) bound reference curvature and \(C_0\) the normal-chart conversion constants. For the case \(t_i\to0\), make the following fresh choices for this target \(\eta\). First choose an arbitrarily small INS output slope \(\upsilon\), then its constants \(\varepsilon,\delta\). Next choose \(r\le1\) below the uniform reach scale so that the reference entrance slope is below \(\varepsilon/4\) and its tangent variation \(Kr\) is as small as required. Put \(s=\delta r\), choose \[\rho<\min\{\iota/100,s/100,\upsilon s/100,\eta/(100C_0)\},\] and only then choose \(h_\eta>0\) satisfying (218) and making the uniform reference tangent modulus \(\omega_{\rm ref}(h_\eta)\) small. Arrange \[C_0\bigl(\upsilon+Kr+\omega_{\rm ref}(h_\eta)\bigr)<\eta/2.\] Only after these choices take \(i\) large enough that the initial error meets all entrance and \(\rho/8\) margins, the additive buffer lies inside the data region, and \(t_i<h_\eta\).

Repeat the shrinking-sphere confinement at this new \(\rho\), rather than using the earlier, wider tube. Equation (219) now places every actual point in a refined output cylinder, including the purported point of failure. Its height over the reference is less than \(\rho\), and its slope in the normal chart is bounded by \(C_0(\upsilon+Kr+\omega_{\rm ref}(h_\eta))\). Both are smaller than the alleged error threshold. This excludes \(t_i\to0\). No relation between \(\delta\) and \(\upsilon\) was needed, and no patch radius was chosen as a function of the failing time.

If instead \(\liminf t_i>0\), recenter at the points of failure. The data domains exhaust, and the complete graphs, area bound, and (217) give a smooth limiting motion for positive time. The integrable curvature-speed bound gives its initial height continuously. The limiting comparison is an exact smooth mean curvature flow: the base multipliers vanish in the bounded-center limit, and the end limits are the shrinking cylinder or stationary plane. We identify these two motions as follows.

Use a normal-velocity parametrization \(F(y,t)\) of the complete comparison, and write the limiting flow as \(F+u\nu\). Its equation in the reference metric has the form \[u_t=a^{ij}(y,t,u,\nabla u)\nabla_{ij}u +B(y,t,u,\nabla u),\qquad B(y,t,0,0)=0.\] The principal coefficients are uniformly elliptic and bounded. The lower function \(B\) has bounded first derivatives in its last arguments, since it involves only bounded reference geometry and motion, the small height, and its first derivative. Retain the actual \(a^{ij}\) and linearize only \(B\): \[B=cu+b^i\nabla_i u,\qquad (c,b)=\int_0^1(B_u,B_{\nabla u})(y,t,\sigma u,\sigma\nabla u) \,d\sigma.\] This gives bounded lower coefficients without multiplying an unknown Hessian by derivatives of \(a^{ij}\), even near time zero. The exact comparison equation is used in \(B(y,t,0,0)=0\). For \(\psi=1+|F|^2\), bounded normal speed and reference geometry give \(|\partial_t\psi|+|\nabla\psi|+|\nabla^2\psi|\le C\psi\). Hence \(\epsilon e^{Ct}\psi\) is a supersolution for both signs of the bounded height \(u\), for one sufficiently large \(C\). Properness supplies compact exhaustions, and continuous zero initial height supplies their initial data. The maximum principle, followed by exhaustion and \(\epsilon\downarrow0\), gives \(u=0\). Normal parametrization has removed the raw homothetic tangential velocity; the needed change of parameters exists on the finite horizon because that tangential field has at most linear growth. This contradicts positive-time failure and proves the modulus on the whole first slab.

On the latter half of a slab, the interior estimates and this identification give smooth closeness to the comparison. Its complete normal graph is the union of the overlapping outputs. A fresh radius-\(4r\) reference patch in the smaller data region is covered by finitely many of their interiors, with a number depending only on the fixed ratio \(r/s\). Thus its fresh radius-\(r\) entrance cylinder contains the whole actual intersection, with the required small entrance slope. Center it at the corresponding actual graph point. A single old radius-\(s\) output is never asked to contain the new larger entrance cylinder.

Keep the original coarse \(r,h\) and weak tolerances for these restarts. The refined choices above serve only to test the modulus at a slab entrance. Finite induction over \(J\le\lceil U/h\rceil+1\) selects initial and multiplier tolerances so that each restart is admissible and its error tends to zero. Every slab consumes a fixed additive buffer \(A\) for enlarged patches, motion, and the confinement collar. Thus the total loss is at most \(JA\), independent of the covering cardinality, \(R\), and the later discrete index. Taking \(JA<\mu R\) proves the asserted complete graph on \(B_{(1-\mu)R}\). Positive-time estimates give all fixed derivative bounds and the corresponding smooth error modulus away from elapsed time zero. ◻

Proof. The chosen initial tangent window supplies the assertion for \(j=0\), including the derivative bounds. Suppose it holds at \(\tau_j\). The localized inequality, the multiplier estimate, and (222) make the core height and compact multiplier small as the minimum radius becomes large. Lemma 45 gives as small a height and slope as desired over \(S_{b_j}\) to radius \((1-\zeta)R_j\), where \(b_j=b(\tau_j)\). In particular this smallness is much stronger than the weak tolerance being propagated.

Normalize the unrescaled time at this slice to \(-1\). For a rescaled duration \(\Delta\) the ordinary elapsed time is \(1-e^{-\Delta}\), and dilation back to the rescaled coordinates has factor \(e^{\Delta/2}\). Consecutive selected times satisfy \(0.8\le\tau_{j+1}-\tau_j\le1.2\). Apply Lemma 46, allocating its additional spatial loss within another \(\zeta R_j\). In the rescaled coordinates it yields a complete weak graph relative to the old \(S_{b_j}\) to radius at least \[ (1-2\zeta)e^{(\tau_{j+1}-\tau_j)/2}R_j, \tag{224}\] and throughout the intervening times on the corresponding interior regions. Its positive-time smoothing supplies the derivative bounds at the next selected slice before any new elliptic improvement is invoked. If an arbitrarily small additional outer margin is needed for those bounds, reserve it in the strict inequality (220).

It remains to change to the new base \(S_{b_{j+1}}\). While the fixed core is a weak graph over \(S\), the graph equation and Cauchy–Schwarz on the compact support of the observations give \[ \left|\frac{d}{d\tau}(b(\tau),x(\tau))\right| \le C d(\tau),\qquad |b_{j+1}-b_j|\le C(a_j+a_{j+1}). \tag{225}\] To see the first inequality, the fixed-core normal graph height \(v\) satisfies \(\partial_\tau v=\Phi\cdot\nu_M/(\nu_S\cdot\nu_M)\); the denominator stays bounded away from zero. Each observation is a fixed compact integral of \(v\), and the inverse coordinate Jacobian is bounded. The second inequality follows by integrating on the portions of \([j,j+1]\) and \([j+1,j+2]\) between the two slices.

Smooth dependence of the rotations, conical links, and decaying remainders gives a chart-change cost in physical height and slope at radius \(R\) bounded by \(C(1+R)|b_{j+1}-b_j|\). Both adjacent cutoffs in (221) are necessary here. Specifically \[R_{j+1}a_j\le Q a_j\sqrt{8\log(1/a_j)},\qquad R_{j+1}a_{j+1}\le a_{j+1}\sqrt{8\log(1/a_{j+1})},\] with the products interpreted as zero at \(a_i=0\). These are uniformly small when \(\sup a_i\le\sqrt{\mathcal E(0)}\) is small. Thus the new-chart graph still lies inside its weak tolerance. Equations (220), (223), and (224) place \(B_{R_{j+1}}\), with its required margins, in that graph. The next localized inequality and improvement can now be used. This proves the induction in the order: previous improvement, parabolic propagation and smoothing, chart change, then new elliptic improvement. ◻

Proof. Call \(j\) good if \(e^{-(1-\eta)R_j^2/8}\le a_j^{1-c}\), and bad otherwise. For a good index with \(a_j>0\), (226) and \(a_j\le1\) imply \[\mathcal E(j+1)^{1-p}\le C_1a_j.\] Write \(u=\mathcal E(j)\) and \(v=\mathcal E(j+1)\). If \(v\ge u/2>0\), concavity and the last inequality give \[u^p-v^p\ge p u^{p-1}(u-v) =p u^{p-1}a_j^2\ge c_1a_j.\] If \(v<u/2\), then, since \(u\le1\) and \(p<1/2\), \[u^p-v^p\ge(1-2^{-p})u^p \ge(1-2^{-p})\sqrt u\ge(1-2^{-p})a_j.\] Zero terms cause no difficulty. Summing the nonnegative energy differences only over good indices gives \(\sum_{j\text{ good}}a_j\le C\mathcal E(0)^p\) on every prefix.

At a bad index, \[ a_j<e^{-hR_j^2} =\max\left\{e^{-hq^jR_{\rm in}^2}, \max_{i\le j}a_i^{\kappa q^{j-i}}\right\}. \tag{228}\] If \(0<a_j<1\), its own term \(a_j^\kappa\) is strictly less than \(a_j\) and cannot account for this maximum. If \(a_j=0\) the following bound is immediate. Thus \[ a_j\le e^{-hq^jR_{\rm in}^2} +\sum_{i<j}a_i^{\kappa q^{j-i}}. \tag{229}\] This exclusion of \(i=j\) is where \(\kappa>1\) is essential.

Let \(\delta_0=\sqrt{\mathcal E(0)}<1\) and \[\rho(\delta_0)=\sum_{\ell\ge1} \delta_0^{\kappa q^\ell-1} \le\frac{\delta_0^{\kappa q-1}} {1-\delta_0^{\kappa(q-1)}}\longrightarrow0.\] Since every \(a_i\le\delta_0\), summing (229) on a prefix and interchanging the finite nonnegative sums bounds its double sum by \(\rho(\delta_0)\sum_{i=0}^n a_i\). Also \(\sum_{j\ge0}e^{-hq^jR^2}\le B(R)\), using \(q^j\ge1+j(q-1)\). Choose \(\mathcal E(0)\) so small that \(\rho(\delta_0)\le1/2\). Absorbing proves (227). Finally \(\int_j^{j+1}d\,d\tau\le a_j\) by (201) and Cauchy–Schwarz. ◻

Proof. Suppose first that the weak core bootstrap has a finite first stopping time \(L\). Use only intervals \([i,i+1]\) whose right endpoints lie strictly before \(L\). If \(i\) is the last such index, then \[ i+1<L\le i+2, \qquad \tau_i\ge i+0.2, \qquad L-\tau_i\le1.8<2. \tag{230}\] Allowing a completed interval ending at \(L\) by continuity only shortens the gap. If no such interval is available, the initial tangent-convergence window supplies the graphs and derivatives through this prospective stopping time.

For a last available slice, propagate directly from its improved graph using Lemma 46 over the fixed horizon \(U=1-e^{-2}<1\). No subsequent selected slice, localized inequality, or future unit energy drop is needed. Even a limiting cylinder retains radius at least \(\sqrt2 e^{-1}\) on this horizon, so all patch constants and buffers were fixed with a genuine pre-extinction margin. Choose the minimum graphical radius so large that this finite additive buffer leaves the fixed bootstrap ball and its surrounding collar strictly inside the propagated domain. This gives graphical coverage there through \(L\).

Uniform derivatives near the last selected time require a preceding positive delay. If \(i\ge1\), use the propagation from \(\tau_{i-1}\) on \[[\tau_i,\min\{L,\tau_i+0.2\}],\] where the rescaled delay lies in \([0.8,1.4]\), and use the propagation from \(\tau_i\) on \([\tau_i+0.2,L]\), where it lies in \([0.2,1.8]\). Both propagations are available on their entire two-unit horizons from the already proved improved graphs; the earlier induction’s use of part of a horizon does not shorten it. On every completed selected interval \([\tau_j,\tau_{j+1}]\) with \(j\ge1\), use \(\tau_{j-1}\) for the first \(0.2\) time units and \(\tau_j\) thereafter. The initial two-unit tangent window supplies the first entrance and the cases \(i=0\) or no complete interval. In these covers the ordinary elapsed times are bounded below by \(1-e^{-0.2}\), and the dilation factors are uniformly bounded. Lemma 46 therefore supplies uniform interior curvature derivatives on the fixed ball and collar. The weak fixed-core graph and the bounded geometry of \(S\) convert these geometric estimates to uniform derivatives of its normal height, as needed for interpolation below. This uses only known selected slices and no future energy drop. The complete graphical coverage also excludes an entering extra sheet or a loss at the boundary of the observation domain at the stopping time.

The length through completed intervals is bounded by Lemma 48. The entire remaining segment from the last selected slice costs at most \[ \int_{\tau_i}^L d(\tau)\,d\tau \le\sqrt2\left(\int_{\tau_i}^Ld(\tau)^2\,d\tau\right)^{1/2} \le\sqrt2 \mathcal E(0)^{1/2}. \tag{231}\] Counting part of a complete interval twice only weakens this upper bound. The initial-window case has the same estimate with its bounded duration. The graph-speed formula used in (225) therefore bounds the change of the fixed-core graph in \(L^2\) by \[C\bigl(\mathcal E(0)^p+B(R_{\rm in})+\mathcal E(0)^{1/2}\bigr).\] The initial graph can be chosen much closer to \(S\) than the bootstrap tolerance. Interpolation with the uniform interior derivatives on the larger collar makes the resulting \(C^1\) distance strictly smaller than that tolerance through \(L\). The already established coverage and these strict estimates extend the bootstrap past \(L\), a contradiction. This proves that the bootstrap and the radius induction hold for all time.

Letting the prefix tend to infinity in (227) gives \[ \int_0^\infty d(\tau)\,d\tau<\infty, \qquad \sum_{j\ge0}a_j<\infty. \tag{232}\] On the fixed observation core the graph functions are Cauchy in \(L^2\), and the uniform derivative bounds make them converge smoothly on a slightly smaller core. Their limit is \(S\), because the original tangent sequence still has a subsequence at arbitrarily large rescaled times and converges there to \(S\). In particular (225) gives \((b_j,x_j)\to(0,0)\) in the fixed compact observation chart.

The radii actually diverge. Since \(a_i\to0\), their logarithmic cutoffs \(r_i\) tend to infinity. Given \(H<\infty\), choose \(I\) such that \(r_i\ge H\) for \(i\ge I\). Every recent term \(Q^{j-i}r_i\), \(I\le i\le j\), is then at least \(H\); each of the finitely many earlier terms and \(Q^jR_{\rm in}\) eventually exceeds \(H\). Taking the minimum in (221) proves \(R_j\to\infty\).

The improved graphs at \(\tau_j\) thus have height and slope tending to zero on every fixed compact subset of \(S_{b_j}\). Their base compact multipliers tend to zero by (206). Apply the error-modulus version of Lemma 46 on every interval \([\tau_j,\tau_{j+1}]\). In the rescaled coordinates its homothetic comparison is exactly the fixed set \(S_{b_j}\), because the comparison shrinkage is canceled by the rescaling. Since \(b_j\to0\), the actual graphs converge to \(S\) at every intermediate time, uniformly on each compact spatial set. The higher derivative upgrade always uses a preceding slice with positive delay. For \(\tau\in[\tau_j,\tau_{j+1}]\), use the slice \(\tau_j\) if \(\tau-\tau_j\ge0.2\), giving rescaled delay in \([0.2,1.2]\). If \(0\le\tau-\tau_j<0.2\), use \(\tau_{j-1}\), giving delay in \([0.8,1.4]\). Both delays are bounded below by \(0.2\) and above by the fixed horizon \(2\), and therefore their ordinary elapsed times are bounded below by \(1-e^{-0.2}>0\) and above by \(1-e^{-2}<1\). The dilation factors are uniformly bounded. Since the preceding graphical radii tend to infinity, every fixed compact set and its interior margin are eventually contained in these propagated domains. Equation (217) supplies uniform bounds in every fixed derivative order there; interpolation with the established \(C^1\) convergence gives smooth convergence, including at each selected endpoint. No curvature estimate at zero elapsed time is used. Thus convergence holds in the original ambient frame at every rescaled time.

Restore the original, unshifted rescaled clock. For every \(s<0\) and every admissible \(\lambda\) there is the exact identity \[ M_s^\lambda =\sqrt{-s}\,M\bigl(2\log\lambda-\log(-s)\bigr). \tag{233}\] The rescaled times on the right tend to infinity, uniformly for \(s\) in any compact interval \([-B,-b]\subset(-\infty,0)\). Consequently the full family of actual surfaces on the left converges locally smoothly and with multiplicity one to \(\sqrt{-s}\,S\). Smooth complete graphical convergence gives both local area-measure and varifold convergence for every individual \(s<0\). The identity uses only the original fixed center and scalar dilation, so it preserves the axes and all ambient positions. This proves the asserted smooth convergence. To pass from convergence of the actual slices to every slice of a weak representative, we use the all-endpoint convention and the weak-limit premise specified below. ◻

Completion of the proof of Proposition 49. Here is the precise weak-limit premise sufficient for the representative assertion. Write \(\mu_s^\lambda=\mathcal H^2\!\llcorner M_s^\lambda\). A tangent Radon family \((\mu_s)_{s<0}\) is assumed to satisfy the local hypotheses and the all-endpoint inequality of Lemma 2, and along its extracted scales \(\lambda_i\to\infty\) to satisfy \[ \int\!\!\int\psi(X,s)\,d\mu_s^{\lambda_i}(X)\,ds \longrightarrow \int\!\!\int\psi(X,s)\,d\mu_s(X)\,ds \quad\bigl(\psi\in C_c(\mathbb R^3\times(-\infty,0))\bigr). \tag{234}\] For each test, the left side is considered once its time support lies in the interval on which the rescaled flow is defined. The local mass condition in the lemma makes the right-hand spacetime weight a locally finite Radon measure. This is the only weak-limit identification used here. Local weak convergence \(\mu_s^{\lambda_i}\rightharpoonup\mu_s\) for almost every \(s\), together with a uniform bound for \(\mu_s^{\lambda_i}(B_R)\) on each compact time interval and for each fixed \(R\), also implies (234) by dominated convergence.

The direct smooth convergence already proved shows that the left side of (234) converges to \(\int\!\!\int\psi\,d\nu_s\,ds\), with \(\nu_s=\mathcal H^2\!\llcorner(\sqrt{-s}\,S)\). To justify the time integration, place the support of \(\psi\) in \(B_R\times J\) with \(J\Subset(-\infty,0)\). The scale-invariant area-ratio bound gives \(\mu_s^{\lambda_i}(B_R)\le\Lambda R^2\), so dominated convergence applies to the slice integrals. Uniqueness of the local weak Radon limit therefore gives \(\mu_s\,ds=\nu_s\,ds\) as spacetime measures.

Choose a countable family of compactly supported smooth spatial tests that determines Radon measures, for example a family dense on each ball in the uniform norm. Test this spacetime equality by the product of each spatial test and an arbitrary compactly supported continuous time test. Local integrability implies equality of the corresponding slice integrals almost everywhere for each spatial test. Intersecting these countably many full-measure sets and using the determining property yields \(\mu_s=\nu_s\) as Radon measures for almost every \(s\). Lemma 2 now gives this equality for every \(s<0\). The argument applies to every all-endpoint representative satisfying (234), so no choice of its exceptional slices can change the negative-time measures. This proves the representative conclusion under the stated convention. ◻

Proof of Theorem 1. Fix a singular point at the first singular time. The preceding construction applies to its tangent section from 3, and Proposition 49 gives the full fixed-frame smooth convergence and the equality of every negative-time measure under the stated tangent convention. The point was arbitrary. ◻