Background and significance

On compact Riemann surfaces, Chruściel proved global forward existence and convergence to a constant-curvature metric (Chruściel 1991, Theorem 2.1). Chen gave a different proof using decreasing functionals and concentration compactness (Chen 2001), and Struwe developed a unified treatment of Calabi and normalized Ricci flows on closed surfaces (Struwe 2002). In higher dimensions, Chen and He proved local existence and uniqueness and established stability near a constant-scalar-curvature metric (Chen and He 2008, Theorems 1.2 and 3.2). He and Zeng developed short-time theory for rougher initial metrics (He and Zeng 2021). These higher-dimensional results concern either short times or restricted initial data.

Curvature estimates give continuation criteria for the smooth flow. Chen and He proved extension under a uniform Ricci-curvature bound (Chen and He 2008, Corollary 5.3), and Chen and Cheng established extension under a uniform scalar-curvature bound (Chen and Cheng 2021, Theorem 1.8). Li, Zhang and Zheng state that a uniform spatial \(L^p\) scalar-curvature bound, measured with respect to the evolving metric, suffices when \(p>\dim_{\mathbb{C}}M\) (Li et al. 2024, Theorem 1.2); Li and Zhang state a further criterion involving spacetime scalar and scalar-Laplacian integrability (Li and Zhang 2025, Theorem 1.1).

On a compact almost homogeneous manifold with two hypersurface ends, Guan proved exponential smooth convergence of the modified Calabi flow for initial metrics invariant under a maximal compact subgroup of the automorphism group (Guan 2007, Theorem A). The two special orbits of the action used here are instead a point and the hyperplane at infinity. The point has complex codimension ten, so the two-hypersurface hypothesis fails.

Streets constructed global weak Calabi flows by minimizing movements (Streets 2014). Berman, Darvas and Lu identify these flows with trajectories in a space of finite-energy Kähler potentials and show that, when the class contains a constant-scalar-curvature metric, every weak trajectory converges to a smooth constant-scalar-curvature potential in the \(d_1\) distance, the extension of the \(L^1\) path-length distance on smooth Kähler potentials (Berman et al. 2017, sec. 6 and Theorem 1.11). The weak and classical flows agree for as long as the latter exists smoothly (Berman et al. 2017, Proposition 6.1). Neither this weak continuation nor the large-time \(d_1\) convergence implies smooth continuation through every finite time.

Structure of the construction

The construction first reduces the flow to a scalar equation, then builds an exact shrinking solution of its flat limit, and finally corrects that solution on the compact manifold. The geometric starting point is the radial ansatz of Calabi (Calabi 1979) and the momentum formulation of Hwang and Singer (Hwang and Singer 2002). Abreu’s toric scalar-curvature formula (Abreu 1998) is expressed in Guillemin’s symplectic-potential framework (Guillemin 1994). For a radial potential \(\Phi(\ell)\), where \(\ell=\log|Z|^2\), set \(x=\Phi'(\ell)\) and \(v=\Phi''(\ell)\). We use \(x\in(0,1)\) as the momentum coordinate and write \[v=x(1-x)+x^2(1-x)^2H.\] Smoothness and positivity at both special orbits are encoded by \(H\in C^\infty[0,1]\) and \(1+x(1-x)H>0\). Section [sec:geometry] derives the scalar fourth-order equation for \(H\), including the affine choice in the Legendre potential needed to recover the Calabi flow with its prescribed normalization. It also realizes the fixed spatial operator on an auxiliary sphere, where the endpoint conditions become ordinary smooth invariance.

The singularity is detected by one value of \(H\). At the fixed affine origin \(p\), the scalar-curvature identity is \[S(t,p)=110\bigl(1-H(t,0)\bigr).\] We construct a smooth positive profile \(G\), bounded above and away from zero, with \(G(0)=1\) and \(G'(0)<0\), and set \[F(t,x)=\rho^{-1}h(x/\rho),\qquad h(y)=\frac{G(y)-1}{y},\qquad \rho=\sqrt{-t},\quad t<0.\] Then \(F(t,0)=\rho^{-1}G'(0)\). Our task is to find an exact compact solution \(H=F+V\) for which positivity persists and \[V(t,0)=O(\rho^{-1+\delta})\qquad\text{for some }\delta>0.\] The correction is therefore smaller than the leading term at the origin, preserving the desired scalar-curvature divergence. A smooth interior time slice of this solution will be the initial metric; no initial metric needs to be fixed before the construction.

Section [sec:profile] constructs \(G\) so that \(F\) solves the scale-invariant flat equation exactly. The profile is obtained by matching a two-parameter analytic solution at the origin to a half-line solution approaching a positive constant. A finite exact-arithmetic calculation, with analytic enclosure and nonlinear error bounds, certifies the matching inequalities. This uses the framework of validated Taylor integration surveyed in (Nedialkov et al. 1999). The finite argument and its complete executable certificate appear in the appendices. The resulting profile also has a differentiable expansion at infinity; this expansion controls the compact error away from the shrinking core.

The correction begins with a right inverse for the flat linearized equation. Section 4 introduces the weighted norms and local estimates, and Section 5 constructs this inverse on all negative times. In similarity variables its time-one evolution is a compact perturbation of a damped constant-coefficient evolution. A spectral split determines finitely many components from future forcing. This choice is available because the initial metric is still free; it requires no stability or nondegeneracy theorem for the profile. The compact-perturbation and spectral-projection tools are classical (Kato 1995). After selecting a spectral weight, an outer annulus estimate upgrades the temporal supremum bound to the full spatially weighted parabolic Hölder estimate. The decisive input there is a scale-invariant \(L^2\) bound that is integrable up to the singular time.

Section [sec:compact] combines this ancient inverse near the core with a short-time forward solution outside it. The outer forcing cancels the inner cutoff commutator exactly; the remaining operator errors are small. For the nonlinear correction one further needs the compact residual to become small after applying the inverse. This holds in a norm controlling all four spatial derivatives and their temporal Hölder seminorms, with the time derivative omitted. The outer estimate uses a strictly higher Hölder exponent for the source, and the fixed-point equation then supplies the missing time derivative. Positivity and the pointwise correction bound above give the geometric hypotheses needed to finish Theorem 1.

A regular auxiliary sphere and its flat limit

The endpoints of the momentum interval can be handled by a smooth equation on a closed auxiliary manifold. Set \[ m=2n+4,\qquad \Sigma=S^{m+5}\subset\mathbb{R}^m\times\mathbb{R}^6,\qquad x=|z|^2. \tag{16}\] We identify functions of \(x\) with \(O(m)\times O(6)\)-invariant functions on \(\Sigma\).

The self-similar core

From now on set \(n=10\), so \(m=24\) and \(\Sigma=S^{29}\). In the flat model the equation for the coefficient is \[ F_t+(1+xF)^2\widehat\mathcal AF=0. \tag{21}\] Let \(D=x\partial_x\). For the function \(g=1+xF\), (19) gives \[ x\widehat\mathcal A\big(x^{-1}(g-1)\big) =x^{-2}P(D)(g-1),\qquad P(D)=(D-2)(D-1)(D+9)(D+10). \tag{22}\] Here \(P\) denotes a polynomial, as distinct from the evolution operator denoted by \(P\) later. One can check (22) on \(g-1=x^j\): both sides equal \((j-2)(j-1)(j+9)(j+10)x^{j-2}\); equality of the differential expressions follows as in Lemma 4. Consequently (21) is equivalent, for \(x>0\), to \[ g_t=-g^2x^{-2}P(D)(g-1). \tag{23}\]

Define the cubic polynomial operator \(Q(D)=(D-1)(D+9)(D+10)\). Suppose that \(G\) is smooth and positive on \([0,\infty)\), with \(G(0)=1\), and for some \(c>0\) satisfies \[ Q(y\partial_y)(G-1)=\frac{y^2}{2}\left(\frac1G-\frac1c\right). \tag{24}\] Then set \[ \rho=\sqrt{-t},\qquad y=x/\rho,\qquad h(y)=\frac{G(y)-1}{y},\qquad F(t,x)=\rho^{-1}h(x/\rho),\qquad t<0. \tag{25}\] The value \(h(0)=G'(0)\) makes \(h\) smooth at zero, since \(h(y)=\int_0^1G'(sy)\,\mathrm{d}s\). Thus \(F\) is smooth for \(t<0\) including at the flat origin. Applying \(D-2\), now with \(D=y\partial_y\), to (24) yields \[P(D)(G-1)=-\frac{y^2DG}{2G^2}.\] On the other hand \(\rho_t=-1/(2\rho)\), so \(\partial_tG(x/\rho)=DG/(2\rho^2)\). These identities prove (23), and hence (21), first for \(x>0\) and then at zero by smoothness. The existence of the required nonconstant positive \(G\) is the subject of Theorem 6.

The same expression \(F(t,x)\) is smooth on the compact sphere for every \(t<0\). Its compact positivity multiplier obeys the useful identity \[ 1+v_*F=x+(1-x)G(x/\rho). \tag{26}\] In particular, if \(G\) is bounded above and below by positive constants, this multiplier has the same property uniformly in \(t<0\) and \(0\leq x\leq1\). This is a bound in the momentum coordinate \(x\), whose relation to the fixed affine coordinate varies under the Legendre transform. It does not imply uniform equivalence to the Fubini–Study metric.

Curvature at the concentrating point

The final geometric implication uses only one value of the correction. It is convenient to state it before constructing that correction.

The Fubini–Study class contains the constant-scalar-curvature form \(\omega_*\), so the hypotheses of Lemma 5 suffice to refute the smooth global-existence assertion. Its requirement of smooth existence at every finite time has already failed. In addition, \(\|S\|_{L^\infty}\) is unchanged by biholomorphic pullback, so the curvature divergence itself is unaffected by allowing automorphisms.

Reduction and the analytic family at the origin

The calculation in Section 2.2 shows that (30) makes \(F\) in (32) an exact solution of the flat equation. It remains to construct a positive \(G\) with the stated behavior at zero and infinity.

Put \[ c=(1+d)^{-1},\qquad s=\left(\frac{27}{16}\right)^{1/3}\left(\frac yc\right)^{2/3}, \qquad G(y)=c(1+p(s)). \tag{33}\] Since \(y\partial_y=\frac23s\partial_s\) and \(G-1=c(p-d)\), (30) becomes \[ \mathcal E_d(p):= p'''+\frac{30}{s}p''+\frac B{s^2}p' +\frac C{s^3}(p-d)+\frac p{1+p}=0, \qquad B=\frac{751}{4},\quad C=-\frac{1215}{4}. \tag{34}\] Primes in this section henceforth denote \(s\)-derivatives, except where another variable is explicitly displayed. For example, the identity \(Q(D)=D^3+18D^2+71D-90\) verifies all the coefficients in (34).

We will construct origin solutions \(p(s)=d+ks^{3/2}+O(s^3)\), with two real parameters: \(d\) fixes \(c=(1+d)^{-1}\), and \(k\) fixes \(G'(0)\). These parameters must be chosen to match \(p\) to a tail tending to zero. The limiting linearized equation \(p'''+p=0\) has roots \(-1\) and \((1\pm i\sqrt3)/2\): one decaying direction and two real growing directions.

The tail argument uses a constant-coefficient comparison equation with the same splitting. After subtracting an explicit decaying approximation, we express the residual jet at \(S_0=56\) in its eigenbasis. For each allowed \(d\), a contraction constructs small bounded tails with one real coordinate free and the other two determined. Choosing the free coordinate from the origin solution leaves two real matching conditions to enforce by varying \((k,d)\).

The affine square below makes the shot’s two remaining coordinates close to its parameter \(z_*\). Together with the tail bound, the certified estimate yields a continuous self-map of the square; Brouwer’s theorem supplies a matching point. All terminating decimals in this section denote exact rational numbers. Define \[ \begin{gathered} R=5\cdot10^{-10},\qquad \mathcal Z=\{z_*\in\mathbb{R}^2:\lVert z_*\rVert_\infty\le R\},\qquad S_0=56,\\ \begin{pmatrix}k\\d\end{pmatrix} =\begin{pmatrix}-.094617116666041\\ .400682721856639\end{pmatrix} +10^{-8} \begin{pmatrix}25190&-63731\\721855&359157\end{pmatrix}z_*. \end{gathered} \tag{35}\] Write \(k_0,d_0\) for the central values. The square implies \[ |k-k_0|\le4.44605\cdot10^{-13},\qquad |d-d_0|\le5.40506\cdot10^{-12}<.011R. \tag{36}\]

The matching coordinates

Define the four-term approximation \[ \begin{gathered} p_{\rm a}(s;d)=\sum_{j=1}^4d_js^{-3j},\qquad a=d_1=Cd,\qquad b=d_2=a^2+567a,\\ d_3=2ab-a^3+\frac{2025}{4}b,\qquad d_4=b^2+2ad_3-3a^2b+a^4+\frac{567}{2}d_3. \end{gathered} \tag{40}\] We choose the constant coefficients in \(D_0\) at the midpoints of the corresponding linearized tail ranges on \([S_0,\infty)\), giving the uniform difference bounds used in Lemma 10. Let \[ \begin{gathered} D_0(\lambda)=\lambda^3+\frac{15}{S_0}\lambda^2 +\frac{B}{2S_0^2}\lambda+1+\frac{C}{2S_0^3},\\ v_i=(1,\lambda_i,\lambda_i^2)^t,\qquad \begin{pmatrix}p-p_{\rm a}\\p'-p_{\rm a}'\\p''-p_{\rm a}''\end{pmatrix}_{s=S_0} =qv_0+\xi v_1+\bar\xi v_2. \end{gathered} \tag{41}\] The roots are ordered so that \(\lambda_0\) is real, \(\Im\lambda_1>0\), and \(\lambda_2=\bar\lambda_1\). The exact real projection matrix \(\Gamma\) takes a real jet to \((q,\Re\xi,\Im\xi)^t\). Its rational approximation used in the finite certificate is \[ \Lambda=10^{-5} \begin{pmatrix} 30756&-27384&33447\\ 34622&13692&-16724\\ -2925&-34040&-28850 \end{pmatrix}. \tag{42}\]

The finite part of the construction is the following consequence of Proposition 24. The origin solution of Lemma 7, for every \(z_*\in\mathcal Z\), continues to \(s=S_0\) and satisfies \[ 1+p(s)>0\quad(0\le s\le S_0),\qquad |q_{\rm shot}(z_*)|\le .25R, \qquad \lVert (\Re\xi_{\rm shot},\Im\xi_{\rm shot})-z_*\rVert_\infty\le .13R. \tag{47}\] The proof in the Appendix includes the analytic interpretation of its rational Taylor enclosures, the propagation of all rounding and truncation errors, and the uniform nonlinear continuation estimate. In particular, (47) is a statement about the exact solutions. Their endpoint jets depend continuously on \(z_*\): this follows first from Lemma 7 at \(s=1\), and then from ordinary continuous dependence for (34) where \(s>0\) and \(1+p>0\). The estimates from this section used in Appendix [sec:certificate] are the origin family and the explicit root and tail-polynomial estimates; all are independent of the matching step.

Construction on the half-line

Matching the two solutions

For a parameter \(z_*\in\mathcal Z\), apply Lemma 10 using the value \(d(z_*)\) and the coordinate \(q_{\rm shot}(z_*)\) supplied by the finite shot. This is allowed by \(|q_{\rm shot}|\le .25R<.26R\). Define a map on the closed square by \[\mathcal B(z_*)=z_* -\bigl(\Re(\xi_{\rm shot}-\xi_{\rm tail}), \Im(\xi_{\rm shot}-\xi_{\rm tail})\bigr).\] It is continuous by the continuous dependence established above and satisfies \[\lVert \mathcal B(z_*)\rVert_\infty\le(.13+.25)R=.38R<R.\] Brouwer’s fixed point theorem gives \(z_*\in\mathcal Z\) with \(\xi_{\rm shot}=\xi_{\rm tail}\). The real coordinate and the parameter \(d\) already agree. The three jet entries in (41) therefore agree, and ordinary uniqueness for (34) joins the shot and tail to one solution on \([0,\infty)\).

The finite shot has \(1+p>0\), and the tail has \(|p|<.001\). Compactness of the finite interval and the uniform tail bound imply that \(1+p\) is bounded and bounded away from zero globally. Since \(1+d>0\), the function \(G\) in (33) has the same properties. Near zero, \[s^{3/2}=\frac{\sqrt{27}}{4c}y, \qquad G(y)=1+\frac{\sqrt{27}}4ky+O(y^2).\] It is therefore analytic there, with \(G'(0)=k\sqrt{27}/4<0\) by (36). This proves all assertions of Theorem 6 except for the differentiable expansion, which we now establish independently of the finite shooting estimates.

Differentiable asymptotics at every order

The distinction between a bounded remainder and a remainder in a prescribed polynomial weight matters here. The following argument first proves uniqueness among bounded solutions and then identifies the actual remainder with the weighted solution.

Norms and scales

On a product \(D=U\times I\subset\mathbb{R}^d\times\mathbb{R}\), put \[d_{\rm p}((z,t),(w,s))=|z-w|+|t-s|^{1/4}.\] Write \[[g]_{\alpha;D}=\sup_{P\ne Q\in D} \frac{|g(P)-g(Q)|}{d_{\rm p}(P,Q)^\alpha}, \quad |g|_{\alpha;D}=\|g\|_{\infty;D}+[g]_{\alpha;D}.\] The joint seminorm is equivalent, with constants depending only on \(\alpha\), to the sum of the spatial \(\alpha\) and temporal \(\alpha/4\) seminorms. We use the following explicit version of the fourth-order parabolic norm: \[ \mathcal N_\alpha(u;D) =\sum_{j=0}^4|D_z^ju|_{\alpha;D}+|u_t|_{\alpha;D},\qquad \mathcal N'_\alpha(u;D)=\sum_{j=0}^4|D_z^ju|_{\alpha;D}. \tag{63}\] Here and below norms of arrays mean sums of component norms. Thus \(\mathcal N'_\alpha\) still includes temporal Hölder seminorms of all four spatial derivatives.

These are local weighted norms; no Hölder quotient between distant boxes is imposed. Their completeness follows by convergence of the functions and indicated derivatives on every compact subset, followed by taking the supremum in the defining inequalities. The derivative relations pass to the limit distributionally. In particular \(X\) and \(X'\) are Banach spaces and \(X\hookrightarrow X'\) has norm at most one.

The radius function \(r\) is Lipschitz on both spaces. The elementary inequalities for Euclidean norms and fourth roots give \[|\sigma(z,t)-\sigma(z_0,t_0)| \le |r(z)-r(z_0)|+|t-t_0|^{1/4}.\] Consequently \(\sigma\) is comparable to its center value on every box in Definition 12, including boxes truncated at \(t=0\). Changing \(a_0\) to another fixed sufficiently small radius gives equivalent norms: cover the enlarged box by finitely many smaller comparable-scale boxes, and split a Hölder quotient whose points are not in a common small box using the supremum norm. The same argument applies to \(s_\beta=\max(\beta,r)\), whose lack of smoothness at \(r=\beta\) is harmless: the scale is frozen at each center, never differentiated. Products satisfy the usual local Hölder inequality \(|fg|_\alpha\le 2|f|_\alpha|g|_\alpha\), with their weights added.

Interpolation without a lower bound for the time length

We first isolate the interpolation used in localization. For fixed nested balls \(B_1\Subset B_2\) and any interval \(I\) of length at most one, let \[U=\|u\|_{\infty;B_2\times I},\qquad M=U+\|D_z^4u\|_{\infty;B_2\times I} +\|u_t\|_{\infty;B_2\times I}.\] Then, for \(0\le j\le3\), \[ |D_z^ju|_{\alpha;B_1\times I} \le C U^{1-(j+\alpha)/4}M^{(j+\alpha)/4} \le \varepsilon M+C_\varepsilon U. \tag{67}\] All constants are independent of \(|I|\). Spatial interpolation on nested balls gives the spatial assertion, including the supremum. For the temporal assertion apply that same interpolation to \(w=u(\cdot,t)-u(\cdot,s)\). With \(b=|t-s|\), \[\|w\|_\infty\le\min(2U,bM),\qquad \|D^4w\|_\infty\le2M,\] and hence \[\|D^jw\|_{\infty;B_1} \le C\min(2U,bM)^{1-j/4}M^{j/4}.\] Divide by \(b^{\alpha/4}\) and split at \(b=U/M\); both resulting bounds give (67). Restricting the available values of \(b\) can only decrease their supremum. The spatial interpolation inequalities themselves follow, for example, by subtracting a Taylor polynomial in a ball of radius \(q\) and minimizing in \(q\); fixed-radius remainders account for the \(U\) term when the minimizer exceeds the gap between the balls. The same argument at order four gives \[ \|D^4u\|_{\infty;B_1\times I} \le\varepsilon[D^4u]_{\alpha;B_2\times I}+C_\varepsilon U. \tag{68}\] Here it suffices to use the spatial part of the seminorm on the right. Indeed, convolution on each spatial slice with a mollifier of radius \(q\) gives \(\|D^4u\|_{B_1}\le Cq^\alpha[D^4u]_{\alpha;B_2} +Cq^{-4}U\); choose \(q\) below the spatial gap and so that \(Cq^\alpha\le\varepsilon\). Neither inequality uses a temporal seminorm of \(u_t\).

Local fourth-order estimates

Consider in a coordinate cylinder the real scalar operator \[ \mathcal L u=u_t+\sum_{|\nu|\le4}a_\nu(z,t)D^\nu u, \qquad \sum_{|\nu|=4}a_\nu(z,t)\xi^\nu\ge\lambda|\xi|^4. \tag{69}\] Assume \(\lambda>0\) and \(\sum_{|\nu|\le4}|a_\nu|_\alpha\le K\). These assumptions are always imposed in the coordinates and units of the estimate. On a fixed smooth manifold they include the coefficients arising from the coordinate expression of the operator. The leading symbols used later are positive multiples of squared positive definite quadratic forms, and satisfy (69).

The outer forward problem

The following formulation permits the localization scale to tend to zero. Fix a compact smooth manifold without boundary and a time-independent positive Lipschitz scale \(s(z)\). Assume that on balls of radius \(4a_0s(z)\) the scale is comparable to its center value, with uniform comparison constants and uniformly bounded coordinate changes in scaled units. We impose the coefficient assumptions of (69) uniformly after scaling space by \(s(z)\) and time by \(s(z)^4\); an original coefficient of spatial order \(j\) is thus multiplied by \(s(z)^{4-j}\). The ellipticity and Hölder constants in these units are denoted again by \(\lambda,K\). These hypotheses hold for \(s=s_\beta\) in the later application.

Similarity coordinates and coefficient bounds

The flat equation for \(F\) in Theorem 6 and the radial operator identity of Lemma 4 yield \[ \frac1{16}\Delta_\zeta^2\bigl(h(|\zeta|^2)\bigr) =-\frac{G'(y)}{2G(y)^2},\qquad y=|\zeta|^2, \qquad F_t+G(|z|^2/\rho)^2\widehat{\mathcal A}F=0. \tag{81}\] In particular, with \[A_f(\zeta)=G(|\zeta|^2)^2,\qquad B_f(\zeta) =\frac{2|\zeta|^2G(|\zeta|^2)}{16} \Delta_\zeta^2\bigl(h(|\zeta|^2)\bigr) =-\frac{yG'(y)}{G(y)},\] one has \(A(z,t)=A_f(z/\sqrt\rho)\) and \(B(z,t)=\rho^{-2}B_f(z/\sqrt\rho)\). These coefficients are smooth at the origin. Positivity and the differentiated asymptotic expansion in Theorem 6 imply \[ \inf A_f>0,\qquad |\partial_\zeta^\nu(A_f-c^2)|+ |\partial_\zeta^\nu B_f| \le C_\nu(1+|\zeta|)^{-4-|\nu|}. \tag{82}\] In particular all derivatives, including any fixed number of applications of \(\zeta\cdot\nabla\), are bounded.

Set \[\tau=-\log(-t),\qquad \zeta=z/\sqrt\rho, \qquad V(z,t)=\rho^{-1}v(\zeta,\tau), \qquad f^\#(\zeta,\tau)=\rho^3f(\sqrt\rho\,\zeta,t).\] Since \(\mathrm{d}\tau/\mathrm{d}t=\rho^{-2}\) and \(\partial_t\zeta=\zeta/(4\rho^2)\), direct differentiation gives \[V_t=\rho^{-3} \left(v_\tau+\tfrac14\zeta\cdot\nabla v+\tfrac12v\right), \qquad \Delta_z^2V=\rho^{-3}\Delta_\zeta^2v.\] Thus (80) becomes the autonomous equation \[ v_\tau+\frac{A_f}{16}\Delta^2v +\frac14\zeta\cdot\nabla v+(\tfrac12+B_f)v=f^\#. \tag{83}\] We shall first solve it in \(E=C_{0,\mathrm{rad}}(\mathbb{R}^m;\mathbb{C})\), with the supremum norm. This denotes continuous radial functions tending to zero at spatial infinity; complex scalars are needed only for spectral theory.

Evolution on continuous functions vanishing at infinity

We give the evolution construction since the unbounded drift in (83) and the change in the principal coefficient both matter. On a time interval beginning at \(\tau_0\), put \[u(\eta,s)=v(e^{s/4}\eta,\tau_0+s),\qquad 0\le s\le1.\] Then \[ u_s+a(\eta,s)\Delta_\eta^2u+b(\eta,s)u=l(\eta,s), \qquad a=\frac{e^{-s}}{16}A_f(e^{s/4}\eta),\quad b=\tfrac12+B_f(e^{s/4}\eta). \tag{84}\] The right side is the corresponding dilation of \(f^\#\). By (82), \(a,b\) have bounded derivatives of every order on this slab, and \(0<\kappa\le a\le\kappa^{-1}\) for a fixed \(\kappa\). The following construction also applies without radial symmetry. It is the scalar whole-space version of the higher-order heat-kernel parametrix; see also (He and Zeng 2021, Appendix A).

Let \[k_\theta(w)=(2\pi)^{-m}\int_{\mathbb{R}^m} e^{iw\cdot\xi-\theta|\xi|^4}\,\mathrm{d}\xi, \qquad \theta>0.\] Its integral is one. Scaling and integration by parts in the Fourier integral show that, uniformly for \(\kappa\le a_*\le\kappa^{-1}\), \[ |\partial_{a_*}^{\ell}\partial_w^\nu k_{a_*q}(w)| \le C_{N,\ell,\nu}\, q^{-(m+|\nu|)/4} (1+|w|/q^{1/4})^{-N},\qquad 0<q\le1. \tag{85}\] Indeed after scaling \(\xi=q^{-1/4}\xi'\) the multipliers and all their derivatives form a bounded set of Schwartz functions. No sign property of \(k_\theta\) is used.

Freeze the principal coefficient at the source point and time: \[K_{s,r}(\eta,y)=k_{a(y,r)(s-r)}(\eta-y),\qquad r<s,\] and also denote by \(K_{s,r}\) the associated integral operator. Applying the left side of (84) at the evaluation point gives the residual kernel \[ J_{s,r}(\eta,y) =\bigl(a(\eta,s)-a(y,r)\bigr) \Delta_\eta^2 k_{a(y,r)(s-r)}(\eta-y) +b(\eta,s)K_{s,r}(\eta,y). \tag{86}\] The global Lipschitz bound for \(a\) and (85) give \[ \|K_{s,r}\|_{C_0\to C_0}\le C, \qquad \|J_{s,r}\|_{C_0\to C_0} \le C\bigl((s-r)^{-3/4}+1\bigr) \le C'(s-r)^{-3/4}. \tag{87}\] For the first residual term, the spatial difference \(|\eta-y|\) contributes \((s-r)^{1/4}\) against a fourth derivative of size \((s-r)^{-1}\); the temporal difference contributes \((s-r)\). The kernel bounds first show that both operators preserve \(C_0\) for compactly supported continuous inputs, by decay in \(\eta\), and then for all \(C_0\) inputs by uniform approximation.

These kernels give a strong approximate identity. To see this explicitly, replace \(a(y,r)\) by \(a(\eta,r)\) in \(K_{s,r}\). The operator norm of the difference is \(O((s-r)^{1/4})\), by the parameter derivative bound in (85). For the replacement kernel, its integral in \(y\) is one at each \(\eta\); uniform continuity of a \(C_0\) function, followed by the uniform kernel tail bound, proves convergence to that function in supremum norm. Consequently \[ K_{s,r}d\longrightarrow d\quad\hbox{in }C_0 \quad\hbox{as }s-r\downarrow0, \tag{88}\] uniformly in the initial time \(r\). This is strong convergence; only the difference between the two frozen kernels was estimated in operator norm.

For \(d\in C_0\) and \(l\in C([0,1];C_0)\) solve \[ \begin{split} j(s)+\int_0^sJ_{s,r}j(r)\,\mathrm{d}r&=l(s)-J_{s,0}d,\\ u(s)&=K_{s,0}d+\int_0^sK_{s,r}j(r)\,\mathrm{d}r. \end{split} \tag{89}\] All integrals are Bochner integrals. The inverse of the first equation is its Volterra series. For completeness, if \(H(s)=Cs^{-3/4}\), its \(k\)-fold convolution on the positive half-line is \[H^{*k}(s) =\frac{C^k\Gamma(1/4)^k}{\Gamma(k/4)}s^{k/4-1} \quad(k\ge1).\] This identity follows by the beta integral and induction. Since the right side of the first equation in (89) is bounded by \(Cs^{-3/4}(\|d\|_\infty+\|l\|_{C([0,1];C_0)})\), the same formula and the growth of the gamma function prove convergence of the series and the estimates \[ \|j(s)\|_\infty\le Cs^{-3/4} (\|d\|_\infty+\|l\|_{C([0,1];C_0)}),\qquad \sup_{0\le s\le1}\|u(s)\|_\infty \le C(\|d\|_\infty+\|l\|_{C([0,1];C_0)}). \tag{90}\] The integral in the formula for \(u\) tends to zero as \(O(s^{1/4})\). Together with (88), this proves \(u\in C([0,1];C_0)\) and \(u(0)=d\). Continuity at positive times follows by dominated convergence, splitting off a short interval adjacent to the diagonal.

The kernel identity (86) proves that \(u\) solves (84) distributionally: apply the equation to the second line of (89), obtaining \(J_{s,0}d+j(s)+\int_0^sJ_{s,r}j(r)\,\mathrm{d}r=l(s)\). This calculation can be made first with a positive separation from the diagonal and smooth compactly supported tests; the diagonal term converges to \(j\) by (88), and (87) justifies the remaining integrals. Lemma 13 and interior regularity then give smoothness for positive times in the homogeneous case, and classical local regularity for locally parabolic Hölder sources.

We also record uniqueness in the class just constructed. If \(u\) is the difference of two such solutions, it solves the homogeneous equation with zero \(C_0\) initial value. Let \(W(\eta)=\exp(-\sqrt{1+|\eta|^2})\). Its derivatives are bounded by constant multiples of \(W\). For \(s\ge\varepsilon>0\), interior regularity supplies globally bounded spatial derivatives of \(u\). Integration by parts, first with compact spatial cutoffs, gives \[\Re\int W\overline u\,a\Delta^2u =\Re\int\Delta(aW\overline u)\Delta u \ge\frac\kappa2\|\Delta u\|_{L^2(W)}^2 -C\|u\|_{L^2(W)}^2.\] Here the mixed terms are absorbed using \[\|\nabla u\|_{L^2(W)}^2 \le\epsilon\|\Delta u\|_{L^2(W)}^2 +C_\epsilon\|u\|_{L^2(W)}^2.\] This last inequality follows by integrating \(\int W|\nabla u|^2\) once by parts and using \(|\nabla W|\le CW\) and Young’s inequality. Exponential decay of \(W\) allows removal of the cutoffs. The bounded zeroth order coefficient therefore gives \[\frac{\mathrm{d}}{\mathrm{d}s}\|u(s)\|_{L^2(W)}^2 \le C\|u(s)\|_{L^2(W)}^2.\] Gronwall’s inequality, followed by \(\varepsilon\downarrow0\), proves \(u=0\), because the zero \(C_0\) initial trace implies a zero weighted \(L^2\) trace. This also proves uniqueness for the inhomogeneous construction by subtraction.

Undoing the dilation now gives a strongly continuous homogeneous semigroup \(S(s)\) on \(C_0(\mathbb{R}^m)\) for (83); concatenation is justified by the uniqueness just proved. Dilation is strongly continuous on \(C_0\), and the equation is autonomous. On \(0\le s\le1\) one has \(\|S(s)\|\le C\). Inhomogeneous evolution for a continuous \(C_0\) source is \[ v(\tau_0+s)=S(s)v(\tau_0) +\int_0^sS(s-r)f^\#(\tau_0+r)\,\mathrm{d}r. \tag{91}\] The formula follows either from the distributional equation and uniqueness or by approximation with smooth sources. Rotation and complex conjugation preserve the equation and the construction. In particular \(S(s)\) preserves \(E\) and its real subspace.

Compactness at positive time

Write \(P=S(1)|_E\), reserving this symbol here for a bounded operator, and let \(P_c\) be the corresponding operator with \(A_f=c^2\) and \(B_f=0\). We prove \[ P-P_c:E\longrightarrow E\quad\hbox{is compact}. \tag{92}\] Both time-one evolutions map their unit balls to locally equicontinuous functions by the interior estimates. It remains to prove uniform decay of their difference at spatial infinity.

Suppose this decay fails. There are \(d_j\in E\) with \(\|d_j\|_\infty\le1\), points \(\zeta_j\) with \(|\zeta_j|\to\infty\), and \(\epsilon_0>0\) such that \(|(P-P_c)d_j(\zeta_j)|\ge\epsilon_0\). In the moving coordinates (84), translate by \(\eta_j=e^{-1/4}\zeta_j\). The two solutions, denoted by \(u_j\) and \(u_{c,j}\) after translation, have the same initial value \(d_j(\eta+\eta_j)\) and uniform supremum bounds. Their coefficients converge, smoothly on compact subsets of \(\mathbb{R}^m\times[0,1]\), to \[a_c(s)=\frac{c^2e^{-s}}{16},\qquad b_c=\tfrac12.\] This follows from (82); it also controls time derivatives in the moving coordinates.

After passing to a subsequence, the translated initial functions converge weakly star in \(L^\infty(\mathbb{R}^m)\) to some \(d_\infty\). Sequential extraction is available because \(L^1(\mathbb{R}^m)\) is separable. Interior estimates and a diagonal extraction give smooth local limits \(u_\infty,u_{c,\infty}\) for \(s>0\), including convergence of their values at \(s=1\). Both limits are bounded on the whole slab. To identify their initial traces, for \(\psi\in C_c^\infty(\mathbb{R}^m)\) integrate the translated equation: \[ \left|\int u_j(s)\psi -\int d_j(\eta+\eta_j)\psi\right| \le Cs\, \sup_{j,\,0\le r\le1} \left(\|\Delta^2(a_j(\cdot,r)\psi)\|_{L^1} +\|b_j(\cdot,r)\psi\|_{L^1}\right) \le C_\psi s. \tag{93}\] All four spatial derivatives have fallen on the coefficient times the test function. Uniform coefficient bounds make the last constant independent of \(j\). The identity extends down to \(s=0\) using the individual \(C_0\) initial traces. The same estimate holds for \(u_{c,j}\). Passing to the limit proves that both limits have distributional initial value \(d_\infty\).

Here is the required whole-space uniqueness for these bounded limits; a \(C_0\) initial trace of the limits is not needed. Their difference \(w\) solves \[w_s+a_c(s)\Delta^2w+\tfrac12w=0\] and has zero distributional initial value. For a fixed \(t_1\in(0,1]\) and \(\psi\in\mathcal S(\mathbb{R}^m)\), use the backward test function \[\phi(s)=e^{-(t_1-s)/2} k_{\int_s^{t_1}a_c(r)\,\mathrm{d}r}*\psi, \qquad 0\le s\le t_1,\] with \(\phi(t_1)=\psi\). It solves \(-\phi_s+a_c(s)\Delta^2\phi+\tfrac12\phi=0\), is continuous in the Schwartz topology, and hence \(\int w(s)\phi(s)\) is constant for \(s>0\). One may justify the pairing by compact cutoffs and then use the Schwartz decay. Boundedness of \(w\) extends its zero initial trace from compactly supported tests to every Schwartz test; \(\phi(s)\to\phi(0)\) in \(L^1\) also permits this changing test. The constant pairing is therefore zero. Since \(\psi\) is arbitrary, \(w(t_1)=0\). This contradicts \(|u_\infty(0,1)-u_{c,\infty}(0,1)|\ge\epsilon_0\).

We have proved uniform vanishing of the tails of \((P-P_c)\) on the unit ball. Together with local equicontinuity, the Arzelà–Ascoli criterion for \(C_0\) proves (92): restrict to a large ball, extract there, and use the common small tails to obtain uniform convergence on the whole space. The closed radial subspace is preserved.

A spectral weight and the two-sided recursion

For the constant coefficient equation, the moving-coordinate formula on an interval of length \(k\) is explicit: \[ (P_c^kd)(\zeta) =e^{-k/2}(k_{\theta_k}*d)(e^{-k/4}\zeta), \qquad \theta_k=\frac{c^2}{16}(1-e^{-k}),\qquad k\ge1. \tag{94}\] The \(L^1\) norm of \(k_\theta\) is independent of \(\theta>0\) by scaling. Thus \(\|P_c^k\|\le Ce^{-k/2}\), and the Neumann resolvent series converges for \(|\lambda|>e^{-1/2}\).

We spell out the spectral consequence of (92). On the connected domain \(\Omega=\{|\lambda|>e^{-1/2}\}\) factor \[\lambda I-P =\left[I-(P-P_c)(\lambda I-P_c)^{-1}\right] (\lambda I-P_c).\] The operator subtracted from the identity is compact and analytic in \(\lambda\). The analytic Fredholm alternative implies that \((\lambda I-P)^{-1}\) is meromorphic on \(\Omega\), with finite rank principal parts, since it is invertible for large \(|\lambda|\). One way to obtain this alternative is to approximate the compact operator at any one parameter by a finite rank operator, invert the remaining operator of norm less than one in a neighborhood, and reduce invertibility to a finite-dimensional analytic determinant. The required approximation is available here: identify radial \(E\) with \(C_0([0,\infty))\), and use bounded polygonal interpolation operators with compact support, which converge strongly to the identity and hence uniformly on compact subsets of \(E\). The determinant either vanishes identically or has isolated zeros; continuation through overlapping neighborhoods and invertibility at a large parameter exclude the first alternative. The same reduction makes every coefficient of a pole finite rank. It follows that the spectrum in \(\Omega\) consists of isolated eigenvalues of finite algebraic multiplicity, with no accumulation inside \(\Omega\); these are the usual compact-operator and resolvent arguments of (Kato 1995, VII, Section 1.3, and the supplementary notes).

Choose \[ e^{-1/2}<q<1,\qquad \{|\lambda|=q\}\cap\mathop{\mathrm{spec}}(P)=\varnothing,\qquad \gamma=-\log q\in(0,\tfrac12),\qquad \delta=2\gamma. \tag{95}\] Such a circle exists by the discreteness just proved. There are only finitely many eigenvalues outside it: they lie in a bounded closed annulus contained in \(\Omega\). Let \(\Pi_u\) be their summed Riesz projection, defined by resolvent integration around these eigenvalues, and put \(\Pi_s=I-\Pi_u\). The resolvent identity gives commuting projections and the invariant decomposition \[E=E_s\oplus E_u,\qquad E_s=\Pi_sE,\quad E_u=\Pi_uE, \qquad P_s=P|_{E_s},\quad P_u=P|_{E_u}.\] The space \(E_u\) is finite dimensional, \(P_u\) is invertible, and the two restricted spectra lie strictly inside and strictly outside the chosen circle, respectively. If \(E_u=\{0\}\), all terms involving it below are omitted. Compactness of the restricted spectra and the spectral radius formula give numbers \(r_s<q<r_u\) such that \[ \|P_s^k\|\le C r_s^k,\qquad \|P_u^{-k}\|\le C r_u^{-k},\qquad k\ge0. \tag{96}\] For example choose \(r_s\) strictly between \(r(P_s)\) and \(q\), and \(r_u\) strictly between \(q\) and the smallest modulus of an eigenvalue of \(P_u\); the spectral radius formula controls all sufficiently large powers, and their finitely many predecessors are absorbed in \(C\). The projections commute with complex conjugation because the enclosed spectral sets are invariant under conjugation.

Fix this \(\delta\). If \(M=\|f\|_Y\), the source weight gives the exact estimate \[ |f^\#(\zeta,\tau)| \le M\rho^\delta(1+|\zeta|^2)^{-3+\delta} =M e^{-\gamma\tau}(1+|\zeta|^2)^{-3+\delta}. \tag{97}\] Indeed \(\sigma^2=\rho(1+|\zeta|^2)\) and \(\rho^3\sigma^{p_0-4}=\rho^\delta(1+|\zeta|^2)^{-3+\delta}\). This proves that \(f^\#(\tau)\in E\). It also proves continuity with values in \(E\): on any compact \(\tau\) interval its spatial tails are uniformly small by (97), while on a fixed spatial ball local Hölder continuity of \(f\) and smoothness of the change of coordinates give uniform continuity in \(\tau\).

Let the one-step source increments be \[l_j=\int_0^1S(1-s)f^\#(j+s)\,\mathrm{d}s,\qquad j\in\mathbb{Z}.\] Then \(\|l_j\|_E\le CMq^j\). Define \[ v_j=\sum_{k=0}^\infty P_s^k\Pi_s l_{j-1-k} -\sum_{k=0}^\infty P_u^{-k-1}\Pi_u l_{j+k}. \tag{98}\] Both series converge absolutely for every integer \(j\). More explicitly, their norms are at most \[CMq^j\left[ q^{-1}\sum_{k=0}^\infty(r_s/q)^k +r_u^{-1}\sum_{k=0}^\infty(q/r_u)^k\right] \le C'Mq^j.\] Shifting the summation indices proves \(v_{j+1}=Pv_j+l_j\). On \([j,j+1]\) define \(v\) by (91) with initial value \(v_j\). The definitions agree at the endpoints and give \[ \|v(\tau)\|_E\le CM e^{-\gamma\tau},\qquad |V(z,t)|\le CM\rho^{-1+\delta}. \tag{99}\] The continuous joining creates no distributional time-boundary term, so (83) holds across every joined face. Interior regularity makes it a classical equation there. The construction is linear and real on real sources. The second sum in (98) selects the outside spectral components using future forcing; its use is permitted precisely because no initial values have been imposed.

The full weighted parabolic estimate

We finish by strengthening (99) to the \(X\) estimate. This step uses \(\delta>0\) as well as \(\delta<1\). For a box centered at \((z_0,t_0)\) with \(r_0^2=|z_0|^2\le\rho_0\), one has \(\sigma_0^2\asymp\rho_0\). On a fixed enlargement of this box, chosen small in units \((\sigma_0,\sigma_0^4)\), the time stays strictly negative, \(\rho\asymp\rho_0\), and \(|z|^2/\rho\) stays bounded. In these units the coefficients \(A\) and \(\sigma_0^4B\) have bounded derivatives of every fixed order, and the principal coefficient has a uniform positive lower bound. The source has scaled \(C^\alpha\) norm at most \(CM\), after division by \(\sigma_0^{p_0-4}\). Also \[|V|\le CM\rho_0^{-1+\delta} \le C'M\sigma_0^{p_0}.\] Lemma 13, with the fixed preceding-time enlargement, therefore gives the required full \(X\) bound on every such box.

For the remaining boxes a separate estimate restores the spatial weight. Fix \(a>0\) and write \[z=ay,\qquad t=a^4s,\qquad U(y,s)=a^{-p_0}V(ay,a^4s),\qquad g(y,s)=a^{4-p_0}f(ay,a^4s).\] Use the fixed annuli \[\mathcal U=\{\tfrac12<|y|<2\},\qquad \mathcal U'=\{\tfrac34<|y|<\tfrac54\}.\] On \(\mathcal U\times(-4,0)\) the rescaled equation is \[ U_s+\frac{G(|y|^2/\sqrt{-s})^2}{16}\Delta_y^2U +\mathfrak b(y,s)U=g,\qquad \mathfrak b(y,s)=(-s)^{-1} B_f\bigl(y/(-s)^{1/4}\bigr). \tag{100}\] All coefficients are independent of \(a\). Their smooth bounds remain uniform as \(s\uparrow0\). To verify the assertion for the zeroth order term explicitly, put \(Y=|y|^2/\sqrt{-s}\) and use \[\mathfrak b(y,s)=-(-s)^{-1}\frac{YG'(Y)}{G(Y)}.\] The expansion of \(G\) in powers of \(Y^{-2}\) gives an expansion of \(-YG'/G\) starting at \(Y^{-2}\). Since \(Y^{-2}=(-s)/|y|^4\), division by \(-s\) leaves a smooth expansion in nonnegative integral powers of \(-s\). Differentiated remainders of arbitrarily high order show that both this coefficient and the principal coefficient extend smoothly to \(s=0\) on the annulus. Away from \(s=0\) their bounds follow from ordinary smoothness of \(G\). The principal coefficient stays uniformly positive throughout.

Furthermore \[\frac{\sigma(ay,a^4s)}a =\bigl(|y|^2+\sqrt{-s}\bigr)^{1/2}\] is bounded above and below on the larger annular slab. A fixed covering by the local boxes of Definition 12 therefore gives \(\|g\|_{C^\alpha(\mathcal U\times(-4,0))}\le CM\). Here and below the norm is parabolic; pairs outside a common small box are controlled by the supremum bound and their fixed positive separation.

The preliminary estimate becomes \[|U(y,s)|\le CM(-s)^{(-1+\delta)/2},\] with no factor of \(a\): indeed \(a^{-p_0}(a^2\sqrt{-s})^{-1+\delta} =(-s)^{(-1+\delta)/2}\). Consequently \[ \|U\|_{L^2(\mathcal U\times(-4,0))}^2 \le C M^2|\mathcal U|\int_0^4 q^{-1+\delta}\,\mathrm{d}q =C M^2|\mathcal U|\frac{4^\delta}{\delta}<\infty. \tag{101}\] The value of \(\delta\) has already been fixed, so the dependence on \(1/\delta\) is harmless.

Apply Lemma 14 to (100) on nested annuli, with the preceding-time buffer from \(-4\) to \(-2\). It gives \[ \sum_{j=0}^4\|D_y^jU\|_{C^\alpha(\mathcal U'\times(-2,0))} +\|U_s\|_{C^\alpha(\mathcal U'\times(-2,0))} \le CM. \tag{102}\] One first applies the estimate with terminal time \(b<0\). On each such truncated slab \(U\) has finite supremum norm by (99); the coefficient, source, and \(L^2\) bounds above are independent of \(b\). Letting \(b\uparrow0\) then proves (102), including Hölder quotients between times arbitrarily close to zero. The estimate uses preceding times and requires no terminal value at \(s=0\).

Finally take an arbitrary remaining center with \(r_0^2>\rho_0\) and choose \(a=r_0\). Its rescaled time satisfies \(-1<s_0<0\) and \(1<\sigma_0/a<\sqrt2\). The entire defining \(X\) box lies in \(\mathcal U'\times(-2,0)\) when the fixed box radius \(a_0\) is small: its spatial radius is at most \(\sqrt2a_0\) and its time radius at most \(4a_0^4\). In particular a box near the dividing time \(s=-1\) has the required preceding-time buffer. Rescaling (102) gives its weighted \(X\) norm at most \(CM\). The core and annular cases cover all centers. Thus \(\|V\|_X\le C\|f\|_Y\), completing the proof of Theorem 18.

Background estimates and the compact error

Write \[\begin{align*} g_c&=1+v_*F=x+(1-x)G(x/\rho),& A_c&=g_c^2,& B_c&=2v_*g_c\mathcal AF,\\ g_f&=1+xF=G(x/\rho),& A_f&=g_f^2,& B_f&=2xg_f\widehat\mathcal AF. \end{align*}\] Here \(B_f\) is in the original flat variables, before the similarity change of variables in Section 5. Set \[ \mathcal P=\partial_t+A_c\mathcal A+B_c,\qquad \widehat\mathcal P=\partial_t+A_f\widehat\mathcal A+B_f,\qquad E_c=F_t+A_c\mathcal AF. \tag{106}\] The profile equation gives \(F_t+A_f\widehat\mathcal AF=0\). Moreover, if \[\mu=\min(1,\inf G)>0,\qquad M=\max(1,\sup G),\] then \[ \mu\le g_c\le M. \tag{107}\]

We make the meaning of all the scaled estimates explicit. In a box with center \((z_0,t_0)\) and center scale \(b=\sigma(z_0,t_0)\), \(O_{\mathrm{sc}}(b^q)\) means that, for every pair of nonnegative integers \(j,k\), the spatial derivatives of order \(j\) and time derivatives of order \(k\) are bounded by \(C_{j,k}b^{q-j-4k}\) on that box. The scale \(b\) is held fixed when differentiating. These bounds also give the corresponding scaled parabolic Hölder bounds of any exponent less than one, on slightly smaller boxes. The notation will only be used for smooth background quantities, not for arbitrary members of \(X\) or \(Y\).

One consequence that we will use repeatedly is that the local \(Y_\vartheta\) size of \(E_c\), for either \(\vartheta=\alpha\) or \(\vartheta=\alpha'\), is at most \[ C b^{6-2\delta}b^{-4}=C b^{2-2\delta} \tag{113}\] near the core. The same statement holds with any fixed finite number of extra scaled derivatives. Globally \(\lVert E_c\rVert_{Y_\vartheta}\le C\). If it is multiplied by a smooth cutoff supported in \(r\le C_0\beta\) and varying only where \(r\) is comparable to \(\beta\), then, whenever \(T\le\beta^4\), \[ \lVert \text{cutoff}\,E_c\rVert_{Y_\vartheta} \le C\beta^{2-2\delta}. \tag{114}\] Indeed every box meeting this support has \(b\le C\beta\); cutoff derivatives have bounded scaled size. The strict inequality \(\delta<1\) is used here.

Localization of the ancient inverse

Multiplying a flat correction by a spatial cutoff produces a commutator on the transition annulus. This error is bounded but need not be small. We will include its negative in the forcing for the outer forward problem, so it cancels in the combined equation. The errors that remain are the geometric mismatch, made small by shrinking the core, and the outer cutoff commutator, made small by shortening the time interval. The nested cutoffs below ensure that these cancellations are exact.

Choose once and for all a smooth function \(\chi:[0,\infty)\to[0,1]\), equal to one on \([0,1]\) and zero on \([2,\infty)\). All its compositions below are smooth at \(r=0\) because they are constant there. Take \(\beta>0\) so small that \(r\le5\beta\) lies in the common coordinate neighborhood above, and assume \(T\le\beta^4\). Define \[ \chi_i=\chi(r/\beta),\qquad \chi_b=\chi(r/(2\beta)),\qquad \chi_o=1-\chi(2r/\beta),\qquad \kappa=1-\chi(4r/\beta). \tag{115}\] Thus the inner input cutoff \(\chi_i\) is supported in \(r\le2\beta\); the inner output cutoff \(\chi_b\) is one there and supported in \(r\le4\beta\). The outer output cutoff \(\chi_o\) is zero for \(r\le\beta/2\) and one for \(r\ge\beta\). The coefficient cutoff \(\kappa\) is already one for \(r\ge\beta/2\) and is zero for \(r\le\beta/4\).

Figure 1 records these nested regions. In particular, the inner output cutoff is one on the support of the inner input, and the modified outer coefficients agree with the compact coefficients where the outer output cutoff can be nonzero. The cutoff commutators are supported on the corresponding transition annuli.

Apply Theorem 18 to obtain \[W_f=\widehat D\mathcal E_i f,\qquad V_i=\chi_b W_f,\] where \(V_i\) is regarded as a function on the sphere and extended by zero. Invariant norm equivalence and the cutoff bounds give \(\lVert V_i\rVert_{X_\alpha}\le C\lVert f\rVert_{Y_\alpha}\). Define \[ R_i f=(\mathcal P-\widehat\mathcal P)(\chi_bW_f),\qquad J_i f=[\widehat\mathcal P,\chi_b]W_f. \tag{117}\] Both expressions are taken in the common coordinates and extended by zero. Since \(\chi_b f_i=f_i\), the exact identity is \[ \mathcal PV_i=f_i+R_i f+J_i f. \tag{118}\] Lemma 20 and \(b\le C\beta\) on the relevant boxes give \[ \lVert R_i f\rVert_{Y_\alpha}\le C\beta^2\lVert f\rVert_{Y_\alpha}, \qquad \lVert J_i f\rVert_{Y_\alpha}\le C\lVert f\rVert_{Y_\alpha}. \tag{119}\] Here \(R_i f\) is supported in \(r\le4\beta\), and \(J_i f\) in \(2\beta\le r\le4\beta\). The commutator has order at most three; on its support \(b\) is comparable to \(\beta\), so every derivative of \(\chi_b\) is uniformly bounded in scale units. The definition of \(R_i\) includes the mismatch applied to the cutoff itself, and hence accounts for all mixed cutoff and mismatch terms.

The outer Cauchy problem and the exact right inverse

Set \[s_\beta=\max(\beta,r),\qquad \eta=T/\beta^4,\] and use the nonsingular weighted spaces with scale \(s_\beta\). Equivalently one can use the smooth comparable scale \((\beta^2+r^2)^{1/2}\); the norms and all following estimates are uniformly equivalent. Define a modified operator on the entire sphere by \[ \mathcal P_\beta=\partial_t+A_\beta\mathcal A+B_\beta, \qquad A_\beta=\kappa A_c+(1-\kappa),\qquad B_\beta=\kappa B_c. \tag{120}\] Its principal multiplier is bounded above and below by positive constants, and \(\mathcal P_\beta=\mathcal P\) for \(r\ge\beta/2\). Lemma 20 shows, for every \(j,k\), uniform bounds for the derivatives of \(A_\beta\) and \(s_\beta^4B_\beta\) in units \((s_\beta,s_\beta^4)\), with the scale fixed at the center of each box. On the transition annulus \(r\asymp\beta\) these are exactly its scaled background bounds and the cutoff bounds. Inside \(r\le\beta/4\) the coefficients are \(1\) and \(0\); outside the transition they are the original coefficients with \(\sigma\asymp s_\beta\). The sphere coefficients of \(\mathcal A\) also have uniform bounds in these units. These facts verify all coefficient and ellipticity hypotheses of Lemma 15, uniformly for \(\eta\le1\).

The outer source is \[ f_o=f-f_i-J_i f. \tag{121}\] It vanishes for \(r<\beta\) and has \[ \lVert f_o\rVert_{Y_\alpha(s_\beta)}\le C\lVert f\rVert_{Y_\alpha}. \tag{122}\] Indeed \(s_\beta\asymp\sigma\) on its support and on every sufficiently small box meeting that support. The latter assertion follows from the Lipschitz bound for \(r\) and the box radius \(a_0\sigma\); it also handles boxes crossing a support boundary. The same comparison holds for outputs supported in \(r\ge\beta/2\).

For each fixed \(\beta,T\), solve the closed-manifold Cauchy problem \[ \mathcal P_\beta Z_o=f_o,\qquad Z_o(-T)=0. \tag{123}\] The modified coefficients extend smoothly to \(t=0\) by (111), and \(f_o\) extends Hölder continuously there: for fixed \(\beta\) its nonsingular weighted norm is an ordinary Hölder bound on a compact cylinder. Thus the solvability assertion in Lemma 15 applies, or one may solve on truncated cylinders and use its uniform bounds. Uniqueness makes \(Z_o\) invariant and the solution operator real linear.

Choose a fixed \(\eta_0>0\) small enough for Lemma 15 at both exponents \(\alpha\) and \(\alpha'\). That Lemma gives, for \(0<\eta\le\eta_0\), \[\begin{align*} \lVert Z_o\rVert_{X_\alpha(s_\beta)} &\le C\lVert f_o\rVert_{Y_\alpha(s_\beta)},\tag{124}\\ \sup s_\beta^{-p_0}\lvert Z_o\rvert &\le C\eta\lVert f_o\rVert_{Y_\alpha(s_\beta)}. \tag{125}\end{align*}\] The constants are uniform in \(\beta,T\) by the coefficient and scale bounds verified above. The small parameter in Lemma 15 is precisely \(\eta=T/\beta^4\), since \(\inf s_\beta=\beta\). That lemma applies to zero initial data even when \(f_o(-T)\) is nonzero.

Put \(V_o=\chi_o Z_o\). On the support of this product the coefficients have already become those of \(\mathcal P\), and \(\chi_o f_o=f_o\). Consequently \[ \mathcal PV_o=f_o+J_o f,\qquad J_o f=[\mathcal P_\beta,\chi_o]Z_o. \tag{126}\] This commutator is supported in \(\beta/2\le r\le\beta\) and involves spatial derivatives of \(Z_o\) of order at most three. For these derivatives, Lemma 16 and (124)–(125) give the scaled parabolic \(C^\alpha\) bound \(C\eta^{(1-\alpha)/4}\lVert f_o\rVert_{Y_\alpha(s_\beta)}\). Since \(s_\beta\asymp\sigma\asymp\beta\) on the commutator annulus, \[ \lVert J_o f\rVert_{Y_\alpha} \le C\eta^{(1-\alpha)/4}\lVert f\rVert_{Y_\alpha}. \tag{127}\] In particular, no fourth derivative needs to be made small for this commutator estimate.

Smallness of the corrected error

The right inverse need not have small norm. What is needed for the nonlinear problem is its small value on the particular error \(E_c\) in the weaker space \(X'_\alpha\).

The nonlinear correction and finite-time obstruction

Integer intervals and the two series recurrences

Put \(Q_{\rm int}=2^{256}\). An integer pair \((a,b)\) represents the closed interval \([a/Q_{\rm int},b/Q_{\rm int}]\). All arithmetic in verification/certificate.py is integer arithmetic. Addition and multiplication by an integer are exact, with endpoint reversal for a negative multiplier. For multiplication, let \(m\) and \(M\) be the minimum and maximum of the four products of integer endpoints. The output pair is \[\left(\left\lfloor\frac{m}{Q_{\rm int}}\right\rfloor, \left\lceil\frac{M}{Q_{\rm int}}\right\rceil\right).\] For a same-sign interval \((a,b)\) excluding zero, the reciprocal pair is \[\left(\left\lfloor\frac{Q_{\rm int}^2}{b}\right\rfloor, \left\lceil\frac{Q_{\rm int}^2}{a}\right\rceil\right).\] These formulas are valid also for negative intervals. Division by a positive integer rounds each endpoint outward in the same way. Python’s integer division is floor division, so the identity \(\lceil x\rceil=-\lfloor-x\rfloor\) implements the required upper roundings. In particular, rat(a,b) encloses the exact rational \(a/b\). The function mag bounds absolute value; the function norm bounds the matrix norm specified above.

The operation widen enlarges an interval at each end by the upper endpoint of a nonnegative error allowance. The operation mid instead selects the grid point \[Q_{\rm int}^{-1}\left\lfloor\frac{a+b}{2}\right\rfloor.\] For an interval of real width \(w\), its distance from every enclosed number is at most \(w/2+Q_{\rm int}^{-1}\). The origin and block estimates below explicitly charge this error; midpoint selection is never used as an enclosure operation.

Here are the coefficient identities underlying the computation. If \(z(\theta)=\sum_{i\ge0}z_i\theta^i\) and \(b(\theta)=1/z(\theta)=\sum_{i\ge0}b_i\theta^i\), then \[ b_0=z_0^{-1},\qquad b_i=-z_0^{-1}\sum_{t=1}^i z_t b_{i-t}\quad(i\ge1). \tag{139}\] Writing a dot for any one parameter variation gives, for \(i\ge0\), \[ \dot b_i=-z_0^{-1} \left(\sum_{t=0}^i\dot z_t b_{i-t} +\sum_{t=1}^i z_t\dot b_{i-t}\right). \tag{140}\] The second sum is empty at \(i=0\). In the first sum the coefficient \(b_i\) has already been computed by (139). This is exactly the ordering in recip; it includes the variation of the constant coefficient \(z_0\).

At the origin, write \(r=s^{3/2}\) locally and \[z=1+p=1+d+kr+\sum_{i\ge2}a_i r^i.\] Equation (34) gives \[ a_i=\frac{b_{i-2}-\mathbf1_{\{i=2\}}} { (27/8)(i-1)(i+9)(i+10)},\qquad i\ge2, \tag{141}\] where now \(b_i\) are the coefficients of the reciprocal series in \(r\). The loop in origin computes \(a_2,\ldots,a_{60}\), together with their two variations under the parameter map (35). The weights for evaluation of \(p,p',p''\) at \(s=1\) are \[1,\qquad \frac{3i}{2},\qquad \frac{3i}{2}\left(\frac{3i}{2}-1\right).\] The code subtracts the constant \(1\) from \(z\) only in the zeroth state entry, as required.

For an ordinary Taylor step, let \(\ell=1/h\) be the block length, \(s_c\) the step’s initial point, \(\theta=h(s-s_c)\), and \(\mathfrak r=hs_c+\theta\). The equation for \(z=1+p\), with \(E=1+d\), is \[ 4\mathfrak r^3 z_{\theta\theta\theta} +120\mathfrak r^2 z_{\theta\theta} +751\mathfrak r z_\theta-1215(z-E) =\frac{4}{h^3}\mathfrak r^3(z^{-1}-1). \tag{142}\] This identity also gives the four tangent equations upon differentiation. For completeness, put \(\mathfrak R=8hs_c\) and \(F_i=b_i-\mathbf1_{\{i=0\}}\), with negative-index coefficients zero. The scalar coefficient update used in evolve is \[\begin{align*} 4(i+1)(i+2)(i+3)\mathfrak R^3z_{i+3} ={}&\frac4{h^3}\sum_{a=0}^3 c_a\mathfrak R^{3-a}F_{i-a}\\ &-(96i+960)\mathfrak R^2(i+1)(i+2)z_{i+2}\\ &-64\mathfrak R(i+1) \bigl(12i(i-1)+240i+751\bigr)z_{i+1}\\ &-512\bigl(4i(i-1)(i-2)+120i(i-1)+751i-1215\bigr)z_i\\ &-1215\cdot512E\mathbf1_{\{i=0\}}, \qquad(c_0,c_1,c_2,c_3)=(1,24,192,512). \tag{143}\end{align*}\] Indeed \(c_a\mathfrak R^{3-a}\) are the coefficients of \((\mathfrak R+8\theta)^3\). The initial three coefficients are \(z_0,z_\theta(0),z_{\theta\theta}(0)/2\). For a tangent column one differentiates every term of (143); the constant \(E\) has variation \(0,0,0,1\) in the four columns. Thus the same update computes both state and tangent coefficients. The values \(i=0,\ldots,43\) produce coefficients through degree \(46\). Horner evaluation at \(\theta=1/8\), after the appropriate one or two differentiations, evaluates the three state entries through degrees \(46,45,44\), respectively. The analytic justification for their common remainder allowance is given in Subsection 7.3.

The complete error at the origin

The parameter family of Lemma 7 is holomorphic on the complex polydisc \[|k-k_0|\le .1,\qquad |d-d_0|\le .1,\] and its coefficients satisfy \[ \sum_{i\ge1}|a_i|3^i<.8,\qquad a_1=k. \tag{144}\] One can check the constants directly from (141): the nonfree denominators exceed \(440\), \(3|k|<.585\), and \(|1+d|>1.30\). The majorant improves itself because \[.585+\frac9{440}\left(1+\frac1{1.30-.8}\right) =\frac{711}{1100}<.8.\] The reciprocal remains nonsingular throughout this construction. In particular the real solutions are positive up to \(s=1\). Also \(|d|<.501\), and (144) gives \[|p(1)|<.501+\frac{.8}3<.768,\quad |p'(1)|\le .8\max_{i\ge1}\frac{3i/2}{3^i}\le .4,\quad |p''(1)|\le .8\max_{i\ge1} \frac{(3i/2)(3i/2-1)}{3^i}<.534.\] In particular the uniform bound \(2\) for every jet entry is available on the parameter polydisc.

Let \[ E_{60}=.8\,\frac{(3\cdot61/2)^2}{3^{61}} <5.267\cdot10^{-26}. \tag{145}\] The sequence \((3i/2)^2/3^i\) decreases for \(i\ge61\). Consequently \(E_{60}\) bounds the omitted tail of each of the three jet entries. Cauchy’s estimate in either parameter bounds the corresponding first derivative of this tail at the center by \(10E_{60}\). For the matrix in (35), the sum of the absolute values of all four entries, including the factor \(10^{-8}\), is \(.01169933\). Thus the row norm of the omitted tail of the two \(z_*\)-variation columns is at most \(.12E_{60}\).

Each of the two original parameter displacements has absolute value at most \(.011R\). The multivariable Cauchy estimate, with \(t_*=.11R\), bounds the terms of total degree at least two in the parameter expansion of any jet entry by \[ E_{\rm par} =2\sum_{j\ge2}(j+1)t_*^j =2\bigl((1-t_*)^{-2}-1-2t_*\bigr) <1.816\cdot10^{-20}. \tag{146}\] The factor \(j+1\) counts the monomials of total degree \(j\) in two parameters.

The width assertion in origin bounds every computed state or variation interval by \(10^{-20}\). Let \(U_0\in\mathbb{R}^3\) be the three selected state midpoints. Let \(Z\) have their two selected variation columns in its first three rows, and in its fourth row the exact second row of the parameter map (35). The code stores intervals containing that exact fourth row; its matrix estimates therefore apply to this choice of \(Z\). Combining the state midpoint error, the state and variation tails, the two variation midpoint errors, and (146), gives \[\begin{align*} \left\| \begin{pmatrix}p(1)\\p'(1)\\p''(1)\\d\end{pmatrix} -\begin{pmatrix}U_0\\d_0\end{pmatrix}-Zz_* \right\|_\infty &\le 5\cdot10^{-21}+E_{60}+E_{\rm par} +.12E_{60}R+10^{-20}R+\frac3{Q_{\rm int}}\\ &<2.316\cdot10^{-20}<2\cdot10^{-19}. \tag{147}\end{align*}\] The fourth component of the error is exactly zero. All uncertainty in the interval construction of the central rational parameters is already included in the width assertion.

Analytic continuation and Taylor remainder on a step

The 288 blocks run from \(s=1\) to \(s=56\), with the following lengths: \[\begin{array}{c|c|c} \text{range}&\ell&\text{number of blocks}\\ \hline {[1,2]}&1/32&32\\ {[2,4]}&1/16&32\\ {[4,8]}&1/8&32\\ {[8,56]}&1/4&192. \end{array}\] Each block consists of eight Taylor steps of length \(\ell/8\). At every step start the interval assertions imply, for the exact reference state, \[ -.76<p(0)<.4,\qquad |p_\theta(0)|<.06,\qquad |p_{\theta\theta}(0)|<.01. \tag{148}\] Every entry in the first three rows of the four-column scaled variation matrix has modulus less than \(6\); its fourth row is \((0,0,0,1)\). The step schedule gives \(hs_c\ge32\).

We now justify an analytic bound on the full complex disk \(|\theta|\le1/2\), using only these starting bounds. On a disk where \(|p(\theta)-p(0)|\le.1\), one has \[|\mathfrak r|\ge31.5,\qquad |1+p|>.14,\qquad |p|<.86,\qquad |p-d_0|<1.3.\] Let \(H_2\) be the supremum of \(|p_{\theta\theta}|\) on any smaller concentric disk. Integrate the equation for \(p_{\theta\theta}\) along radial line segments and apply the integral Gronwall inequality. Since \[\frac{30}{31.5}<.954,\qquad \frac{B}{31.5^2}<.190,\qquad \frac{|C|}{31.5^3}<.011,\qquad h^{-3}\le.015625,\] and \(|p_\theta|\le .06+H_2/2\), this gives \[ H_2\le e^{.954/2}\left[ .01+\frac12\left\{.190(.06+H_2/2) +.011\cdot1.3+.015625\frac{.86}{.14}\right\}\right]. \tag{149}\] The coefficient of \(H_2\) on the right is less than one. Solving this linear inequality yields \(H_2<.123602<.13\). Therefore \[ |p(\theta)-p(0)|\le .06/2+.13/8=.04625<.1. \tag{150}\] This strictly improves the bootstrap. On every disk of radius at most \(1/2\), the state and its first two derivatives are bounded and both denominators \(\mathfrak r\) and \(1+p\) stay uniformly away from zero. The local holomorphic ODE theorem then continues the solution across any putative boundary of its disk of existence. Uniqueness makes the continuations agree. Thus the solution is holomorphic on a neighborhood of the closed disk of radius \(1/2\), with the displayed bounds.

In the scaled state \((p,p_\theta,p_{\theta\theta},d)\), the Jacobian is \[ \mathcal J(\theta)= \begin{pmatrix} 0&1&0&0\\ 0&0&1&0\\ -C/\mathfrak r^3-h^{-3}/(1+p)^2& -B/\mathfrak r^2&-30/\mathfrak r&C/\mathfrak r^3\\ 0&0&0&0 \end{pmatrix}. \tag{151}\] On the complex disk its norm is bounded by the maximum of \(1\) and \[\frac{30}{31.5}+\frac{751/4}{31.5^2} +\frac{1215/2}{31.5^3}+\frac{1/64}{.14^2}<2.1.\] Each actual variation column, with initial norm at most \(6\), consequently has norm at most \(6e^{2.1/2}<17.146<32\) throughout the disk. The state entries themselves also have modulus less than \(32\). These estimates apply to the actual block variation already propagated from the block’s initial identity matrix; a new identity is not assumed at each substep.

If a scalar holomorphic function is bounded by \(32\) on this disk, Cauchy’s coefficient estimate gives a tail after degree \(44\), at \(\theta=1/8\), bounded by \[ 32\sum_{i\ge45}\left(\frac{1/8}{1/2}\right)^i =\frac{128}{3}\,4^{-45}<48\,4^{-45}. \tag{152}\] Every state entry and every tangent entry is approximated to at least degree \(44\). Hence the allowance \(48\,4^{-45}\) added by evolve encloses all their Taylor tails. The interval recurrences then inductively enclose the exact reference solution and its exact first variation through all eight substeps.

On a real substep, the sharper displacement bound is \[ |p(\theta)-p(0)| \le .06/8+\frac{.13}{2\cdot8^2} =.008515625<.01. \tag{153}\] Accordingly, the smallest lower starting endpoint for \(1+p\), reduced by \((\lfloor Q_{\rm int}/100\rfloor+1)/Q_{\rm int}>.01\), is a valid lower bound \(m_j\) on the entire reference block. This is the variable mini divided by \(Q_{\rm int}\).

The finite inequalities and their exact interpretation

Write the block endpoints as \(s_0=1,\ldots,s_N=56\), \(N=288\). On block \(j\), the reference solution has initial physical jet \(U_j\) and parameter \(d_0\); it is evolved exactly on that block. The rounded jet selected for the next block is \(U_{j+1}\). Let \(M_j\) be the exact \(4\times4\) transfer of the linearized four-component physical state along this reference block.

The conversion from scaled derivatives to physical derivatives is essential. With \[S_j=\mathop{\mathrm{diag}}(1,\ell_j,\ell_j^2,1),\] a scaled transfer \(T\) becomes \(S_j^{-1}TS_j\). Thus its entry in row \(a\), column \(b\), for \(a,b\in\{0,1,2\}\), is multiplied by \(h_j^a/h_j^b\); column \(3\) is multiplied by \(h_j^a\). The fourth row remains \((0,0,0,1)\). The code performs this conversion before asserting the widths. Consequently its assertions imply \[ \big\|U_{j+1}- (p_{\rm ref},p'_{\rm ref},p''_{\rm ref}) (s_{j+1})\big\|_\infty<10^{-19}, \qquad |(M_j-\widetilde M_j)_{ab}|<10^{-15}, \tag{154}\] where \(\widetilde M_j\) is the selected matrix of transfer midpoints. The state and transfer widths are, respectively, less than \(10^{-19}\) and \(10^{-15}\); the stated looser midpoint bounds follow from the midpoint estimate in Subsection 7.1. All fourth-component errors in (154) are zero.

Define the products of these point matrices by \[\widetilde T_{i,j}= \widetilde M_{i-1}\cdots\widetilde M_j\quad (i>j), \qquad \widetilde T_{i,i}=I_4.\] The exact integer comparisons in check give \[\begin{align*} m_j&>.23,\qquad \|Z\|<.02,\\ \max_{i\ge j}\|\widetilde T_{i,j}\|&\le8500, &\max_i\sum_{j<i}\|\widetilde T_{i,j+1}^{(3)}\|&\le570000,\\ \max_i\sum_{j<i}\frac{\ell_j}{m_j^3} \|\widetilde T_{i,j+1}^{(3)}\|&\le246000, &\max_i\|\widetilde T_{i,0}Z\|&\le2.7. \tag{155}\end{align*}\] Here and below a maximum over endpoints includes endpoint \(0\); the assertion for it follows from \(\|Z\|<.02\).

To explain the summation indices, for each endpoint \(i\) the inner loop starts with the identity and takes \(j=i-1,\ldots,0\). Immediately before multiplying by \(\widetilde M_j\), its current interval matrix encloses \(\widetilde T_{i,j+1}\). The update T=mm(T,allm[j]) therefore produces the desired forward product in the correct order. If \(r_j=Q_{\rm int}m_j\) and \(\mathfrak d\) is the integer interval bound for the leading \(3\times3\) norm, then \[\frac{\mathfrak d Q_{\rm int}^{\,2}}{h_jr_j^3} =\frac{\ell_j}{m_j^3}\frac{\mathfrak d}{Q_{\rm int}}.\] The expression used for K is its floor plus one, hence an upper bound even when the quotient is integral. Thus the interval product arithmetic and the weighted sum both round in the required direction.

Let \(P_{\rm a}(d)=(p_{\rm a},p'_{\rm a},p''_{\rm a})(56)\), and put \(Y=(\widetilde T_{N,0}Z)_{\mathrm{top}\,3}\). The final comparisons in check give \[\begin{align*} \|U_N-P_{\rm a}(d_0)\|_\infty&<.04R,& \|\Lambda(U_N-P_{\rm a}(d_0))\|_\infty&<.006R,\\ \|(\Lambda Y)_{1,\cdot}\|_{\rm row}&<.116,& \|(\Lambda Y)_{\mathrm{bottom}\,2}-I_2\|&<.001. \tag{156}\end{align*}\] The jet factors used for the term \(d_m s^{-3m}\) are \(1,-3m,3m(3m+1)\), with denominators \(56^{3m}\), \(56^{3m+1}\), \(56^{3m+2}\), respectively. After multiplication by \(d_m\), these give its value and first two derivatives at \(s=56\). The index test in the last assertion subtracts the \(2\times2\) identity from the two nonreal-coordinate rows.

The listed comparisons are finite integer statements: the displayed source constructs their operands and asserts them. Subsection 7.8 gives the command for checking them with assertions enabled. The preceding recurrence and disk arguments prove that these finite statements bound the specified analytic trajectories and variations.

Transfer errors and estimates within a block

Let \(T_{i,j}=M_{i-1}\cdots M_j\) be the exact products. By (154), \(\|M_j-\widetilde M_j\|<4\cdot10^{-15}\). Expand a product of at most \(288\) factors, choosing the error factor at \(k\) positions. Each intervening, possibly empty, product of approximate factors has norm at most \(8500\), by (155). Summing the bounds for \(k\ge1\) gives \[ \|T_{i,j}-\widetilde T_{i,j}\| \le8500\left[(1+4\cdot10^{-15}\cdot8500)^{288}-1\right] <.001. \tag{157}\] In particular \(\|T_{i,j}\|<8501\). Since the lengths of all the blocks sum to \(55\), (155) implies \[ \max_i\sum_{j<i}\|T_{i,j+1}^{(3)}\|\le580000,\qquad \max_i\sum_{j<i}\frac{\ell_j}{m_j^3} \|T_{i,j+1}^{(3)}\|\le250000, \tag{158}\] using the elementary comparisons \[570000+288\cdot.001<580000,\qquad 246000+\frac{55\cdot.001}{.23^3}<250000.\] Also \[ \|T_{i,0}Z\|\le2.7+.001\cdot.02<2.72. \tag{159}\]

We need two estimates for transfers between any two ordered points of one reference block, including points in different Taylor substeps. The real scaled Jacobian (151) has norm at most \[L_{\rm real}= \max\left\{1,\frac{30}{32}+\frac{751/4}{32^2} +\frac{1215/2}{32^3} +\frac{1/64}{.23^2}\right\}.\] An interval of at most one block unit therefore has scaled transfer norm at most \(e^{L_{\rm real}}<4.3\). Let \(\Phi_j(s,t)\) denote its physical transfer. Conjugation by \(S_j\) does not increase either the first-row norm or the maximum norm of the third column: in the first row the scaling factors are \(1,\ell_j,\ell_j^2,1\), and in the third column they are \(\ell_j^2,\ell_j,1,\ell_j^2\). Thus, with \(e_3=(0,0,1,0)^t\), \[ \|(\Phi_j(s,t))_{1,\cdot}\|_{\rm row}\le4.3,\qquad \|\Phi_j(s,t)e_3\|_\infty\le4.3 \quad(s_j\le t\le s\le s_{j+1}). \tag{160}\] Only these two parts of a physical transfer are needed. Its full norm need not satisfy the same estimate.

All exponential comparisons just used, and the one in (149), can be checked by rational arithmetic. For \(0\le x<82\), let \[E_{80}(x)=\sum_{j=0}^{80}\frac{x^j}{j!} +\frac{x^{81}}{81!}\frac1{1-x/82}.\] The positive exponential tail has successive ratios at most \(x/82\), so \(e^x\le E_{80}(x)\). Substituting the displayed rational values in this formula verifies all the strict numerical comparisons. The supplementary verification/arithmetic_allowances.py performs these rational checks, including (147), (157), and the remaining comparisons below.

Uniform nonlinear error and closure of the bootstrap

Fix \(z_*\in[-R,R]^2\). On each block compare its actual shot with the reference solution on that block, and write \(\delta p=p-p_{\rm ref}\). For the only nonlinear term in the physical third-component equation, \[-\frac{p}{1+p} +\frac{p_{\rm ref}}{1+p_{\rm ref}} +\frac{\delta p}{(1+p_{\rm ref})^2} =\frac{(\delta p)^2}{(1+p_{\rm ref})^2(1+p)}.\] It follows that the difference equation, beyond the reference linearization, has forcing \(e_3\mathcal N\), where \[ |\mathcal N|= \frac{|\delta p|^2}{|1+p_{\rm ref}|^2|1+p|}. \tag{161}\] The \(d\)-dependence is linear and is already included in the fourth variation component.

Temporarily assume \(|\delta p|\le14R\) throughout the portions of the blocks reached so far. Since \(m_j>.23\) and \(m_j/(m_j-14R)<1.001\), (161) implies \[|\mathcal N|\le\frac{1.001(14R)^2}{m_j^3}.\] The error accumulated by the end of block \(j\), before multiplication by subsequent complete-block transfers, has zero fourth component. Its norm is bounded by \[10^{-19} +4.3\,\ell_j\frac{1.001(14R)^2}{m_j^3}.\] The first term is the reference jump (154); the second follows by variation of constants and the third-column estimate in (160).

Define the endpoint error \(e_i\) by \[\begin{pmatrix}p(s_i)\\p'(s_i)\\p''(s_i)\\d\end{pmatrix} =\begin{pmatrix}U_i\\d_0\end{pmatrix} +T_{i,0}Zz_*+e_i.\] Its fourth component is zero. By (147), (158), and the preceding local error estimate, \[\begin{align*} \|e_i\|_\infty &\le580000\cdot10^{-19}+8501\cdot(2\cdot10^{-19}) +4.3\cdot250000\cdot1.001(14R)^2\\ &=.1055747504R<.11R. \tag{162}\end{align*}\] For an interior point of a block, the first-row estimate in (160), (159), and the forcing estimate give \[\begin{align*} |\delta p| &\le4.3(2.72+.11)R +4.3\frac{1.001(14R)^2}{.23^3}\\ &<12.169035R<14R. \tag{163}\end{align*}\] Here replacing the remaining block length by \(1\) is a harmless overestimate. This improves the bootstrap strictly both at block starts and in block interiors, including the reference jumps. At \(s=1\) it starts from \(\|Z\|<.02\) and (147). Continuation over the finitely many blocks therefore proves (162) at every block endpoint.

For clarity, existence is part of this continuation argument. As long as the bootstrap holds, \(1+p\ge .23-14R>0\). On the finite interval \(1\le s\le56\), the equations for \(p'\) and \(p''\) form a linear system with bounded coefficients and bounded forcing once \(p\) stays in this range. Gronwall’s inequality bounds these derivatives on every such interval. Thus no other finite-time escape can interrupt continuation. Positivity before \(s=1\) was established in Subsection 7.2. This proves the existence and positivity statements of Proposition 24.

Projection at the matching point

Let \(\Gamma\) denote the exact real coordinate matrix of (41). Lemma 8 gives \[|\Gamma_{ab}-\Lambda_{ab}|<.001, \qquad \|\Gamma_{1,\cdot}\|_{\rm row}<.92,\qquad \|\Gamma_{a,\cdot}\|_{\rm row}<.68\quad(a=2,3).\] Thus \(\|\Gamma-\Lambda\|<.003\). The polynomial \(P_{\rm a}\) also satisfies \[ \|P_{\rm a}(d)-P_{\rm a}(d_0)\|_\infty \le .01|d-d_0|\le .01\cdot.011R; \tag{164}\] this follows by differentiating the four explicit coefficients in (40), as detailed in the proof of Lemma 9.

Set \(A_*=U_N-P_{\rm a}(d_0)\). The endpoint error and transfer error give \[\begin{pmatrix}p\\p'\\p''\end{pmatrix}(56)-P_{\rm a}(d) = A_*+Yz_*+\epsilon_* -\bigl(P_{\rm a}(d)-P_{\rm a}(d_0)\bigr), \qquad \|\epsilon_*\|_\infty\le \bigl(.11+.001\cdot.02\bigr)R.\] In fact \(\epsilon_*\) is the sum of the top three entries of \(e_N\) and \(((T_{N,0}-\widetilde T_{N,0})Zz_*)_{\mathrm{top}\,3}\). Apply one row of \(\Gamma\). The contribution of \(A_*\), using (156), is at most \((.006+.003\cdot.04)R\). For the first row the linear contribution is bounded by \((.116+.003\cdot2.7)R\). For row \(a=2,3\), after subtracting the corresponding coordinate of \(z_*\), it is bounded by \((.001+.003\cdot2.7)R\). The remaining contribution is bounded by the row norm of \(\Gamma\) times \[\bigl(.11+.001\cdot.02+.01\cdot.011\bigr)R.\] Consequently, in units of \(R\), the two required estimates are \[\begin{align*} |q|/R &\le .006+.003\cdot.04+.116+.003\cdot2.7 +.92(.11+.001\cdot.02+.01\cdot.011)\\ &=.2315396<.25,\\ \big\|(\Re\xi,\Im\xi)-z_*\big\|_\infty/R &\le .006+.003\cdot.04+.001+.003\cdot2.7 +.68(.11+.001\cdot.02+.01\cdot.011)\\ &=.0901084<.13. \end{align*}\] These are exact rational comparisons and establish (138), completing the proof of Proposition 24.

Executable source and reproducibility

The complete source printed in Appendix 8 is verification/certificate.py, comprising \(6594\) UTF-8 bytes. Its SHA-256 digest is

where the line break is for typesetting only. It uses no imported libraries, floating-point operations, external data, or platform-dependent numerical routines. From the paper directory, the verification command is

python3 verification/certificate.py

with Python assertions enabled (no -O or -OO, and no PYTHONOPTIMIZE setting that disables assertions). The program is silent when every assertion succeeds.

The supplementary rational check

python3 verification/arithmetic_allowances.py

verifies the analytic allowances used above by exact fraction arithmetic; its decimal report is descriptive only. The optional script verification/tail-check.py checks the explicit root and tail estimates in Section [sec:profile] by exact rational arithmetic. The finite integer assertions, the analytic enclosure argument in Subsections 7.2–7.3, and the nonlinear argument in Subsections 7.5–7.7 together prove the finite shooting proposition.