Proof. Restrict the foliation parameter to a compact interval close to zero. Continuity of its parametrizing homeomorphism gives uniform convergence of the leaf curves to the outer boundary. Support functions are continuous for this convergence. Hence the convex bodies bounded by the leaves contain the origin for sufficiently small parameters. Two disjoint convex boundary curves surrounding the origin are nested: otherwise their radial functions change order and the curves intersect. Their compact disjointness makes the nesting strict, including strict ordering of their support functions in every direction. It follows, using the limiting outer boundary, that \(g_b(\theta)\) is continuous and strictly decreasing in \(b\), for each \(\theta\). The intervals it fills, as \(\theta\) varies, form a phase collar.

Every positively oriented supporting line of a leaf in this collar reflects to another positively oriented supporting line of the same leaf. Indeed the reflected line is tangent by the caustic property. For a sufficiently small collar both the incoming and outgoing lines are close to the corresponding positive outer supporting lines. The other orientation of a leaf support has coordinate \(-g_b(\theta+\pi)\), which stays negative because the bodies contain a fixed neighborhood of the origin. Thus this orientation cannot occur. The same argument applies to the inverse reflection. The billiard map therefore restricts to a circle homeomorphism of each graph. It is close to the identity and consequently has degree one. Its lift \(F_b\) is increasing, has strictly positive increments, and depends continuously on \(b\). By [ana:twist], these lifts are strictly ordered as \(b\) increases.

We recall the elementary rotation facts needed here. For an increasing lift \(F\) commuting with translation by \(2\pi\), the oscillation of \(F^m(x)-x\) is at most \(2\pi\). The maximum and minimum of this displacement are respectively subadditive and superadditive in \(m\). It follows that \[\tau(F)=\lim_{m\to\infty}\frac{F^m(x)-x}{m}\] exists, is independent of \(x\), and satisfies \[\left|\tau(F)-\frac{F^m(x)-x}{m}\right|\le\frac{2\pi}{m}.\] This bound gives continuity of \(\tau\) under uniform convergence of lifts. Ordering lifts orders their iterates and hence their translation numbers. A lift with translation number zero has a fixed point: if its displacement never vanishes, its sign is constant and its absolute value has a positive minimum. Applied to \(F^q-2\pi l\), this says that \(\tau(F)=2\pi l/q\) implies the existence of a \(q\)-periodic point. Conversely, such a point forces that translation number.

These facts show that \(\tau(b)\) is continuous and nondecreasing. Its positivity follows from the positive minimum of \(F_b(\theta)-\theta\) on each fixed graph; its limit at zero follows from uniform convergence of the map to the identity.

There cannot be a nontrivial interval of parameters with the same rational translation number \(2\pi l/q\). Otherwise the closed annulus between two distinct graphs in that interval is invariant and has positive finite area. Almost every point in it is recurrent for the \(q\)-th iterate, by finite-measure recurrence. On a graph of the specified rotation number, \(F_b^q-2\pi l\) has fixed points. Every point in a component of their complement moves strictly monotonically towards an endpoint of that component and is not recurrent. Thus the \(q\)-th iterate is the identity almost everywhere in the annulus, and hence everywhere by continuity. This contradicts strict ordering of the \(q\)-th iterates on its two boundary graphs.

Continuity now supplies a unique graph at each sufficiently small positive rational rotation, with smaller and larger rotations arbitrarily close on its two sides. A lift with rotation greater than \(2\pi l/q\) has \(F^q(\theta)-\theta-2\pi l>0\) everywhere: a zero would force that rational rotation, and a strictly negative displacement would force a smaller one. The corresponding inequality is reversed below that rotation. Passing to the graph in question proves [ana:rational-identity]. ◻

Proof. We give the composition estimates, including their width dependence. For a trial mesh in the ball [ana:mesh-norm], set \[X=X_j,\qquad s=X_j-X_{j-1},\qquad t=X_{j+1}-X_j.\] Then \(s,t=\delta+O(q^{-3})\) in \(C^p\). For \(T=t\) or \(T=-s\), and \(0\le\lambda\le1\), write \(\mu=X+\lambda T z_\lambda\). The midpoint equation is \[ \int_0^{z_\lambda} e^{u(X+\lambda Tv)-u(X)}\,\,\mathrm dv =\frac12\int_0^1e^{u(X+\lambda Tv)-u(X)}\,\,\mathrm dv . \tag{11}\] Moving the integrals of the exponential minus one to the other side expresses \(z_\lambda-1/2\) as a contraction on a \(C^p\) ball of radius \(C/q\). Indeed the exponential minus one is \(O(q^{-1})\); its derivative with respect to \(z_\lambda\), after writing the left integral as an integral over \(v=\alpha z_\lambda\), has the same bound. Products and compositions in this argument are estimated by the ordinary finite-order chain rule.

All its evaluation points lie in the original strip. Their maps from \(x\) are real symmetric and have derivative \(1+O(q^{-2})\): \[X'=1+O(q^{-2}),\qquad T'=O(q^{-3}),\qquad z_\lambda'=O(q^{-1}).\] For a real-symmetric map \(x+\rho+e(x)\), with \(\rho\) real and \(\|e'\|\le Cq^{-2}\), integration along the vertical segment from \(\Re x\) gives \[|\Im(x+\rho+e(x))|\le(1+Cq^{-2})|\Im x|.\] Taking \(K\) sufficiently large therefore puts every such point, and every segment used in the integrals, inside \(\mathcal A_\sigma\). This is a relative loss \(K\sigma/q^2\), with no reciprocal power of \(\sigma\).

Implicit differentiation of [ana:midpoint-equation] gives, for \(1\le i\le3\), \[ \|\partial_\lambda^i z_\lambda\|_{p,\sigma'}\le Cq^{-i}. \tag{12}\] To see the scaling, the coefficient of the derivative being solved for is \(e^{u(X+\lambda Tz_\lambda)-u(X)}=1+O(q^{-1})\). Each explicit \(\lambda\)-derivative contributes a factor \(T\); the lower implicit derivatives satisfy the same induction. For a mesh variation, \[\|\dot X\|_{p,\sigma'}\le q^{-2}\|\dot w\|_*, \qquad \|\dot T\|_{p,\sigma'}\le q^{-3}\|\dot w\|_*.\] Differentiating the same equations therefore gives, for \(1\le i\le3\), \[ \|\partial_\lambda^i\dot z_\lambda\|_{p,\sigma'} \le Cq^{-i} \bigl(q^{-2}\|\dot w\|_*+\|\dot u\|_{r_0,\sigma}\bigr). \tag{13}\] For \(i=0\) the right hand side has an additional factor \(q^{-1}\). For example, a difference \(u(B)-u(X)\) is written as the integral of \(u'\) from \(X\) to \(B\). Its variation in a changed basepoint uses a difference of \(u'\), hence retains the factor \(B-X=O(q^{-1})\). This proves the extra factor also for variations, rather than merely for the values.

The rescaled expression for [ana:D1-integral] is \[\begin{split} F(\lambda) &=(\lambda T)^{-2}D_1(X,X+\lambda T)\\ &=e^{u(X)}\int_0^{z_\lambda} \frac{\sin\left(\lambda T\int_0^v e^{u(X+\lambda Tv')}\,\,\mathrm dv'\right)}{\lambda T} e^{-2u(X+\lambda Tv)}\,\,\mathrm dv . \end{split}\] The quotient at zero is removable: write it as \(A_\lambda(v)\operatorname{sinc}(\lambda T A_\lambda(v))\), where \(A_\lambda(v)=\int_0^v e^{u(X+\lambda Tv')}\,\,\mathrm dv'\) and \(\operatorname{sinc}z=\sin z/z\). Thus [ana:midpoint-jets,ana:midpoint-variation] apply to this expression and its first variations. The calculation following [ana:action-expansion] gives \[F(0)=\frac18,\qquad F'(0)=0,\qquad F''(0)=2a(X)T^2.\] More explicitly, its third derivative has \(C^p\) norm at most \(Cq^{-3}\), and the third derivative of its first variation has norm at most \[Cq^{-3}\bigl(q^{-2}\|\dot w\|_*+\|\dot u\|_{r_0,\sigma}\bigr).\] Taylor’s integral remainder gives \[ D_1(X,X+T) =T^2\left(\frac18+a(X)T^2+T^3b(X,T)\right), \tag{14}\] where the evaluated functions \(a(X)\), \(b(X,T)\) have bounded \(C^p\) norms. Their first variations are bounded by \[C\bigl(q^{-2}\|\dot w\|_*+\|\dot u\|_{r_0,\sigma}\bigr).\] In dividing the Taylor remainder by \(T^3\), no small denominator is lost: \(qT\) is \(C^p\)-close to \(2\pi\) or \(-2\pi\), while \(\dot T/T=O(q^{-2}\|\dot w\|_*)\).

We record a finite derivative budget for these statements. Third \(\lambda\)-derivatives of the midpoint and integrand use derivatives of \(u\) through order \(p+3\); their first variations use order \(p+4\). The estimates for differences follow from \[f\circ A-f\circ B =\int_0^1 f'(B+\alpha(A-B))(A-B)\,\,\mathrm d\alpha\] and the \(C^p\) chain rule, with the straight segment inside the input strip. Allowing two further derivatives covers all the coefficient and composition bounds just used. Thus \(p+6\) derivatives already suffice here, fewer than \(r_0=p+13\). There has been no Cauchy estimate on \(u\), so these constants are uniform also as \(\sigma\downarrow0\).

Antisymmetry of \(D\) makes the Euler equation at \(X_j\) \[D_1(X,X+t)-D_1(X,X-s)=0.\] Using [ana:D1-uniform-expansion] and factoring \(t^2-s^2\) rewrites it as \[ w_{j+1}-2w_j+w_{j-1} =-\frac{t^5b(X,t)+s^5b(X,-s)} {(t+s)\bigl(1/8+(t^2+s^2)a(X)\bigr)}. \tag{15}\] The right hand side is \(O(q^{-4})\) in \(C^p\), and its Lipschitz constant in the starred mesh norm is \(O(q^{-6})\). For example its numerator is \(O(q^{-5})\), its denominator has size comparable to \(q^{-1}\), and a unit starred variation changes each coefficient and each relative step by \(O(q^{-2})\). These statements hold for complex values as well, since the denominator is uniformly separated from zero.

The Dirichlet inverse of the second difference has norm at most \(Cq^4\) from uniform \(C^p\) forcing into the starred norm. Indeed, if the forcing is \(f_j\), set \(d_j=w_j-w_{j-1}\). Then \[d_{j+1}=d_j+f_j,\qquad d_1=-\frac1q\sum_{i=1}^{q-1}(q-i)f_i.\] It follows that \(\max\|d_j\|_{C^p}\le Cq\max\|f_j\|_{C^p}\) and \(\max\|w_j\|_{C^p}\le Cq^2\max\|f_j\|_{C^p}\). Equation (15) is consequently a contraction with constant \(Cq^{-2}\). Choose \(R\) larger than its uniform image bound, then choose the width constant and \(N\). The image bound can be taken independently of this fixed \(R\): once \(q\) is large depending on \(R\), all scaled steps and derivatives in the preceding estimates lie in fixed bounded sets.

Differentiating the contraction gives the asserted input derivative bound; comparing its fixed points gives the corresponding Lipschitz bound. The same argument with order-zero real norms proves pointwise uniqueness and works in any larger fixed real ball after increasing \(N\). It follows in particular that constructions made with different fixed smoothness bounds or widths agree on their real domains for all sufficiently large \(q\). ◻

Proof. Put \(q=|n|\) and \(p=m+2\). In varying the critical action, all interior mesh-variation terms vanish by stationarity. The endpoints \(x\) and \(x+2\pi\) are fixed. We may therefore keep the vertices fixed and differentiate \(D\) directly with respect to \(u\).

On the left half of a cell \((X,X+t)\), differentiation of [ana:normalized-action] and reversal of the two integrations give the kernel \[ \begin{split} \mathcal K_-(z)={}& 2\bigl(1-\cos(\phi(z)-\phi(X))\bigr)e^{-2u(z)}\\ &-e^{u(z)} \int_z^\mu\sin(\phi(\zeta)-\phi(X))e^{-2u(\zeta)}\,\,\mathrm d\zeta , \end{split} \tag{20}\] integrated against \(v(z)\,\,\mathrm dz\). On the right half, the corresponding formula is \[\begin{split} \mathcal K_+(z)={}& 2\bigl(1-\cos(\phi(X+t)-\phi(z))\bigr)e^{-2u(z)}\\ &-e^{u(z)} \int_\mu^z\sin(\phi(X+t)-\phi(\zeta)) e^{-2u(\zeta)}\,\,\mathrm d\zeta . \end{split}\] The moving-midpoint terms cancel because the two original integrands agree at \(\mu\).

Parametrize the two halves, always from left to right, by \[z_-(x,\alpha)=X+\alpha(\mu-X),\qquad z_+(x,\alpha)=\mu+\alpha(X+t-\mu),\qquad 0\le\alpha\le1,\] and set \(\tau_-=\alpha/2\), \(\tau_+=(1+\alpha)/2\). Uniformly in \(j,\alpha\), the mesh and midpoint estimates give \[ \begin{split} z_\pm(x,\alpha) &=x+(j+\tau_\pm)\delta+O_{C^p}(q^{-2}),\\ (z_\pm)_\alpha&=\frac{\pi}{q}+O_{C^p}(q^{-2}),\\ \mathcal K_\pm(z_\pm(x,\alpha)) &=q^{-2}k_0(2\pi\tau_\pm)+O_{C^p}(q^{-3}), \end{split} \tag{21}\] where \[ k_0(\xi)=\frac32\min(\xi,2\pi-\xi)^2-\frac{\pi^2}{2}, \qquad 0\le\xi\le2\pi. \tag{22}\] Each remainder in [ana:half-cell-estimates] has the corresponding Lipschitz bound, with the same power of \(q\), in \(\|u-\widetilde u\|_{r_0,\sigma}\).

Here is a verification of both the leading terms and the remainder orders. On a cell, replace each occurrence of \(u\) by its value at \(X\), using the integrated identity for differences of \(u\). Then replace \(\sin a\) by \(a\) and \(1-\cos a\) by \(a^2/2\); every phase difference is \(O(q^{-1})\). On the left this gives \[\mathcal K_-(z) =(z-X)^2-\int_z^\mu(\zeta-X)\,\,\mathrm d\zeta+O_{C^p}(q^{-3}) =\frac32(z-X)^2-\frac12(\mu-X)^2+O_{C^p}(q^{-3}).\] The constant exponential factors cancel. The reflected expression gives the right half. Since \[z-X=\delta\tau_-+O_{C^p}(q^{-2}),\qquad \mu-X=\delta/2+O_{C^p}(q^{-2}),\] and similarly from the right endpoint, these formulas give [ana:leading-kernel]. An integrated difference gains one factor \(q^{-1}\) also after \(p\) derivatives. Applying the composition-difference identity used in 5 proves the same gain for input variations. For completeness, the mesh coefficient estimates used at most \(p+6\) input derivatives; one further derivative for an integrated spatial difference and one for its moving-point variation bound all the kernel estimates by \(C^{p+8}=C^{m+10}\) input norms. This is still below \(r_0\).

For each \(j,\alpha\) and sign, the real map \(x\mapsto z_\pm(x,\alpha)\) is an increasing lift close to a real translation. Write its inverse as \(x=x_\pm(z,\alpha)\). On a strip \(\mathcal A_{\sigma''}\), with \(\sigma''=(1-K/q^2)\sigma\) and a larger \(K\), it satisfies \[ x_\pm(z,\alpha) =z-(j+\tau_\pm)\delta+O_{C^{m+1}}(q^{-2}). \tag{23}\] This estimate and its Lipschitz version use no additional input derivatives. In fact the forward error is \(O(q^{-2})\) in \(C^{m+2}\). Its inverse equation is a contraction in \(C^{m+1}\), since a composition difference there uses just one additional derivative of the forward error. Real symmetry gives exactly the relative imaginary displacement used in the mesh construction, so a larger relative strip loss suffices for this inverse as well.

Change the Fourier integration variable from \(x\) to \(z\). The integration interval may start at a different point, but its length is \(2\pi\), and all integrands are periodic. Formula (17) follows with \[ b_n(z)= -\frac{q^2}{6\pi} \sum_{j=0}^{q-1}\sum_{\pm} \int_0^1 \mathcal K_\pm(z)\, (z_\pm)_\alpha(x_\pm(z,\alpha),\alpha)\, (x_\pm)_z(z,\alpha)\, e^{-\mathrm in(x_\pm(z,\alpha)-z)}\,\,\mathrm d\alpha . \tag{24}\] In this formula the kernel is that of the cell belonging to the indicated \(j,\alpha,\pm\), evaluated after the inverse substitution.

The phase in [ana:bn-formula] has leading value \[e^{\mathrm in(j+\tau_\pm)\delta} =e^{2\pi\mathrm i\mathop{\mathrm{sgn}}(n)\tau_\pm}.\] Its error, and the error in each of its first \(m\) derivatives, is \(O(q^{-1})\), since \(|n|=q\) multiplies the \(O_{C^{m+1}}(q^{-2})\) inverse displacement. The same statement holds for input differences. Substituting the leading terms in [ana:bn-formula] thus gives \[-\frac13\int_0^1 k_0(2\pi\tau)e^{2\pi\mathrm i\mathop{\mathrm{sgn}}(n)\tau}\,\,\mathrm d\tau .\] The integral equals \(-3\). Indeed symmetry reduces it to \[\frac1\pi\int_0^\pi \left(\frac32\xi^2-\frac{\pi^2}{2}\right)\cos\xi\,\,\mathrm d\xi=-3.\] The leading coefficient is therefore \(1\).

Each cell contributes \(O(q^{-1})\) to the leading coefficient and \(O(q^{-2})\) to its error: the kernel, prefactor, and cell Jacobian have respective sizes \(q^{-2},q^2,q^{-1}\), and their errors have one further factor \(q^{-1}\). These are absolute estimates, including at zeros of the leading kernel \(k_0\). Summation over \(q\) cells proves [ana:coefficient-estimate]. Since the leading term is independent of \(u\), the same computation with differences proves [ana:coefficient-lipschitz]. The finite derivative budgets above and the relative domain losses prove the stated uniformity in \(\sigma\), including \(\sigma=0\). ◻

Proof. The rational graphs of rotation \(2\pi/q\) tend to the boundary: otherwise their parameters have a positive subsequential lower bound, whereas every fixed positive leaf has positive rotation. By 4, the successive \(x\)-coordinates on each such graph form a positive stationary mesh of \(q\) steps whose sum is \(2\pi\). All its steps tend uniformly to zero.

Let \(s,t\) be consecutive steps. The real Taylor expansion (14) first gives \(s\asymp t\), and then, after factoring \(t^2-s^2\), gives \(t-s=O(s^4)\). Consequently \[|\log t-\log s|\le Cs^3.\] Summing along any part of a circuit costs at most \[C\sum_j s_j^3\le C(\max_j s_j)^2\sum_j s_j =2\pi C(\max_j s_j)^2.\] All steps are therefore uniformly comparable. Their sum implies each is \(O(q^{-1})\), so adjacent differences are \(O(q^{-4})\). It follows that \[s_j=\frac{2\pi}{q}+O(q^{-3}),\qquad X_j-x-\frac{2\pi j}{q}=O(q^{-2}).\] These are exactly the real position and step bounds in a sufficiently large fixed uniqueness ball of 5. Thus this geometric mesh is the mesh defining \(W_q\), pointwise for every initial \(x\). In particular it has the smooth dependence on \(x\) supplied by that construction.

The geometric orbit satisfies the Euler equation also at the glued endpoints. In differentiating \(W_q(x)\), its interior terms vanish by stationarity and its two endpoint derivatives sum to zero. Therefore \(W_q'(x)=0\), proving [ana:constraint-vanishing]. ◻

Proof. Contour displacement and two integrations by parts give \[|[b_n(u)-1]_j| \le\frac{C}{|n|}(1+|j|)^{-2} e^{-(1-K/n^2)\sigma|j|}.\] At \(\sigma=0\) only the integrations by parts are used. For a positive strip one first displaces to a smaller closed strip and then lets its width tend to the stated width. The same estimate for coefficient differences has the additional factor \(\|u-\widetilde u\|_{r_0,\sigma}\).

Take \(N^2\ge 2K\). For all \(k\in\mathbb Z\) and \(|n|\ge N\), \[ |n|-|k|-(1-K/n^2)|n-k|\le\frac{2K}{|n|}. \tag{28}\] If \(|k|\le|n|\), the triangle inequality leaves at most \((K/n^2)|n-k|\le2K/|n|\). If \(|k|>|n|\), the entire left hand side is nonpositive. The weighted matrix entry for \(R(u)\) is consequently bounded by \[\frac{C}{|n|}(1+|n-k|)^{-2}e^{2K\sigma/|n|}.\] Its column sum is at most \[\frac{C'}N\sum_{j\in\mathbb Z}(1+|j|)^{-2}\le\frac{C''}N.\] This proves [ana:wiener-near-identity]. The difference estimate follows from the same calculation. A Neumann series gives the high-mode inverse. Finally \(C_{-q}(u)=\overline{C_q(u)}\) for real inputs, so the operators and their inverse preserve real Fourier symmetry. ◻

Proof of 3. Let \(u\) be the actual smooth function in [ana:normalization]. For the remainder of the proof set \(m=2\) and \(r_0=m+15\). Choose \[M>4e\bigl(1+|u|_{0,r_0}\bigr),\] and choose \(N\) large enough for [ana:constraint-vanishing,ana:wiener-estimate], with \(\|R\|\le1/2\). These choices are fixed before the approximation parameter below.

Let \(u_0\) be the Fourier truncation of \(u\) to \(|k|\le P\), where \(P>N\), and put \(\sigma_0=1/P\). For each fixed integer \(a\), \[ |u_0|_{\sigma_0,a}\le e|u|_{0,a}. \tag{29}\] We first prove that its residual is arbitrarily small in powers of \(P\): \[ |C(u_0)|_{\sigma_0}=O(P^{-A}) \quad\text{for every fixed }A. \tag{30}\] The constants and the lower threshold for \(P\) may depend on \(A\).

For \(N\le|n|\le P\), the exponential weight is at most \(e\). Integrate the derivative on the real segment from \(u\) to \(u_0\), use \(C(u)=0\), and apply 8 at width zero. The sum of these weighted coefficients is bounded by \[C|u_0-u|_0=O(P^{-A})\] for every \(A\), since \(u\) is smooth. For the remaining tail, \(C(0)=0\): the zero input has a uniform mesh and a constant critical action. Thus \[C_n(u_0)=\int_0^1(\,\mathrm dC_n)_{\lambda u_0}(u_0)\,\,\mathrm d\lambda.\] Use 6 here with one fixed integer \(m>A+2\). By [ana:truncation-bound], its higher input norms are uniformly bounded in \(P\). Once \(P\) exceeds the corresponding threshold, the estimate applies to every \(|n|>P\). Real mesh uniqueness identifies these constraints with the ones already chosen using \(m=2\). The weighted convolution estimate in the preceding lemma also holds with a factor \((1+|n|)^A\): use \[(1+|n|)^A\le (1+|n-k|)^A(1+|k|)^A\] and sum the resulting power \((1+|n-k|)^{A-m}\). It follows that \[\sum_{|n|>P}(1+|n|)^A|C_n(u_0)|e^{\sigma_0|n|} \le C_A|u_0|_{\sigma_0,A}\le C'_A.\] Dividing by \((1+P)^A\) proves the tail bound and hence [ana:initial-residual]. Only one larger fixed smoothness order is used for each requested power; no threshold uniform over all smoothness orders is needed.

We now solve the high-mode equations while keeping every mode \(|k|<N\) fixed. Set \[\sigma_j=\frac{1+2^{-j}}{2P}, \qquad \sigma_\infty=\frac1{2P}, \qquad \varepsilon_j=|C(u_j)|_{\sigma_j}.\] At step \(j\), let \[ \eta_j=-A_H(u_j)^{-1}C(u_j),\qquad u_{j+1}=u_j+\eta_j. \tag{31}\] The correction has only high modes and satisfies \[|\eta_j|_{\sigma_j}\le2\varepsilon_j.\] For \(a=\sigma_j-\sigma_{j+1}=2^{-j-2}/P\), the elementary inequality \(\sup_{\xi\ge0}\xi^{r_0}e^{-a\xi}\le C_{r_0}a^{-r_0}\) gives \[ \|\eta_j\|_{r_0,\sigma_{j+1}} \le C P^{r_0}2^{r_0j}\varepsilon_j. \tag{32}\] Thus the correcting segment has the finite input norm required for the derivative estimate on the smaller strip.

As long as the inputs on that segment stay within the fixed bound \(M\), integrate the difference of derivatives on it. The linear term cancels exactly by [ana:newton-correction]. Although its inverse was computed at width \(\sigma_j\), it is the same coefficient-defined derivative on the smaller strip, by the compatibility following 8. Therefore \[\begin{split} |C(u_{j+1})|_{\sigma_{j+1}} &\le C\|\eta_j\|_{r_0,\sigma_{j+1}} |\eta_j|_{\sigma_{j+1}}\\ &\le C P^{r_0}2^{r_0j}\varepsilon_j^2. \end{split}\] Equivalently, \[ \varepsilon_{j+1}\le C P^{r_0}2^{r_0j}\varepsilon_j^2. \tag{33}\] This calculation can first be performed coefficientwise on the real axis, where every variation is smooth, and then estimated in the smaller-strip norm. It does not assume differentiability on a Wiener space with insufficient boundary derivatives.

Choose \(P\) sufficiently large that \(\varepsilon_0\le P^{-2r_0}\), as permitted by [ana:initial-residual]. Increase it further so that the fixed constant in [ana:newton-recurrence] satisfies \(CP^{-r_0}2^{r_0+1}<1\). Induction gives \[ \varepsilon_j\le P^{-2r_0}2^{-(r_0+1)j}. \tag{34}\] Indeed the ratio of the right hand side of [ana:newton-recurrence], with this bound substituted, to the proposed bound at \(j+1\), is at most \(CP^{-r_0}2^{r_0+1}2^{-j}\). Moreover, [ana:correction-smoothing,ana:newton-decay] give \[\sum_{j\ge0}\|\eta_j\|_{r_0,\sigma_{j+1}} \le CP^{-r_0}\sum_{j\ge0}2^{-j}.\] For sufficiently large \(P\) this is smaller than \(M/4\). By [ana:truncation-bound], \(u_0\) has norm less than \(M/4\). Every iterate and every correcting segment therefore has the required norm with slack. This proves the inductive bounded-input condition rather than assuming it. Formally, the induction is simultaneous: before applying [ana:newton-recurrence] at step \(j\), the already known bound on \(\varepsilon_j\) and [ana:correction-smoothing] put that entire correcting segment within the same cumulative norm bound; the recurrence then proves the next error bound.

The corrections converge in \(C^{r_0}(\mathcal A_{\sigma_\infty})\) to a real-symmetric holomorphic function \(u_*\). Its low Fourier modes agree with those of \(u\). For each fixed constraint, continuity of the real mesh construction and \(\varepsilon_j\to0\) give \(C_n(u_*)=0\). The real segment from \(u\) to \(u_*\) remains in the original bounded input set. Put \(v=u_*-u\), which has only high Fourier modes. At width zero, \[0=C(u_*)-C(u) =v+\int_0^1R(u+\lambda v)v\,\,\mathrm d\lambda .\] The operator norm of the integral is at most \(1/2\). Hence \(|v|_0\le|v|_0/2\), so \(v=0\). Thus the original \(u\) is real analytic.

By [ana:normalization], \(\phi'=e^u>0\) on the real axis. The primitive \(\phi\) and its inverse are real analytic. The same equation then gives analytic \(r\) in the normal-angle coordinate. Finally the smooth support function solves \(h''+h=r\), an analytic ordinary differential equation, and is therefore analytic. Its boundary parametrization \(h n_\theta+h'e_\theta\) is analytic and regular. ◻

Proof. Holomorphic extension follows from the analytic \(u\), the midpoint implicit equation, and the integral formula. The identities and expansions are [ana:action-symmetries,ana:action-expansion]; the final assertion is 4. ◻

Proof. By antisymmetry, \[\mathcal R=D_1(U,T_tU)-D_1(U,T_{-t}U).\] Put \(v_+=T_tU-U=tA_+\) and \(v_-=T_{-t}U-U=-tA_-\), where \[A_\pm=\int_0^1 U'(y\pm\lambda t,t)\,d\lambda.\] The leading part of \(t^{-3}\mathcal R\) is \[\frac18(A_++A_-)\frac{A_+-A_-}{t}.\] The last quotient is holomorphic, because \[\frac{A_+-A_-}{t} =\int_0^1\int_{-\lambda}^{\lambda}U''(y+\mu t,t)\,d\mu\,d\lambda.\] The remaining part, using (35), is \[t\left[A_+^4B(U,tA_+)-A_-^4B(U,-tA_-)\right].\] This proves regularity and the formula for \(E(y,0)\). Differentiating (35) in its second argument gives \(D_{12}(x,x+b)=b/4+O(b^3)\), proving the formula for \(a\). The same integral formulas show that, on smaller domains, the first two variations of \(E\) are bounded by fixed polynomials in the appropriate norms of the variations and their first two \(y\)-derivatives. There is no negative power of \(t\) in these bounds.

Antisymmetry interchanges the forward and backward terms when \(t\) is replaced by \(-t\), giving evenness. For real \(t\), the mean action \[\frac{1}{2\pi}\int_0^{2\pi}D\bigl(U(y+\eta,t),U(y+\eta+t,t)\bigr)\,dy\] does not depend on \(\eta\). Differentiation at \(\eta=0\) gives \(\langle U'\mathcal R\rangle=0\). Holomorphic continuation gives \(\langle E\rangle=0\) for complex \(t\) as well.

For (42), subtract from the off-diagonal Hessian terms those in which \(v\) is evaluated at \(y\). The remaining terms are \[t^{-3}\left[c_+(T_tv-v)+c_-(T_{-t}v-v)\right],\] where \[c_+=D_{12}(U,T_tU)U'T_tU',\qquad c_-=D_{12}(T_{-t}U,U)T_{-t}U'U'.\] Since \(T_{\pm t/2}a=c_\pm/t\), this is \(\partial(a\partial v)\). The factored terms, including the variation of the prefactor \(U'\), are \((Ev)'\).

Finally, at an exact solution the derivative of \(t^{-3}\) contributes zero. Differentiating only the explicit shifts gives \((c_+-c_-)/t_0^3=\partial a/t_0\). The assigned jet contributes \(\partial(a\partial z)\) by (42). This proves (43). ◻

Proof. We first take \(t_0>0\). Write \(U_j(y)=U(y+jt_0,t_0)\) for \(0\le j\le q\), and look for stationary nodes \[\widetilde X_j=U_j+U'_j\xi_j, \qquad \xi_0=\xi_q=0.\] The endpoints differ by \(qt_0=2\pi l\). Scale the interior stationarity equations by \(t_0^{-3}U'_j\). Their residual at \(\xi=0\) is \(T_{jt_0}E\), and their linearization is a divergence matrix plus a diagonal potential: \[\begin{align*} (A_0\xi)_j &=\frac{a_{j+1/2}(\xi_{j+1}-\xi_j) -a_{j-1/2}(\xi_j-\xi_{j-1})}{t_0^2},\\ V_j&=T_{jt_0}\left(E'-\frac{U''}{U'}E\right),\qquad a_{j+1/2}=a(y+(j+1/2)t_0,t_0). \tag{48}\end{align*}\] All translations here are real and preserve the \(y\)-strip.

For completeness, the complex Dirichlet inverse has the same polynomial bound as the real one. For \(A_0\xi=F\), introduce the fluxes \(J_{j+1/2}=a_{j+1/2}(\xi_{j+1}-\xi_j)\). Then \[J_{j+1/2}=J_{1/2}+t_0^2\sum_{i=1}^j F_i.\] The condition \(\sum_{j=0}^{q-1}(\xi_{j+1}-\xi_j)=0\) determines \(J_{1/2}\). Since \(a_{j+1/2}^{-1}\) is uniformly close to 4, \[\left|\sum_{j=0}^{q-1}a_{j+1/2}^{-1}\right|\ge c_0q.\] Consequently \[ \|A_0^{-1}\|_{\infty\to\infty}\le Cq^2t_0^2\le CQ^2. \tag{49}\] Cauchy’s estimate gives \(\|V\|_{Y_1}\le CQ\varepsilon\). A Neumann perturbation therefore gives the same inverse bound for \(A_0+V\) if \(CQ^3\varepsilon\) is small.

There is also a uniform polynomial bound for the nonlinear remainder. The scaled equations contain only three adjacent nodes, fixed bounded derivatives of \(D\), and the factor \(t_0^{-3}\). Since \[ |t_0|^{-1}\le\frac{N}{2\pi}, \tag{50}\] their second derivatives in \(\xi\) have norm at most \(CQ^3\). The fixed-point equation obtained with \((A_0+V)^{-1}\) thus preserves a ball of radius \(CQ^2\varepsilon\) and contracts there if \(CQ^7\varepsilon\) is small. This gives holomorphic corrections on \(Y_1\), with \(\|\xi\|\le CQ^2\varepsilon\). Uniqueness preserves periodicity and real symmetry. Further derivatives follow by Cauchy estimates on subsequent strips.

On real \(y\), the node increments are positive and small: their uncorrected increments are bounded below by a fixed multiple of \(t_0\), and the correction is smaller than that by (50) and the polynomial smallness condition. The stationary nodes therefore describe actual billiard trajectories whose initial lines lie in the given phase collar. Such a line belongs to one of the invariant caustic graphs. At the q-th step its angle has increased by \(2\pi l\); invariance of the graph forces its momentum also to return. Its rotation is \(l/q\). By 4, this is the unique graph of that rotation, and every point of it is q-periodic.

Use the local inverse of \(U(\,\cdot\,,t_0)\) near each \(U_j\) to write \[\widetilde X_j(y)=U(\widetilde y_j(y),t_0),\qquad \widetilde y_j=y+jt_0+O(P\varepsilon).\] These are holomorphic on \(Y_2\). A contraction for the displacement from \(y+jt_0\), using \(U'\) close to one, gives this inverse without any global inverse-domain assumption. On real points, the graph just identified implies \(\widetilde y_j=\widetilde y_1^{\,j}\) as iterates and \(\widetilde y_q=y+qt_0\). Hence \[G(y)=\frac1q\sum_{j=0}^{q-1}(\widetilde y_j(y)-jt_0)\] satisfies \(G(\widetilde y_1(y))=G(y)+t_0\). It is close to the identity, and Cauchy’s estimate makes \(G'\) close to one on \(Y_3\). Its inverse exists on \(Y_4\), again by contraction. The function \[ U_*(y)=U(G^{-1}(y),t_0) \tag{51}\] is an exact solution at \(t_0\), and \(U_*-U(\,\cdot\,,t_0)=O(P\varepsilon)\). The identities first obtained on the real axis extend throughout the stated complex strips.

We next assign a compatible parameter jet to \(U_*\). Start with \(\dot U_*=U_t(\,\cdot\,,t_0)\). Its total residual jet is \(O(P\varepsilon)\) on \(Y_6\). Indeed \(E_t=O(P\varepsilon)\) by Cauchy’s estimate, and replacing \(U\) by \(U_*\), with the same assigned jet, changes the explicit differentiated formula for \(E\) by \(O(P\varepsilon)\). Only finitely many derivatives occur, and every denominator \(t_0\) is controlled by (50). More explicitly, for any assigned jet \(V\), the raw derivative is \[\begin{align*} \dot E={}&-3t^{-4}U'\mathcal R+t^{-3}V'\mathcal R\\ &+t^{-3}U'\bigl[ D_{11}(U,T_tU)V +D_{12}(U,T_tU)(T_tV+T_tU')\\ &\hspace{35mm} +D_{21}(T_{-t}U,U)(T_{-t}V-T_{-t}U') +D_{22}(T_{-t}U,U)V\bigr]. \tag{52}\end{align*}\] This formula makes both the fixed derivative count and the worst explicit denominator \(t^{-4}\) transparent.

Let \[\Pi v=\frac1q\sum_{j=0}^{q-1}T_{jt_0}v\] be the orbit-average projection. Equation (43) shows that \(\Pi\dot E_*=0\). This uses only \(\partial=T_{-t_0/2}(T_{t_0}-1)/t_0\) and \(\Pi T_{t_0}=\Pi\); the half-shift need not act trivially. To write the cyclic inverse explicitly, put \(T=T_{t_0}\) and \[R_q=\frac1q\sum_{j=0}^{q-1}jT^j.\] Since \(T^q=1\), one has \((T-1)R_q=1-\Pi\). Thus on \(\ker\Pi\), \(\partial^{-1}=t_0R_qT_{t_0/2}\), and \[\|\partial^{-1}\|\le Cq|t_0|.\] The polynomial smallness condition makes \(U_*\) close enough to \(U\) through the required derivatives that \(4a_*\) is still close to one. The operator \((1-\Pi)a_*\) on \(\ker\Pi\) is therefore invertible: its difference from multiplication by \(1/4\) has norm at most \(2\|a_*-1/4\|\). Factoring through these three operators proves \[ \left\|(\partial a_*\partial)^{-1}\right\|_{\ker\Pi} \le Cq^2t_0^2. \tag{53}\] Correcting \(\dot U_*\) by \(U_*'z\), with \(z\in\ker\Pi\), now makes \(\dot E_*=0\), at cost \(O(P\varepsilon)\).

Define \(w=(U_*-U)/U'\) and assign its derivative by differentiating this quotient with the corrected jet of \(U_*\). Both \(w\) and \(\dot w\) are \(O(P\varepsilon)\). Taylor expansion and (42) give \[0=E(U_*)=E+\partial(a\partial w)+(Ew)'+O(P\varepsilon^2).\] Differentiating this Taylor identity once in the assigned parameter jets gives the second assertion in (47). The term \((Ew)'\) and its jet are quadratic, because \(E,E_t,w,\dot w\) are all bounded by a polynomial times \(\varepsilon\). The same is true of the differentiated Taylor remainder.

Here is a concrete accounting for the fixed-polynomial assertion. The mesh inverse and correction cost at most \(CQ^2\); the nonlinear mesh Hessian costs \(CQ^3\). Using subsequent strip losses for the fixed number of derivatives, comparison of the initially assigned residual jet costs at most \(CQ^{10}\varepsilon\), and its correction costs at most \(CQ^{12}\varepsilon\). In a first parameter derivative of a quadratic remainder, either one correction is replaced by its assigned jet, or one coefficient is differentiated. The latter introduces at most one additional \(t_0^{-1}\), fixed derivatives of \(D\), and bounded \(U_t,U'_t\). Before further y-derivatives, the value remainder is bounded by \(CQ^8\varepsilon^2\) and its first jet by \(CQ^{18}\varepsilon^2\): a correction and its first y-derivative cost \(Q^2\varepsilon,Q^3\varepsilon\), whereas the assigned correction jet and its first y-derivative cost \(Q^{12}\varepsilon,Q^{13}\varepsilon\). Multiplication by \(t_0^{-3}\) gives these bounds; differentiating that factor or a coefficient costs less. Allowing six additional y-derivatives on a fixed smaller strip therefore bounds both remainders by \(CQ^{24}\varepsilon^2\), more than is needed below. Enlarging to \(CQ^{40}\) covers all bounds and smallness conditions in this Lemma. These exponents are deliberately inessential upper bounds; in particular they do not depend on any formal approximation order.

For negative \(t_0\), apply the positive construction at \(-t_0\) and use the even symmetry, reversing the sign of the assigned parameter jet. ◻

Proof. At two distinct divisor zeros, the sinc kernel in (54) is zero; at its own center it is one. Thus \(R_n(t_j)=F_n(t_j)\), so the zeros in the quotient are removable. There are at most \(CN\) interpolation points for each mode, because \(d<1\). The identity \[\frac{\sin z}{z}=\int_0^1\cos(uz)\,du\] gives \[ |R_n(t)|+N^{-1}|R_n'(t)| \le P\eta e^{-|n|(s-a\delta-|\Im t|/2)}. \tag{57}\] Outside disks centered at the divisor zeros with radius \(c_1\min(|n|^{-1},\delta)\), elementary estimates for the sine give \[|d_n(t)|^{-1}\le Pe^{-|n\Im t|/2}.\] Here and below a harmless fixed constant can be absorbed into \(P\). For \(|nt|\le1\), the function \(\sin(nt/2)/(nt/2)\) is bounded away from zero. Thus \(|d_n(t)|^{-1}\le C/|n|\), which has the required bound since \(e^{-|n\Im t|/2}\ge e^{-1/2}\). On the remaining region, write \(nt/2=u+\mathrm iv\) and use \(|\sin(u+\mathrm iv)|^2=\sin^2u+\sinh^2v\). Outside the nonzero-root disks, \(|\sin(u+\mathrm iv)|\) is bounded below by a fixed multiple of \(\min(1,|n|\delta)e^{|v|}\): if the nearest sine zero is zero, the condition \(|nt|>1\) supplies the required fixed separation. Multiplication by \(|t|<1\) then gives a polynomial bound in \(Q\). Inside the nonzero-root disks, apply maximum modulus to the holomorphic quotient. The change in any exponential weight is bounded by \(e^{C|n|\min(|n|^{-1},\delta)}\), a fixed constant. The parameter buffer ensures that every disk meeting the target domain is contained in the domain of \(F_n\) and has its center among the interpolation points.

The input strip gives \[|F_n(t)|\le M e^{-|n|(s-j_0\delta-\nu|\Im t|)}.\] After division its imaginary-parameter loss is \((\nu-1/2)|\Im t|\). The interpolant contribution after division has no such loss. Hence \[|H_n(t)|\le P(M+\eta) e^{-|n|(s-a\delta-(\nu-1/2)|\Im t|)}.\] Summing on the smaller y-strip proves the first bound, with a geometric sum of size at most \(C/\delta\). The retained-mode residual is \(-R_n\), and the omitted tail of \(F\) gains at least \(e^{-N\delta}\); summing similarly proves the second bound. Each fixed y-derivative costs only a fixed power of \(N\) or \(\delta^{-1}\). Parameter derivatives follow from the explicit bound on \(R_n'\) or from Cauchy’s estimate on a further buffered domain.

For real-symmetric input, conjugate modes have conjugate output, because \(d_{-n}(\bar t)=\overline{d_n(t)}\). If the input is even, the divisor zeros occur in opposite pairs and the interpolant is even; so is the quotient. These properties survive the symmetric truncation. ◻

Proof of 12. First divide \(f=-E\). Its mean is zero by 11. For \(|n|\le N\), every nonzero zero of \(d_n\) has reduced denominator at most \(N\). At those zeros with \(|t_j|<d-3\delta\), the first equation of (47) gives \[|f_n(t_j)|\le P\varepsilon^2e^{-|n|(s-10\delta)}.\] Apply 14 with \(\nu=1\), \(j_0=0\), \(a=10\), \(b=3\), and \(J=11\). Call its output \(h\). Then \[ h=O(P\varepsilon)\ \hbox{on }\mathcal D_{11}^{1/2},\qquad \partial h+E=O(P\varepsilon^2)\ \hbox{on }\mathcal D_{11}^{1}. \tag{58}\] The term involving the interpolation values is absorbed by the smallness condition. Define \[ b=a^{-1}(h+\lambda),\qquad \lambda(t)=-\frac{\langle a^{-1}h\rangle}{\langle a^{-1}\rangle}. \tag{59}\] The denominator is bounded away from zero because \(a^{-1}\) is close to four. Thus \(b\) has mean zero and is \(O(P\varepsilon)\) on \(\mathcal D_{12}^{1/2}\).

We need new compatibility conditions before dividing \(b\). Fix a retained rational \(t_0\) with \(|t_0|<d-14\delta\), and put \(g=a\partial w\), using 13. For retained nonzero modes the exact first-division identity is \[d_n(h_n-g_n)=-(E+\partial g)_n-R_n.\] Its value and its first parameter derivative at \(t_0\) are bounded by \(P\varepsilon^2e^{-|n|(s-10\delta)}\), by the two compatibility conditions and (57). If \(d_n(t_0)\ne0\), reducedness of \(l/q\) gives \[|\sin(\pi nl/q)|\ge c/q, \qquad |d_n(t_0)|^{-1}\le Cq|t_0|.\] If \(d_n(t_0)=0\), then \[|d_n'(t_0)|=|n|/|t_0|,\] and differentiation of the product gives the same bound for \(h_n-g_n\). In particular, this comparison does not require the resonant modes of \(g\) to vanish.

For the omitted modes, \(h_n=0\), whereas the analytic function \(g\) has norm \(P\varepsilon\) on \(Y_{10}\). Its tail on \(Y_{13}\) is bounded by \(P\varepsilon e^{-3N\delta}\), hence by \(P\varepsilon^2\). Summing retained and omitted modes gives \[h-(g-g_0)=O(P\varepsilon^2)\quad\hbox{on }Y_{13}.\] Since \(\langle a^{-1}g\rangle=\langle\partial w\rangle=0\), the mean choice in (59) gives \(\lambda=g_0+O(P\varepsilon^2)\). Consequently \[ b-\partial w=O(P\varepsilon^2)\quad\hbox{on }Y_{13}. \tag{60}\] At a nonzero divisor zero for a retained mode, \((\partial w)_n=0\). Thus (60) supplies the values needed for the second division, with bound \(P\varepsilon^2e^{-|n|(s-13\delta)}\).

Apply 14 to \(b\), now with \(\nu=1/2\), \(j_0=12\), \(a=13\), \(b=15\), and \(J=17\). To avoid ambiguity, the letter \(b=15\) here denotes the integer buffer in that Lemma, not the function in (59). Call the output \(v\). The remaining half of the imaginary-parameter loss is canceled, giving \[ v=O(P\varepsilon)\ \hbox{on }\mathcal D_{17}^{0},\qquad \partial v-b=O(P\varepsilon^2)\ \hbox{on }\mathcal D_{17}^{1/2}. \tag{61}\] Both divisions, multiplication by \(a^{-1}\), and the mean adjustment preserve real symmetry and evenness.

Because \(ab=h+\lambda(t)\) and \(\partial\lambda=0\), (58) and (61) imply \[E+\partial(a\partial v)=O(P\varepsilon^2) \quad\hbox{on }\mathcal D_{18}^{1}.\] Here application of \(\partial\) is bounded after the indicated strip loss: use \(\partial F=\int_{-1/2}^{1/2}F'(y+\mu t,t)\,d\mu\). Take \(\Delta U=U'v\). Cauchy’s estimate on the remaining domains gives the required bounds through three y-derivatives. Equation (42) gives a quadratic linearized residual, since \((Ev)'=O(P\varepsilon^2)\). The normalized integral formulas in the proof of 11 give \[E(U+\Delta U)-E(U)-E_U(\Delta U)=O(P\varepsilon^2) \quad\hbox{on }\mathcal D_{20}^{1},\] uniformly at \(t=0\) as well.

Every cost used in this proof is a fixed number of additions, multiplications, bounded inverses, Cauchy losses, sums over at most \(CN\) roots per mode, and Fourier sums bounded by a power of \(\delta^{-1}\). The rational part has the fixed ledger given in 13. The two divisions add only finitely many powers of \(Q\), and the normalized remainder adds a fixed derivative cost. Choosing one larger exponent \(B_0\) proves (45)–(46), with all required smallness conditions included. No step involves a number of derivatives depending on \(N\) or on the initialization order. ◻

Proof. The coefficient of \(t^{2j}\) involving the newly chosen \(u_j\) is \(u_j''/4\), by (41); all other terms at that order depend only on earlier coefficients. Their mean is zero, because the full residual has mean zero for every candidate. Solve the periodic equation for \(u_j\), choosing its mean to be zero. Integration, or division of its nonzero Fourier modes by \(-n^2\), preserves holomorphy on every strictly smaller strip and real symmetry. At every fixed order the forcing uses only finitely many derivatives of holomorphic functions on the original open strip. We may therefore fix one smaller strip first and carry out any finite number of recursions there, with constants allowed to depend on \(m\).

Evenness is built into the construction. On the fixed smaller strip, the first three y-derivatives of \(U^{(m)}-y\) are \(O(C_m d_0^2)\), and the residual bound follows from its first omitted power. Taking \(d_0\) smaller gives all required slack. ◻

Construction of the function in 10. Choose a fixed sufficiently small y-width \(s_0\). The constants in 12 are now fixed. Start with 15, choosing its order below, and let the initial parameter radius be \(d_0<s_0/4\). At step \(i\), use losses \[\delta_i=2^{-i}d_0/200,\] and reduce both current radii by \(20\delta_i\). The limiting parameter radius is at least \(4d_0/5\), and the limiting y-width is at least \(s_0-d_0/5\). If the current error is \(\varepsilon_i>0\), choose \[N_i=\left\lceil\delta_i^{-1}\log(1/\varepsilon_i)\right\rceil.\] If it is zero, the construction has already finished. Enlarge one fixed integer \(B\) if necessary. Since every power of \(1+\log(1/\varepsilon_i)\) is bounded by a constant times \(\varepsilon_i^{-1/4}\), the step estimates imply \[\begin{align*} \varepsilon_{i+1}&\le C d_0^{-B}2^{Bi}\varepsilon_i^{7/4}, \tag{62}\\ \|\Delta U_i\|_{C_y^3} &\le C d_0^{-B}2^{Bi}\varepsilon_i^{3/4}. \tag{63}\end{align*}\] The norms in the second line are on the next domains.

Choose first \(m\) so that \(2m+2>4B+4\), then \(d_0\) sufficiently small. Its initial error is at most \(d_0^{4B+4}\). Direct substitution in (62) shows inductively that \[ \varepsilon_i\le d_0^{4B+4}2^{-(4B+4)i}. \tag{64}\] Indeed the ratio of the resulting right-hand side of (62) to the next bound in (64) is at most \(C d_0^{2B+3}2^{4B+4-(2B+3)i}\), which is less than one for small \(d_0\). Equation (63) then bounds the corrections by \[C d_0^{2B+3}2^{-(2B+3)i}.\] They are summable and preserve the fixed smallness hypotheses with slack. The same estimates ensure all polynomial smallness conditions needed at every step. This order of choices is important: the exponent \(B\) is fixed before \(m\), whereas the possibly large constant \(C_m\) is absorbed by choosing \(d_0\) afterward.

The corrections converge uniformly, together with their first three y-derivatives, on the limiting domains. Their limit is holomorphic, real-symmetric, even, and periodic modulo \(y\). The residual tends to zero on the limiting \(\mathcal D^1\), so the limit solves (36). At \(t=0\), (41) and \(U'\ne0\) give \(U''=0\). Periodicity forces \(U(y,0)=y+c_0\), with a real constant \(c_0\). Replacing \(y\) by \(y-c_0\) removes this constant without changing any of the other properties. Finally choose a smaller product domain with enough y-margin for the two shifts. This gives the first assertion of 10. ◻

Completion of the proof of 10. The change from \(\mathcal S\) to \(D\) subtracted a telescoping endpoint integral and changed the boundary coordinate. Thus (36) is equivalent to \[\mathcal S_1(\Theta,T_t\Theta) +\mathcal S_2(T_{-t}\Theta,\Theta)=0.\] The momentum formula in (37) therefore matches at successive vertices, giving the claimed billiard dynamics. All steps are positive and small for positive small real \(t\).

The angle \(\Theta\) is even. Antisymmetry of \(\mathcal S\) and the last stationarity equation give \[p(y,-t)=-\mathcal S_1(\Theta,T_{-t}\Theta) =\mathcal S_2(T_{-t}\Theta,\Theta)=p(y,t).\] Hence both oriented line coordinates are jointly analytic and even. Taylor expansion of the support-point formula for the momentum gives \[ p(y,t)=h(\Theta(y,t)) -\frac18 r(\Theta(y,t))\phi'(y)^2t^2+O(t^4). \tag{65}\] For example, if \(\beta=T_t\Theta-\Theta\), that formula first gives \(p=h(\Theta)-r(\Theta)\beta^2/8+O(\beta^3)\). Since \(\beta=\phi'(y)t+O(t^2)\), it determines the quadratic term; evenness then removes the cubic term from the full expression.

An even holomorphic function of \(t\) is holomorphic in \(t^2\). The derivative of \(\Theta\) with respect to \(y\) at zero is \(\phi'(y)>0\), and the coefficient of \(t^2\) in the inward momentum displacement in (65) is strictly positive. The analytic inverse function theorem, uniformly around the compact real period circle, consequently gives local collar coordinates. They are globally one-to-one modulo the period after shrinking the collar: for each fixed small real \(t^2\), \(\Theta_y>0\) determines a unique \(y\) from the angle, and at a fixed angle the inward momentum displacement has strictly positive derivative in \(t^2\). Thus the coordinates give a full one-sided phase collar, and the shift preserves the nondegenerate analytic coordinate \(t^2\).

Write the pulled-back area form as \(B_0(y,t)\,dy\wedge dt\). It is holomorphic and periodic in \(y\). The billiard map preserves \(dp\wedge d\Theta\), and in these coordinates it is \((y,t)\mapsto(y+t,t)\). Therefore \(B_0(y+t,t)=B_0(y,t)\). For every small real \(t\) whose ratio to \(2\pi\) is irrational, density of the translation orbit makes \(B_0(\,\cdot\,,t)\) constant. Its nonzero Fourier coefficients vanish on this set of parameters and hence identically by holomorphy. Thus \(B_0(y,t)=A(t)\).

Finally, (65) gives the coefficient of \(dy\wedge dt\) as \[\frac14 r(\phi(y))\phi'(y)^3t+O(t^3) =\frac{c}{4}t+O(t^3).\] The coefficient is odd because the line coordinates are even. This proves (38) and completes the theorem. ◻

Proof. Intersection in (66) gives \[h_X(t)\ge h_L(t)-\frac{|\Im t|}{2}.\] For the opposite inequality suppose first that \(\Im t\ge0\). The line through \(X(y-3t/2,t)\) and \(X(y-t/2,t)\) is the preceding billiard line. Reflecting it at the second point gives \(L(y,t)\). Both points, and the tangent at the second one, are available whenever \(\Im y<h_X(t)+\Im t/2\), at each strict subheight and on a sufficiently small parameter neighborhood. This rational formula agrees with the primary line in the basic strip. The initially wider domain supplies any lower-strip overlap needed by the shifts. For \(\Im t<0\), use the succeeding two impacts and backward reflection instead. This proves the opposite inequality and hence (68). ◻

Proof. Locally choose a coordinate direction transverse to all the finitely many foliations; write it as the \(a\)-direction. Put \[A_i=\partial_a\psi_i,\quad B_i=\partial_\delta\psi_i, \quad \rho_i=B_i/A_i,\quad F_{i,l}=f_i^{(l)}(\psi_i).\] The \(A_i\) are nonzero and the \(\rho_i\) are distinct. Differentiate (69) by \(\partial_a^{m-r}\partial_\delta^r\), \(0\le r<m\). The terms containing \(F_{i,m}\) have coefficient matrix \[\big(w_i A_i^m\rho_i^r\big)_{\substack{0\le r<m\\1\le i\le m}},\] which is invertible by the Vandermonde determinant. All remaining terms are linear in \(F_{i,l}\), \(l<m\), with analytic coefficients and an analytic inhomogeneous term. Thus each \(F_{i,m}\) is an analytic affine-linear expression in the finite vector \[V=(F_{i,l}:1\le i\le m,\ 0\le l<m).\] The chain rules \[\partial_a F_{i,l}=A_iF_{i,l+1},\qquad \partial_\delta F_{i,l}=B_iF_{i,l+1}\] then give a full first-order system \[ \partial_aV=M_aV+c_a,\qquad \partial_\delta V=M_\delta V+c_\delta. \tag{70}\]

To prove convergence, solve the first coordinate linear ODE on the base section and then the second with analytic dependence on the first coordinate. The Taylor series of this analytic solution is the unique formal solution of those two selected ODE problems. It therefore agrees with the original formal vector. The residual in the unused equation of (70) is analytic and has zero Taylor series, so it vanishes. The web relation and all its differential identities hold near the basepoint.

Linear ODEs with analytic coefficients continue along finite coordinate segments in the interior. Solving first along a horizontal section and then along vertical segments, and using analytic uniqueness on overlaps, gives continuation throughout any smaller rectangle. The other equations of (70) persist by analytic continuation. Coordinate changes handle portions where a different transverse direction was used. In particular the continued \(F_{i,0}\) still depends locally only on \(\psi_i\).

For the edge assertion, shrink the interval \(I\) and fix a small positive section \(\delta=\delta_1\). For \(\psi\) in a neighborhood of the smaller interval, the equation \(\psi_i(a,\delta)=\psi\) has a solution \(a=a_i(\psi,\delta)\) inside the rectangle for every \(0<\delta\le\delta_1\). Along that path, \[\frac{\,\mathrm d}{\,\mathrm d\delta}F_{i,0}(a_i(\psi,\delta),\delta) = -\frac{B_i}{A_i}A_iF_{i,1}+B_iF_{i,1}=0.\] Thus its value is represented by the analytic one-variable germ on the fixed positive section. That germ extends to \(\delta=0\), regardless of any singular behavior of \(M_a,M_\delta,c_a,c_\delta\) there. ◻

Proof. We work first near real glancing trajectories. The proof has four parts: elimination of high support jets, convergence of formal variations, isolation by a coalescing limit, and specialization back to meromorphic functions.

Proof.

Proof. Fix \(w_0\in D_w\). There is an intermediate circle \(|\zeta|=r\), \(\rho<r<R\), on a neighborhood of which \((w-w_0)^m F\) is jointly holomorphic for some integer \(m\ge0\). To see this, write local meromorphic quotients on a compact subannulus at \(w=w_0\). Remove from their denominators the powers of \(w-w_0\) that divide them identically along that slice. The remaining denominator zeros on the slice are locally isolated in \(\zeta\). A circle avoiding these zeros exists, and a finite cover of that circle gives one common power \(m\).

On a sufficiently small disk \(D'\) about \(w_0\), let \(c_{-j}(w)\), \(j\ge1\), be the negative Laurent coefficients of \((w-w_0)^mF\) on this circle. They are holomorphic in \(w\). Suppose that \(D'\) meets an open set of parameters already reached by meromorphic extension to the full \(\zeta\)-disk. For generic parameters in that open set, the slice has finitely many poles inside the circle. Multiplication by a polynomial in \(\zeta\) removes them. Equivalently, the Hankel matrix \[\big(c_{-(i+j+1)}(w)\big)_{i,j\ge0}\] has finite rank there.

The rank bound can be made independent of \(w\). For each fixed integer \(r_0\), the condition that all minors of size \(r_0+1\) vanish is an analytic condition on \(D'\). Unless it holds identically, its solution set is contained in the discrete zero set of one nonzero minor and is therefore countable. A countable union of such sets cannot contain the open set of generic extended slices. Hence all minors of one fixed size vanish identically on \(D'\).

Over the field of meromorphic functions on \(D'\), finite Hankel rank gives a finite recurrence for the negative coefficients. Here is the elementary reason. A dependence among the first finitely many columns with nonzero last coefficient expresses that last column in terms of its predecessors. Shifting the row index gives the same dependence for all later columns. Thus, for a nonzero polynomial \(Q(w,\zeta)\) with meromorphic coefficients in \(w\), every negative Laurent coefficient of \(Q(w,\zeta)(w-w_0)^mF\) vanishes. Clear the finitely many coefficient denominators locally. The remaining Laurent series has a holomorphic extension inside the circle, given by its Cauchy integral on that circle. Dividing back by the polynomial and by the cleared factors gives the desired meromorphic extension on a neighborhood of \(w_0\) times the disk.

The set of reached parameters is therefore open and relatively closed in \(D_w\): at a limit point the preceding local argument uses any overlapping reached open set. It is nonempty and hence is all of \(D_w\). The extensions glue by uniqueness on their annular overlaps, and the original annulus completes them to radius \(R\).

In the holomorphic case every negative Laurent coefficient itself vanishes on \(D_0\). The identity theorem makes it vanish on \(D_w\); the same Cauchy integral gives a holomorphic extension without a polynomial denominator. ◻

Proof. Cover the compact part of the center path under consideration by finitely many coordinate disks. On each one the pulled-back data satisfy 21; the full disks already reached give the nonempty open seed for the next one. Uniqueness on the boundary collars gives agreement. In the holomorphic case, additional holomorphic parameters can be retained: the same Laurent coefficients and Cauchy integrals are holomorphic in them, and the identity theorem applies on their common neighborhoods. In the holomorphic case the final assertions are the maximum principle and its harmonic real-part version. ◻

Proof. Let \(t_0+r\overline{\mathbb D}\subset B\). Choose a nonnegative trigonometric polynomial \(q\) satisfying \[q(\zeta)<h_F(t_0+r\zeta),\qquad |\zeta|=1,\] and let \(g\) be a holomorphic polynomial whose boundary imaginary part is \(q\). Thus \[\Im g(0)=\frac1{2\pi}\int_0^{2\pi}q(e^{\mathrm i\theta})\,\,\mathrm d\theta.\] For real \(x_0\), \(0\le\lambda\le1\), and a small complex variable \(w\), consider \[ t=t_0+r\zeta,\qquad y=x_0+\lambda g(\zeta)+w. \tag{89}\] On \(|\zeta|=1\), these points lie strictly below the appropriate width. Lower semicontinuity and compactness in \((x_0\bmod 2\pi,\lambda,\zeta)\) provide a common annular neighborhood in \(\zeta\) and a common disk \(|w|<d_w\) on which the pulled-back functions are jointly meromorphic.

At \(\lambda=0\), the whole disks are in the initial basic strip. Increase \(\lambda\) in fixed sufficiently small increments satisfying \[\sup_{|\zeta|\le1+d_\zeta} |(\lambda_{\rm new}-\lambda_{\rm old})g(\zeta)|<d_w/2.\] The full-disk data at the old value provide a full-disk extension at the new value for \(|w|<d_w/2\). Product continuation fills the original \(w\)-disk. Repeat finitely many times.

For each \(\lambda\), uniqueness on the annulus glues the extensions as \(x_0\) varies. Adjacent values of \(\lambda\) also glue, because their overlapping strips contain the full-disk seeds just described. At \(\zeta=0\) these are strips with ordered centers \(\Im y=\lambda\Im g(0)\). Consequently the continued branch is attached to the basic strip throughout the ascent and reaches slightly above height \(\Im g(0)\) at \(t_0\). We obtain \[h_F(t_0)\ge \frac1{2\pi} \int_0^{2\pi}q(e^{\mathrm i\theta})\,\,\mathrm d\theta.\] Approximate the positive lower semicontinuous boundary function from below by continuous functions and then, with a strict margin, by trigonometric polynomials. This gives the supermean inequality, including infinite integrals. Together with lower semicontinuity it gives the asserted superharmonic comparison. ◻

Proof. We first prove the assertion for \(L\). Suppose a patch exists. Choose a horizontal arc and a vertical band about \(H_1\) on which joint data are known for all \(t\) in a neighborhood of \(t_0\). The patch agrees across the bottom of the band with the lower primary tube. Fix a narrower concentric band and \(\eta>0\) such that \(H_1+\eta\) lies strictly inside it. These choices are independent of the nearby irrational real shifts used below. Shrink the parameter neighborhood so that a common strip below \(H_1\), reaching the bottom of the narrower band, is available jointly.

Proof. At each strict upper subheight of \(H\), \(X\) has a joint meromorphic extension near \(t=0\). In generic affine coordinates there is no polar component along the entire zero slice: such a component would meet the initial collar, where \(X(y,0)=\Gamma(y)\) is regular. Restrictions to that slice are therefore meromorphic, with only isolated exceptional points to remove in the one-variable projective map. They agree on overlaps by the identity theorem. Real symmetry supplies the lower half, proving the first assertion.

On the original real arc the curvature normalization is \(r(\phi(y))(\phi'(y))^3=c\). Since the unit tangent is \(e_\phi\), it gives \[\Gamma'=\frac{c}{(\phi')^2}e_\phi,\qquad \Gamma'\cdot\Gamma'=\frac{c^2}{(\phi')^4},\qquad \det(\Gamma',\Gamma'')=\frac{c^2}{(\phi')^3}.\] These equations prove (93) on the real arc, hence meromorphically everywhere on \(D_H\) and on any continuation patch.

Suppose \(H<\infty\) and a patch exists. By real symmetry we may use the upper edge. Choose a regular point \[s_{\rm edge}=x_*+\mathrm iH\] of the patch where \(\Gamma'\cdot\Gamma'\) and \(\det(\Gamma',\Gamma'')\) are both nonzero. Such a point exists after a small move along the edge: all exceptional sets are isolated, since the corresponding meromorphic functions are nonzero on the original real arc. Shrink the patch so that \(\Gamma\), its normal, a local normal angle \(\phi\), the inverse angle coordinate, and the curvature radius are holomorphic and nondegenerate. The normal is obtained from \(\Gamma'\) by a local square root of its bilinear squared length; the angle then comes from a local logarithm. The intrinsic identity (93) shows that \(r(\phi)(\phi')^3\) is a nonzero constant on this patch, equal to \(c\) up to the local sign choices. The zero slice \(L(y,0)\) is its tangent, by continuation from below.

Proof. Write \(z=s-p\). By (98), \(f=n/(\mathrm iz)+O(1)\). Integration of (97) therefore gives \[ Z(s)-Z(p)\sim c_Z z^{n+3},\qquad W(s)-W_0\sim c_W z^{3-n}, \tag{101}\] with nonzero constants \(c_Z,c_W\). Here \(W_0=W(p)\) when \(W\) is finite, and any fixed constant may be used when \(W\) has a pole. If \(n=3\), the leading term of \(W'\) is a nonzero multiple of \(z^{-1}\), which cannot be the derivative of a meromorphic function. This proves \(n\ne3\).

At sufficiently close punctured points of the zero-parameter slice, \(\Gamma\) is finite, regular and nonisotropic. Its local inverse, together with the rank-one identity for \(X\), gives \[ X(y,t)=\Gamma(v(y,t)),\qquad v(y,t)=y+\sum_{j\geq1}t^{2j}v_j(y). \tag{102}\] The even powers follow from evenness of \(X\). Possible joint poles or indeterminacies at points where the zero-slice restrictions are regular form a discrete set on that slice. After shrinking the punctured disk, there are no such exceptions. The coefficients in (102) glue there. Recursively inverting one of the coordinates of \(\Gamma\) shows that every \(v_j\) is meromorphic at \(p\): at each order the new coefficient is divided only by a nonzero meromorphic derivative of that coordinate.

To take a leading limit with \(z=s-p\) and \(t\) on the same scale, meromorphy of the coefficients alone is not enough. We need the pole bound \[ \mathop{\mathrm{ord}}_p v_j\geq1-2j. \tag{103}\] Under \(z=tw\), this keeps every formal term \(t^{2j}v_j(p+tw)\) at order at least \(t\), just as for the leading displacement \(z\). We prove the bound using the arc-minus-chord action, divided by \(c\), between shape parameters \(s\) and \(s+h\): \[ \mathcal D(s,s+h)= \int_s^{s+h}f^{-2} -\sqrt{\left(\int_s^{s+h}f^{-2}a\right) \left(\int_s^{s+h}f^{-2}a^{-1}\right)} =\frac{h^3}{24}+O(h^5). \tag{104}\] The square root initially has the arc-length branch. To check the cubic coefficient, expand each integral at its midpoint. The difference between the normalized second derivative of \(f^{-2}\) and the average of those of \(f^{-2}e^{\pm\mathrm i\phi}\) is \(f^2\). Multiplication by \(f^{-2}\) leaves the constant coefficient \(1/24\).

Write \(\mathcal D(s,s+h)=\sum_m d_m(s)h^m\). Substituting \(s=p+z\), \(h=z\eta\), the two chord integrals factor as \(z^{3+n}\eta\) and \(z^{3-n}\eta\), respectively, times functions holomorphic near \((z,\eta)=(0,0)\) and nonzero there after division by \(\eta\). Their product has the square root \(z^3\eta\) times a holomorphic unit, with the chosen glancing sign. Thus \[ \mathop{\mathrm{ord}}_p d_m\geq3-m. \tag{105}\] This argument concerns short segments away from the center; the resulting coefficient identities are meromorphic identities at the center.

Reflection implies stationarity of the sum of two consecutive actions \(\mathcal D\) on the lifted impacts in (102). The arc terms telescope, and the chosen square roots have velocities tending to the same unit tangent, so their signs give the ordinary reflected branch. Divide this stationary equation by \(t^3\). For a general function \(v\), its leading expression is \[-\frac14v_yv_{yy}.\] Indeed the cubic action contributes one eighth of the difference of the squared previous and next increments. At order \(t^{2j}\), the new coefficient consequently enters as \(-v_j''/4\).

Assign weight one to each of \(z,t\). Assuming (103) at preceding orders, the formal truncation of \(v-p\) through order \(t^{2j-2}\) and its increments have lower total degree at least one. Taylor shifts by \(\pm t\) preserve this property. Composition of \(d_m\) with \(v\) preserves its lower degree \(3-m\), because each derivative lowers the allowed order by at most one. The endpoint derivatives of the unnormalized action therefore have total lower degree at least two. After division by \(t^3\), the forcing at order \(t^{2j}\) has order at least \(-1-2j\) in \(z\). Only finitely many terms enter any fixed coefficient, since every step has positive \(t\)-order. Hence \(\mathop{\mathrm{ord}}_p v_j''\geq-1-2j\), which proves (103) because \(v_j\) is meromorphic.

Let \(P(z,t)\) be a constant-coefficient polynomial combination of the affine coordinates of \(X(p+z,t)\). Suppose that its zero-slice restriction has order \(b\in\mathbb Z\) and leading coefficient \(c_P\ne0\). Composition with (102) and (103) show that its coefficient of \(t^l\) has order at least \(b-l\). Consequently \[z^{-b}P(z,z\lambda)\] is holomorphic near \((z,\lambda)=(0,0)\). Here is the convergence point in this assertion. The expression is a meromorphic germ and its formal expansion in \(\lambda\), with Laurent coefficients in \(z\), belongs to \(\mathbb C[[z,\lambda]]\) by the order bound. A meromorphic germ whose quotient is formally a power series is holomorphic: apply Weierstrass preparation and division, after a generic linear change of coordinates, to its numerator and denominator. More explicitly, replace the denominator by a Weierstrass polynomial times a unit, and divide the analytic numerator by that polynomial. The analytic remainder has degree below the polynomial degree. Uniqueness of formal division, compared with the given formal quotient, makes its formal remainder zero. The analytic remainder is consequently zero as well, and the analytic quotient is the desired germ; see (Hörmander 1990, Theorem 6.1.1). Thus formal nonnegative orders imply genuine holomorphy in this instance.

Its restriction to \(z=0\) is rational in \(\lambda\), since it is the ratio of the first nonzero homogeneous parts of a local numerator and denominator after the substitution \(t=z\lambda\). Formally that restriction equals \[c_P\mathcal U(\lambda)^b,\qquad \mathcal U(0)=1,\] where \(\mathcal U\) collects the total-degree-one terms of \(v-p\). Taking \(P=Z-Z(p)\), for which \(b=D_+>0\), supplies an analytic \(D_+\)-th root near one and proves convergence of \(\mathcal U\). Set \(\xi(w)=w\mathcal U(1/w)\). For sufficiently large \(R_0\), \[ t^{-b}P(tw,t)\longrightarrow c_P\xi(w)^b \tag{106}\] locally uniformly, with derivatives, on \(|w|>R_0\); the limit is rational. Applying this to the two coordinates in (101) proves rationality of the stated powers.

In isotropic coordinates reflection says that the product of the two incident chord slopes, expressed as \(Z\)-increment divided by \(W\)-increment, is the square of the tangent slope. Substitute (106) at \(w,w-1,w+1\). The constants \(c_Z/c_W\) cancel, and the derivative of \(\xi\) cancels from the tangent ratio. This gives (100), first near infinity and then as an identity of rational functions. Interchanging the isotropic coordinates treats a pole of \(a\). ◻

Proof. It suffices to consider a zero of order \(n\). Use the preceding lemma and put \[d=\gcd(D_+,|D_-|),\quad R=\xi^d,\quad \alpha=D_+/d,\quad\beta=D_-/d,\quad \rho(w)=\frac{R(w+1)}{R(w)}.\] Bezout’s identity makes \(R\) rational from the rationality of the two powers of \(\xi\). Since \(R(w)\sim w^d\), \(\rho(w)=1+d/w+O(w^{-2})\). Thus \(\rho\) is a nonconstant rational map of some degree \(s\geq1\), with \(\rho(\infty)=1\). Define \[F(x)=\frac{x^\alpha-1}{x^\beta-1}.\] After cancellation of the common powers of \(R(w)\), (100) becomes \[ F(\rho(w+1))= \frac{(\alpha/\beta)^2}{F(1/\rho(w))}. \tag{108}\] We use the Riemann–Hurwitz formula: a rational map of degree \(s\) has total ramification \(2s-2\); see (Forster 1981).

If \(n=2\), then \(\alpha=5,\beta=1\), so \(F\) is a polynomial of degree four with simple zeros at the four nontrivial fifth roots of unity. At each preimage by \(\rho\) of one of these four values, the right side of (108) has a pole whose order is the local multiplicity of that preimage. On the left, a pole has order divisible by four. Thus all these multiplicities are divisible by four. Each of the four fibers contributes at least \(3s/4\) to ramification, giving total at least \(3s>2s-2\), a contradiction.

Suppose next that \(n>3\). Write \(\gamma=-\beta>0\) and \(m=\alpha-\gamma=6/d\geq1\). The function \[F(x)=-x^\gamma\frac{x^\alpha-1}{x^\gamma-1}\] has a pole of order \(\alpha\) at infinity. The function on the right of (108), considered as a function of \(\rho\), has a pole of order \(\gamma\) at infinity and simple poles at the nontrivial \(\alpha\)-th roots of unity. These roots are distinct from its other possible exceptional values, since \(\gcd(\alpha,\gamma)=1\).

Group the finitely many poles of \(\rho\) into maximal runs \(w_0,w_0+1,\ldots,w_0+\ell\), with respective orders \(p_0,\ldots,p_\ell\). Comparing pole orders at the points of the run gives \[\alpha p_{j+1}=\gamma p_j\quad(0\leq j<\ell).\] At the preceding point \(w_0-1\), which is not a pole of \(\rho\), \(\rho\) must take a nontrivial \(\alpha\)-th root with multiplicity \[ \alpha p_0=m\sum_{j=0}^\ell p_j+\gamma p_\ell. \tag{109}\] The preceding points of different runs are distinct. Their ramification contributions are their displayed multiplicities minus one. If \(m\geq2\), (109) shows that these points alone contribute at least \(ms\geq2s\), a contradiction.

If \(m=1\) and \(\gamma\geq2\), coprimality in the recurrence implies \(\gamma^\ell\mid p_\ell\). Therefore \(\gamma p_\ell-1\geq\ell+1\). If \(N_P\) is the number of distinct poles, the preceding points contribute at least \(s+N_P\), while the poles themselves contribute \(s-N_P\). Again the total is at least \(2s\). The only remaining possibility is \(m=\gamma=1\), \(\alpha=2\). All preceding points then take the single value \(-1\), and their total multiplicity, by (109), is strictly larger than \(s\). This is impossible for one fiber of a degree-\(s\) map. Together with the exclusion of \(n=3\), these arguments leave only \(n=1\).

For \(n=1\) we have \(d=2,\alpha=2,\beta=1\). Equation (108) reads \[(1+\rho(w+1))(1+1/\rho(w))=4.\] Equivalently, \(2/(\rho-1)\) increases by one under translation. A rational one-periodic function is constant, so \[\frac{2}{\rho(w)-1}=w+b_0, \qquad R(w)=(w+b_0)(w+b_0+1).\] The second identity follows by comparing the shift ratios and the asymptotic \(R(w)\sim w^2\). In particular \(\xi\) is not rational, since \(\xi^2\) has two simple zeros.

Choose a local coordinate \(\zeta=z+O(z^2)\) such that \(W-W_0=c_W\zeta^2\). If the expansion of \(Z\) in \(\zeta\) had a first odd term of order \(j\), subtracting the preceding even terms as a polynomial in \(W-W_0\) would give a polynomial combination \(P\) with order \(j\). By (106), \(\xi^j\) would be rational. Since \(j\) is odd and \(\xi^2\) is rational, Bezout’s identity would make \(\xi\) rational, a contradiction. Thus both coordinates of \(\Gamma\) are even in \(\zeta\). The involution \(\zeta\mapsto-\zeta\) induces a local symmetry \(\sigma\) of the shape parameter, fixing \(p\) with derivative \(-1\).

By (99), \(\sigma'^6=1\) on regular points. Its derivative is therefore constant, and \(\sigma(s)=2p-s\). The tangent-ratio identity gives \(a(\sigma(s))^2=a(s)^2\). The simple zero fixes the sign to be negative. This proves (107); the pole case is identical after interchanging \(Z,W\). ◻

Proof. A center is nonreal, since \(|a|=1\) on the real axis. If \(H<\infty\), the symmetry (107) extends by identity on the overlap of \(D_H\) with its reflection about the center. That overlap is connected. The reflected strip extends past one of the edges of \(D_H\), giving a meromorphic continuation of \(\Gamma\) through an edge arc. This contradicts 25. Hence \(H=\infty\).

Real symmetry supplies the conjugate center, and composition of the two center reflections supplies a nonreal period, in addition to \(2\pi\). Thus \(a\) is a nonconstant elliptic function. Let \(\Lambda\) be its full period lattice. Composing reflections about any two centers shows that twice their difference belongs to \(\Lambda\). Consequently all zeros and poles lie in the four classes of one translate of \(\frac12\Lambda/\Lambda\). They are all simple by 27. The degree of \(a\) on \(\mathbb C/\Lambda\) is therefore at most two. It cannot be one, because a degree-one meromorphic map from a torus to the sphere would be a biholomorphism. Its degree is exactly two.

Riemann–Hurwitz now gives four simple ramification points. Their branch values \(\beta_1,\ldots,\beta_4\) are distinct and finite: a fiber of degree two cannot contain two ramification points, and the poles of \(a\) are simple. The quotient \[\frac{a'^2}{\prod_{j=1}^4(a-\beta_j)}\] has neither zeros nor poles on the torus. At a ramification point both numerator and denominator have order two; at a simple pole both have order minus four. Hence the quotient is a nonzero constant. We have obtained \[ a'^2=P(a),\qquad \deg P=4. \tag{110}\] Differentiating \(a(2p-s)=-a(s)\) gives \(a'(2p-s)=a'(s)\), so \(P(-a)=P(a)\). Thus \(P\) is even. On the real curve, \[f^2=-\frac{a'^2}{a^2} =C+B\cos(2\phi)+D\sin(2\phi)>0\] for real constants \(B,C,D\). Therefore \(f^2=n_\phi^{\mathsf T}Nn_\phi\) for a real positive-definite symmetric matrix \(N\). Positivity follows on all directions since \(\phi\) runs once around the circle.

For \(q_N(\phi)=n_\phi^{\mathsf T}Nn_\phi\), direct differentiation gives \[\left(\sqrt{q_N}\right)''+\sqrt{q_N} =\frac{\det N}{q_N^{3/2}}.\] For example, use \(q_N'=2e_\phi^{\mathsf T}Nn_\phi\) and \(q_N''=2\operatorname{tr}N-4q_N\), and evaluate the determinant of \(N\) in the orthonormal basis \((n_\phi,e_\phi)\). It follows that the support \[h_0(\phi)=\frac{c}{\det N}\sqrt{n_\phi^{\mathsf T}Nn_\phi}\] has curvature radius \(c/f^3\), the radius of the given boundary. This is the support of an ellipse. The difference \(h-h_0\) solves \((h-h_0)''+(h-h_0)=0\), so it is a linear combination of \(\cos\phi,\sin\phi\), precisely a translation of the support. ◻

Proof. Near real glancing, the regular parametrization of the curve and the rank-one identity give the asserted lift. Its differential satisfies \[ c^2\mathfrak s_y^6 =\frac{\det(X_y,X_{yy})^4}{(X_y\mathbin{\cdot}X_y)^3}, \qquad \frac{\mathfrak s_t}{\mathfrak s_y} =\frac{X_t\mathbin{\cdot}X_y}{X_y\mathbin{\cdot}X_y}. \tag{114}\] Remove the proper analytic sets on which the meromorphic quantities needed here have poles, vanish in a denominator, or make the chosen sixth root zero. The complement containing the initial germ is connected: removing complex analytic sets of positive codimension does not disconnect a domain. Along any path in this complement the sixth root, the closed differential it determines, and its local primitive continue. Closedness continues from the initial germ. On every compact regular path the differential is finite, and its \(y\)-component is nonzero.

The composition identity persists for as long as the continued primitive lies in \(D_H\). If \(H<\infty\), it cannot first reach an edge of that strip. At such a first point the primitive is still finite and locally submersive. Complete it to local product coordinates \((\mathfrak s,z)\), and restrict the meromorphic map \(X\) to a generic constant-\(z\) section. The identity on the inner side would then continue the primary \(\Gamma\) meromorphically through its edge, contradicting 25. Thus the primitive remains in the strip throughout continuation. For \(H=\infty\) there is no edge to encounter.

After return along a loop, the continued primitive has the form \[\mathfrak s\longmapsto\omega\mathfrak s+b, \qquad \omega^6=1.\] It consequently induces the symmetry \[ \Gamma(\omega s+b)=\Gamma(s) \tag{115}\] on the overlap of the two relevant strips. That overlap is connected, being the intersection of two strips. If \(H<\infty\), a nonreal \(\omega\) makes the transformed strip cross an edge of \(D_H\). If \(\omega=\pm1\), the same is true when \(\Im b\ne0\). In either case (115) would give an edge patch. Thus only \(\omega=\pm1\), \(b\in\mathbb R\) remain.

For \(H=\infty\), a nonreal \(\omega\) gives a nonreal period by conjugating the real translation \(s\mapsto s+2\pi\) with the affine symmetry in (115). A nonreal translation is already such a period. If \(\omega=-1\) and \(\Im b\ne0\), compose that reflection with its conjugate, which is also a symmetry by reality, to obtain the nonreal period \(b-\bar b\). A period of \(\Gamma\) is a period of \(a\) by (98). Such a nonreal period, together with \(2\pi\), would make the entire holomorphic function \(a\) doubly periodic and hence constant. This contradicts its real winding. Again only the real possibilities remain.

Real embeddedness rules out a real reflection in (115) and makes a real translation an integer multiple of \(2\pi\). We conclude that \[v=e^{\mathrm i\mathfrak s}\] is single-valued on the regular complement. When \(H<\infty\), both \(v\) and \(1/v\) are locally bounded, in fact between the bounds given by \(e^{\pm H}\). They extend holomorphically across the removed analytic sets by the removable-singularity theorem. The bounds remain strict by the maximum principle, since \(v\) is nonconstant on the zero slice. Hence the extension is nonvanishing and still has \(|\log|v||<H\).

For completeness we address the unbounded-strip case separately; bounded removability is not available there. The differential \(\,\mathrm d\mathfrak s\) is already single-valued, because all its monodromy is translational. It extends meromorphically by (114). Indeed, if a single-valued holomorphic function off an analytic set has a meromorphic sixth power, multiply it by a local denominator of that power; the result is bounded, since its sixth power is holomorphic, and is therefore removable. Apply this to \(\mathfrak s_y\), then use the second identity in (114) for the other component.

Write \(\Gamma(s)=G(e^{\mathrm is})\), where \(G\) is meromorphic on \(\mathbb C^*\). At a generic smooth point of a possible polar divisor, restrict to a transverse disk, with the center deleted. The single-valued function \(v\) on this punctured disk cannot have an essential singularity. If it did, the image of every punctured neighborhood would be dense; choosing sequences approaching any regular value \(w\in\mathbb C^*\) would force \(G(w)\) to equal the projective limit of the meromorphic map \(X=G(v)\). The nonconstant map \(G\) would be constant.

Suppose instead that \(v\) has a zero of finite order \(k\). In a local coordinate, \(v(z)=z^k u(z)\), \(u(0)\ne0\). Meromorphic affine coordinates of \(X\) grow at most like \(|z|^{-N}\). Every sufficiently small nonzero \(w\) has preimages satisfying \(|z|\asymp|w|^{1/k}\). The identity \(G(w)=X(z)\) gives polynomial growth of the affine coordinates of \(G\) at zero, hence meromorphic extension there. A pole of \(v\) gives the analogous extension at infinity. Reality, \(G(1/\bar w)=\overline{G(w)}\) in Euclidean coordinates, supplies extension at the other end. Consequently \(G\) is rational. Equations (98) then make \(a\) rational in \(w=e^{\mathrm is}\). A rational function with no zeros or poles on \(\mathbb C^*\) is \(Cw^k\); its real winding is one, so \(k=1\). This makes \(f\) constant and the boundary a circle, the excluded case.

It follows that \(v\) extends with a nonzero finite value on every generic transverse disk. Therefore \(\,\mathrm d\mathfrak s=\,\mathrm dv/(\mathrm iv)\) has no pole on such a disk. Generic transverse directions detect all leading polar coefficients of a meromorphic one-form: a nonzero coefficient covector cannot annihilate every direction having nonzero normal component. Thus \(\,\mathrm d\mathfrak s\) has no polar divisor. Removability in codimension two makes it holomorphic throughout \(\mathcal T\).

Finally, \(\mathcal T\) is simply connected. Its vertical fibers in \(y\) are intervals containing \(\Im y=0\), so vertical contraction retracts it to \(\mathbb R\) times the parameter disk. For finite \(H\), take a holomorphic logarithm of the nonvanishing extension of \(v\). For infinite \(H\), take a primitive of the holomorphic closed one-form \(\,\mathrm d\mathfrak s\). Normalize by the original germ. This gives the required single-valued lift. The composition and all symmetries in (113), including real symmetry, follow by identity from the initial domain. On the real phase circle, \(\mathfrak s_y=1+O(t^2)\) uniformly. Shrinking the disk makes this derivative positive for real \(t\); the degree-one periodicity then gives the asserted circle diffeomorphism. ◻

Proof. Suppose that \(f\) vanishes at \(s_0\) to order \(m\geq1\). Use fixed tangent and normal coordinates for the angle \(\phi(s_0)\), and write \(z=s-s_0\). Since \(\phi(s)-\phi(s_0)\) has order \(m+1\), the tangent and normal components of \(\Gamma'\) in (97) have leading orders \(-2m\) and \(1-m\), respectively, with nonzero coefficients. If \(m=2\), the normal derivative has a nonzero residue, contrary to meromorphy. If \(m\geq3\), integration gives tangent and normal coordinate poles of orders \(2m-1\) and \(m-2\).

By 29, the lift is holomorphic near \((s_0,0)\). The equation \(\mathfrak s(y,t)=s_0\) defines \(y=s_0+O(t^2)\). For sufficiently small \(t\ne0\), the two neighboring shape coordinates at phase shifts \(\pm t\) are \(s_0\pm t+O(t^2)\); their impacts are finite. Along the central curve, the chord to the previous impact therefore has normal-to-tangent slope of order at least \(m+1\). Its normal angle \(\theta\) can be chosen holomorphically, and \[d=\theta-\phi(\mathfrak s) =O\big((\mathfrak s-s_0)^{m+1}\big).\] For its signed offset \(p_{\rm chord}\), differentiation of \(p_{\rm chord}=n_\theta\cdot\Gamma(\mathfrak s)\) gives \[ \,\mathrm dp_{\rm chord}\wedge\,\mathrm d\theta =c f(\mathfrak s)^{-2}\sin d\, \,\mathrm d\mathfrak s\wedge\,\mathrm dd =\pm A(t)\,\,\mathrm dy\wedge\,\mathrm dt. \tag{117}\] The final identity is the continued area identity (38); choosing a local orientation can change only its sign. Since \(A(t)=ct/4+O(t^3)\), it is nonzero for small \(t\ne0\).

Set \(z=\mathfrak s-s_0\) and write \(d=z^{m+1}b\). Then \(\,\mathrm d\mathfrak s\wedge\,\mathrm dd =z^{m+1}\,\mathrm d\mathfrak s\wedge\,\mathrm db\). The left side of (117) thus vanishes to order at least \(-2m+(m+1)+(m+1)=2\), a contradiction. Hence \(m=1\). At a simple zero write \(f=Az+Bz^2+O(z^3)\), \(A\ne0\). Since \(a'=\mathrm ifa\), we have \(a=a(s_0)+O(z^2)\). The residue of \(a/f^2\) is \(-2a(s_0)B/A^3\). It vanishes because \(\mathrm ic a/f^2=Z'\) is a derivative of a meromorphic function. Thus \(B=0\), which is \(f''(s_0)=0\). The asserted removability of \(q\) follows. ◻

Proof. It suffices to calculate on real nonsingular data and then continue by identity. In (104) put \(g=f^{-2}\), \(g_\pm=g e^{\pm\mathrm i\phi}\). Midpoint integration through the fourth derivative gives \[ \mathcal D(s,s+h) =\frac{h^3}{24}+d_5(s+h/2)h^5+O(h^7). \tag{121}\] For clarity, define \[B_j=\frac12\left(\frac{g_+^{(j)}}{g_+} +\frac{g_-^{(j)}}{g_-}\right),\qquad C_2=2(g'/g)f+f'=-3f'.\] Multiplying the two midpoint integral expansions and taking the glancing square root gives \[d_5=g\left[ \frac{g''''/g-B_4}{1920} -\frac{C_2^2}{2\cdot24^2}\right].\] Differentiating \(g_\pm\) explicitly yields \[g''''/g-B_4=-8ff''+15(f')^2-f^4.\] The two \((f')^2\) contributions cancel, since \(15/1920=9/(2\cdot24^2)\). Therefore \[ d_5=-\frac{f^2+8q}{1920}. \tag{122}\]

At the middle impact \(s=\mathfrak s(y,t)\), the previous and next shape increments are \[h_-=t+t^3V'-\tfrac12t^4V''+O(t^5),\qquad h_+=t+t^3V'+\tfrac12t^4V''+O(t^5).\] The cubic term in the stationary action contributes \(-t^5V''/4\). The fifth-order terms contribute \(-5t^5d_5'+t^5d_5'\): the first part comes from differentiating the increments and comparing the two midpoint values, and the second from differentiating the midpoints. Hence stationarity at order \(t^5\) reads \[-V''/4-4d_5'=0.\] Substitute (122), integrate once, and use periodicity of \(V\) to determine the integration constant. This proves (120). The zero mean of \(V\) follows from (118). ◻

Proof. The integral equation \[k_j(u)=u+\int_0^u(u-v)Q_j(v)k_j(v)\,\,\mathrm dv\] proves analytic dependence and uniform bounds by its uniformly convergent Picard series. The same argument applies to \(F\) and to a moving endpoint \(J\), provided the segments and their small thickenings remain in the fixed domain.

Suppose first that \(F(0)=0\). If \(F\not\equiv0\), uniqueness gives \(g_j=g_j'(0)k_j\), and \(g_+'(0)g_-'(0)=F'(0)^2\). Since \(g_+(J)g_-(J)=F(J)^2=F'(0)^2k_0(J)^2\), it follows that \(k_+(J)k_-(J)=k_0(J)^2\). The conclusion also holds when \(F\equiv0\). If instead \(F(J)=0\), apply the same argument to kernels started at \(J\). The value at \(0\) of such a backward kernel is \(-k_j(J)\): this follows by evaluating the constant Wronskian of the forward and backward kernels at both endpoints. The two signs cancel in their product, and the same identity follows.

At zero initial data, \(k_+k_-=k_0^2\ne0\). Consequently the square root agreeing with \(k_0\) is analytic nearby. On each of the two zero hyperplanes just considered it agrees with \(k_0\), by uniqueness of that local branch. The map from initial data to endpoint data is \[(F(0),F'(0))\longmapsto(F(0),F(J)),\] whose determinant is \(k_0(J)\). Its coordinates are therefore independent. An analytic function vanishing on both coordinate hyperplanes is divisible by their product, as is seen by its convergent power series. This proves the claimed extension. Cauchy estimates in these two coordinates prove locally uniform bounds for the quotient and its derivatives, also when \(Q\) and \(J\) vary.

Replacing \(F\) by \(-F\) interchanges \(k_+\) and \(k_-\), so the quotient is even. At \(F=Q=0\), its value is \(J^3/24\), as follows by taking \(F\) to be a small constant in the formula computed below. Evenness removes the terms linear in the initial data; analytic interpolation from \(0\) to \(Q\) bounds the remaining constant term by \(O(\|Q\|)\). This proves (127) and its derivative versions. ◻

Proof. In the first case, \(p_{JJ}=\sin z/(2\lambda)\); hence its nonvanishing excludes the zero of \(k_\pm\), while \(k_0=J\ne0\). The ordinary quotient formula and analytic dependence of kernels apply. In the second case, \[p_{JJ}=\frac{\tanh z\,\operatorname{sech}^2z}{2\lambda},\qquad k_0=k_+=k_-=\frac{\sinh(2z)}{\lambda}.\] A zero of this last kernel either has \(\sinh z=0\), when \(p_{JJ}=0\), or \(\cosh z=0\), when the displayed branch is not finite. Thus the division lemma applies. The limiting case \(\lambda=0\) has \(p_{JJ}=J/4\). A compact path can be covered by finitely many of these analytic neighborhoods. Uniqueness of the square root on each overlap tracks the chosen branch, and Cauchy estimates give convergence of all derivatives. ◻

Proof. We give separately the recurrence argument in steep parameter directions, illustrated in 3, and the analytic-disk argument that fills the remaining directions.

Fix a small number \(\gamma>0\). Suppose first that \(|\Im t|\ge\gamma|t|>0\). By evenness in \(t\), use whichever of \(t,-t\) moves upward. Start in a fixed slab about depth \(\delta_b/2\), where joint analyticity and the phase normalization give \[\mathfrak s-y=O(t^2),\qquad \frac{\mathfrak s(y+t,t)-\mathfrak s(y,t)}{t}=1+O(t^2)\] with derivative bounds uniform in the real coordinate modulo its period. The largest parameters in the construction tend to zero, since \(r d^{-\alpha}\le Kd^{1-\alpha}\to0\).

Follow the exact recurrence toward the target, denoting successive unperturbed depths by \(x_j\). They decrease by \(|\Im t|\). We allow here the slightly larger range \(x_j\ge d/2\), \(|t|\le10r(x_j/d)^\alpha\), which will be useful in the disk construction. Suppose inductively that position errors are a small fraction of the depths and that normalized steps are close to \(1\). The shape fields on every required segment and a small thickening satisfy \[|F(0)|+|F'(0)|\le C_K|t|Bx_j^{-\alpha},\qquad \|Q\|\le C_K|t|^2B^2x_j^{-2\alpha}.\] To see the derivative estimate, apply Cauchy on a disk of radius a small fixed fraction of \(x_j\). Although \(|t|/x_j\) need not tend to zero, it is bounded by a constant depending on \(K\). Consecutive depths are therefore comparable, with constants depending on \(K\), and the thickenings may be taken with smaller such fractions. They remain above depth \(d/10\). Their horizontal displacements lie in the direction cone of slope at most \(1/\gamma\), with a fixed extra margin. Increasing the input aperture by this fixed geometric amount suffices; reducing the thickness when \(K\) increases does not require a larger aperture.

The small parameter in the kernel estimates is bounded by \[|t|Bx_j^{-\alpha} \le C\bigl(\varepsilon+rd^{-\alpha}\bigr).\] Thus 32 applies uniformly. If \(a(s,J;t)=p(s,J;t)-J^3/24\), then \[ |\partial_J^\nu\partial_s^\mu a| \le C_{K,\mu,\nu}|t|^2B^2x_j^{-2\alpha-\mu} \tag{136}\] for the finitely many derivatives used below. The \(s\)-derivative bounds follow by Cauchy on the same thickenings.

Evaluate (128) with the trial new step \(J=J_-\). The cubic contributions cancel. The difference of \(a_J\) at the two adjacent basepoints, together with \(ta_s\), is at most \(C_K|t|^3B^2x_j^{-2\alpha-1}\). The derivative of the equation in the new \(J\) is \(J/4+o(1)\), uniformly bounded away from zero. The analytic implicit function theorem therefore supplies a unique nearby step, with \[ |J-J_-|\le C_K|t|^3B^2x_j^{-2\alpha-1}. \tag{137}\] For \(\beta>1\), comparison with an integral on the depth mesh gives \[|t|\sum_{x_j\ge x}x_j^{-\beta}\le C_{K,\gamma,\beta}x^{1-\beta}.\] The possible final summand is controlled because \(|t|/x\) is bounded. Summing (137) first, and then summing the position increments \(t(J-1)\), gives \[ |J-1|\le C_K|t|^2B^2x^{-2\alpha},\qquad |\mathfrak s-y|\le C_K|t|^2B^2x^{1-2\alpha}. \tag{138}\] The second sum uses \(2\alpha>1\). Initial errors from the fixed slab are absorbed because \(B\ge1\). Moreover \[\frac{|\mathfrak s-y|}{x}+|J-1| \le C_K\bigl(\varepsilon+rd^{-\alpha}\bigr)^2,\] which closes the induction after the prescribed order of choices.

These recurrences define analytic germs in \((y,t)\). Locally choose a fixed integer number of steps back into the starting slab, where both initial neighbors are known. Two such choices agree by uniqueness of the near-identity recurrence. They consequently give analytic continuation from below on every steep sector under discussion.

We now fill all parameter directions. Call a germ reached if it is obtained by continuing from the fixed lower row along the vertical path with \(\Re y\) and \(t\) fixed. Reached germs form a single-valued holomorphic function on their open domain: nearby parallel paths are compared through a finite chain of the same germs from below. The steep continuations just constructed agree with these germs. Work with \[U(y,t)=\frac{\mathfrak s(y,t)-y}{t^2},\] which is regular at \(t=0\) on the initial domain and on its continuations.

Choose a large fixed \(D_0\) with \(D_0^\alpha>10\), then a sufficiently small fixed \(\gamma\). For a target \((y_c,t_c)\) of depth \(\delta\), put \(R=r(\delta/d)^\alpha\). Since \(|t_c|<R\), the circle \(t=t_c+3R\zeta\), \(|\zeta|=1\), meets the nonsteep sector only on two uniformly short arcs. There is a polynomial \(g\), with \(g(0)=0\), such that \[ |g(\zeta)|\le C_0\delta, \quad \Im g\le\delta/4\ \text{on the circle}, \quad \Im g\le-D_0\delta\ \text{on those arcs and a collar}. \tag{139}\] Here \(C_0\) is fixed independently of the target aperture. For completeness, choose smooth negative bumps equal to a sufficiently large negative constant on the two arcs and supported in slightly larger short intervals. Add a constant to make their mean zero. The intervals can be made so short that the added constant is less than \(1/8\). Approximate the resulting smooth function, with a fixed strict margin in its inequalities, by a real trigonometric polynomial of mean zero. It is the boundary imaginary part of an analytic polynomial vanishing at zero. Multiplication by \(\delta\) gives (139). The centers of the arcs and \(t_c/(3R)\) range over a compact set with modulus at most \(1/3\), so the bump widths and polynomial degree can be fixed uniformly. This also bounds the conjugate real part by \(C_0\delta\).

Increase an auxiliary target aperture \(m\ge m_0\) so that \[m+C_0<mD_0.\] Choose \(L\) to contain this aperture, the shifts by \(C_0\delta\), and the steep paths and thickenings above, all with fixed slack. We induct from a sufficiently small fixed depth down through successive bands whose depth thresholds are halved. In a new band use the disks \[ t=t_c+3R\zeta,\qquad y=y_c+g(\zeta)+w, \qquad |\zeta|\le1, \tag{140}\] where \(w\) is a small complex parameter. On the nonsteep boundary arcs the depth is at least \(D_0\delta\). They are therefore in the previous all-direction domain; the inequality \(D_0^\alpha>10\) supplies the parameter margin, and \(m+C_0<mD_0\) supplies the aperture margin. On the remaining arcs, steep continuation applies at depth at least \(\delta/2\), with the larger parameter range already allowed. Slight collars of the arcs have the same properties.

Start \(y_c\) lower by a fixed sufficiently large multiple of \(\delta\); then the entire disk and a neighborhood lie in the previous domain. Raise \(y_c\) to its target value in small steps. The holomorphic disk-continuation principle in [lem:meromorphic-hartogs,cont:disc-sweep], with the parameter \(w\), continues \(U\) over the disks and identifies the resulting central germ with vertical continuation from below.

There is no growth of the estimate with the number of bands. If the inductive bound is \(A B^2x^{1-2\alpha}\), its value on a previous-domain arc of depth \(x\ge D_0\delta\) is at most \(A D_0^{1-2\alpha}B^2\delta^{1-2\alpha}\), with the fixed factor \(D_0^{1-2\alpha}<1\). On a steep arc (138) gives an independent constant times \(B^2\delta^{1-2\alpha}\). Choose \(A\) once larger than that constant and the initial lower-strip constant. Maximum modulus on each disk retains the same \(A\). Starting the induction at a sufficiently small fixed depth keeps its lower dips below \(\delta_b\); a partial last band reaches any \(\delta>d\). This proves (135).

The constants in the depth sums, field bounds, and maximum-modulus argument depend only on the fixed local margins. They do not depend on how far the cone extends horizontally. Initial data are uniform modulo the real period. Thus increasing a target aperture only increases the input aperture needed to contain the indicated geometric enlargements; it does not enlarge the estimate constant or change the smallness choice. The relative-error estimate already proved keeps the image below the upper shape edge and, after reducing \(\delta_*\), above the lower edge. ◻

Proof. For (i), apply 34 with its starting scale a fixed large multiple of \(d_n\) and with \(r\) a sufficiently small fixed multiple of \(d_n/b_n\). Then its \(B\) is bounded by a constant times \(b_nd_n^{\alpha-1}\), which gives (142). Increasing the fixed starting multiple permits any prescribed bounded interval of \(T\).

For (ii), choose \(\beta\) with \(1/2<\beta<\alpha\). On a rescaled cone put \(D=\delta_s/d_n\). The weighted norm to be made small is \(D^\beta(d_n/b_n)M\). On a compact range of \(D\) it tends to zero by (143); on the tail \(D\ge R\), (141) bounds it by \(CR^{\beta-\alpha}\). First take \(R\) large, then \(n\) large. This proves a vanishing weighted bound down to every fixed positive depth ratio. Apply 34 with a small fixed multiple of \(c_1d_n\) as its scale and a large enough fixed multiple of \(T_1d_n/b_n\) as its radius. Its ratio \(r/d\) is bounded because \(b_n\ge1\). The resulting relative error tends to zero. The real no-patch transfer gives \(h_X(t)\ge H-c_1d_n\) uniformly in the stated real interval. Since \(c_1\) is arbitrary and \(h_X\le H\), this proves (144).

For (iii), boundedness of \(b_n\) turns the vanishing weighted bound into the same statement with normalization \(d_nM\). Use 34 with starting scale \(d_n/20\), radius \(2d_n\), and target aperture at least \(50\). Its hypotheses need only hold down to a fixed smaller depth ratio. We obtain the shifted impact data needed on the boundaries of the disks \[ t=d_n\zeta,\qquad y=x_n+\mathrm i\sigma-\frac{\mathrm id_n}{4}\zeta^2+w, \quad |\zeta|\le1, \tag{145}\] where \(w\) is small and \(\sigma\) rises from \(H-4d_n\) to \(H\). On upper imaginary parameter arcs, recover \(L(y,t)\) from the preceding two impacts, at phases \(y-t/2\) and \(y-3t/2\), and reflection at the nearer impact. On lower arcs use the two forward impacts. These are meromorphic identities in the line and point data, so exceptional affine configurations are covered by continuation. At \(\sigma=H\), \(w=0\), and \(\zeta=e^{\mathrm i\theta}\), the nearer of the two selected impacts has depth \[d_n\left(\frac{1-2\sin^2\theta}{4} +\frac{|\sin\theta|}{2}\right)\ge\frac{d_n}{4}.\] The farther one is deeper, and all horizontal shifts fit the fixed aperture. On overlaps near real parameter directions both constructions agree with the primary line by continuation from below. At the starting height the full disks have the necessary impact data. The meromorphic disk-continuation principle therefore fills (145) up to \(\sigma=H\). At their centers it gives a joint patch of \(L\) through \((x_n+\mathrm iH,0)\), attached to the primary zero-parameter data, contrary to 24. ◻

Proof. Choose \(\alpha\) with \(\max\{p_0,1/2\}<\alpha<1\). On any cone with \(\delta_s\ge d\), \[dM(s)\le C d^{1-p_0}(d/\delta_s)^\alpha (\delta_s/d)^{\alpha-p_0}.\] More directly, the weighted quantity \(dM(s)(\delta_s/d)^\alpha\) is bounded by \(C d^{1-\alpha}\delta_s^{\alpha-p_0}\), which tends to zero uniformly for depths between any fixed multiple of \(d\) and the fixed starting row. Thus 36(iii) applies with \(b=1\), giving a contradiction. ◻

Proof. Fix \(1/2<\alpha<1\) and the cone aperture needed in the bounded-scale quiet contradiction. Choose \(C_2\) so that \(e^{-C_2}d\) is below all the fixed depth ratios required there, and then choose \(C_1>A/(1-\alpha)\). If no uniform \(c_*\) exists, there are \(k_n\to\infty\) and centers such that \(\sup\delta_sM(s)\to0\) in their windows. On that window, \[d_nM(s)(\delta_s/d_n)^\alpha =\delta_sM(s)(d_n/\delta_s)^{1-\alpha}\longrightarrow0\] uniformly, because the last factor is at most \(e^{(1-\alpha)C_2}\). Below the window, where \(\delta_s/d_n\ge k_n^{C_1}\), the global polynomial bound gives \[\delta_sM(s)(d_n/\delta_s)^{1-\alpha} \le C'(1+k_n)^A k_n^{-C_1(1-\alpha)}\longrightarrow0.\] The remaining fixed lower-strip portion has bounded \(M\) and is smaller still. This is the vanishing cone bound forbidden by 36(iii), with \(b_n=1\). ◻

Proof. Write the limsup in (147) as \(\rho\), and fix \[1<\alpha_1<\alpha_2<\rho<P.\] We first select scales with useful bounds on both sides. There are \(k_n\to\infty\) such that \(g(k_n)\ge\alpha_2k_n\) and \[ g(k)\le g(k_n)+ \begin{cases} \alpha_1(k-k_n),&k_0\le k\le k_n,\\ P(k-k_n),&k\ge k_n. \end{cases} \tag{148}\] Here \(k_0\) is fixed and sufficiently large. To prove this, take arbitrarily late running records \(r_n\) of \(g(k)-\alpha_1k\) for which \(g(r_n)/r_n>\alpha_2\). Such records exist: a time with \(g(k)>\alpha_2k\) forces the preceding running maximum to occur at a time \(r\le k\) with \(g(r)-\alpha_1r>(\alpha_2-\alpha_1)k\ge(\alpha_2-\alpha_1)r\). The maximizing times tend to infinity. Since \(P>\rho\), the function \(g(k)-Pk\) tends to \(-\infty\); let \(k_n\ge r_n\) maximize it on \([r_n,\infty)\). For \(k\ge k_n\) this gives the second inequality in (148). If \(r_n\le k\le k_n\), it gives \(g(k)\le g(k_n)+P(k-k_n)\le g(k_n)+\alpha_1(k-k_n)\). For \(k\le r_n\), combine the record bound with \(g(k_n)\ge g(r_n)+P(k_n-r_n)\), obtaining the first inequality again. The same last inequality gives \(g(k_n)\ge\alpha_2 k_n\).

Set \[ d_n=e^{-k_n},\qquad m_n=e^{g(k_n)},\qquad b_n=d_nm_n. \tag{149}\] Then \(b_n\to\infty\), and in fact \[ \frac{b_n}{\log m_n}\longrightarrow\infty, \tag{150}\] because \(b_n\ge e^{(\alpha_2-1)k_n}\), whereas \(\log m_n=O(k_n)\).

Compact profiles and a trivial pair. For any choice of real centers \(x_n\), introduce the rescaled shape coordinate \(S=(s-x_n-\mathrm iH)/d_n\) and fields \[ F_n(S)=\frac{f(x_n+\mathrm iH+d_nS)}{m_n},\qquad Q_n(S)=\frac{q(x_n+\mathrm iH+d_nS)}{m_n^2}. \tag{151}\] Their domains exhaust the lower half-plane. The envelopes give, with \(D=-\Im S\), \[ \max\{|F_n(S)|,|Q_n(S)|^{1/2}\} \le \begin{cases} D^{-\alpha_1},&D\ge1,\\ D^{-P},&0<D\le1, \end{cases} \tag{152}\] on every fixed compact set for all large \(n\). These bounds are uniform in all real centers. Montel’s theorem gives subsequential holomorphic limits \(F,Q\). The rescaled differential equation is \[ F_n''=b_n^2Q_nF_n. \tag{153}\] Cauchy estimates bound the left side locally, so \(FQ=0\). Since the half-plane is connected, either \(F\equiv0\) or \(Q\equiv0\). Every limit also satisfies the deeper bound \(D^{-\alpha_1}\) in (152).

We claim that some sequence of centers, after passage to a subsequence, has \(F=Q=0\). Suppose otherwise. Fix a small closed disk about \(-\mathrm i\) inside the half-plane and take the supremum norms of the two fields on it. Compactness then gives a constant \(c>0\) such that, for all centers and large \(n\), the larger of the two norms is at least \(c\). Otherwise a sequence with both norms tending to zero would have the zero pair as its limit everywhere, by analytic uniqueness. Purity and compactness likewise imply that the smaller norm tends to zero uniformly over centers: any contrary sequence would have a limiting pair with both norms positive. The norms depend continuously on the center. Once their smaller value is less than \(c/3\), the two possible dominant types give disjoint open sets covering the connected circle of centers. Thus one type holds at every center. Passing to a subsequence, it is the same type for every \(n\).

Suppose first that it is the \(F\)-type. All local limiting functions \(F\) obtained from arbitrary centers are nonzero. This compact family has a uniform bound on the number of zeros in any fixed smaller disk about \(-\mathrm i\), and, for every fixed small \(\eta>0\), a uniform positive lower bound away from the \(\eta\)-neighborhoods of those zeros. Here is a direct compactness justification. A convergent subsequence has a nonzero holomorphic limit; choose a slightly larger circle without zeros of that limit. The argument principle bounds the number of zeros inside, and uniform convergence bounds the function below on a smaller compact set away from the zeros. A failure of either uniform assertion would produce a contrary convergent subsequence. The same assertions hold on the slightly enlarged scaled neighborhoods needed to cover a narrow band about the row \(D=1\).

Cover the physical periodic row at depth \(d_n\) by \(O(d_n^{-1})\) cells of size comparable to \(d_n\). Around its nearby zeros remove disks of radii between \(\eta d_n\) and \(2\eta d_n\), with generic radii to avoid boundary tangencies. Choose \(\eta\) sufficiently small in terms of the uniform local zero count. Every connected cluster meeting the row then has a uniformly bounded number of disks and diameter much smaller than \(d_n\). Indeed, a minimal chain escaping a fixed enlarged cell would require more zeros in that cell than the uniform bound allows. Fill holes in the clusters and discard nested obstacles. Detour the row along the outer boundary arcs of these clusters. The resulting closed contour \(C_n\) winds once around the periodic cylinder, stays in a fixed narrow band about depth \(d_n\), and satisfies \[ \operatorname{length}(C_n)=O(1),\qquad |f|\ge c_\eta m_n\quad\hbox{on }C_n. \tag{154}\] The length estimate follows because the total number of relevant zero disks is \(O(d_n^{-1})\), each with circumference \(O(d_n)\).

All poles of \(f^{-2}\) in the shape strip are at simple zeros of \(f\). At such a zero \(z\), 30 gives \(f''(z)=0\), and hence \[\operatorname{Res}_{z}(f^{-2}) =-\frac{f''(z)}{f'(z)^3}=0.\] Deforming \(C_n\) to the real period, with residues included, therefore gives \[\int_{C_n}\frac{\,\mathrm ds}{f(s)^2} =\int_0^{2\pi}\frac{\,\mathrm ds}{f(s)^2}>0.\] The right side is a fixed positive number because \(f\) is positive on the real curve. The absolute value of the left side is \(O(m_n^{-2})\) by (154). This is a contradiction.

Suppose next that the common type is the \(Q\)-type. There are centers \(x_n\) with \[ |f(x_n+\mathrm i(H-d_n))|\ge c_0>0. \tag{155}\] Indeed the horizontal mean of \(\log|f|\) is convex as a function of height, by subharmonicity and periodicity. It is even by real symmetry, so it is at least its finite real-row value. At least one point on each row has modulus no smaller than the exponential of that mean. Zeros cause no difficulty because \(\log|f|\) is locally integrable. At these centers pass to a subsequence with \(Q_n\to Q\not\equiv0\).

Near a point where \(Q\ne0\), choose a unit complex number \(e\) and a fixed short segment in direction \(e\) on which \(\Re(e^2Q_n)\ge c'>0\) for all large \(n\). On that segment, with real parameter \(v\), the nonnegative function \(u(v)=|F_n(z+ev)|^2\) satisfies \[u''=2|eF_n'|^2+2b_n^2\Re(e^2Q_n)|F_n|^2 \ge 2c'b_n^2u.\] The endpoint values are uniformly bounded by normality. Comparison with the solution of \(v''=2c'b_n^2v\) having those endpoint bounds shows, on a shorter central segment, \(u\le C e^{-c b_n}\). To justify comparison, subtract that solution; a positive interior maximum of the difference contradicts \(w''-2c'b_n^2w\ge0\). The same estimate holds uniformly on a small neighborhood of the chosen point.

Choose a circle centered at \(-\mathrm i\), contained in the half-plane, and avoiding the discrete zero set of \(Q\). A finite cover by these neighborhoods gives \(|F_n|\le C e^{-c b_n}\) on the circle, after changing \(c\). Maximum modulus gives the same bound at its center. Therefore \[|f(x_n+\mathrm i(H-d_n))|\le C m_ne^{-c b_n}\longrightarrow0\] by (150), contradicting (155). Both common types are impossible, proving the existence of trivial profiles.

A global width bound and a nontrivial limiting lift. Use the centers with the trivial pair in 36(ii). The bound (152) is stronger on the coarse tail than its hypothesis, since \(\alpha_1>1\); it may be weakened to any exponent between \(1/2\) and \(1\). Since \(d_n/b_n=1/m_n\), we obtain \[ h_X(T/m_n)\ge H-o(d_n) \tag{156}\] uniformly for \(T\) in each fixed bounded real interval. This is a global row width. We may therefore choose new centers \(x_n\) attaining the row maximum that defines \(m_n\). At these centers extract limits \(F,Q\). They are pure, and they are not both zero because \[\max\{|F_n(-\mathrm i)|,|Q_n(-\mathrm i)|^{1/2}\}=1.\]

Fix a real \(T\ne0\). Define, now as a function of the rescaled phase coordinate \(Y\), \[S_n(Y)=\frac{\mathfrak s(x_n+\mathrm iH+d_nY,T/m_n)-x_n-\mathrm iH}{d_n}.\] By (156), its phase domains exhaust \(\Im Y<0\). By the shape constraint in 29, \(\Im S_n<0\). For any \(1/2<\alpha<1\), the coarse cone estimate anchors these half-plane-valued functions on deep compact sets: \[ |S_n(Y)-Y|\le C_TD^{1-2\alpha},\qquad D=-\Im Y\ge D_T, \tag{157}\] on the fixed compact portions under consideration, with constants uniform in horizontal position. To obtain such uniform constants, apply the cone theorem at each real center; the row bounds and the lower-strip data are the same at every center.

Map the target half-plane conformally to the unit disk. Montel’s theorem then gives a subsequential limit. The anchor excludes convergence to a boundary constant or infinity, so the limit is a holomorphic map \(S\) of the entire lower half-plane into itself. Passing to the limit in (157) gives the same anchor for \(S\).

The limiting energy equation. Set \(h_n=T/b_n\), which tends to zero. Then \[J_n(Y)=\frac{S_n(Y+h_n)-S_n(Y)}{h_n}\longrightarrow S_Y(Y)\] locally analytically. On a sufficiently deep half-plane, the cone estimates and Cauchy estimates keep these steps close to \(1\) and the fast fields small. The exact recurrence holds there with its continued glancing root, also jointly for \(T\) along the segment from \(0\) to the fixed value chosen. This segment, begun at \(T=0\) in the deep small-field regime, selects the fixed sheet required by 33; no additional loop is used. Let \(p_n(A,J)\) denote the normalized action with rescaled shape basepoint \(A\). Its fast fields are \[T F_n(A+h_nu),\qquad T^2Q_n(A+h_nu).\] They converge analytically to the frozen fields \(TF(A)\), \(T^2Q(A)\). By 33, the actions and their derivatives converge analytically to \[p(A,J)= \begin{cases} p_F(TF(A),J),&Q\equiv0,\\ p_Q(\lambda(A),J),\quad\lambda(A)^2=T^2Q(A),&F\equiv0. \end{cases}\] The second expression depends analytically on \(\lambda^2\) near zero, so no choice of a square root is needed there.

In these coordinates (128) reads \[p_{n,J}(S_n(Y),J_n(Y)) -p_{n,J}(S_n(Y-h_n),J_n(Y-h_n)) =h_n p_{n,A}(S_n(Y),J_n(Y)).\] Division by \(h_n\) is legitimate without any rate of convergence assumption. Indeed, if \(G_n(Y)=p_{n,J}(S_n(Y),J_n(Y))\), then \[\frac{G_n(Y)-G_n(Y-h_n)}{h_n} =\int_0^1G_n'(Y-\theta h_n)\,\,\mathrm d\theta,\] and analytic convergence gives convergence of \(G_n'\). The limit is consequently the Euler–Lagrange equation \[\partial_Yp_J(S,S_Y)=p_A(S,S_Y).\] For \(E(A,J)=Jp_J(A,J)-p(A,J)\), the chain rule now gives \[\partial_Y E(S,S_Y) =S_Y\bigl(\partial_Yp_J(S,S_Y)-p_A(S,S_Y)\bigr)=0.\] The deep decay of the fields and the anchor, with Cauchy estimates, give \(S_Y\to1\) as \(D\to\infty\). Hence the constant is \[ E(S,S_Y)=\frac1{12}. \tag{158}\]

The stronger exponent \(\alpha_1\) now improves the anchor. At \((\lambda,J)=(0,1)\), both frozen energies have \(E_J=1/4\); their first field correction is quadratic by (131)–(132). The analytic implicit function theorem applied to (158), together with (152), yields \[|S_Y-1|\le C_TD^{-2\alpha_1}\] on a deeper half-plane. The argument of the fields is \(S(Y)\), whose depth is comparable to \(D\) by the original anchor. Integrate along vertical lines from infinity, where the original anchor makes \(S-Y\) tend uniformly to zero. We obtain \[ |S(Y)-Y|\le C_TD^{1-2\alpha_1}=o(D^{-1}). \tag{159}\]

Half-plane rigidity and the contradiction. Put \(v(Y)=\Im(Y-S(Y))\). It is harmonic and satisfies \(v\ge-D\), since \(S\) maps into the lower half-plane. We verify carefully that \(v\ge0\). On a slab \(\varepsilon<D<R\), its top boundary is bounded below by \(-\varepsilon\), and its bottom boundary is bounded below by \(-\eta_R\), where \(\eta_R=\sup_x|S(x-\mathrm iR)-(x-\mathrm iR)|\to0\). Compare on finite rectangles with the harmonic function interpolating these two constants on the horizontal sides. The extra negative contribution needed on the vertical sides is bounded by a constant depending on \(R\) times their harmonic measure, because \(v\ge-R\) in the slab. Their harmonic measure tends to zero when the sides move to infinity. Thus \[v(Y)\ge-\varepsilon\frac{R-D}{R-\varepsilon} -\eta_R\frac{D-\varepsilon}{R-\varepsilon}.\] First let \(\varepsilon\downarrow0\), then \(R\to\infty\), to get \(v\ge0\). Harnack’s inequality in the half-plane gives, for \(R\ge1\), \[v(-\mathrm iR)\ge R^{-1}v(-\mathrm i).\] For example, this is the ordinary disk Harnack inequality after mapping \(-\mathrm i\) to the origin, since \(-\mathrm iR\) has disk modulus \((R-1)/(R+1)\). By (159) its left side is \(o(R^{-1})\). Hence \(v(-\mathrm i)=0\), and the strong minimum principle gives \(v\equiv0\). The holomorphic function \(S-Y\) is therefore real constant; the anchor makes that constant zero. We have proved \(S(Y)=Y\).

Finally, set \(J=1\) in (158). Deep down the field parameter is small, and the energy expansions are \[E_F(\lambda,1)-\frac1{12} =-\frac{\lambda^2}{480}+O(\lambda^4),\qquad E_Q(\lambda,1)-\frac1{12} =-\frac{\lambda^2}{60}+O(\lambda^4).\] Each right side can vanish near zero only when \(\lambda^2=0\). Since \(T\ne0\), the surviving pure field vanishes on a deep open set, and hence identically. This contradicts its normalization at \(-\mathrm i\), proving the Theorem. ◻

Proof. We first make a uniformity observation. With \(D=\delta/\ell_n\), weakening the exponent in (162) gives \[D^\alpha M/m_n\le C_LD^{-(\alpha_0-\alpha)}.\] For each fixed \(L\), this is bounded by an absolute constant beyond a finite depth threshold depending on \(L\). On the remaining finite cone, (161) supplies the same bound eventually: the limiting value of \(M/m_n\) is \(|c_0|e^{-D}\), independent of the horizontal position. Consequently the weakened cone bound can have a constant independent of \(L\), provided its eventual sequence threshold is allowed to depend on \(L\). This permits the choice of \(D_0\) before a large aperture.

Suppose first that \(H<\infty\). Choose \(D_0\ge10\), so the required input depth \(d/10\) is at least \(\ell_n\). Apply 34 and its reference-height formulation in 35 with reference height \(h_n\), rather than \(H\), and with \[d=\ell_nD_0,\qquad r=\frac{2T_0}{m_n},\qquad \varepsilon=C'T_0D_0^{-\alpha}.\] The cone hypothesis is exactly \(rM\le\varepsilon(d/\delta)^\alpha\). Since \(r/d=2T_0/(b_nD_0)\to0\), the fixed ratio allowance in that theorem can be chosen to be one. Its auxiliary bound satisfies \[B=1+(\varepsilon/r)d^\alpha\le C m_n\ell_n^\alpha.\] Here \(m_n\ell_n^\alpha\to\infty\), since \(b_n\to\infty\) and \(\ell_n\to0\). The physical displacement estimate, divided by \(\ell_n\), is therefore (165). Increasing \(D_0\) makes the required smallness conditions hold. The starting slab lies in a fixed strict substrip of the original shape strip. The relative displacement is small compared with the depth below \(h_n\); in particular the image remains in \(D_H\). This use of an auxiliary reference height needs no no-patch assertion at that height. The aperture assertion of 34, together with the preceding weakening argument, gives the stated independence of \(D_0,C\) from \(A\).

We provide the initializer needed when \(H=\infty\), because the starting height is then unbounded. Retain the same \(d,r\), and put \(B=Cm_n\ell_n^\alpha\). All horizontal ranges may now be used. Condition (163) implies \[ r(h_n/d)^\alpha\longrightarrow0, \qquad Bh_n^{-\alpha}\longrightarrow\infty. \tag{166}\] Thus the known data near \(\Im y=0\) have uniformly small parameters over the entire range that will be used.

Consider first \(|\Im t|\ge\gamma|t|\), with a fixed steep cutoff \(\gamma>0\). Use the upward increment \(t\) or \(-t\), as appropriate. The normalized chord recurrence and kernel estimates in 34 apply along the segment from the real-row slab to depth \(\delta\ge d/2\), for \(|t|\le10r(\delta/d)^\alpha\). Indeed the fast fields are small there after increasing \(D_0\), while \(|t|/\delta\) is bounded. Local Cauchy disks have radius a fixed fraction of the depth. Any part of a thickening slightly below the real row is covered by real symmetry and the comparable bound above that row. The recurrence estimates, with \(x\) denoting the current depth, are \[|J-1|\le C|t|^2B^2x^{-2\alpha},\qquad |\mathfrak s-y|\le C|t|^2B^2x^{1-2\alpha}.\] To see the accumulation directly, the change of normalized step is \(O(|t|^3B^2x^{-2\alpha-1})\). Summation over rows separated by a constant multiple of \(|t|\) gives the first estimate, and a second summation gives the second because \(2\alpha>1\). Initial step and position errors of order \(|t|^2\) contribute at most \(C|t|^2(1+h_n)\) to the latter. This is absorbed, uniformly at all \(\delta\le h_n\), because \[B^2h_n^{1-2\alpha}/(1+h_n)\longrightarrow\infty\] by (166). The small relative-error induction consequently closes. Choosing the number of backward steps locally constantly, with two starting entries inside the known slab, gives holomorphic germs. Different choices agree by the exact recurrence and uniqueness of its nearby root.

It remains to initialize all parameter directions at large depths. Fix a large constant \(K_0\). At a target depth \(h_n/K_0\le\delta\le h_n\), set \(R_t=r(\delta/d)^\alpha\). For \(|t_c|<R_t\) consider \[ t'=t_c+3R_t\zeta,\qquad y=x+\mathrm i\sigma+\sigma g(\zeta),\qquad 0\le\sigma\le h_n-\delta, \quad g(0)=0. \tag{167}\] Choose a smaller fixed steep cutoff for this initializer. On the unit circle the nonsteep arcs for \(t'\) are uniformly short: the circle has radius \(3R_t\), its center has modulus at most \(R_t\), and its intersections with a sufficiently narrow horizontal sector are transverse. We choose a smooth real boundary function of mean zero which equals \(-1\) on these arcs and slightly larger collars, is nowhere below \(-1\), and is at most \(1/(6K_0)\). This is possible by making the arcs sufficiently short and balancing the negative bumps by a small positive part. Approximate it, with its mean removed, by a real trigonometric polynomial with error \(o(1/h_n)\). It is the boundary imaginary part of a holomorphic polynomial \(g\) with \(g(0)=0\). In particular, \[1+\Im g=o(1/h_n)\ \hbox{on the nonsteep collars},\qquad -o(1/h_n)\le1+\Im g\le1+1/(3K_0)\] on the whole circle. The smooth bumps can be chosen with uniform smooth bounds before approximation, so \(g\) is uniformly bounded; its degree need not be fixed.

On the nonsteep collars, (167) lands in the original fixed real-row neighborhood. On the other arcs its depth is at least \(\delta-h_n/(3K_0)\ge2\delta/3\), and the steep estimates just proved apply, or the point is still in the base neighborhood. All parameter sizes tend to zero by (166). These boundary germs glue as vertical continuations from below.

Here and in subsequent disk fillings we use the following explicit holomorphic continuation argument. Pull the boundary data back to a thin annulus about \(|\zeta|=1\), times a complex neighborhood of the center path. Their negative Laurent coefficients in \(\zeta\) are holomorphic functions of the center variables. Initially the full disks lie in known data, so all those coefficients vanish. The identity theorem makes them vanish along the whole connected center path. The nonnegative Laurent series fills each disk and gives joint holomorphic center germs. Applied to \((\mathfrak s-y)/(t')^2\), which is regular at \(t'=0\) in the initial data, the maximum principle transfers the boundary estimate to the disk center. Small complex changes of \(\sigma,t_c\) are allowed by the strict margins above, so these really are joint germs. This proves the all-direction estimate for \(h_n/K_0\le\delta\le h_n\), with constants independent of \(h_n\).

The dyadic disk construction in 34 now continues it from \(h_n/K_0\) to \(d\). Choose \(K_0\) first so large that its previous-band dips and starting slabs remain at positive heights. No factor accumulates with the number of bands: the exponent \(1-2\alpha<0\) makes the estimates on the deeper, previously known arcs strictly smaller, whereas the steep-arc estimates have an independent fixed constant. Fix both steep cutoffs and \(K_0\), and only then increase \(D_0\) to meet their smallness requirements. After scaling by \(\ell_n\), the result is (165). In the infinite-width case the shape constraint is automatic. This completes both cases. ◻

Proof. The deep estimates and Cauchy’s inequalities give locally normal families and the asserted anchors for any subsequential limit. Fix \(T\ne0\) and put \(a_n=T/b_n\). On a compact set with a deep margin, \[J_n(Y)=\frac{S_n(Y+a_n,T)-S_n(Y,T)}{a_n} \longrightarrow\partial_Y S_*(Y,T).\] The fast fields used in the normalized action are \(TF_n(S+a_nu)\) and \(T^2Q_n(S+a_nu)\). They converge analytically to the appropriate frozen pair in (168). The action \(p_n(S,J;T)\) and its derivatives converge on these neighborhoods by 33. In particular the case \(F_n\to0\) uses the analytic division in 32, rather than division of two limiting zeros.

The exact recurrence takes the form \[ p_{n,J}(S,J)-a_np_{n,S}(S,J) =p_{n,J}(S_-,J_-),\qquad S=S_-+a_nJ_-. \tag{170}\] It holds on the passages by analytic continuation. To fix its root choice, first take \(T\) sufficiently small, for each fixed \(n\), so that all physical points in question lie in known strict substrips with the glancing branch. Continue in the deep \(T\)-disk. Its shifted arguments remain in the domain with margin, and Cauchy estimates keep \(J_n\) near one. The fast fields are uniformly small after increasing \(D_0\), so the relevant kernels and roots remain nondegenerate throughout that continuation.

Divide the difference in (170) by \(a_n\). Both terms are evaluations of the same analytic function \(p_{n,J}\) at neighboring states. Analytic convergence therefore gives \[\frac{\,\mathrm d}{\,\mathrm dY}p_J(S_*,J)=p_S(S_*,J).\] This step requires no rate of convergence compared with \(a_n\). Since \(S_{*,Y}=J\), differentiating \(Jp_J-p\) shows that it is constant in \(Y\). The deep anchor gives its value \(1/12\). All statements hold on arbitrary compact sets by exhaustion, and at \(T=0\) by removability.

For uniqueness write \(w=Tc_0e^{-\mathrm iY}\) and \(z=\lambda J/2\). The energy equation is \[ \lambda^3=24(\sin z-z\cos z) \quad\hbox{or}\quad \lambda^3=12(\tanh z-z\operatorname{sech}^2z). \tag{171}\] At \((\lambda,J)=(0,1)\), \(\partial_J E=1/4\); thus it determines \(J\) analytically as a function of \(\lambda\) near that point. Moreover \[w\frac{\,\mathrm d\lambda}{\,\mathrm dw}=\lambda J=2z, \qquad \frac{\lambda}{w}\longrightarrow1.\] Integrating this differential relation gives \[ w=\lambda\exp\left(\int_0^\lambda \frac{1/J(v)-1}{v}\,\,\mathrm dv\right), \qquad S_*-Y=\mathrm i\log(\lambda/w). \tag{172}\] The integrand is analytic at zero, the inverse has derivative one there, and the anchor fixes the constant and logarithm. Hence there is a unique deep model. Normality now gives convergence of the whole sequence on its compact domains, with derivatives by Cauchy’s inequalities. ◻

Proof. In the first case of (171) put \(v=z\) and define \[ g(v)=\frac{3(\sin v-v\cos v)}{v^3},\qquad P(v)=\frac{v g(v)^{2/3}}{\sin v},\qquad v_c=\pi. \tag{173}\] In the second case put \(v=\tanh z\) and define \[ g(v)=\frac{3}{2v^3} \bigl[v-(1-v^2)\operatorname{arctanh}v\bigr],\qquad P(v)=g(v)^{2/3},\qquad v_c=1. \tag{174}\] All branches at zero are normalized by \(g(0)=P(0)=1\). Direct differentiation of (171) gives, in both cases, \[ \lambda=2v g(v)^{1/3},\qquad wv'(w)=vP(v),\qquad v(w)\sim w/2. \tag{175}\] For example, in the first case \(\lambda_v=2\sin v/(vg(v)^{2/3})\); in the second it is \(2\operatorname{arctanh}v/(vg(v)^{2/3})\).

We verify the positivity and zero-freeness needed to control the analytic region. In (173), every nonzero zero of \(\sin v-v\cos v\) is real. Indeed \(u(x)=\sin(vx)\) solves \(-u''=v^2u\), \(u(0)=0\), \(u'(1)=u(1)\), and integration by parts gives the nonnegative quadratic form \[\int_0^1|u'|^2\,\,\mathrm dx-|u(1)|^2\ge0;\] the inequality follows from \(u(1)=\int_0^1u'\). Hence \(v^2\) is real and nonnegative. The positive roots \(r_k\) are simple and satisfy \(k\pi<r_k<k\pi+\pi/2\), by \(\tan v=v\). Hadamard factorization for the even entire function of order one (McMullen 2017, Theorem 3.7, p. 43) therefore gives \[g(v)=\prod_{k\ge1}(1-v^2/r_k^2).\] Comparison with the sine product yields \[\log P(v)=\sum_{j\ge1}\frac{v^{2j}}{j} \sum_{k\ge1}\left((k\pi)^{-2j}-\frac23r_k^{-2j}\right).\] All its coefficients are strictly positive, and it is analytic for \(|v|<\pi\). In particular \(P\) has nonnegative even coefficients, with a positive quadratic coefficient.

In (174), \[g(v)=\sum_{j\ge0}a_jv^{2j},\qquad a_j=\frac{3}{(2j+1)(2j+3)},\qquad a_0=1.\] Write its reciprocal formally as \(1-\sum_{j\ge1}b_jv^{2j}\). The ratios \(a_j/a_{j-1}=(2j-1)/(2j+3)\) increase. The recurrence \[b_n=a_n-\sum_{j=1}^{n-1}b_ja_{n-j}\] shows by induction that \(0\le b_n\le a_n\). For completeness, with \(r_n=a_n/a_{n-1}\), the induction hypothesis gives \[\sum_{j=1}^{n-2}b_ja_{n-j} \le r_n(a_{n-1}-b_{n-1}),\qquad b_n\ge b_{n-1}(r_n-a_1)\ge0.\] These series converge in the unit disk. For real \(0\le r<1\), \(\sum b_jr^{2j}=1-1/g(r)<1\); thus its modulus is less than one throughout that disk. Consequently \(g\) is zero-free there and \(P=(1-\sum b_jv^{2j})^{-2/3}\) has nonnegative even coefficients, again with a positive quadratic coefficient.

The inverse of (175) is analytic at zero: \[w(v)=2v\exp\left(\int_0^v \frac{1/P(\xi)-1}{\xi}\,\,\mathrm d\xi\right).\] The differential recurrence for its inverse shows that \(v(w)\) is odd and has nonnegative Taylor coefficients. Its linear coefficient is \(1/2\), and its cubic coefficient is strictly positive. On \(0<v<v_c\), \(w(v)\) increases and \[\frac{\,\mathrm d\log w}{\,\mathrm dv}=\frac{1}{vP(v)}.\] It tends to a finite positive limit \(R\) as \(v\uparrow v_c\): the last integrand is integrable at \(\pi\) in the first case and has a finite limit at one in the second.

The power series of \(v(w)\) reaches radius \(R\). Otherwise, at a smaller radius \(\rho\), positivity bounds its sum of absolute coefficients by the real solution \(v(\rho)<v_c\). It converges uniformly on the closed circle and the regular differential equation (175) continues it through every point of that circle, a contradiction. At radius \(R\) its positive sum is \(v_c\). At phases other than zero and \(\pi\), the modulus is strictly smaller than \(v_c\): equality in the triangle inequality for the strictly positive linear and cubic terms would require \(e^{2\mathrm i\theta}=1\). The differential equation therefore gives continuation through those phases.

For \(0<|w|<R\), the solution \(v(w)\) never vanishes, since uniqueness for the regular differential equation at a nonzero \(w\) would otherwise make it identically zero. It follows that \(\lambda/w\) is nonzero and holomorphic on the whole disk, with value one at zero. Thus (172) defines the impact model there. Its regularity tests can be made explicit. In the first case, \[J=g(v)^{-1/3},\qquad p_{JJ}=\frac{\sin v}{2\lambda},\qquad k_0=J,\quad k_+=k_-=\frac{2\sin v}{\lambda}.\] In the second case, with \(z=\operatorname{arctanh}v\), \[J=\frac{2z}{\lambda},\qquad p_{JJ}=\frac{v(1-v^2)}{2\lambda},\qquad k_0=k_+=k_-=\frac{2v}{(1-v^2)\lambda}.\] They are nonzero in the inner disk, with removable values \(J=k_j=1\), \(p_{JJ}=1/4\) at zero.

We finally verify that continuation at \(R\) actually changes the impact model. In the first case \(w(v)\) is analytic at \(v=\pi\) and has a simple quadratic critical point there. If \(\lambda_c=(24\pi)^{1/3}\), then \(w''(\pi)=-4R/\lambda_c^2\ne0\). Hence \(v(w)\) has a nonzero square-root term. The identity \(w\lambda_w=2v\) shows that \(\lambda(w)\), though its square-root term cancels, has a nonzero term of order \((w-R)^{3/2}\). More explicitly, on the branch from \(0<w<R\), with \(\eta=1-w/R\), \[\lambda=\lambda_c-2\pi\eta +\frac{2\sqrt2\lambda_c}{3}\eta^{3/2}+O(\eta^2).\] A turn around \(R\) changes this value.

In the second case let \(x=v-1\), \(G=3/2\), and \(\Lambda=2G^{1/3}\). On any fixed bounded range of turning on the logarithmic cover, \[g(v)=G\bigl[1-x\log(-x)+(\log2-2)x +O(x^2(1+|\log|x||))\bigr].\] Integration of \(w_v=w/(vg(v)^{2/3})\) gives \[ w(v)=R+a_0x+O(x^2(1+|\log|x||)),\qquad a_0=R/G^{2/3}\ne0. \tag{176}\] The inverse on a lifted sector exists with \(x=(w-R)/a_0+O(|w-R|^2(1+|\log|w-R||))\). For example, Rouche’s theorem on disks of a sufficiently small fixed relative radius gives this inverse and its continuation; the derivative tends to \(a_0\). Substitution in \(\lambda=2vg(v)^{1/3}\) gives, with \(\delta=w-R\), \[\lambda=\Lambda-\frac{\delta}{R}\log(-\delta/a_0) +\frac{1+\log2}{R}\delta +O\bigl(|\delta|^2(1+|\log|\delta||^2)\bigr).\] The same coefficients apply over any fixed bounded amount of turning. The logarithmic coefficient is nonzero, so a turn changes the value; the remainder is smaller than that change on a sufficiently small fixed bypass.

These branches attach to the inner model along the real interval \(w<R\). On a sufficiently small compact bypass, in the first case \(v\) stays near but not at \(\pi\). In the second it stays near but not at one, with \(v\ne0,\pm1\), and the chosen \(\operatorname{arctanh}v\) is finite and nonzero on each compact lifted corridor. The displayed kernel and twist formulas are therefore finite and nonzero, also after the turn. Since \(\lambda\) changes and remains nonzero, \(\mathrm i\log(\lambda/w)\) changes as well. This proves the Lemma. ◻

Proof. Cover the compact model states by finitely many analytic action charts with kernels and \(p_{JJ}\) bounded away from zero. The analytic action convergence in 33 applies on these charts, following the branches initialized in the known data. Given the old state \((S_-,J_-)\), set \(S=S_-+a_nJ_-\). Equation (170) has a unique nearby root for \(J\) by the implicit function theorem. Evaluating its left side at \(J=J_-\), and subtracting before taking the limit, gives \[J-J_-=a_n\frac{p_S-J_-p_{SJ}}{p_{JJ}}(S_-,J_-)+o(|a_n|).\] The error is uniform with state derivatives on the compact charts. Indeed the corresponding expression with \(p_n\) has a Taylor remainder \(O(|a_n|^2)\), and its coefficient converges with derivatives to the displayed one. The state map consequently is \[ (S_-,J_-)\longmapsto(S_-,J_-) +a_n\left(J_-,\frac{p_S-J_-p_{SJ}}{p_{JJ}}(S_-,J_-)\right) +o(|a_n|), \tag{177}\] and has local Lipschitz bound \(1+C|a_n|\).

The model solves the ordinary differential equation in (177): its second component follows by expanding \(\,\mathrm d_Yp_J=p_S\). In \(O(1/|a_n|)\) steps, the initial error and the local \(o(|a_n|)\) errors sum, with a bounded stability factor, to \(o(1)\). This proves tracking and keeps the continued states in the chosen regular charts. The shape arguments remain in the exhausting shape domains. The implicit roots are holomorphic also in the initial point and parameter; the same estimates on slightly larger compact charts give uniform convergence on thickenings. For the negative increment use the fast fields with \(-T\). The frozen actions are even in \(\lambda\), so their limit is unchanged, and the same calculation applies.

At an initialization obtained by previous regular passage, the required exact recurrence and its branch choice persist by analytic identity from the deep data. Cauchy’s integral formula gives convergence of the first divided differences whenever the two-entry functions converge on neighborhoods. Thus the construction can be reinitialized in every regular patch it reaches. ◻

Proof of 40. Let \[T_* = |c_0|^{-1},\qquad r_0=T_*/32,\] so that \(T_*\) is real and \(|T_*c_0|=1\). Take the parameter bound in 41 larger than all the parameter disks below. We first reach, vertically and with joint convergence, the inner regular region \[ |Tc_0e^{-\mathrm iY}|\le Re^{-\nu},\qquad |T-T_*|\le2r_0, \tag{178}\] for any fixed \(\nu>0\) and every needed compact horizontal range.

Fix a center \(T_c\) in this parameter disk and use \[ T'=T_c+8r_0\zeta,\qquad Y'=Y_c-\mathrm i\log(T'/T_c)+g(\zeta),\qquad g(0)=0. \tag{179}\] The parameter disk is bounded away from zero, so this logarithm is single-valued. Its sign is important: \[T'e^{-\mathrm i[Y_c-\mathrm i\log(T'/T_c)]}=T_ce^{-\mathrm iY_c}.\] Choose a small fixed \(\eta>0\). On the unit circle require the boundary imaginary part of \(g\) to be a sufficiently large negative constant on the arcs \(|\Im T'|<\eta\) and slightly larger collars, and to be at most \(\nu/2\) everywhere. It must also have mean zero. Such a choice is possible: the bad arcs have arbitrarily small total length as \(\eta\downarrow0\), uniformly in the allowed centers, by transversality of the parameter circles. Use smooth negative bumps on those arcs, balance their mean by a positive part smaller than \(\nu/2\), then approximate by a trigonometric polynomial with strict margins. The corresponding holomorphic polynomial can be normalized by \(g(0)=0\).

On the bad arcs the adjusted point lies in deep initialized data. On the other arcs, use the upward increment \(\pm T'/b_n\) to track from a deep slab. Every intermediate point of its backward slanted segment has smaller \(|T'c_0e^{-\mathrm iY}|\) than the target, so the whole segment lies in the inner regular model region. The horizontal range needed for these paths, including the shift by \(g\), can be large, but it is fixed after \(\eta\) is chosen. The depth \(D_0\) in 41 was independent of that aperture, so the initialization remains available. The boundary germs agree as vertical continuations and converge to the model on compact neighborhoods.

Raise \(\Im Y_c\) from a deep initial position to the desired target in (178). By the maximum principle for \(\Im g\), the full model disks in (179) stay within radius \(Re^{-\nu/2}\). Initially the full disks lie in known data. The Laurent-coefficient argument used in the proof of 41 fills the disks as the center rises, giving joint center germs. Apply the maximum principle to the difference between the filled lift and the analytic model. Boundary convergence gives convergence at the centers, and on neighborhoods by retaining the strict margins. This proves (178). The exact chord identities, including neighboring arguments, continue through these passages for all sufficiently large \(n\).

We next reach two regular patches just beyond the model circle, near the phases \[ w=Re^{u+\mathrm i\theta}\quad\hbox{and}\quad w=Re^{u-\mathrm i\theta},\qquad u\le\nu, \quad T=T_*. \tag{180}\] First fix a sufficiently small \(\theta>0\); then choose \(\nu>0\) sufficiently small in terms of \(\theta\). Use parameter circles of radius \(r_0\), so that their initial points remain in the parameter range already reached, and use the same logarithmic adjustment as in (179). This time choose a polynomial with \[\Im g\le-4\nu\ \hbox{on }|\Im T'|<\eta_1 \hbox{ and collars},\qquad \Im g\le\nu,\qquad |g|\le C_1\nu.\] The cutoff \(\eta_1>0\) and constant \(C_1\) can be fixed independently of \(\nu\): first choose the sufficiently short bad arcs, construct one mean-zero adjustment, and multiply it by \(\nu\).

Raise the centers from \(u=-C\nu\), with \(C\) a sufficiently large fixed constant, to \(u=\nu\). The nonsteep arcs remain in the known inner region. On the steep arcs, tracking starts just below \(u=-\nu\), and the additional vertical distance is \(O(\nu)\). The horizontal drift is therefore \(O(\nu/\eta_1)\). Choose \(\nu\) so small that this drift and the real displacement of \(g\) stay in the chosen regular phase corridor about each sign in (180). The full model disks stay in that corridor as well. Disk filling and maximum modulus, exactly as above, give vertical joint passages and compact convergence to the two radial model continuations.

At the real physical parameter \(t=T_*/m_n\), one of these passages reaches the physical height \[h_n+\ell_n(\log R+\nu).\] It remains in the shape strip: this was proved in the deep part and follows on the remaining compact path from model convergence and the exhaustion of scaled shape domains. Consequently it gives joint meromorphic germs for \(X=\Gamma(\mathfrak s)\), attached to the primary branch from below. If the real-parameter joint width ended at an earlier height, the passage would give a prohibited joint patch at its first wall, contrary to 24. The same argument excludes equality at the endpoint, because the passage has a neighborhood there. Thus, for all sufficiently large \(n\), \[ h_X(T_*/m_n)>h_n+\ell_n(\log R+\nu). \tag{181}\] This is the one place where a local passage is promoted to an entire horizontal band, and the parameter here is real. By 29, \(S_n(Y,T_*)\) is a single-valued holomorphic function on a neighborhood of the full row \(\Im Y=\log R+\nu/2\). The two vertical passages agree with this actual lift, by analytic identity from their lower starting segments.

Follow this row from phase \(+\theta\) to phase \(-\theta\), increasing \(\Re Y\), with real steps \(a_n=T_*/b_n>0\). Initialize in the first of the patches (180). The continued model follows the outside of \(w=R\). At its endpoint it differs from the independent radial continuation to the other patch: close the comparison path through the inner disk, and the resulting loop winds once around \(R\), while avoiding \(-R\). This is the nontrivial continuation proved in 43. All states of the two paths and their small thickenings are regular if \(\theta,\nu\) have been chosen small enough.

For clarity, we verify why recurrence tracking on this horizontal path must agree with the actual single-valued lift. Take a small fixed complex ball \(B_*\) in the initial patch, and run the tracking construction for each initial offset in \(B_*\). Its successive output germs live on \(B_*+ja_n\). For an offset \(Y\) and the offset \(Y+a_n\), their first two common entries coincide by the actual data and the exact recurrence in the initial patch; take \(B_*\) with room for two extra steps. Uniqueness of every subsequent nearby root makes the chains coincide on their common entries. Hence the germs on adjacent translated balls agree on overlap. They glue, and by the identity theorem agree throughout with the actual holomorphic function on the band supplied by (181). Tracking gives convergence to the horizontally continued model on these balls.

Choose a number of steps whose endpoint tends to a point in the second regular patch; the remaining offset tends to zero. Compact convergence on neighborhoods makes this harmless. At that point the same actual function would converge both to the horizontally continued model and to the distinct radial model value. This contradiction proves the Theorem. ◻

Proof. Put \(U=K-\log|f|\ge0\) and \(A=1+\int_{\mathbb D}U\). We may assume that this integral is finite. The distribution \(-\Delta U\) is \(2\pi\) times the zero-counting measure, including multiplicities. Testing it against a nonnegative smooth cutoff equal to one on \(|z|\le0.8\) and supported in the unit disk bounds the number of those zeros by \(CA\). If they are \(z_j\), listed with multiplicity, then \[V=U+\sum_j\log|z-z_j|\] is harmonic on \(|z|<0.8\). On a slightly smaller disk it has \(L^1\) norm \(O(A)\), since each logarithm has uniformly bounded \(L^1\) norm there. Interior harmonic estimates bound its first and second derivatives by \(CA\), with a further fixed margin.

Equivalently, divide \(f\) by its listed zero factors. The zero-free quotient has a holomorphic logarithm in the disk, whose first two derivatives are controlled by the derivatives of \(V\). Off the zeros, differentiating the logarithm of \(f\) gives \[|f''/f|^{1/2}\le CA+C\sum_j|z-z_j|^{-1}\] on a fixed smaller disk. Here the second derivative of the logarithm contributes square roots of sums of \(|z-z_j|^{-2}\), bounded by the displayed sum. Each summand is locally integrable in two real dimensions with a uniform integral bound, and there are at most \(CA\) of them. The resulting \(L^1\) bound for \(|q|^{1/2}\), followed by submean for that subharmonic function, proves (182). Its subharmonicity across the zeros uses the assumed holomorphic extension of \(q\). Rescale the variable to obtain the final assertion. ◻

Proof. We may exclude constant \(f\), which gives a circle. Suppose first that \(H<\infty\) and the displayed limsup is infinite. The same is true with \(M\) replaced by \(|f|\). Otherwise \(\log\max|f|=O(k)\) on those rows; disks with radius comparable to \(e^{-k}\), and with a fixed extra depth margin, have a global log upper bound \(O(k)\). Formula (184) then gives a finite power bound in inverse depth for \(|q|^{1/2}\), contradicting the assumed infinite order of \(M\).

Treat this case and \(H=\infty\) together. Put \[\rho(h)=\log\max_x|f(x+\mathrm ih)|.\] The function \(\rho\) is convex in height by the three-lines theorem, and even by symmetry. In the finite-width case just identified it tends to infinity and has infinite upper order relative to \(\log(1/(H-h))\). In the entire case a nonconstant real-symmetric periodic function has a growing Fourier mode: if a nonconstant coefficient is nonzero, its conjugate coefficient is also nonzero, and one grows as \(h\to\infty\). Hence \(\rho(h)\ge c h-C\) for some \(c>0\). In either case \(\rho\to\infty\), and it is eventually strictly increasing with slope bounded below by a positive constant.

Let \(h(R)\) denote its inverse at large values, let \(N(R)\) be its right slope there, and put \(\mu(R)=\log N(R)\). The function \(\mu\) is nondecreasing and bounded below. The inverse is locally absolutely continuous and \[h'(R)=e^{-\mu(R)}\quad\hbox{for almost every }R.\] We claim that \[ \liminf_{R\to\infty}\mu(R)/R=0. \tag{185}\] If \(\mu(R)\ge cR\) eventually for some \(c>0\), inverse integration would give a finite limiting height and \(H-h(R)\le Ce^{-cR}\). This is impossible if \(H=\infty\). For finite \(H\) it implies \(R/\log(1/(H-h(R)))\le1/c+o(1)\), contradicting infinite upper order. Since \(\mu\) is bounded below, the liminf cannot be negative. This proves (185).

Choose \(R\to\infty\) along values with \(\mu(R)=o(R)\), and fix a lower base value \(R_b\). The positive measure \(\,\mathrm d\mu\) on \([R_b,R]\) has mass \(o(R)\). The weak maximal-interval inequality gives a number \(\varepsilon_R\to0\) and a continuity point \(v\in[R/3,2R/3]\) such that \[ |\mu(z)-\mu(v)|\le\varepsilon_R|z-v| \qquad (R_b\le z\le R). \tag{186}\] Here is a direct measure justification of the choice. The set of points belonging to an interval on which the mass divided by its length exceeds \(\varepsilon_R\) has length at most a constant times the total mass divided by \(\varepsilon_R\), by the disjoint-interval covering argument. Choose, for example, a constant multiple of \(\sqrt{(1+\mu(R)-\mu(R_b))/R}\) for \(\varepsilon_R\). The bad set has length \(o(R)\), so cannot cover \([R/3,2R/3]\). Increasing the constant accounts for intervals with one endpoint at the selected point. Atoms and discontinuity points may be discarded. This gives both one-sided secant bounds in (186).

Set \[ h_n=h(v),\qquad \ell_n=N(v)^{-1},\qquad m_n=e^v. \tag{187}\] For bounded \(u\), integrate the inverse derivative using (186), then invert the result. It gives, uniformly on each fixed bounded real range, \[ \rho(h_n+\ell_nu)=v+u+o(1). \tag{188}\] These heights stay in the shape strip: bounded intervals of values about \(v\) lie in \([R_b,R]\) with distances tending to infinity from the endpoints, and the inverse integration applies there. In particular the upper-wall distance in units of \(\ell_n\) tends to infinity.

The slope lower bound, and \(\mu(v)\le\mu(R)=o(R)\), also give \[ 0<\ell_n\le C,\qquad h_n\le Cv, \qquad |\log\ell_n|=o(v),\qquad b_n=\ell_nm_n=\exp(v-o(v))\longrightarrow\infty. \tag{189}\] The centers satisfy \(h_n\to H\), with the infinite value allowed. For finite \(H\), convex divergence at that height forces the slopes to infinity, so \(\ell_n\to0\). For infinite \(H\), \(h_n\to\infty\) and \(\ell_n\le C\). Thus the lower wall also escapes in scaled coordinates, and the shape domains exhaust \(\mathbb C\).

Choose \(x_n\) realizing the row maximum at \(h_n\). By (188), the entire subsequential limits of \[F_n(S)=m_n^{-1}f(x_n+\mathrm ih_n+\ell_nS)\] satisfy \(|F(S)|\le e^{\Im S}\) and \(|F(0)|=1\). The maximum principle applied to \(e^{\mathrm iS}F(S)\) therefore gives \[ F(S)=c_0e^{-\mathrm iS},\qquad |c_0|=1. \tag{190}\] Normal-family extraction, including the phase at zero, supplies such locally uniform convergence.

The normalized second field tends to zero. In fact, for each fixed \(C_1\), use disks of radius a fixed small multiple of \(\ell_n\) throughout \(|\Im s|\le h_n+C_1\ell_n\), with an extra margin of a few such radii. Symmetry and (188) give the global log bound \(K=v+O_{C_1}(1)\) on their heights. Equations (183)–(184) and (189) imply \[ |q(s)|^{1/2}\le C(\ell_n^{-1}+\ell_n^{-2})(1+v)=\exp(o(v)). \tag{191}\] Consequently \(Q_n=q/m_n^2\to0\) on every scaled compact set.

We finish by verifying the coarser bounds, including their full range to a fixed physical depth. For \(R_b\le v'\le v\), inverse integration and (186) give \[ \frac{h(v)-h(v')}{\ell_n} =\int_{v'}^v e^{\mu(v)-\mu(z)}\,\,\mathrm dz \le\int_{v'}^v e^{\varepsilon_R(v-z)}\,\,\mathrm dz \le (v-v')e^{\varepsilon_R(v-v')}. \tag{192}\] Fix any \(\alpha_0\in(1/2,1)\). At height \(h(v')\), multiply the row bound \(|f|/m_n\le e^{-(v-v')}\) by the \(\alpha_0\)-power of the left side of (192). The result is at most \[(v-v')^{\alpha_0} \exp\bigl(-(1-\alpha_0\varepsilon_R)(v-v')\bigr),\] which is bounded uniformly. Hence the required power bound for \(|f|/m_n\) holds globally in the horizontal variable, at \(\ell_n\le\delta=h_n-\Im s\le h_n-h(R_b)\). The remaining lower compact range has bounded \(f,q\). Furthermore \[ \log(1+h_n/\ell_n)=o(v) \tag{193}\] by (189). This absorbs that compact range and, using (191), the entire \(q\) contribution: their normalized size is \(e^{-v+o(v)}\), whereas the largest power of \(\delta/\ell_n\) in question is only \(e^{o(v)}\). We have therefore proved \[M(s)/m_n\le C(\ell_n/\delta)^{\alpha_0} \qquad(\ell_n\le\delta\le h_n),\] uniformly in horizontal position. In the finite-width case retain any fixed smaller range \(\delta\le\delta_b<H/2\). In the entire case (193) is exactly the additional condition (163).

All hypotheses of 40 now hold, with the first profile in (161), by (190) and (191). Its contradiction proves both assertions. ◻

Proof. At a fixed center the cone theorem gives a joint holomorphic vertical passage at the row point for \[|t|<r_{x,k}:=c\frac{d_k}{N(k)b(x,k)}.\] By decreasing \(c\), and using a fixed slightly shallower target in the application of that theorem, it gives \(|\mathfrak s-y|\le C d_k\) on this parameter disk and on a fixed neighborhood of the row point. The passage is attached to the actual lift from below and stays inside the shape strip. Evenness in \(t\) and Cauchy’s estimate therefore give \[ \left|\mathfrak s(y,t)-y-t^2V(y)\right| \le C d_k(zb(x,k))^4 \quad\text{when }|t|\le r_{x,k}/2. \tag{209}\] The fixed factors in \(c\) are included in the constant here.

By (205) there is a center with \(b(x,k)\le C\). The real no-patch transfer from 24 then implies \[ \Delta(t)\le d_k \quad\text{for real }|t|\le c\,d_k/N(k). \tag{210}\] This holds for all sufficiently large \(k\); decreasing \(c\) handles the remaining fixed compact range by joint analyticity near \(t=0\).

For \(j=k+h\), (200) gives \[\frac{d_j}{N(j)}\ge C_N^{-1} \frac{d_k}{N(k)}e^{-(1+a)h}.\] For small \(z\), choose \(h\ge0\) so that \(e^{-h}=C'z^{1/(1+a)}\), where the fixed constant \(C'\) is large enough for (210) to apply at level \(j\). This proves (206). Rounding \(h\) to an integer, if desired, changes only a fixed constant.

For the negative part, take \(z=c'b^{-4/(4-\gamma)}\), with \(c'>0\) fixed sufficiently small. Since \(b\ge1\), this makes both \(zb\) and \(\Delta(t)/d_k\le Cz^\gamma\) small. In particular the row lies strictly inside the actual strip. Its actual lift agrees with the vertical passage in (209). Combining that estimate with (199) gives \[\frac{d_kP^-}{N(k)^2} \le C\left(z^{\gamma-2}+z^2b^4\right).\] The two powers of \(b\) coincide and equal the exponent in (207). Since \(\gamma>0\), that exponent is strictly less than two. ◻

Proof. It is enough to prove the assertion on nested disks. In disk coordinates put \(g=d_kf\) and \(\widetilde W=d_k^2W\). Then \[g''=\frac18g(\widetilde W-g^2).\] Suppose that no universal bound \(\sup_{\mathbb D}|g|\le C(1+\sup_{2\mathbb D}|\widetilde W|)^{1/2}\) holds. For a counterexample sequence choose in an intermediate disk a point maximizing \(|g(z)|\) times its distance to that disk’s boundary. Write \(m_n\) for the value of \(|g|\) there and \(r_n\) for that distance. Failure of the proposed estimate implies \(r_nm_n\to\infty\) and \(\sup|\widetilde W|/m_n^2\to0\). On the disk of radius \(r_n/2\) about the chosen point, maximality gives \(|g|\le2m_n\). Rescale space by \(m_n^{-1}\) and divide the function by \(m_n\). After passage to a subsequence, the resulting functions converge on every compact subset of \(\mathbb C\) to an entire function \(G\) with \(|G|\le2\), \(|G(0)|=1\), and \(G''=-G^3/8\). Liouville’s theorem makes \(G\) constant, which contradicts this equation and \(|G(0)|=1\). This bounds \(f\). The identity \(8q=W-f^2\) bounds \(q\) on the same inner disk. A finite disk covering proves (211). ◻

Proof of the quadratic and higher-moment assertions. Let \(C_* =\sup_{j\ge k_0}\mu(j)/N(j)<\infty\). By (205), \(\|b(\cdot,k)\|_2\le C(1+C_*)\). Hence (208) gives \[\mathbb E\frac{d_kP^-}{N(k)^2}\le C(1+C_*^m).\] The horizontal mean of \(P\) is zero, so its positive and negative parts have equal integrals. Thus the same bound, with a factor two, holds for \(|P|\).

Interior harmonic estimates bound the supremum of \(|\nabla P|\) on a tile by \(C d_k^{-3}\) times the integral of \(|P|\) on a fixed enlargement. Averaging in the horizontal center and using the bounds on neighboring rows gives \[\mathbb E\sup_{\mathcal T_A(k,x)} \frac{d_k^2|W|}{N(k)^2} \le C_A(1+C_*^m).\] Here (198) accounts for the fixed additive constant \(\langle W\rangle\), which is absorbed because \(N\ge1\). The disk bound now yields \(C_*^2\le C(1+C_*^m)\). Since \(m<2\), this bounds \(C_*\) by a fixed constant and proves (212), including the tile version of its first assertion.

We give the concentration estimate that improves the moment. Put \(p=2/m>1\) and \(\vartheta=1-1/p>0\). The bound for \(b\) and (208) give, on all rows in any fixed range of depth ratios, \[ \left\|\frac{d_kP^-}{N(k)^2}\right\|_p\le C. \tag{215}\] Partition the horizontal circle into intervals of lengths comparable to \(d_k\). Select a nonempty collection of fractional total length \(\sigma\), and let \(E\) be its enlargement by a fixed multiple of \(d_k\). In particular \(\sigma\ge c d_k\), and \(|E|/(2\pi)\le C\sigma\). We estimate \(P\) on \(E\) on a row of depth \(d'\) comparable to \(d_k\).

Choose \(R\ge R_0\), with \(R_0\) fixed sufficiently large, and suppose \(Rd_k\) is small. Harmonic extension from that row to depth \(Rd_k\), with zero data on the real row, is convolution with a positive periodic strip kernel \(K\) of mass at most one. Before periodization its density is \[ \frac1{2h'} \frac{\sin(\pi l/h')} {\cosh(\pi u/h')-\cos(\pi l/h')}, \qquad h'=H-d',\quad l=Rd_k-d'. \tag{216}\] It is at least \(c/(Rd_k)\) when \(|u|\le Rd_k\). Let \(E_R\) be the enlargement of \(E\) by a sufficiently large multiple of \(Rd_k\); it has fractional length at most \(C\min\{1,R\sigma\}\). Integrating first in the destination variable shows that a fixed fraction of the positive mass of \(P\) on \(E\) contributes to \(\int_{E_R}K*P^+\). As \(K*P^+=P_{\mathrm{coarse}}+K*P^-\), we obtain \[ \int_E P^+\,\frac{\,\mathrm dx}{2\pi} \le C\left(\int |P_{\mathrm{coarse}}|\,\frac{\,\mathrm dx}{2\pi} +\int_{E_R}K*P^-\,\frac{\,\mathrm dx}{2\pi}\right). \tag{217}\] The normalized first term is at most \(CR^{2\beta-1}\): at the coarse level \(j=k-\log R\), use \(N(j)\le C R^\beta N(k)\) and \(d_j=Rd_k\) in (212). The convolution is an \(L^p\) contraction, so Holder’s inequality and (215) bound the second normalized term by \(C(R\sigma)^\vartheta\). The negative mass directly on \(E\) has the same bound. Consequently \[ \int_E\frac{d_k|P(x+\mathrm i(H-d'))|}{N(k)^2} \,\frac{\,\mathrm dx}{2\pi} \le C\left(R^{2\beta-1}+(R\sigma)^\vartheta\right). \tag{218}\]

The interior harmonic estimates used above, followed by 49, now give \[ \sum_{I\ \mathrm{selected}} |I| \sup_{x\in I}\sup_{s\in\mathcal T_A(k,x)} \frac{d_k^2M(s)^2}{N(k)^2} \le C_A\left(\sigma+R^{2\beta-1}+(R\sigma)^\vartheta\right). \tag{219}\] To see that no concentration is lost in this passage, enlarge the selected intervals by the fixed horizontal margins needed for the harmonic estimates and integrate those estimates over the fixed range of neighboring depths. The enlarged intervals have bounded overlap and still total length \(O(\sigma)\), so (218) applies at every such depth.

Take \(R=R_0\sigma^{-1/2}\). This is allowed for all sufficiently large \(k\), since \(\sigma\ge c d_k\) gives \(Rd_k\le C\sqrt{d_k}\). The remaining compact range of \(k\) is harmless. There is therefore \(0<\zeta<1\), depending only on \(\beta\) and \(p\), such that the right side of (219) is at most \(C_A\sigma^\zeta\). For example one may take any positive \(\zeta\) no larger than \(\min\{(1-2\beta)/2,\vartheta/2\}\).

Let \(A_k(I)\) be the supremum appearing in the sum in (219). Selecting the cells for which \(A_k(I)>b_0\ge1\) gives \[\mathbb P\{A_k>b_0\}\le C b_0^{-1/(1-\zeta)}.\] The exponent is greater than one. Integrating this distribution bound shows that \(A_k\) has a uniformly bounded moment of some order \(1+\eta/2>1\). This proves (213), including its fixed-tile versions. ◻

Proof of the global assertions in 50. Suppose \(N\) is nondecreasing. We first prove that, for a fixed small \(c_2>0\), \[ \Delta(t)\le d_k/4\qquad \left(|t|\le \rho_k:=\frac{c_2d_k}{(1+k)N(k)}\right). \tag{220}\] Choose a fixed \(\varepsilon_0>0\) so small that (206) bounds \(\Delta\) by \(d_k/16\) on the real interval \(|u|\le u_0:=\varepsilon_0d_k/N(k)\). For a fixed small parameter half-disk, harmonic upper comparison for the bounded subharmonic function \(\Delta\) gives, when \(|t|\le u_0/2\), an upper bound consisting of this near-diameter contribution, an \(O(|t|)\) contribution from the curved boundary, and \[ C|t|\int_{u_0}^{r_0} \max\{\Delta(u),\Delta(-u)\}\,\frac{\,\mathrm du}{u^2}. \tag{221}\] Here \(r_0>0\) is fixed. The kernel bound follows directly from the upper-half-plane Poisson kernel on the diameter; the harmonic measure of the distant curved arc is \(O(|t|)\). Upper semicontinuous boundary data cause no difficulty: apply the comparison first to continuous majorants and decrease them to the given data. The lower half-disk is handled in the same way.

We claim that the integral in (221) is at most \(C_{\varepsilon_0}(1+k)N(k)\). For earlier levels \(j\le k\), monotonicity gives \(N(j)\le N(k)\), so (210) implies \(\Delta(u)\le d_j\) up to \(|u|\le c d_j/N(k)\). Divide the interval of integration into the corresponding ranges for \(j=k,k-1,\ldots\), down to a fixed level. On each such range \(\Delta(u)\le C d_j\) and its \(u^{-2}\) integral is at most \(C N(k)\). The first range costs \(C_{\varepsilon_0}N(k)\). Beyond the last one the boundedness of \(\Delta\) costs at most \(C N(k)\). There are \(O(1+k)\) ranges, proving the claim. Choosing \(c_2\) small now proves (220).

For \(y\) at depth comparable to \(d_k\), with a fixed neighborhood of that size, the actual lift is consequently holomorphic for \(|t|<\rho_k\) and satisfies \(\Im(\mathfrak s(y,t)-y)\le C d_k\). It vanishes at \(t=0\). The coefficient estimate for an analytic function with positive real part, applied to \(C d_k+\mathrm i(\mathfrak s-y)\), gives \[|V(y)|\le C d_k\rho_k^{-2}.\] This elementary estimate also follows by integrating the nonnegative real part on concentric circles and taking its second Fourier coefficient. It proves the asserted bound for \(P\). Cauchy’s estimate in \(y\), followed by (198), gives \(|W|\le C\rho_k^{-2}+C\) on a slightly smaller tile. The disk bound therefore gives \(d_kM\le C d_k/\rho_k+C\), which is the other assertion in (214). ◻

Proof. It suffices first to bound row norms. Let \(C_*\) be their supremum ratio to \(N\) on the fine levels, enlarged to include the fixed coarse constants. This number is finite. For example, the density of \(\nu\) is bounded for each fixed ancestor, so finiteness follows from the unweighted finite-ratio hypothesis. This preliminary bound may depend on \(d_a\); none of the estimates below will depend on it.

Tile averaging on levels whose depths are at most a fixed multiple of \(d_a\) is valid for \(\nu\), by its translation-comparison property. In the logarithmic summation defining \(b\), use these fine tile bounds on that part of the cone, and use (223) on the coarser part. This gives \[ \|b(\cdot,j)\|_{L^2(\nu)}\le C(1+C_*). \tag{226}\] Fixed ranges immediately adjacent to the coarse cutoff are already covered by the assumed fixed tile enlargements at levels at most \(k_a+3\). In particular, choosing larger fixed interior-estimate neighborhoods does not create a gap at the cutoff.

For \(d_j<d_a/2\), let \(\kappa_j\) be the mixed unnormalized harmonic-measure law from \(\omega\) to the row at depth \(d_j\). Since \(P=0\) on the real bottom, \[\int P(x+\mathrm i(H-d_j))\,\,\mathrm d\kappa_j(x) =\int P(x+\mathrm i(H-d_a))\,\,\mathrm d\omega(x).\] The laws \(\kappa_j\) and \(\nu\) are comparable. Moreover the absolute value of the right side, in fine-row normalization, is at most \[ C\frac{d_j}{d_a}\frac{N(k_a)^2}{N(j)^2} \le C e^{-(1-2\beta)(j-k_a)}\le C. \tag{227}\] The negative-part estimate (208), integrated using (226), bounds the normalized negative integral by \(C(1+C_*^m)\). Taking the signed integral under \(\kappa_j\) first, and then using comparison of the two positive measures, therefore gives \[\int\frac{d_j|P(x+\mathrm i(H-d_j))|}{N(j)^2}\,\,\mathrm d\nu(x) \le C(1+C_*^m).\] The strict inequality \(2\beta<1\) gives uniform decay in (227).

Weighted interior harmonic estimates now follow by the same local averaging argument, since the required horizontal translations are of size \(O(d_j)\). Combining them with 49 gives \(C_*^2\le C(1+C_*^m)\), away from the already controlled fixed cutoff range. This bounds \(C_*\) uniformly. One more application of weighted tile averaging proves (225). ◻

Proof. Suppose first that (228) holds. For any fixed \(c,T_0>0\), apply 34 at a depth which is a fixed small multiple of \(c d_n\), and with parameter radius \(T_0d_n/B_n\). The ratio of this radius to the target depth tends to zero. Thus its fixed \(K=1\) regime suffices, and its aperture can be chosen once, independently of \(c,T_0\). The weighted smallness in (228) supplies its small parameter. The real no-patch transfer gives \[\Delta(Td_n/B_n)\le c d_n \quad (|T|\le T_0)\] for all sufficiently large \(n\). Since \(c>0\) is arbitrary, \[ \sup_{|T|\le T_0}\frac{\Delta(Td_n/B_n)}{d_n}\longrightarrow0 \qquad(T\text{ real}). \tag{231}\]

We show that this width estimate alone contradicts \(B_n=\mu(k_n)\). Let \(j-k_n\) vary in an arbitrary fixed compact interval. The envelope inequalities imply \(N_n(j)\le C B_n\), with a constant uniform on that interval. By the quadratic bootstrap and (205), the cone costs \(b(x,j)\) have uniformly bounded second moments. Also (208) gives \[ \frac{d_nP^-(x+\mathrm i(H-d_j))}{B_n^2}\le C b(x,j)^m, \qquad m<2. \tag{232}\] These variables are uniformly integrable: their \(2/m\)-moments are bounded, and \(2/m>1\).

Fix a cutoff \(A>1\) and work where \(b(x,j)\le A\). Take \(t=T d_n/B_n\) with a fixed nonzero \(T\) small enough for the Taylor estimate at all these points. By (231), the relevant rows lie in the actual strip for large \(n\). Combining (199) and (209), now in this normalization, gives \[\frac{d_nP^-}{B_n^2} \le C\frac{\Delta(Td_n/B_n)}{T^2d_n}+C T^2 A^4.\] Given a positive threshold, first choose \(A\) so that the complement has small probability, then choose \(|T|\) so that the second term is below that threshold, and finally let \(n\to\infty\). This proves convergence to zero in probability. The uniform integrability in (232) gives convergence in mean, uniformly on every fixed range of \(j-k_n\).

The signed horizontal mean of \(P\) is zero, so the same conclusion holds for \(d_n|P|/B_n^2\). Interior harmonic estimates, using a fixed range of neighboring depths, imply \[\mathbb E\sup_{\text{fixed tile}} \frac{d_n^2|W|}{B_n^2}\longrightarrow0.\] The disk bound and \(B_n\to\infty\) now imply \[\frac{\mu(k_n)^2}{B_n^2}\longrightarrow0,\] contradicting its value one.

For the stated consequence, choose \(1/2<\alpha<\min\{1,\alpha_0\}\). On the part of the cone with \(\delta/d_n\ge R\), (230) bounds the expression in (228) by \(C R^{\alpha-\alpha_0}\). On each remaining compact part with \(c\le\delta/d_n\le R\), the assumed profile convergence makes it tend to zero. Letting \(R\to\infty\) proves (228), which was just excluded. In fact compact convergence within this fixed cone is enough for the argument.

Finally the scaled differential identity is \[\partial_S^2F_n=B_n^2Q_nF_n.\] Compact holomorphic convergence bounds its left side. Dividing by \(B_n^2\) and passing to the limit gives \(FQ=0\). On the connected lower half-plane, a nonzero analytic factor then forces the other factor to vanish identically. ◻

Proof. Suppose otherwise. We first choose levels \(\kappa_n\to\infty\) and values \(b_n=\mu_f(\kappa_n)\) such that \[ \begin{split} b_n&\ge\kappa_n^{1-o(1)},\\ \mu_f(j)&\le b_n\quad(k_0\le j\le\kappa_n),\\ \mu_f(j)&\le b_n e^{\eta_n(j-\kappa_n)} \quad(j\ge\kappa_n),\qquad \eta_n\to0. \end{split} \tag{239}\] Here is an explicit selection. Failure of (238) provides arbitrarily large levels with \(\mu_f\ge k^{1-1/n}\). Replace such a level by a maximum of \(\mu_f\) over all earlier levels; denote the resulting record level by \(u_n\). The record values tend to infinity, so \(u_n\to\infty\), and they still exceed \(u_n^{1-1/n}\). Choose \(\eta_n\to0\) with \(\eta_n\ge1/u_n\). Maximize \(\mu_f(j)e^{-\eta_n(j-u_n)}\) on \(j\ge u_n\), and call a maximizer \(\kappa_n\). It exists by (195). It is still a record among all earlier levels. Its value satisfies \[b_n\ge u_n^{1-1/n}e^{\eta_n(\kappa_n-u_n)} \ge\kappa_n^{1-1/n},\] where we used \(\log(\kappa_n/u_n)\le(\kappa_n-u_n)/u_n\). The maximizing property gives the last inequality in (239).

For each fixed \(\eta>0\), and all sufficiently large \(n\), set \[N_{n,\eta}(j)=(b_n+\kappa_n) e^{\eta(j-\kappa_n)_+}.\] This nondecreasing envelope controls the first moment uniformly in \(n\). For \(|f|\) use (239); for \(|q|^{1/2}\) use (196) and \(1+j\le C_\eta(1+\kappa_n)e^{\eta(j-\kappa_n)}\) on the future side. Its finite-ratio hypothesis follows from its exponential tail. 50 consequently gives \[ \mu(j)\le C_\eta(b_n+\kappa_n)e^{\eta(j-\kappa_n)_+},\qquad d_j\max M\le C_\eta(1+j)(b_n+\kappa_n) e^{\eta(j-\kappa_n)_+}. \tag{240}\] Fix one positive \(\eta\) to use the higher-moment assertion, and let \(p>2\) be the resulting exponent. On the row \(\kappa_n\) put \[E_n=\{x:d_{\kappa_n}|f(x+\mathrm i(H-d_{\kappa_n}))|\ge b_n/2\}, \qquad p_n=\mathbb P(E_n).\] Holder’s inequality gives \[\frac34 b_n^2 \le C(b_n+\kappa_n)^2p_n^{1-2/p}.\] Therefore \[ p_n\ge c\left(\frac{b_n}{b_n+\kappa_n}\right)^{2p/(p-2)} =\kappa_n^{-o(1)}. \tag{241}\]

We next select a normal level for the full moment. For any fixed small \(\varepsilon>0\), maximize \(\mu(j)e^{-\varepsilon|j-\kappa_n|}\) over \(j\ge k_0\). Let \(k\) be a maximizer and \(B=\mu(k)\). The triangle inequality gives \[ \mu(j)\le B e^{\varepsilon|j-k|},\qquad B\ge b_n. \tag{242}\] Using (240) with rate \(\varepsilon/2\) also gives \[|k-\kappa_n|\le\frac2\varepsilon \log\left(\frac{C_\varepsilon(b_n+\kappa_n)}{b_n}\right).\] For fixed \(\varepsilon\), the right side is \(o(\log\kappa_n)\), by (239). A slow diagonal choice thus gives \(\varepsilon_n\downarrow0\) and maximizing levels \(k_n\) such that \[ B_n^{\mathrm{new}}:=\mu(k_n)\ge b_n, \qquad |k_n-\kappa_n|=o(\log\kappa_n), \tag{243}\] while (242) holds with \(\varepsilon=\varepsilon_n\). For the remainder of this proof write \(B_n=B_n^{\mathrm{new}}\); the record value remains denoted by \(b_n\).

Choose \(s_n\to\infty\), \(s_n=o(\log\kappa_n)\), so slowly that it dominates both \(|k_n-\kappa_n|\) and \(\log(1/p_n)\). Such a choice is possible by (241) and (243); for example take the geometric mean of \(\log\kappa_n\) and \(1+|k_n-\kappa_n|+\log(1/p_n)\). Set \(k_{a,n}=k_n-s_n\), let \(\omega_n\) be uniform horizontal measure conditioned on \(E_n\), and let \(\nu_n=K_{d_{a,n}}*\omega_n\). Both \(\omega_n\) and \(\nu_n\) have densities at most \(p_n^{-1}=e^{o(s_n)}\) relative to uniform measure.

We check all ancestor bounds. Fix \(0<\beta<1/2\), a tile enlargement, and \(0<\beta_0<\beta\). Primary normality and tile averaging give the unweighted bound \[\left\|\sup_{\mathcal T_A(j,x)}d_jM\right\|_2 \le C_A B_n e^{\beta_0(k_n-j)} \quad(j\le k_{a,n}+3)\] for large \(n\). Passing to \(\nu_n\) costs at most \(p_n^{-1/2}\). This is absorbed by \(e^{(\beta-\beta_0)(k_n-j)}\), uniformly on that range because \(k_n-j\ge s_n-3\). For the anchor, apply the \(P\) estimate of 50 to the envelope \[B_n\exp\bigl(\beta_0(k_n-j)_++a_0(j-k_n)_+\bigr)\] with any fixed \(a_0>0\). Primary normality supplies its hypotheses. Conditioning costs \(p_n^{-1}\), which is absorbed by \(e^{2(\beta-\beta_0)s_n}\). This proves (234) and (235), with constants independent of \(n\) for every fixed choice.

It remains to prove (236). Fix a compact set \(\mathcal K\) in the lower half-plane and a fixed \(A>1\). For a center \(x_0\in E_n\), consider all target disks needed to control \(q\) on \(x+\mathrm iH+d_n\mathcal K\), where \(|x-x_0|_{\mathrm{per}}\le A d_{a,n}\). One can connect the initial point at level \(\kappa_n\) to these disks by first moving to the ancestor row, moving horizontally there, and then moving to the target depth. The number of depth-sized disks required is \(O_A(1+s_n+|k_n-\kappa_n|)=o(\log\kappa_n)\).

On a fixed enlargement of this entire range, (240) implies \[\log|f|\le U_n^*:=\kappa_n+\log b_n+C\log\kappa_n.\] Here \(C\) can be fixed: all level changes are \(o(\log\kappa_n)\), and \(\log((b_n+\kappa_n)/b_n)=o(\log\kappa_n)\). At the initial point in \(E_n\), the nonnegative superharmonic deficit \(U_n^*-\log|f|\) is \(O(\log\kappa_n)\). The disk-chain estimate (237) bounds its average on every target disk by \(\kappa_n^{o(1)}\), uniformly in the indicated centers. 45, with fixed inner margins, therefore gives \[\sup_{S\in\mathcal K}d_n |q(x+\mathrm iH+d_nS)|^{1/2}\le\kappa_n^{o(1)}.\] Since \(B_n\ge b_n\ge\kappa_n^{1-o(1)}\), this is negligible relative to \(B_n\).

Under \(\nu_n\), a center can be sampled by first choosing \(x_0\) with law \(\omega_n\) and then adding the kernel increment. The probability that its periodic distance exceeds \(A d_{a,n}\) is at most \(C/A\), uniformly in \(n\). For every fixed \(A\), the preceding estimate holds on the complement. First let \(n\to\infty\), and then let \(A\to\infty\). This proves (236). We have constructed all the data of 53, a contradiction. ◻

Proof. By 54 and (196), \(\mathbb Ed_jM\le C(1+j)\). At a target level \(K\), use the nondecreasing envelope \[N_K(j)= \begin{cases} 1+j,&k_0\le j\le K,\\ (1+K)e^{j-K},&j\ge K. \end{cases}\] Its envelope and first-moment constants are independent of \(K\), and its exponential tail makes the bootstrap ratio finite. Apply 50 at \(K\). Since \(N_K(K)=1+K\), all three assertions follow. Fixed tile versions follow by applying the same row estimates on their fixed ranges of depth ratios. ◻

Proof. The function \(G_f\) is nondecreasing: apply the maximum principle between two opposite rows and use real symmetry. Suppose the claimed constants do not exist. We may choose \(\kappa_n\to\infty\) and \(C_n\to\infty\), with \(C_n\log\kappa_n=o(\kappa_n)\), such that \(b_n:=\mu_f(\kappa_n)\ge\kappa_n^\alpha\) and \[ G_f(\kappa_n+C_n\log\kappa_n) -G_f(\kappa_n-C_n\log\kappa_n)=o(b_n). \tag{246}\] For clarity one can first set \(C_n=n\), require the increment to be less than \(b_n/n\), and then choose the violating level larger than any prescribed threshold.

Maximize \(\mu(j)e^{-\varepsilon_n|j-\kappa_n|}\), where \(\varepsilon_n\downarrow0\) sufficiently slowly, and denote the maximizer by \(k_n\), with \(B_n=\mu(k_n)\ge b_n\). The maximizing argument in (242) gives primary normality. The polynomial bound \(\mu(j)\le C(1+j)\) shows that \[|k_n-\kappa_n|=O(\varepsilon_n^{-1}\log\kappa_n).\] Indeed, on the future side use \((1+j)/(1+\kappa_n)\le e^{(j-\kappa_n)/(1+\kappa_n)}\) and choose \(\varepsilon_n\ge2/(1+\kappa_n)\); the past side is immediate. We can therefore arrange \[|k_n-\kappa_n|=o(C_n\log\kappa_n),\qquad \log\kappa_n+|k_n-\kappa_n|\ll s_n \ll C_n\log\kappa_n.\] Here both ratios denoted by \(\ll\) tend to zero. For example, after a further slow choice of \(C_n\), one may use \(\varepsilon_n=C_n^{-1/3}\) and \(s_n=C_n^{2/3}\log\kappa_n\). Set \(k_{a,n}=k_n-s_n\); then \(k_{a,n}\to\infty\). Choose a horizontal maximizer \(x_{a,n}\) of \(|f|\) on the ancestor row, take \(\omega_n\) to be its point mass, and put \(\nu_n=K_{d_{a,n}}*\omega_n\).

For every fixed \(\beta>0\), the coarse and anchor hypotheses follow directly from (244). Indeed all required coarser tile suprema are \(O_A((1+k_n)^2)\), whereas \(B_n e^{\beta(k_n-j)}\) is at least \(\kappa_n^\alpha e^{\beta(s_n-3)}\) for \(j\le k_{a,n}+3\). The latter dominates every fixed polynomial in \(\kappa_n\), since \(s_n/\log\kappa_n\to\infty\). Similarly the normalized ancestor bound for \(P\) is \(O((1+k_{a,n})^4)\), and is dominated by \(B_n^2e^{2\beta s_n}\). These are supremum estimates, so no bound on the density of the point mass is needed.

Fix a compact target set \(\mathcal K\subset\{\Im S<0\}\), and choose a fixed \(C'>0\) so that the row \(k_n+C'\) is above all the target disks and their fixed enlargements. On the periodic strip from the real row to that row, put \[U_n(s)=G_f(k_n+C')-\log|f(s)|\ge0.\] At the chosen ancestor point, monotonicity and (246) give \[ U_n(x_{a,n}+\mathrm i(H-d_{a,n})) =G_f(k_n+C')-G_f(k_{a,n})=o(b_n)=o(B_n). \tag{247}\] Both levels in this difference lie inside the interval in (246), for sufficiently large \(n\).

For any row in the fixed target range, apply positive superharmonic comparison to \(U_n\) from the ancestor, using the real bottom. The upper-boundary harmonic-measure law is comparable to \(\nu_n\), because its depth is \(o(d_{a,n})\). The bottom contribution is nonnegative. It follows that \[\int U_n(x+\mathrm i(H-d_j))\,\,\mathrm d\nu_n(x)=o(B_n)\] uniformly over that fixed range of \(j-k_n\). The comparison can be justified first for the superharmonic truncations \(\min\{U_n,R\}\) and then by monotone convergence. Averaging over the target disks has the same bound: their horizontal shifts are \(O(d_n)\), and the density of \(\nu_n\) changes by bounded factors under those shifts.

Apply 45 on a finite covering of the target compact set by disks with fixed inner margins. It yields \[\int \sup_{S\in\mathcal K} d_n|q(x+\mathrm iH+d_nS)|^{1/2}\,\,\mathrm d\nu_n(x) \le C+o(B_n).\] Since \(B_n\to\infty\), Markov’s inequality proves (236). Together with the preceding normality and ancestor estimates this constructs a prohibited sequence, the required contradiction. ◻

Proof. Fix \(0<\alpha<1/2\) and split the sufficiently large levels into the high set \(\mathcal A=\{k:\mu_f(k)\ge k^\alpha\}\) and its complement. By 56, \[\int_{\mathcal A\cap[K_0,K]}\mu_f(k)\,\,\mathrm dk \le C\int_{K_0}^K [G_f(k+C\log k)-G_f(k-C\log k)]\,\,\mathrm dk.\] The polynomial bound gives \(G_f(v)\le v+C\log(2+v)\). Since \(G_f\) is nondecreasing, its positive Stieltjes measure can be used to evaluate the last integral. Each point of that measure is counted for a set of \(k\)’s of length at most \(2C\log K\). Its total mass over the relevant interval, whose upper endpoint is \(K+C\log K\), is \(O(K)\). The lower endpoint can be kept fixed by increasing \(K_0\). Hence \[\int_{\mathcal A\cap[K_0,K]}\mu_f(k)\,\,\mathrm dk=O(K\log K).\] Multiplying by the bound from 54 gives \[\int_{\mathcal A\cap[K_0,K]}\mu_f(k)^2\,\,\mathrm dk \le C K^{2-\varepsilon_1}\log K.\] On the complementary set the integral is at most \(\int_{K_0}^K k^{2\alpha}\,\,\mathrm dk=O(K^{1+2\alpha})\). Both have a positive power saving from \(K^2\); decreasing that power slightly absorbs the logarithm and the initial compact range. This proves (248). ◻

Proof. For a periodic holomorphic function, its horizontal \(L^p\) norm, \(p\ge1\), is log-convex in the height. One way to see this is to regard horizontal translates of the function as a Banach-space-valued analytic function and apply the three-lines theorem to bounded linear functionals. Equivalently, apply the scalar three-lines theorem followed by the dual description of the \(L^p\) norm. We use this for the \(L^2\) norm of \(f\) and the \(L^1\) norm of \(q\). If \(A\) and \(C\) are nonnegative log-convex functions, then their sum is log-convex: the required interpolation inequality follows from Holder’s inequality applied to \(A_0^{1-\theta}A_1^\theta+C_0^{1-\theta}C_1^\theta\). Consequently \(\log m_n\) is convex. A field which is identically zero can simply be omitted from this argument.

The normalization gives \(m_n(1)=1\), and (249) gives \[m_n(D)\le D^{-1}\exp(\varepsilon_n|\log D|).\] Thus \(\log m_n\) is uniformly bounded above on every compact subinterval. Convexity and its value zero at \(1\) give a lower bound there as well. For example, if \(D<1<D_+\), use the convexity inequality at \(1\), viewed as a convex combination of \(D\) and \(D_+\), and the upper bound at \(D_+\). For \(D>1\), use a point \(D_-<1\). Bounds on a slightly larger compact interval bound the slopes of a convex function on the smaller one. Arzela–Ascoli and a diagonal subsequence therefore give a finite locally uniform limit for \(\log m_n\). Exponentiating proves positivity, and all the properties in (251) pass to the limit. ◻

Proof. Since \(\rho(v)-cv\ge(1-c)v\), the infimum in (254) is attained. The same inequality gives \[x\le\rho_c(x)\le\rho(x).\] The running infimum is locally Lipschitz: when its lower endpoint is moved, its value can increase only because a minimizer lay in the swept interval, where the local Lipschitz bound for \(\rho-cx\) applies. On an open interval containing no contact the minimizer remains to the right, so \(\rho_c\) is affine of slope \(c\). Everywhere, \(\rho_c(v)-\rho_c(u)\ge c(v-u)\) for \(v\ge u\).

First suppose there are contacts \(z_j\to\infty\) with \(p(z_j)\to\infty\). By (253), \[p(x)\ge e^{-1}p(z_j)\qquad(z_j-1\le x\le z_j).\] For large \(j\), \(\rho(x)-cx\) is strictly increasing on this interval. Each point in it is itself a contact: values between that point and \(z_j\) are larger, and values beyond \(z_j\) are at least the value at \(z_j\) because \(z_j\) is a contact.

Set \(g=p^{-1/2}\) on this interval. Inequality (253) says that \(e^{x/2}g(x)\) is nondecreasing. If \(\sigma_j\) is the supremum of \(g\) on the interval, then \(\sigma_j\to0\), and the total variation of \(g\) there is \(O(\sigma_j)\). To choose a point where this small total variation is locally well distributed, let \(\nu_j\) be its variation measure. The elementary weak maximal inequality for intervals gives \[\left|\left\{x: \sup_{h>0}\frac{\nu_j((x-h,x+h))}{2h}> \sqrt{\sigma_j}\right\}\right|=O(\sqrt{\sigma_j}).\] Here intervals are restricted to the original unit interval. This inequality follows by selecting disjoint bad intervals and covering their union by their fivefold enlargements. Choose \(x_j\) outside this bad set in the middle third of the unit interval. Then, whenever \(|h|\) is smaller than its distance to the endpoints, \[|g(x_j+h)-g(x_j)|\le 2\sqrt{\sigma_j}|h|.\] Taking \(|h|\le g(x_j)=P_j^{-1/2}\) shows that \[\frac{p(x_j+h)}{P_j}\longrightarrow1 \quad\hbox{uniformly for }|h|\le P_j^{-1/2}.\] Integration, with \(\log(1-u/P_j)=-u/P_j+O(P_j^{-2})\), proves (255).

For the tail estimate write \(z=\log(1+v/P_j)\). If \(0\le z\le P_j^{-1/2}\), integration of the last slope estimate gives \(\rho(x_j+z)-\rho(x_j)\ge P_jz/2\), which is at least a fixed positive multiple of \(v\) in this range. At \(z=P_j^{-1/2}\) the increase is at least a fixed multiple of \(\sqrt{P_j}\). This endpoint is still a contact. For larger \(z\), the contact inequality adds at least \(c(z-P_j^{-1/2})\). Since \[\log(1+v)\le \log P_j+z+O(1)\] and \(\alpha_0<c\), the increase is at least \(\alpha_0\log(1+v)-O(1)\), uniformly for all \(v\ge1\). Exponentiating proves (256).

Now suppose that \(p\) is bounded on contacts in a tail. Almost everywhere on the contact set the derivatives of \(\rho\) and \(\rho_c\) agree, since the two locally Lipschitz functions agree there. Off that set the derivative of \(\rho_c\) equals \(c\). Consequently \(\rho_c\) has a uniform Lipschitz constant \(K_1\) on some tail. Extend it to the left with a fixed Lipschitz constant. Consider the anchored translates \[u\longmapsto\rho_c(x+u)-\rho_c(x).\] They lie in a compact space of anchored Lipschitz functions, with the topology of uniform convergence on compact intervals. Average their point masses over \(x\in[x_0,x_0+R]\). A weak limit as \(R\to\infty\) is invariant under every anchored translation \[(\mathcal T_h q)(u)=q(u+h)-q(h).\] Indeed translating the averaging interval changes its normalized measure by at most \(2|h|/R\).

The mean increment at \(1\) of this invariant law is at least \(1\). To check this, telescope the integral: \[\frac1R\int_{x_0}^{x_0+R} \bigl(\rho_c(x+1)-\rho_c(x)\bigr)\,\,\mathrm dx = \frac1R\left( \int_{x_0+R}^{x_0+R+1}\rho_c- \int_{x_0}^{x_0+1}\rho_c\right),\] and use \(\rho_c(x)\ge x\). Birkhoff’s pointwise ergodic theorem (Durrett 2019, Theorem 6.2.1) for the invertible map \(\mathcal T_1\), applied in both directions to the bounded unit increments, now provides a path \(q_*\) in the support of the law for which \[ \frac{q_*(u)}u\longrightarrow\lambda\ge1 \qquad(u\to+\infty\text{ or }u\to-\infty). \tag{261}\] The two limits agree because they are the same invariant conditional expectation of the unit increment. Bounded increments between successive integers extend the conclusion to real \(u\). Such a path is approximated locally uniformly by translates with centers \(x_j\to\infty\): a neighborhood visited only for bounded \(x\) has zero mass in every limiting averaged measure.

We next replace \(\rho_c\) by \(\rho\). To the right of any fixed compact \(u\)-interval, (261) gives a fixed interval \([v,w]\) on which the increment of \(q_*\) exceeds \(c(w-v)\). The approximating translates of \(\rho_c\) must therefore have a contact in \([x_j+v,x_j+w]\). Monotonicity of \(\rho\), together with \(\rho\ge\rho_c\), bounds \(\rho(x_j+u)-\rho_c(x_j)\) above and below on every compact interval. It also bounds the local slopes of \(\rho\): from (253), \[p(x)\le e\int_{x-1}^x p(v)\,\,\mathrm dv =e\bigl(\rho(x)-\rho(x-1)\bigr).\] Thus a subsequence of the anchored translates of \(\rho\) has a locally uniform limit \(\rho_*\), and \(\rho(x_j)-\rho_c(x_j)=O(1)\).

For completeness, this limit has the same two-ended slope as \(q_*\). Fix a small \(\eta>0\). For every sufficiently large \(|u|\), the increment of \(q_*\) on \([u,u+\eta|u|]\) exceeds \(c\eta|u|\), by (261) and \(\lambda>c\). The same contact argument, followed by passage to the limit, bounds the excess of \(\rho_*(u)\) over \(q_*(u)\), up to a fixed additive constant, by \(K_1\eta|u|\). The lower bound \(\rho\ge\rho_c\) bounds the deficit by a fixed constant. Letting \(\eta\downarrow0\) proves (258).

Finally, the lower increment bound for \(\rho_c\), its uniform Lipschitz bound, and the bounded difference at \(x_j\) give (259) and (260) as long as \(v\) remains in the tail. For the remaining left range, use \(\rho(v)\ge v\), increasing \(P'>\max(K_1,1)\) and the additive constant if necessary. The same Lipschitz bound gives \(\rho(x_j)=O(1+x_j)\). ◻

Proof. For each fixed \(j\), take the original index \(n\) sufficiently large and set \[ h_n=H-dD_j,\qquad \ell_n=\frac{dD_j}{P_j},\qquad m_0=\frac Bd\,m_n(D_j). \tag{262}\] The diagonal can be chosen so that \[h_n\to H,\qquad \ell_n\to0,\qquad m_0\ell_n\to\infty.\] For the last assertion, first keep \(j\) fixed: \(m_0\ell_n=B D_jm_n(D_j)/P_j\to\infty\). The distance from \(h_n\) to the upper shape wall, in units of \(\ell_n\), is \(P_j\to\infty\); the lower wall also escapes in these units.

Both (255) and (256) can be transferred to the actual moments on the required ranges. Here is the tail detail. Fix \(\alpha_0\in(1/2,c)\), and require \(\varepsilon_n<(1-\alpha_0)/2\). For each fixed \(j\), choose a finite cutoff \(D_{\rm cut}\) large enough that the primary bound \(m_n(D)\le D^{-1+\varepsilon_n}\) is below a fixed multiple of the right side of (256), with the normalizing factor \(m_n(D_j)\), whenever \(D\ge D_{\rm cut}\). Such a cutoff exists because \(1-\varepsilon_n>\alpha_0\) and \(m_n(D_j)\to m(D_j)>0\). On the finite interval below the cutoff use locally uniform moment convergence. Choose the fixed physical cutoff \(0<\delta_b<H/2\) small enough that the enlarged depth range used for submean disks has logarithmic level at least \(k_0\). Thus all the coarser depths up to \(\delta_b\) are controlled, with a constant independent of \(j\).

On the row of height \(h_n\), define \[H_x=\frac{|f(x+\mathrm ih_n)|^2+|q(x+\mathrm ih_n)|}{m_0^2}.\] Its horizontal mean is \(1\). Sample \(x\) with probability density \(H_x\) relative to normalized horizontal measure, and set \(m_x=m_0\sqrt{H_x}\); the zero-density set \(H_x=0\) can be discarded. In the coordinate \[S=\frac{s-x-\mathrm ih_n}{\ell_n},\] let \[\mathcal F_x(S)=\frac{f(s)}{m_x},\qquad \mathcal Q_x(S)=\frac{q(s)}{m_x^2}.\] There is an exact cancellation in the expectation: \[ \mathbb E_H\bigl(|\mathcal F_x(S)|^2+|\mathcal Q_x(S)|\bigr) = \frac{\mathbb E\bigl(|f|^2+|q|\bigr) \text{ on the row }h_n+\ell_n\Im S}{m_0^2}. \tag{263}\] The denominator \(H_x\) cancels against the sampling density, and the remaining integral is invariant under the horizontal translation \(x\mapsto x+\ell_n\Re S\). The transferred local moment law therefore makes (263) converge locally uniformly to \(e^{2\Im S}\).

Submean estimates applied before taking expectation give the same bounds for suprema on smaller tiles. In a cone of fixed aperture \(L\), a dyadic band with depth \(D\ge1\) requires only a number of depth-sized tiles depending on \(L\). The expected squared size of the fields on this band is at most \(C_LD^{-2\alpha_0}\). Choose one \(\alpha\in(1/2,\alpha_0)\), independently of \(L\). Summation of \(D^{2\alpha}C_LD^{-2\alpha_0}\) over dyadic bands proves tightness of the random constants in the pointwise bounds \[ \max(|\mathcal F_x(S)|,\sqrt{|\mathcal Q_x(S)|}) \le C_L(-\Im S)^{-\alpha} \tag{264}\] for \(1\le-\Im S\le\delta_b/\ell_n\) and \(|\Re S|\le L(-\Im S)\). Also \[\mathbb P_H(H_x\le a)=\mathbb E\bigl(H_x\mathbf 1_{\{H_x\le a\}}\bigr)\le a.\] Since \(m_0\ell_n\to\infty\), it follows that \(m_x\ell_n\to\infty\) in probability.

Now consider the nonnegative subharmonic function \[U_x(S)=|e^{\mathrm iS}\mathcal F_x(S)|^2 +|e^{2\mathrm iS}\mathcal Q_x(S)|.\] Its mean tends locally uniformly to \(1\). If a nonnegative smooth cutoff equals \(1\) on a fixed compact set, integration against its Laplacian shows that the expected Riesz mass of \(U_x\) on that set tends to zero: the limiting integral of the Laplacian of the cutoff is zero. The mass itself is nonnegative, so it tends to zero in probability.

We may therefore select deterministic centers and a subsequence with all the following properties: local analytic bounds on an exhaustion of the plane; the bounds (264) for every fixed aperture; \(m_x\ell_n\to\infty\); and vanishing Riesz mass on every fixed compact set. To make the quantifier explicit, assign summable failure probabilities to countably increasing apertures and compact sets, choose their bound constants once, and then take the sequence index sufficiently large for the remaining probability conditions. The union of the bad events has probability less than one. This gives fixed eventual constants for each aperture, not constants growing with the sequence index.

The resulting entire limits \((\mathcal F,\mathcal Q)\) obey \[|e^{\mathrm iS}\mathcal F(S)|^2 +|e^{2\mathrm iS}\mathcal Q(S)| \quad\hbox{harmonic}.\] Each summand is subharmonic, so each is harmonic. An analytic function with harmonic squared modulus is constant. An analytic function with harmonic modulus is also constant: away from its zeros its Laplacian is \(|G'|^2/|G|\), and the assertion follows by analytic continuation. Thus \[\mathcal F(S)=c_1e^{-\mathrm iS},\qquad \mathcal Q(S)=c_2e^{-2\mathrm iS}.\] At \(S=0\) the normalized sum is exactly \(1\), so the limit is nonzero. Moreover \[\mathcal F_x''=(m_x\ell_n)^2\mathcal Q_x\mathcal F_x.\] Compact convergence and \(m_x\ell_n\to\infty\) imply \(\mathcal Q\mathcal F=0\). The limit is consequently one of the two nonzero pure exponential profiles in 40. All its finite-wall hypotheses were checked above, including a single exponent \(\alpha>1/2\) for every fixed aperture. That theorem gives the contradiction. ◻

Proof. For each fixed \(j\), take the original index \(n\) sufficiently large and put \[d'=d e^{x_j},\qquad K=k-x_j,\qquad B'=\mu(K).\] The identity \[\frac{B'}B=e^{x_j}m_n(e^{x_j})\] shows that \(K,B'\to\infty\) are compatible with this order of choices. We also impose \(s_n>x_j+3\), so the ancestor cutoff lies strictly below the new target level. The new normalized moment law is \[\frac{m_n(e^{x_j}D)}{m_n(e^{x_j})},\] which tends, after the diagonal selection, to (266).

We spell out the global envelope because its constants must remain uniform after diagonalization. With \(v=k-l\), \[ \frac{\mu(l)}{B'}= e^{v-x_j}\frac{m_n(e^v)}{m_n(e^{x_j})}. \tag{269}\] On a finite \(v\)-range, the limiting estimates (259) and (260) imply an upper exponent \(1-c\) for \(v\ge x_j\) and \(P'-1\) for \(v\le x_j\). Choose \(\beta>1-c\) and \(a>\max(P'-1,0)\). Since \(\rho(x_j)=O(1+x_j)\), primary normality bounds the logarithm of (269) by \[C(1+x_j)+\varepsilon_n|v|.\] If \(\varepsilon_n\) is small compared with \(\beta\) and \(a\), this is absorbed by the chosen envelope for \(|v|\ge C_0(1+x_j)\), with \(C_0\) fixed sufficiently large. On the remaining finite interval take \(n\) large enough for moment convergence with a fixed multiplicative error. This proves (267) with one constant \(C\).

Let \(k_a=k-s_n\) and \(\omega,\nu\) be the ancestors and measures in 53. To apply 51, take an old coarse exponent \(0<\beta_0<\beta\). For \(l\le k_a+3\), the ratio of the new coarse normalization to the old one equals \[ \frac{B'e^{\beta(K-l)}}{Be^{\beta_0(k-l)}} = \frac{B'}B e^{-\beta x_j} e^{(\beta-\beta_0)(k-l)} \ge \frac{B'}B e^{-\beta x_j} e^{(\beta-\beta_0)(s_n-3)}. \tag{270}\] For fixed \(j\), the last expression tends to infinity. We can therefore make it at least one before proceeding to the next \(j\). The same comparison, squared, applies to the absolute ancestor integral of \(P\). Thus the coarse hypotheses of the weighted bootstrap hold with constants independent of \(j\). The global envelope already has this uniformity, so the resulting weighted \(L^2\) bounds do as well.

For each fixed \(j\), the new \(q\)-field on a fixed compact set is the primary \(q\)-field on another fixed compact set, multiplied by the finite factor \(m_n(e^{x_j})^{-2}\). The primary convergence to zero in \(\nu\)-probability can therefore be imposed to any prescribed tolerance before advancing \(j\). Hence the new \(q\)-field tends to zero in probability on compact sets.

Weighted submean estimates now give tightness of the analytic fields on every compact subset of \(\mathbb H_-\). They also give pointwise cone tightness. At depth ratio \(D\ge1\), the coarse part of (267) bounds the weighted field norm by \(CD^{-(1-\beta)}\). Summing depth-sized tile suprema over dyadic bands, with any fixed \(\alpha\in(1/2,1-\beta)\), proves tightness of the constants in (268). Fine tiles use the bounded change of the density of \(\nu\) on shifts of order the ancestor depth. For tiles coarser than that depth, the necessary tile estimates are precisely the explicit coarse hypotheses of the weighted bootstrap. Thus no translation invariance of \(\nu\) is required.

Choose deterministic centers on events with these compact and cone bounds and with the \(q\)-field small, using summable exceptional probabilities as in the preceding proof. Extract a locally uniform limit \((F,0)\). If \(F\) were zero, its compact smallness and the coarser bound with exponent \(\alpha>1/2\) would violate 52; the global normalization at the target is \(B'=\mu(K)\), as that theorem requires. Therefore \(F\not\equiv0\).

Each quotient \(F_n''/F_n=(d')^2q\) is holomorphic. Around any zero of \(F\), choose a small circle containing no other zero and no zero on the circle. The quotients converge on that circle to \(F''/F\), so maximum modulus bounds them in its interior. Their limit removes the apparent singularity of \(F''/F\). This proves the last asserted regularity. In particular all zeros of \(F\) are simple and \(F''\) vanishes at them: otherwise the analytic equation \(F''=(F''/F)F\) with initial data \(F=F'=0\) would force \(F\equiv0\). ◻

Proof. For a bounded complex \(T\)-range, apply 34 to (268) at a sufficiently large fixed depth ratio. Its smallness condition holds because \(|T|C_AD^{-\alpha}\) is small when \(D\) is large. The physical parameter is \(t=Td'/B'\); relative to a fixed positive multiple of \(d'\), its size tends to zero. Thus the constant bounding the parameter-to-depth ratio in that theorem can be fixed, for example at one. The resulting passages are attached vertically to the original lift and satisfy \[S_n-Y=O\bigl(|T|^2D^{1-2\alpha}\bigr)\] on each fixed deep cone. They stay in the shape strip. The bound and holomorphic compactness give joint subsequential limits.

We derive their equation before proving uniqueness. Write \(h=T/B'\), and use the rescaled step \[J_n(Y)=\frac{S_n(Y+h,T)-S_n(Y,T)}h.\] On the fast segment based at a scaled shape coordinate \(S\), the fields entering the normalized chord action are \[T F_n(S+hu),\qquad T^2Q_n(S+hu).\] They converge analytically, including on small thickenings, to the constant fast pair \((TF(S),0)\). The kernel division in 32 and the frozen first-field formula (129) therefore give the local action limit \[p(S,J)=\frac{J}{(TF(S))^2} -\frac{2\sin(TF(S)J/2)}{(TF(S))^3},\] with its removable interpretation at \(TF(S)=0\). This passage is initially made where the deep near-identity branches have nonvanishing kernels.

In scaled coordinates the exact stationary equation reads \[p_{n,J}(S,J)-h\,p_{n,S}(S,J) =p_{n,J}(S_-,J_-).\] The preceding and current states are evaluations of the same holomorphic functions at phase arguments differing by \(h\). Dividing their difference by \(h\) and using analytic convergence gives \[\frac{\partial}{\partial Y}p_J=p_S,\qquad J=S_Y.\] In particular no prescribed rate of convergence relative to \(h\) is needed. Differentiating \(E=Jp_J-p\) along the limiting solution gives \[E_Y=J\frac{\partial p_J}{\partial Y}-p_SS_Y=0.\] The deep anchor implies \(J\to1\), while \(F(S)\to0\), so \(E=1/12\). The frozen energy identity \[E=\frac{2(\sin z-z\cos z)}{(TF(S))^3}\] is exactly (274).

Near zero field, (274) determines an analytic branch \(J=j(TF(S))\) with \(j(0)=1\), since its \(J\)-derivative at zero field and \(J=1\) is \(3\). It has \(j(w)^{-1}-1=O(w^2)\). Define the inverse coordinate \(\mathcal Y_T\) deep in the cone by \[\mathcal Y_T'(S)=j(TF(S))^{-1},\qquad \mathcal Y_T(S)-S\longrightarrow0 \quad\hbox{down the cone}.\] The normalized primitive exists because its derivative minus one is \(O(|T|^2D^{-2\alpha})\) and \(2\alpha>1\). It can be constructed by integration from a point at depth \(R\), then letting \(R\to\infty\); changing the horizontal coordinate of the starting point contributes \(O(R^{1-2\alpha})\), which vanishes. Along every limiting solution, \[\frac{\partial}{\partial Y}\mathcal Y_T(S(T,Y))=1.\] The anchor fixes the additive constant to zero. Hence the solution is the unique nearby inverse of \(\mathcal Y_T\) sufficiently deep. All deep subsequential limits agree, and the convergence and holomorphic dependence asserted in (i) follow.

We next obtain the domains for real \(T\). Fix \[\frac12<c_1<1-\beta\] and a sufficiently large fixed number \(C_0\). For a small fixed depth ratio \(D>0\), the global moment laws on ratios \(v\ge D/C_0\) have the bound \[ C_\varepsilon D^{-\lambda-\varepsilon}(D/v)^{c_1}. \tag{276}\] For \(D/C_0\le v\le1\), first use \(m_*(v)\le C_\varepsilon v^{-\lambda-\varepsilon/2}\), then take the original index sufficiently large at this fixed \(D\). For \(v\ge1\), use (267), whose field exponent is \(1-\beta>c_1\). These give (276) with constants independent of \(D\), although the required sequence index may depend on \(D\).

Submean and dyadic tile summation, with a slightly smaller exponent still greater than \(1/2\), provide at least one horizontal center whose actual fields obey the corresponding pointwise cone bound. This step uses uniform horizontal sampling and the global moments; the good center need not be \(\xi_n\). Apply 34 at that center with \[|T|\le c_\varepsilon D^{\lambda+\varepsilon}.\] After choosing its small constant and, if necessary, replacing the base depth by a fixed smaller multiple, the passage reaches phase depth \(d'D\). The real no-patch transfer, using 24 as after 34, consequently gives the global width bound \[ \Delta(Td'/B')\le d'D \quad\hbox{for }|T|\le c_\varepsilon D^{\lambda+\varepsilon}, \tag{277}\] eventually for each fixed \(D\). Recall that \(\Delta(t)=H-h_X(t)\). This is the point where a successful local passage is converted to a full horizontal domain; local analytic continuation alone would not give (277).

The full-width domains in (277) exhaust (275), with a strict margin on compact sets. The shape constraint gives \(\Im S_n<0\), and the deep anchor gives a fixed interior normalization. Holomorphic maps to a half-plane with such an anchor form a normal family, as is seen by mapping the half-plane to the unit disk. Every limit agrees with the unique deep solution. This defines \(S(T,\cdot)\) on (275); the energy identity propagates there by the identity theorem. The same argument for a sequence of real parameters \(T_j\to T\) proves local uniform continuity: on a common strict interior, every convergent subsequence has the same deep germ and hence the same limit. This also treats \(T=0\).

It remains to justify joint germs at all real parameters. Equation (274) first shows that \(J\ne0\). At fixed \(w=TF(S)\), its \(J\)-derivative is \[ \frac{\partial}{\partial J}\bigl(J^3g(wJ/2)\bigr) =3J^2\frac{\sin z}{z}. \tag{278}\] It can vanish only when \(z=n\pi\), \(n\in\mathbb Z\setminus\{0\}\). The equivalent entire equation \[ (TF(S))^3=24(\sin z-z\cos z) \tag{279}\] then has a nonzero right side. Differentiating in \(Y\) gives \[3T^3F(S)^2F'(S)J=24z\sin z\,z_Y,\] and hence \(F'(S)=0\) at such a point. The function \(F\) is nonconstant, because it is nonzero and decays down the cones. Its critical points are therefore countable. At each such point and each nonzero integer \(n\), (279) allows at most three values of \(T\). Thus the exceptional real parameters form at most a countable set.

This countable exception is removable. Fix an exceptional real \(T\), a strict interior point, and a nearby sequence of nonexceptional real parameters. On a common connected interior \(Y\)-domain, continuity gives local uniform convergence of \(z(Y,T_j)\) to \(z(Y,T)\). For each fixed nonzero integer \(n\), Hurwitz’s theorem says that \(z(Y,T)-n\pi\) is either zero-free or identically zero. The latter alternative is impossible on the common domain: analytic continuation to its deep part gives \(z\to0\). Exhausting the strict interior proves omission of all the exceptional values for every real \(T\).

Consequently (278) is nonzero along every compact vertical path in the real-parameter domain. Solve the regular implicit equation as an analytic first-order ODE in \(Y\), starting with the joint deep data. A finite chain of local ODE neighborhoods along the path gives a joint germ in \((T,Y)\) at its endpoint. The constructed germ agrees with the given real-parameter functions by their continuity and local ODE uniqueness. Germs obtained on overlapping vertical paths agree by the same uniqueness, which proves (iv). ◻

Proof. For every integer \(n\ge0\), the moment \[\int_\Gamma T^n A(T,w)\,\,\mathrm dT\] is holomorphic in \(w\). It vanishes in the initial open set and hence throughout \(U\). For an exterior point \(z\) of large modulus, expand \((T-z)^{-1}\) in its geometric series. The vanishing moments show that the exterior Cauchy transform is zero there, and analytic continuation makes it zero on the whole exterior. The interior Cauchy integral \[\frac{1}{2\pi\mathrm i}\int_\Gamma \frac{A(T,w)}{T-z}\,\,\mathrm dT\] is jointly holomorphic in \(z,w\). Deforming the contour a short distance inside and outside any collar point and using the zero exterior transform shows that it agrees there with \(A\). It is the required extension. ◻

Proof. Fix \(\varepsilon>0\), use 63 with \(r=1/(\lambda+\varepsilon)\in(0,1)\), and take a small fixed upper half-disk \(|T|<t_0\), \(\Im T>0\). For a target \(Y\) with \(\Re Y\in I\), \(-\Im Y=D\), and for \(|T_c|\le c_*D^{1/r}\) in this half-disk, use the graph with phase argument \[ Y+G(T)-G(T_c),\qquad G(T)=-\mathrm iC_*(\eta_0-\mathrm iT)^r,\qquad 0<\eta_0\le c_*D^{1/r}. \tag{282}\] The fractional power is the branch on the right half-plane; \(\Re(\eta_0-\mathrm iT)>0\) in the upper half-disk. The auxiliary \(\eta_0\) is distinct from the fixed growth exponent \(\varepsilon\).

On the real diameter, \[-\Im G(T)=C_*\Re(\eta_0-\mathrm iT)^r \ge C_*\cos(\pi r/2)|T|^r.\] Choose \(C_*\) large compared with the constant in (275), and then \(c_*\) so small that \(|G(T_c)|<D/2\). Every diameter point then lies strictly in its real-parameter domain. On the semicircle \(|T|=t_0\), \(\Im T\ge0\), \[-\Im G(T)\ge c_rC_*t_0^r,\qquad |\Re G(T)|\le C_r(-\Im G(T)).\] Fix the required cone aperture first and then increase \(C_*\) further to put this semicircle in the deep joint domain. The bounded interval \(I\) and the correction \(G(T_c)\) cost only fixed additional margins.

The graph has coherent joint germs on a boundary collar. On the diameter these are the germs of 63(iv); on the semicircle they are deep joint data. They agree at the corners because both are attached to the same deep branch. Overlapping real-parameter neighborhoods agree first for real parameters and then by holomorphic identity.

Lower the center \(Y\) until the whole graph lies in deep joint data, then raise it vertically to the target. Boundary membership and strict margins persist. On each compact center segment a finite covering gives a holomorphic function on a neighborhood of the boundary contour times a connected complex neighborhood of that segment. Coherence follows from vertical-path uniqueness. Initially the graph interior is filled by deep data, so 64 fills it throughout the motion. Its boundary values have negative imaginary part; the harmonic maximum principle therefore gives \(\Im S<0\) on the filled interior.

For the lower half-disk use \(G(T)=-\mathrm iC_*(\eta_0+\mathrm iT)^r\). The two constructions agree on the joint real-parameter germs. The germs obtained for different \(T_c\) also agree on overlaps: at a fixed parameter they follow the same vertical path from the unique deep branch. Thus the model is holomorphic on the whole parameter disk \[ |T|<c_*D^{1/r} \tag{283}\] at the target, still with \(\Im S<0\). Shrinking \(c_*\) once makes this construction uniform for \(Y\) in a disk of radius a small fixed multiple of \(D\).

Write \[S(T,Y)=Y+T^2A(Y)+O(T^4).\] Evenness follows from the deep branch and continuation. The positive harmonic function \(-\Im S(T,Y)\) equals \(D\) at \(T=0\). Its Fourier coefficient estimate on circles in (283), or the Caratheodory coefficient estimate, gives \[|A(Y)|\le C D^{1-2/r}.\] This also holds on the depth-sized \(Y\)-neighborhood, so Cauchy’s estimate yields \[ |A'(Y)|\le C D^{-2/r}. \tag{284}\] The energy equation determines the derivative: \[g(z)=1-\frac{z^2}{10}+O(z^4),\quad J=1+T^2A'(Y)+O(T^4),\quad z=\frac{TF(Y)}2+O(T^3).\] Its coefficient of \(T^2\) is \(3A'(Y)-F(Y)^2/40=0\). Hence \[ A'(Y)=F(Y)^2/120. \tag{285}\] Combining these bounds gives \(|F(Y)|\le C D^{-1/r}=C D^{-\lambda-\varepsilon}\). Arbitrariness of \(\varepsilon>0\) gives (281). ◻

Proof. The coarse part of (267) gives \(L(v)\le C+(1-\beta)v\) for \(v\le0\). Thus \(L(v)-c_1v\to-\infty\) as \(v\to-\infty\). For a large \(w\), choose a running record \(u_0\le w\) of \(L(v)-c_1v\) on \((-\infty,w]\). Its positive asymptotic slope \(\lambda-c_1\) implies \(w-u_0=o(w)\): for every fixed \(\eta>0\), all values at \(v\le(1-\eta)w\) are eventually below the value at \(w\).

Next maximize \(L(v)-P_1v\) over \(v\ge u_0\), and call a maximizer \(u\). It exists because \(P_1>\lambda\). The negative asymptotic slope shows \(u-u_0=o(u_0)\). Its maximality gives the future bound in (287). For \(v\le u_0\), the past record and \(L(u)-L(u_0)\ge P_1(u-u_0)\ge c_1(u-u_0)\) give the past bound at \(u\). For \(u_0\le v\le u\), maximality gives \[L(v)-L(u)\le P_1(v-u)\le c_1(v-u).\] The epochs obtained this way are within \(o(w)\) of every sufficiently large \(w\). ◻

Proof. Fix \[\frac12<\alpha'<\min(\alpha,c_1,1)\] and choose \(L_0\) large enough for the quiet-cone form of 52 with this exponent. If the conclusion fails, some fixed small \(\eta>0\), epochs \(u_j\to\infty\) in \(E\), and centers \(x_j\) in a fixed compact interval \(I\) satisfy \[ \log|F(p-\mathrm ie^{-v})|<\lambda u_j-\eta u_j \tag{289}\] for every \[|p-x_j|\le L_0e^{-v},\qquad |v-u_j|\le\eta u_j.\] Pass to a subsequence with \(u_j\ge j^2\).

Return to the actual data of 62, taking their sequence index sufficiently large after each fixed \(u=u_j\). Set \[ d''=d'e^{-u},\qquad K''=K+u,\qquad B''=\mu(K+u), \tag{290}\] with physical center \(\xi_n+d'x_j\). We may require \(K'',B''\to\infty\): for fixed \(u\), \(B''/B'\) tends to the positive number \(e^{-u}m_*(e^{-u})\).

First retain a uniform global moment envelope at these scales. Write \[M_n(v)=e^v\frac{\mu(K+v)}{B'},\qquad M_n(v)\longrightarrow e^{L(v)}\] locally uniformly. The old envelope says \[ M_n(v)\le \begin{cases} C e^{(1-\beta)v},&v\le0,\\ C e^{(1+a)v},&v\ge0. \end{cases} \tag{291}\] Choose fixed \(1/2<c_2<c_1\) and \(P_2>\max(P_1,1+a)\). On the finite range \(0\le v\le C_0u\), take the actual index so large that the ratios \(M_n(v)/M_n(u)\) differ from their limits by at most a factor two. The epoch bounds imply \[\begin{align*} \frac{M_n(v)}{M_n(u)} &\le2e^{c_2(v-u)}&& (0\le v\le u),\tag{292}\\ \frac{M_n(v)}{M_n(u)} &\le2e^{P_2(v-u)}&& (u\le v\le C_0u). \tag{293}\end{align*}\] For all \(v\le0\), (291) supplies the first bound up to one fixed constant, since \(1-\beta>c_2\) and \(L(u)=\lambda u+o(u)\), \(\lambda\ge1>c_2\). For all \(v\ge C_0u\), it supplies the second bound if \(C_0\) is a sufficiently large fixed constant: \[(1+a-P_2)v+P_2u-L(u)\le0.\] In terms of \(\mu\), this is a uniform new envelope with coarse exponent \(1-c_2<1/2\) and fine exponent \(P_2-1>0\). Only a finite range chosen before the actual index used moment convergence.

We next prove quietness on the whole needed cone, including the unbounded coarser tail. At old log depth \(v\), the denominator for the new weighted quiet-cone assertion is \[ M_n(u)e^{\alpha'(v-u)}. \tag{294}\] Indeed the new depth ratio is \(e^{u-v}\), and \[\frac{d''}{B''}M_{\rm shape} = \frac{\max(|F_n|,\sqrt{|Q_n|})}{M_n(u)},\] where \(M_{\rm shape}=\max(|f|,\sqrt{|q|})\).

Increase the actual index further so that \[ |F_n-F|\le1,\qquad \sqrt{|Q_n|}\le1 \tag{295}\] throughout \[0\le v\le u_j+j,\qquad |p-x_j|\le L_0e^{-v}.\] For each \(j\), a small enlargement of this set is a compact subset of \(\mathbb H_-\), so compact convergence allows the choice. Throughout the set, (294) is at least \[\exp\bigl((\lambda-\alpha')u_j+o(u_j)\bigr).\] The errors in (295) are therefore negligible uniformly.

For \(u_j-\eta u_j\le v\le u_j+j\), the upper endpoint is in the failed-witness window for large \(j\). Equation (289) bounds the logarithmic ratio by \[-(1-\alpha')\eta u_j+o(u_j).\] For \(0\le v<u_j-\eta u_j\), use (281) on the fixed horizontal enlargement of \(I\) containing this part of the cone. Its errors over \(0\le v\le u_j\) are \(o(u_j)\): separate a fixed compact \(v\)-range, then use the uniform asymptotic estimate on the remainder. The logarithmic ratio is at most \[(\lambda-\alpha')(v-u_j)+o(u_j) \le-(\lambda-\alpha')\eta u_j+o(u_j).\]

For every \(v<0\) back to fixed physical depth, no convergence transfer is necessary. If \(I\subset[-R,R]\), the cone at old depth \(e^{-v}\ge1\) lies inside the old cone of aperture \(L_0+R\), because \[|p|\le R+L_0e^{-v}\le(L_0+R)e^{-v}.\] The inherited bound (268) gives \[\frac{\max(|F_n|,\sqrt{|Q_n|})} {M_n(u_j)e^{\alpha'(v-u_j)}} \le C_{L_0+R} \exp\bigl(-(\lambda-\alpha')u_j+o(u_j)\bigr) e^{(\alpha-\alpha')v}.\] It tends uniformly to zero since \(\alpha>\alpha'\). Both the aperture and its constant remain fixed despite movement of the centers.

The three ranges give quietness down to \(e^{-j}d''\), and hence down to every fixed positive multiple of \(d''\) eventually. The normalization is exactly \(B''=\mu(K'')\), with the uniform moment envelope proved above. These are the hypotheses of the quiet-cone form of 52. For clarity, at a fixed depth ratio and bounded scaled parameter, the parameter-to-depth ratio tends to zero because \(B''\to\infty\). The local cone theorem uses a fixed ratio bound and a fixed aperture. Its quiet passage forces the global real width deficit divided by \(d''\) to vanish, and the one-sided coefficient and moment argument in that theorem is contradictory.

Thus, for each fixed small \(\eta>0\), all sufficiently large common epochs have the required witnesses uniformly over \(x\in I\). A decreasing diagonal choice of \(\eta\) yields \(\eta_I(u)\to0\) in (288). ◻

Proof. Let \(B\) be Brownian motion started at \(w\), stopped when it exits \(R\). Superharmonic comparison, applied first up to the first hit of the union of the disks and then up to the exit, gives \[ \mathbb P_w\{B\text{ hits some }D_T\text{ before exiting }R\} \le U(w)/H. \tag{300}\] This comparison allows logarithmic poles of \(U\): one can first stop away from those poles and then use monotone exhaustion. Alternatively, the harmonic probability of hitting the disk union, multiplied by \(H\), is a subfunction of \(U\) by the minimum principle on the complement of the disks.

For each \(T\), consider the event that the motion hits \(D_T\) and subsequently exits \(R\) through its top side at a horizontal coordinate in \(I_T\). We claim that its probability is at least \(c\rho_T/d\). First stop at the horizontal line of depth \(d_T\), or at the other sides of the part of \(R\) below that line. The harmonic measure density on the segment \[|x-\Re c_T|<\rho_T/32, \qquad -\Im z=d_T,\] is bounded below by \(c/d\). To see the uniformity, divide all lengths by \(d\). The starting point has depth one, the bottom has depth two, and the new top has depth in \([0,1/2]\); the segment lies a fixed positive distance from the lateral sides. The interior-to-side Poisson density of a rectangle is positive and continuous in these parameters, including top depth zero. Its minimum on this compact family is therefore positive. Thus the probability of this first hit inside \(D_T\) is at least \(c\rho_T/d\).

From any point of that segment, the probability of exiting the original rectangle through its top within \(I_T\) has a fixed positive lower bound. Indeed, use a smaller rectangle with horizontal center \(\Re c_T\), horizontal half-width a fixed large multiple of \(d_T\), top depth \(\tau\), and bottom depth \(2d_T\). It lies in \(R\) after increasing \(M\). After division by \(d_T\), its starting points, top depth in \([0,1/4]\), and target intervals of length \(\rho_T/d_T\in[1/b,1/a]\) again range over a compact family with nondegenerate margins. Positivity and continuity of harmonic measure give the claimed lower bound. Exiting this smaller rectangle on the specified top interval also exits \(R\) there. The strong Markov property proves the claim.

Choose a subfamily of the intervals \(I_T\) with disjoint interiors whose total length is at least one third of the length of their union. For completeness, order the finite family by decreasing length, retain an interval if its interior is disjoint from those already retained, and discard it otherwise. Each discarded interval is contained in the threefold concentric enlargement of an intersecting retained interval of at least its length. This proves the asserted covering estimate. The events just constructed for the retained intervals are disjoint, because their final exit coordinates are in disjoint intervals; individual endpoints have harmonic measure zero. Each event is contained in the event in (300). Summing their lower bounds gives (299). In particular, no union bound over the number of tiles is being used. ◻

Proof. Choose \(q\in Q\) with \(U(q)\le2T\). A disk about \(q\) of radius \(40\rho\) lies in the given domain and contains \(B(c,30\rho)\). The super-mean inequality, integrated over radii, and nonnegativity give \[ \int_{B(c,30\rho)}U\,\,\mathrm dA\le C\rho^2T. \tag{302}\] The distributional identity \(-\Delta U=2\pi\sum_a m_a\delta_a\) records the zeros of \(F\) with their multiplicities. A nonnegative smooth cutoff equal to one on \(B(c,20\rho)\), supported in \(B(c,30\rho)\), and with second derivatives bounded by \(C\rho^{-2}\) yields \[N:=\sum_{a\in B(c,20\rho)}m_a \le C\rho^{-2}\int_{B(c,30\rho)}U\,\,\mathrm dA\le CT.\] List these zeros with multiplicity as \(a_1,\ldots,a_N\). The function \[H(z)=U(z)+\sum_{\nu=1}^N \log\frac{|z-a_\nu|}{\rho}\] extends harmonically across the zeros on \(B(c,20\rho)\). Each logarithm has bounded absolute integral in scaled coordinates, so (302) gives \(\int_{B(c,20\rho)}|H|\,\,\mathrm dA\le C\rho^2T\). The mean-value inequality for the absolute value of a harmonic function then gives \(|H|\le CT\) on \(B(c,10\rho)\).

Here is the precise elementary covering estimate needed for the logarithmic terms. For points \(b_1,\ldots,b_N\in\mathbb C\), outside disks of sum of radii at most \(3h\), \[ \prod_{\nu=1}^N|z-b_\nu|\ge(h/e)^N. \tag{303}\] There is nothing to prove when \(N=0\). Otherwise, consider all disks \(B(z,hi/N)\) whose interiors contain at least \(i\) of the listed points, for \(1\le i\le N\). Starting with the largest of the finitely many possible radii, select a maximal disjoint family at each radius, discarding disks that intersect ones already selected. Only finitely many disks can be selected: their centers lie in a bounded set and their radii are at least \(h/N\). Each unselected disk intersects a selected one of at least its radius, and is contained in that disk’s threefold enlargement. If the selected radii are \(hi_k/N\), disjoint interiors show that \(\sum_k i_k\le N\), by counting the points they contain. The enlarged disks therefore have total radius at most \(3h\). Outside them, the \(i\)-th smallest distance to the points is at least \(hi/N\), for every \(i\). Their product is at least \(h^N N!/N^N\ge(h/e)^N\), proving (303).

Apply this estimate to \(b_\nu=(a_\nu-c)/\rho\). Outside the resulting disks, \[U\le CT+N\log(e/h)\le C_hT \quad\text{on }B(c,10\rho).\] This implies (301).

It remains to justify actual routing, including at corner contacts. If \(Q,Q'\) touch and have comparable sizes, choose a square centered at a contact point of side \(\min(\rho_Q,\rho_{Q'})\). This square lies in \(Q^*\cap(Q')^*\). The sums of the lengths of the horizontal and vertical projections of all exceptional disks for both tiles are at most \(C h(\rho_Q+\rho_{Q'})\). Fix \(h\) so small that each of these sums is less than half the side of the contact square. One can then choose an \(x\)-coordinate and a \(y\)-coordinate in that square, each outside the corresponding projections of both disk families. Their intersection is a junction point safe for both tiles. Choose the two endpoint coordinates in the same fashion in the prescribed endpoint squares. Within a tile, join its two chosen points by one horizontal and one vertical segment. Both coordinates at each endpoint avoid all exceptional projections for that tile, so these segments avoid all of its exceptional disks. They lie in \(Q^*\), and their combined length is at most \(C\rho_Q\). Concatenation proves the routing assertion. The same construction works when a chain consists of a single tile. ◻

Proof of 68. We work in fixed horizontal intervals contained in \([-5,5]\). All errors below are uniform on these intervals. Fix the aperture from (298) for this compact interval.

Proof of 58. Assume that a sequence in 53 exists. Its primary moments have a limit by 59. Apply 60 to its logarithmic law. The steep-contact alternative contradicts 40, as proved in 61. The bounded-contact alternative gives the fields in 62. Their nonzero first field has holomorphic quotient \(F''/F\). The limiting impact equation and parameter continuation, [prop:pure-model,prop:model-growth], give its uniform upper growth. [model:normal-epochs,prop:model-witnesses] give common relatively dense epochs and uniform witnesses at every horizontal center. These are precisely the hypotheses contradicted by 68. Both alternatives are impossible, proving the Theorem. ◻

Proof. The sole forward assumption in the critical reduction was the nonexistence of the sequences in 53. This is 58. Apply the two stated results. Their conclusions were not used anywhere in the proof of that theorem. ◻

Proof. First work on the real collar. If \(\delta\mathrm{arc}_j\) is an arc-length variation at vertex \(j\), put \(z_j=\sin d_j^0\,\delta\mathrm{arc}_j\). The normal projections on edge \(j\) of the two endpoint variations are \(z_j\) and \(-z_{j+1}\). The second variation of that edge length is therefore \((z_j+z_{j+1})^2/\ell_j\), apart from the acceleration of the endpoint paths. The latter acceleration is \(-n_{\psi_j}f_j^3/c\) per squared arc-length increment. Pairing it with the length gradient \(e_{\theta_{j-1}}-e_{\theta_j}\) gives \(-2f_j^3\sin d_j^0/c\) per squared arc-length increment, or \(-\mathcal V_j z_j^2\). Thus \(\mathcal H\) is the length Hessian in these variables. Differentiation along the periodic family gives \(\mathcal H w=0\). Since \(\mathcal V_jw_j=2f_js_j'\), this also gives the second identity in [odd:jacobi-angle]. Analytic continuation proves the meromorphic identities.

The real collar is sufficiently small that \(s_j'>0\) and \(\sin d_j^0>0\), so \(w_j>0\). Let \(D_w=\mathop{\mathrm{diag}}(w_j)\) and put \(\gamma_j=w_jw_{j+1}/\ell_j>0\). The off-diagonal entries of \(D_w\mathcal H D_w\) are \(\gamma_j\); the identity \(\mathcal Hw=0\) fixes its diagonal entries as \(-\gamma_{j-1}-\gamma_j\). Consequently \[\langle D_w\mathcal H D_w z,z\rangle =-\sum_j\gamma_j(z_j-z_{j+1})^2.\] The cycle is connected. Its only null vectors are constant, and fixing any one coordinate to zero removes the nullspace. This proves the two claims concerning \(\mathcal H\). A null vector for \(M_0\) would satisfy \(z_{j+1}=-z_j\) for every \(j\); oddness then forces \(z=0\).

For the last assertion, arrange the retained coordinates as \(I,F\). The truthful diagonal block is \(\mathcal H_{I,I}<0\), and the flat diagonal block is \((M_0)_{F,F}>0\). When both are nonempty, the Schur complement of the flat block is \[\mathcal H_{I,I} -(M_0)_{I,F}(M_0)_{F,F}^{-1}(M_0)_{F,I}<0.\] Hence the full matrix is invertible. If one of the two blocks is empty, one of the already proved definite-matrix statements applies. ◻

Proof. On the collar let \(v_j\) be the variation obtained by differentiating \(\mathfrak s(y+jt,t)\) with respect to \(t\), including its argument \(jt\), and multiplying by \(c\sin d_j^0 f_j^{-2}\), at \(t=t_0\). Periodicity gives \[v_{j+N}=v_j+Nw_j.\] The endpoint normal differentials on edge \(j\) are \[n_{\theta_j}\cdot\,\mathrm dX_j=w_j\,\,\mathrm dy+v_j\,\,\mathrm dt, \qquad n_{\theta_j}\cdot\,\mathrm dX_{j+1} =-w_{j+1}\,\,\mathrm dy-v_{j+1}\,\,\mathrm dt.\] Differentiating \(X_{j+1}-X_j=\ell_je_{\theta_j}\) and taking the normal component yields \[\,\mathrm d\theta_j =\frac{(w_j+w_{j+1})\,\,\mathrm dy+(v_j+v_{j+1})\,\,\mathrm dt}{\ell_j}.\] Since \(\,\mathrm dp_j\wedge\,\mathrm d\theta_j =(n_{\theta_j}\cdot\,\mathrm dX_j)\wedge\,\mathrm d\theta_j\), the area identity gives \[\frac{w_jv_{j+1}-v_jw_{j+1}}{\ell_j}=A(t_0).\] The shift of the line phase by \((j+1/2)t\) does not alter the coefficient of \(\,\mathrm dy\wedge\,\mathrm dt\). Summing the successive differences of \(v_j/w_j\) therefore gives \[ \sum_j\frac{\ell_j}{w_jw_{j+1}}=\frac{N}{A(t_0)}. \tag{328}\]

The deleted cofactor of the negative weighted cycle Laplacian is \[(-1)^{N-1}\sum_k\prod_{i\ne k}\gamma_i =(-1)^{N-1}\left(\prod_i\gamma_i\right) \left(\sum_i\gamma_i^{-1}\right).\] For completeness, write the positive Laplacian as an incidence matrix, times \(\mathop{\mathrm{diag}}(\gamma_i)\), times its transpose, and delete one vertex. Cauchy–Binet expands the determinant over the omitted edge. Each remaining edge set is a path tree, whose incidence minor has determinant \(\pm1\); this gives precisely the displayed sum. Undoing the conjugation by \(D_w\), and then using [odd:Wronskian-sum], proves [eq:cofactor] on the collar. Both sides are meromorphic functions of \(y\), so the identity holds throughout the slice. ◻

Proof. By 29, each \(e^{\mathrm is_j}\) is bounded and bounded away from zero. The boundary limit theorem for bounded holomorphic functions gives nontangential limits almost everywhere (Garnett 2007, I, Theorem 5.3). Near a nonzero limiting value, choose a local logarithm. On a connected truncated approach cone it differs from \(\mathrm is_j\) by one fixed integral multiple of \(2\pi\mathrm i\). Thus \(s_j\) itself has a finite nontangential limit.

Suppose the limiting states are all strictly inside the shape strip on a set of positive measure. At a point of this set we can require, in addition, that every \(f_j\), \(\sin d_j^0\), \(\ell_j\), and \(\det\mathcal H_{\widehat j}\) have a nonzero limit. To justify the requirement, first exclude limiting zeros of the \(f_j\). Each \(f_j\) is holomorphic on the phase strip and nonzero on the real collar. If its angular limit were zero on a positive-measure subset, the same boundary uniqueness theorem used below would force \(f_j\equiv0\), a contradiction. Thus the finite union of these exceptional sets has measure zero. The sine and length evaluations are then regular at the interior limiting state tuple. Exclude their zero limits next; all cofactor evaluations are regular after those exclusions. If any one of these meromorphic functions had limiting value zero on a positive-measure set, Privalov’s nontangential boundary uniqueness theorem for meromorphic functions would make it identically zero (Garnett 2007, II, p. 91). This is impossible by its nondegenerate real values. There are only finitely many functions to exclude. To apply the disk boundary theorems to this strip, restrict to a compact boundary subarc of positive measure and use a conformal map to the disk. The map extends analytically across that subarc with nonzero derivative, preserving null sets and nontangential approach regions.

Write \(\mathbf s=(s_0,\ldots,s_{N-1})\). Near such a limiting tuple the cofactor identity and [odd:w-definition] express \[(s_j')^2=G_j(\mathbf s),\] where each \(G_j\) is analytic and nonzero. Choose the square roots with the signs of the actual functions on a truncated approach cone. We obtain a regular analytic vector equation \(\mathbf s'=\mathbf g(\mathbf s)\). Local solutions starting sufficiently near the limiting tuple exist in disks of a common positive radius. Starting at an interior point sufficiently close to the boundary, uniqueness identifies that solution with the actual slice on their overlap. This continues the whole slice arc across the chosen boundary point.

It remains to explain why an extension of this single slice yields a joint extension, as required for 24. We give the argument at the upper boundary. The states \[Z=(s_0,s_1)\] on the extended arc form a regular analytic curve, parameterized by \(y\). All incidence angles and chords along its orbit are nondegenerate. Thus the local billiard return \(R\), after \(N\) reflections and subtraction of the known \(2\pi\) shifts, is analytic near these states. At each reflection, this follows by solving the next incidence equation on the regular table arc; its derivative in the next intersection coordinate is nonzero because the sine of incidence is nonzero. The return fixes the extended curve, by analytic identity from below.

The determinant of its linearization at a fixed state is one. One can see this without choosing symplectic coordinates: successive linearized stationarity equations in the Jacobi variables have transfer determinants equal to ratios of consecutive off-diagonal entries. The ratios telescope over the cycle. The coordinate factors converting \(\delta s_j\) to \(c\sin d_j^0 f_j^{-2}\delta s_j\) are also periodic along the fixed orbit. Choose local coordinates \((u,v)\) with the fixed arc given by \((u,v)=(y,0)\). The return consequently has the form \[ R(u,v)= \bigl(u+\sigma(u)v+O(v^2),\ v+\tau(u)v^2+O(v^3)\bigr). \tag{329}\]

In a lower part of this neighborhood the original joint data are known. For \[Z(y,t)=\bigl(\mathfrak s(y,t),\mathfrak s(y+t,t)\bigr), \qquad b=N(t-t_0),\] the reflection identities give \(R(Z(y,t))=Z(y+b,t)\). Expanding there in the coordinates just chosen, write \[u=y+O(b),\qquad v=b p_0(y)+O(b^2).\] The first component of the return identity at order \(b\), and its second component at order \(b^2\), give respectively \[ \sigma p_0=1,\qquad p_0'=\tau p_0^2, \qquad \sigma'=-\tau. \tag{330}\] These identities continue along the extended arc. By moving the boundary point slightly within this arc if necessary, take a smaller neighborhood where \(\sigma\ne0\), and set \(p_0=1/\sigma\).

For small nonzero \(b\) with \(|\Im b|\ge |b|/3\), the return and its inverse continue the joint states upwards in steps \(\pm b\). Here are the relevant uniform estimates. In the coordinates \((u,p)=(u,v/b)\), [odd:return-expansion] reads \[(u,p)\longmapsto(u,p) +b\bigl(\sigma(u)p,\tau(u)p^2\bigr)+O(b^2)\] on fixed compact subsets; the inverse has the negative first increment. The analytic differential equation with that vector field has the regular solution \((u,p)=(y,p_0(y))\), by [odd:return-compatibility]. Choose the arc neighborhood and the height to be crossed sufficiently small. The steepness condition keeps all the slanted paths from the lower starting slab inside that neighborhood. There are \(O(1/|b|)\) steps; each has error \(O(|b|^2)\) and stability factor at most \(1+C|b|\). The continued states therefore stay within \(O(|b|)\) of this regular solution. All shape coordinates remain strictly inside the shape strip.

All intermediate site coordinates are recovered by the regular analytic billiard steps along the orbit. Thus their continuations agree with the original \(s_j\)’s wherever the latter were already defined.

Use locally constant numbers of backward steps to place initial states in the lower slab. On overlaps, the exact return identity in that slab and uniqueness of each forward step identify the constructions. They give joint holomorphic germs in the steep sectors, attached by vertical continuation to the lower data.

Finally choose a small \(\eta>0\), then a much smaller \(r>0\), and use the analytic disks \[ t=t_0+r\zeta,\qquad y=x_*+\mathrm i\varsigma-\mathrm i\eta\zeta^2+w, \qquad |\zeta|\le1, \tag{331}\] as \(\varsigma\) rises from \(h-2\eta\) to \(h\); \(w\) varies in a sufficiently small complex disk. On the boundary arcs where \(\zeta\) is near the real axis, the phase point is a fixed amount below the wall. For small \(r\), these arcs lie in the original joint tube, by lower semicontinuity of the width. On the remaining arcs the parameter is in a fixed steep sector, so the preceding continuation applies. The collar germs agree because both are vertical continuations of the lower data. The full disks are initially inside known data. Holomorphic product continuation, as in 21, fills them as the center is raised. Equivalently, pull the germs back to a collar of the unit circle and a complex neighborhood of the center path: the negative Cauchy moments vanish in the initial open part and hence throughout that connected neighborhood. More explicitly, combine \(x_*+\mathrm i\varsigma+w\) into one complex center variable \(a\). The map \((a,\zeta)\mapsto(a-\mathrm i\eta\zeta^2,t_0+r\zeta)\) has Jacobian determinant \(r\ne0\), so the filling supplies a joint neighborhood in \((y,t)\).

The maximum principle preserves \(|\Im s_j|<H\) on the filled disks. Composition with \(\Gamma\) gives a joint meromorphic patch for \(X\) at \((x_*+\mathrm ih,t_0)\), agreeing with the original tube below the wall. This contradicts 24, which applies also to the rational parameter \(t_0\). The lower boundary is treated in the same way, or by real symmetry. The supposed positive-measure set therefore cannot exist. ◻

Proof. The real and imaginary parts of a holomorphic function of planar Brownian motion are local martingales. The Cauchy–Riemann equations give equal quadratic variations \(\,\mathrm dQ_j\) and zero mutual quadratic covariation. The Brownian time-change theorem therefore applies. To use it up to a given interior phase stop, append independent Brownian increments in intrinsic time after that stop. The extension cannot have exited the shape strip before the part of the path already used, because all those states lie in the strip. Approximation by compact interior stops proves the same domination for nonnegative occupations over the whole phase interval.

On the real collar \(y\mapsto s_j(y)\) is an increasing analytic circle diffeomorphism whose derivative is bounded above and below. This proves the initial-density assertion. Under a uniform real start, the periodic shape-strip Green weight is a constant times \(\delta=H-|\Im s|\). It has finite integral, proving the total-clock claim. At either wall, the contribution of \(|f|^2\) up to log depth \(K\) is bounded by a constant times \[\int_{e^{-K}}^{\delta_*}\delta\,\mathbb E|f|^2\,\,\mathrm d\delta =\int_{k_*}^{K}\delta^2\mathbb E|f|^2\,\,\mathrm dk =\int_{k_*}^{K}\mu_f(k)^2\,\,\mathrm dk.\] The fixed interior region has bounded contribution. Now use [odd:angle-budget]. Domination of first-hit laws follows from the same Brownian extension; if the original path never reaches the row, it contributes no mass.

Starting at distance \(\delta\) from one horizontal wall, the exit time from the strip is no greater than the first time the vertical Brownian coordinate hits that particular wall. The one-dimensional reflection principle bounds survival to time \(T\) by \(C\min(1,\delta/\sqrt T)\). Applying this after the hit, with the strong Markov property in intrinsic time, proves (iii).

For (iv), at the first hit of each integer level \(m\), the probability of reaching depth \(R e^{-m}\) before hitting the adjacent wall is at most \(1/R\), by the linear one-dimensional hitting probability. Take \(R\) a sufficiently large fixed power of \(L\), and sum over the \(O(L)\) integer levels up to \(L\), and the fixed number of sites. If a retreat larger than \(C\log L\) occurred, it would contain one of these integer-level retreats, with an immaterial factor from rounding. The union probability tends to zero. ◻

Proof. Away from the polar zero and pole set, \(\log|g(y_u)|\) is a continuous local martingale. Before the indicated cut, \(B_0-\log|g(y_u)|\) is nonnegative, hence is a supermartingale after localization. Stop additionally at its first hit of \(B_0+B_1\). The supermartingale inequality gives the first bound. Exhaustion removes the auxiliary compact localizations. ◻

Proof. On the stopped interval, [odd:amplitude-upper] bounds \(a_i\) by \(C(1+U)\) for \(i\in I\). The geometry cap bounds the reciprocal lengths, the reciprocal incidence sines, and the angles. Hence every entry, determinant, and cofactor of \(B_{I,F}\) has modulus at most \(\exp(C(1+U))\). There is no amplitude bound required at flat or deleted sites: their curvature terms do not appear. By 72, the determinant has a positive minimum in modulus on the compact real starting period. Apply 76 to that determinant with upper bound \(C(1+U)\) and lower-log threshold \(-V\). The adjugate formula gives [odd:inverse-bound] outside the resulting exceptional event. Empty matrices require no estimate.

There are at most \(3^N\) choices. Each matrix is defined and estimated from the original real start, with its own fixed stopping rule. If at a later time all the histories selected by that matrix are at most \(U\), none of them has crossed its cap previously. Thus the estimate already secured for that fixed candidate is still valid as long as the common geometry cut has not occurred. This is the reason for using history levels rather than current depths. No estimate is restarted when the current selection changes. ◻

Proof. The angle differential is \(\,\mathrm d\psi_j=f_j\,\,\mathrm ds_j\). Up to \(\tau_L\) all current log depths are at most \(L\), so [odd:angle-clock] and the maximal inequality bound all angles by \(L^{\rho+\varepsilon}\) with probability tending to one, for each fixed \(\varepsilon>0\). The linear equations (318) give the same bound for \(\theta_j,d_j^0\). They imply an upper bound of this order for \(\log|\sin d_j^0|\). The sine has a positive minimum on the real starting collar. Applying 76 with a slightly larger power gives, for some fixed \(\rho<\rho'<1\), simultaneous angle and sine cost at most \[r_*=L^{\rho'}.\] We stop at the first failure of these bounds for the estimates that follow, and add the already vanishing failure probability at the end.

Choose \[G=L^\gamma,\qquad \frac{1+\rho'}2<\gamma<1,\] and a small fixed \(b>0\). At each site use two statuses, initially moderate. Switch from moderate to tiny at a downcrossing of \(a_j=-2bG\), and switch back at an upcrossing of \(a_j=-bG\). The hysteresis gap is \(bG\). By [odd:amplitude-upper], \(CL-a_j\) is a nonnegative local martingale, hence a supermartingale, up to \(\tau_L\) and the preceding cuts. Its upcrossing levels are \(CL+bG\) and \(CL+2bG\). The upcrossing inequality, and alternation of the two switch directions, give an expected number of switches \(O(L/G)\). One can also obtain this bound by holding supermartingale increments during attempted passages between the two levels: each completed passage earns \(bG\), while an unfinished passage has possible loss bounded by \(CL+2bG\). Thus, for any sufficiently small fixed \(\varepsilon_1>0\), the total number of switches over the fixed number of sites is at most \[ L^{1+\varepsilon_1}/G \tag{341}\] with probability tending to one.

Let \[\mathcal R(u)=1+\sup_{0\le v\le u,\,j}|X_j(y_v)|.\] During a moderate interval, \(a_j\ge-2bG\). From [odd:support-differentials], the coefficient of \(\,\mathrm ds_j\) in \(\,\mathrm dX_j\) has modulus at most \(e^{5bG}\), for large \(L\), because \(r_*=o(G)\). During a tiny interval, \(a_j\le-bG\), and the coefficient of \(\,\mathrm ds_j\) in \(\mathcal R^{-1}\,\mathrm dH_j^0\) has modulus at most \(e^{-bG/2}\). The latter estimate uses \[\mathcal R^{-1}|X_j\cdot e_{\psi_j}|\,|f_j| \le \exp(Cr_*-bG).\] Mask these two differentials by their corresponding status indicators. The total expected shape clocks are bounded by 75. The maximal inequality, with the cuts described above, shows that the cumulative moderate-position integrals and tiny-support integrals have respective suprema at most \[ e^{6bG},\qquad e^{-bG/3} \tag{342}\] with probability tending to one.

On a tiny interval, write \(\,\mathrm dH_j^0=\mathcal R\,\,\mathrm dM_j\), where the appropriate masked martingale \(M_j\) is bounded as in [odd:masked-integrals]. Integration by parts against the continuous increasing function \(\mathcal R\) bounds the support increment on any such interval by \(C e^{-bG/3}\mathcal R\) at its endpoint. For example, after subtracting the starting value of \(M_j\), both the endpoint term and the integral against \(\,\mathrm d\mathcal R\) are bounded by that quantity.

Between successive switches of the joint status vector, a moderate site’s position differs from its starting position by at most \(2e^{6bG}\), and a tiny site’s support has the preceding increment bound. If \(\mathcal R_{\rm start}\) is the running maximum at the start of such an interval, every support during it is therefore at most \[ e^{Cr_*}\bigl(\mathcal R_{\rm start}+C e^{6bG}\bigr) +C e^{-bG/3}\mathcal R. \tag{343}\] Use [odd:reconstruction] and absorb the last term; its coefficient after reconstruction tends to zero because \(r_*=o(G)\). Each interval then multiplies the maximum of its starting bound and \(C e^{6bG}\) by at most \(e^{Cr_*}\). By [odd:switch-count], its accumulated logarithmic cost is at most \[C r_*L^{1+\varepsilon_1}/G=o(G),\] provided \(\varepsilon_1\) is small enough that \(2\gamma>1+\rho'+\varepsilon_1\). Hence \(\log\mathcal R\le C G\) with probability tending to one.

The identity \(\ell_j=(X_{j+1}-X_j)\cdot e_{\theta_j}\) now gives upper logarithmic length bounds of the same order. Apply 76 to each nonzero length, allowing a further small increase of the exponent, to obtain lower logarithmic bounds. Choose a fixed \(\sigma\) strictly between all these exponents and \(1\). Constants are then absorbed by \(L^\sigma\) for large \(L\), proving the claim. Each previous good bound was enforced by its first-failure cut; only the sum of their vanishing exceptional probabilities has been added. ◻

Proof. Apply 78 through \(\tau_{2L}\). The entries of \(\mathcal H\) have upper logarithmic size \(O(L)\), by [odd:amplitude-upper]; the incidence sines and lengths have only sublinear logarithmic cost. The cofactor formula [eq:cofactor] consequently gives \[ |w_j|+|s_j'|\le e^{C_1L} \tag{345}\] through that stop on the good paths. The second bound follows from [odd:w-definition], using the amplitude upper bound again.

Let \(T\) be the first phase depth \(r_y=e^{-C_2L}\), with \(C_2\) fixed and sufficiently large. By the one-dimensional hitting estimate, after \(T\) the remaining phase time is at most \(e^{-C_2L}\), except with probability at most \(C e^{-C_2L/2}\). Suppose \(T<\tau_L\). At time \(T\), all shape depths are at least \(e^{-L}\), whereas at \(\tau_{2L}\) one of them is \(e^{-2L}\). On the good derivative paths, the quadratic time of each shape increment from \(T\) to \(\tau_{2L}\) is bounded by \(e^{2C_1L-C_2L}\). Enforce this by the derivative cut and by a physical time cut of \(e^{-C_2L}\) after \(T\). The maximal inequality then shows that, for sufficiently large \(C_2\), all these increments are \(o(e^{-L})\) with probability tending to one. This contradicts the required change of one imaginary part by at least \(e^{-L}-e^{-2L}\). Thus \(T<\tau_L\) has probability tending to zero, which proves the first bound.

Under the uniform real start, the phase-strip Green weight at a wall is proportional to \(r_y\). Therefore \[\mathbb E\int_0^\zeta \mathbf 1_{\{r_y\ge e^{-C_2L}\}}r_y^{-2}\,\,\mathrm du \le C(1+L).\] On the already established good depth event, the integral through \(\tau_L\) is part of this truncated occupation. Markov’s inequality, for example at threshold \(L^2\), gives a polynomial bound with probability tending to one. Increasing the single constant \(C\) if necessary yields [odd:phase-clock-bounds]. ◻

Proof. It suffices to improve an exponent \(0<\sigma<1\) already known to work. Choose a small fixed \(\varepsilon>0\) with \(\sigma+\varepsilon<1\), and put \[H_0=L^{\sigma+\varepsilon}.\] The old estimate, applied at level \(H_0\), gives geometry at most \(H_0^\sigma\) through \(\tau_{H_0}\), with probability tending to one. We shall show that, after this time, angles and supports change by amounts exponentially smaller than this improved initial scale.

Choose a sufficiently small fixed \(\eta>0\), and a finite family of disjoint intervals \((\xi_m,\xi_m+\eta)\) inside \((\sigma+\varepsilon/3,\sigma+\varepsilon)\), with more than \(N\) members. At every time after \(\tau_{H_0}\), at least one history is \(\ge H_0\). There is one choice \(\xi=\xi_m\) for which the corresponding power interval \((L^\xi,L^{\xi+\eta})\) contains none of the \(N\) history levels. For that choice, partition the sites as \[I=\{j:K_j\le L^\xi\},\qquad I^c=\{j:K_j\ge H_*\},\qquad H_*=L^{\xi+\eta}.\] Endpoint equalities can be assigned to their respective sets; they do not affect any estimate. The set \(I\) is proper because one history is \(\ge H_0\). By [odd:retreat], every deep site’s current log depth satisfies \(k_j\ge H_*/2\), on an event of probability tending to one.

We secure all matrix inverses before making any state-dependent selection. For each fixed \(\xi_m\), proper set \(I\), and flat subset \(F\subseteq I^c\), use 77 with, for instance, \[U=L^{\xi_m+\eta/4},\qquad V=L^{\xi_m+\eta/2}.\] The old geometry cap \(L^\sigma\) and the selected truthful history cap \(L^{\xi_m}\) both lie below \(U\) for large \(L\). The failure probability is \(O(L^{-\eta/4})\) for each candidate. The number of candidates is fixed, so outside one event of probability tending to zero, all required inverses have norm \[ \|B_{I,F}^{-1}\|\le \exp\bigl(C L^{\xi+\eta/2}\bigr) =\exp(o(H_*)). \tag{347}\] These bounds hold whenever the corresponding histories are below their caps, regardless of how often the empty gap and the flat subset change.

Set \[z_j=r_y w_j.\] Schwarz–Pick, transported from the disk by conformal maps of the two strips, gives \(r_y|s_j'|\le C\delta_j\) (Garnett 2007, I, Lemma 1.2); the strip hyperbolic densities are comparable with reciprocal boundary distance. Hence [odd:w-definition] and the old geometry bound imply \[ \log|z_j|+2a_j+k_j\le C L^\sigma, \tag{348}\] with the usual interpretation when \(z_j=0\). Fix a sufficiently small positive constant \(c_1\). Among the deep sites retain as flat exactly those for which \[|z_j|>e^{-c_1H_*};\] call this set \(F\), and delete the other deep sites. At a retained deep site, [odd:Schwarz-suppression] and \(k_j\ge H_*/2\) imply \(a_j<-c_2H_*\) for a fixed \(c_2>0\), since \(L^\sigma=o(H_*)\). Consequently its actual curvature term obeys \[|\mathcal V_j|\le e^{-c_3H_*}\] for another fixed positive constant. Every deleted coordinate already has modulus at most \(e^{-c_1H_*}\).

Apply the equation \((M_0-\mathcal V)z=0\) to the retained coordinates \(I\cup F\). Its left matrix is \(B_{I,F}\) minus the tiny curvature diagonal on \(F\); its right side is the coupling to the deleted coordinates. Entries of \(M_0\) cost at most \(e^{CL^\sigma}\). By [odd:gap-inverse], the tiny diagonal is absorbable, and the right side gives \[ \max_j|z_j|\le e^{-c_4H_*} \tag{349}\] for a fixed \(c_4>0\), uniformly over all the finite choices. The same argument includes an empty deleted set, in which case the retained homogeneous system has only the zero solution.

The important consequence is suppression at every site, including the shallow ones. From [odd:jacobi-angle], \[ 2r_y\partial_y\psi_j =\mathcal V_j z_j=(M_0z)_j. \tag{350}\] Since \(H_*\ge L^{\sigma+\varepsilon/3}\), the right side has modulus at most \(\exp(-cL^{\sigma+\varepsilon/3})\), with a fixed \(c>0\), after decreasing that constant if necessary. The phase-clock bound now makes the quadratic time of each angle increment after \(\tau_{H_0}\) exponentially small. Cut at that deterministic quadratic-time bound and apply the maximal inequality. We obtain, with probability tending to one, \[ \sup_{\tau_{H_0}\le u\le\tau_L} |\psi_j(y_u)-\psi_j(y_{\tau_{H_0}})| \le \exp(-c' L^{\sigma+\varepsilon/3}). \tag{351}\] All choices of history gaps and flat sets have disappeared from this martingale statement: they were used only to prove the uniform pointwise differential bound on the fixed good event.

The angles at \(\tau_{H_0}\) have size at most \(H_0^\sigma\). Their incidence sines have modulus at least \(e^{-H_0^\sigma}\). The absolute change of each sine is at most the angle change times \(\exp(CL^\sigma)\), using the old angle bound along the interval. By [odd:late-angle-increments], this is negligible relative to its improved initial lower bound. Thus the angles and both logarithmic sine bounds keep cost \(O(H_0^\sigma)\) throughout the late interval.

The supports obey the same improvement. By [odd:support-differentials], \[\mathcal R^{-1}\,\mathrm dH_j^0 =\mathcal R^{-1}(X_j\cdot e_{\psi_j})\,\,\mathrm d\psi_j.\] Its coefficient relative to the angle differential has modulus at most \(e^{CL^\sigma}\) by the old bound. Combining [odd:all-angle-suppression] with the phase clock shows that the martingale integral of this differential over the single late interval has exponentially small supremum. Integration by parts against the increasing \(\mathcal R\), as in the initial geometry argument, gives \[|H_j^0(y_u)-H_j^0(y_{\tau_{H_0}})| \le e^{-c''L^{\sigma+\varepsilon/3}}\mathcal R(u)\] on a further event of probability tending to one. Reconstruct positions with [odd:reconstruction], now using the improved angle and sine cost \(O(H_0^\sigma)\). The initial positions and supports have this same logarithmic cost. The coefficient of \(\mathcal R(u)\) in the last display remains exponentially small after reconstruction and is absorbed. It follows that \[\log\mathcal R(u)\le C H_0^\sigma =C L^{\sigma(\sigma+\varepsilon)} \qquad(u\le\tau_L)\] with probability tending to one.

The length upper logs have the same improved bound. For their lower logs, cut at the first failure of this upper bound. Choose \(\varepsilon\) small enough that \[\sigma(\sigma+\varepsilon)<\frac{\sigma+\sigma^2}{2}<\sigma.\] Choose a power strictly between the first two quantities as the lower threshold in 76; its failure probability tends to zero. All fixed constants and all needed lower-log slack are then absorbed by the new exponent \((\sigma+\sigma^2)/2\). This proves an improvement of the old assertion.

Starting with 78, iterate this improvement a finite number of times. The sequence generated by \(\sigma\mapsto(\sigma+\sigma^2)/2\), starting below one, decreases to zero. Given any desired positive exponent, choose the number of iterations in advance, independently of \(L\). There are then only finitely many exceptional events to discard, each with probability tending to zero. This proves [odd:all-power-geometry]; 79 supplies the simultaneous phase estimates. ◻

Proof. Use the increasing convex function \[\Phi(x)=\sqrt{h^2+x^2}+x.\] Its derivative is between \(0\) and \(2\), it is nonnegative, and \(\Phi(x)\le h+2x^+\). Moreover, \(\Phi''(x)\ge c/h\) on \([-2h,2h]\). Integrating this second derivative along the first \(\min\{|y-x|,h\}\) of the segment from \(x\) to \(y\) proves \[ \Phi(y)-\Phi(x)-\Phi'(x)(y-x) \ge \frac c h\mathbf 1_{\{|x|\le h\}}\min\{(y-x)^2,h^2\}. \tag{355}\] This includes a displacement that leaves \([-2h,2h]\), because the remaining convex remainder is nonnegative.

Let \(T\) be the first index at which \(X_m>R\), capped at \(M\). If \(X_0>R\), set \(T=0\). Sum (355) only on \(m<T\). Conditional expectation removes the linear term with the correct sign, since \(\Phi'\ge0\) and the conditional drift is nonnegative. The resulting sum is at most \(C h\mathbb E[\Phi(X_T)-\Phi(X_0)]\). Paths starting above \(R\) contribute zero. On the others the terminal positive part is at most \[R+\max_{m<M}|X_{m+1}-X_m|,\] including the crossing increment. Cauchy–Schwarz therefore bounds the pre-\(T\) cost by \(C h(h+R+\sqrt V)\).

Write \[b_m=\mathbb E[X_{m+1}-X_m\mid\mathcal F_m]\ge0,\qquad Z_n=\sum_{m<n}\mathbf 1_{\{T\le m\}} (X_{m+1}-X_m-b_m).\] This is a square-integrable martingale starting at zero, with terminal second moment at most \(V\). If a chain which starts above \(R\), or crosses \(R\), subsequently returns to \([-h,h]\), then \(\sup_n|Z_n|\ge R-h\); accumulated drift cannot assist this downward move. The martingale maximal inequality bounds the probability of any such return by \(C\min\{1,V/(R-h)^2\}\). After \(T\), the entire cost is at most \(Mh^2\) on that event. Adding the two estimates proves (354). In particular, no conditional estimate given the crossing event has been used. ◻

Proof. Choose a small fixed \(c>0\), with \(c<1/4\), and maximize \(\mu(l)e^{-c|l-j|}\) over \(l\ge k_0\). The maximum exists by (352). A maximizer \(k\) satisfies \[|k-j|=O(\log K),\qquad B:=\mu(k)\ge K^\alpha, \qquad \mu(l)\le B e^{c|l-k|}.\] For a uniform random center \(x\), use the normalized profiles \[ F_{K,x}(Z)=\frac{d_k}{B}f(x+\mathrm iH+d_k Z),\qquad Q_{K,x}(Z)=\frac{d_k^2}{B^2}q(x+\mathrm iH+d_k Z). \tag{357}\] Their domains exhaust the lower half-plane. At a depth in the fixed upper collar, \(0<D\le e^{k-k_0}\), the moment envelope gives \[ \mathbb E\bigl(|F_{K,x}(X-\mathrm iD)|^2+|Q_{K,x}(X-\mathrm iD)|\bigr) \le C D^{-2}e^{2c|\log D|}. \tag{358}\] The bound is uniform in the horizontal coordinate \(X\). Submean estimates on fixed disks give tightness in the topology of locally uniform convergence. More explicitly, take a countable compact exhaustion, view the restrictions in the product of the corresponding continuous-function spaces, and put a zero placeholder on compact sets not yet contained in a domain. Tightness and diagonal extraction give consistent holomorphic limits. Uniformity of the original center makes each limiting law stationary under all real translations.

The scaled equation is \[F_{K,x}''=B^2Q_{K,x}F_{K,x}.\] Consequently each limit \((F,Q)\) is pure: \(FQ=0\), so at least one of the two holomorphic functions vanishes identically. The limit law has no atom at \((0,0)\). To check precisely the extra hypothesis needed for 52, choose \(1/2<\eta<1-c\). In a fixed enlarged cone, subdivide the coarser depths \(1\le D\le\delta_b/d_k\) into dyadic bands. The choice of \(\delta_b\) in 8 keeps even the fixed tile enlargements within \(d_kD\le e^{-k_0}\), the range of (358). Each band contains only a bounded number of tiles of size comparable to its depth. By (358) and submean, the second moments of \[D^\eta\bigl(|F_{K,x}|+\sqrt{|Q_{K,x}|}\bigr)\] on these tiles have a summable bound \(C D^{-2(1-c-\eta)}\). Thus the required cone bound has a tight random constant. Positive mass at the trivial pair would allow selection of centers converging to that pair while this constant stayed bounded: first discard a set of probability smaller than half the alleged atom, then use successively smaller compact neighborhoods of zero. The moment envelope has coarser exponent \(c<1/2\) and finer exponent \(c>0\), so this selection contradicts 52.

There is one more degeneracy to exclude. With probability one, a nonzero limiting \(Q\) is not real on the entire line \(\Im Z=-1\). Otherwise condition on the event that it is real there and nonzero; this event is translation invariant. Schwarz reflection across that line extends \(Q\) to an entire function. For each fixed \(D\ge1\), (358) holds for all sufficiently large \(k\), so Fatou’s lemma gives \[\mathbb E|Q(X-\mathrm iD)|\le C D^{-2(1-c)}.\] Conditioning increases the constant only by the reciprocal probability of the event. Reflection supplies the same bound in the reflected half-plane. Every point is represented either at depth at least one or by its reflection at such a depth; hence the conditional mean modulus is bounded on the entire plane. Cauchy’s derivative estimate on circles of radius \(R\), followed by expectation, gives \(\mathbb E|Q'(z)|\le C/R\). Letting \(R\to\infty\) makes \(Q\) constant almost surely; the decay at large depth makes that constant zero. This contradicts the event on which we conditioned.

We now examine any deterministic convergent sequence of profiles with one of the allowed limits. If \(F\not\equiv0\), then, away from its discrete zeros on a fixed row segment, \[\log|f(x+\mathrm iH+d_kZ)|=k+\log B+\log|F_{K,x}(Z)|.\] The right side exceeds \(K^{\alpha/2}\) uniformly on compact subsets away from those zeros. Arbitrarily small neighborhoods of the zeros therefore contain all the exceptional points in the limit.

Suppose instead that \(Q\not\equiv0\). Since its imaginary part on \(\Im Z=-1\) is a nonzero real-analytic function, its zeros on compact row intervals are discrete. Apart from arbitrarily small neighborhoods of these points, cover the row by finitely many short intervals with complex neighborhoods on which consistent branches \(p_K=\sqrt{Q_{K,x}}\) satisfy \[\Re p_K\ge c_1>0\] and have uniformly bounded derivatives. In the scaled variable the original function \(g(Z)=f(x+\mathrm iH+d_kZ)\) satisfies \(g''=B^2Q_{K,x}g\). A basis on each such interval has the form \[ g_\pm(u)=Q_{K,x}(u-\mathrm i)^{-1/4} \exp\left(\pm B\int_{u_0}^u p_K(v-\mathrm i)\,\,\mathrm dv\right) (1+O(B^{-1})). \tag{359}\] Here and below the error is uniform on the chosen closed interval. For completeness, the expressions without the final factor solve the same equation with an added uniformly bounded potential; their Wronskian has modulus comparable to \(B\). Variation of constants, from the left for the growing expression and from the right for the decaying expression, gives Volterra equations for the relative factors. After division by the leading expressions the nonconstant kernel contains a factor \[\exp\left(-2B\int_v^u p_K(r-\mathrm i)\,\,\mathrm dr\right),\qquad v\le u,\] whose modulus is at most \(e^{-2Bc_1(u-v)}\); the remaining kernel has size \(O(B^{-1})\). Successive approximation, or the integral Gronwall inequality, therefore gives relative error \(O(B^{-1})\). The two solutions are independent, since their modulus ratio changes by an exponential factor along the interval. This proves (359).

Write \(g=A_+g_++A_-g_-\), with coefficients allowed to depend on \(K\). If both coefficients are nonzero, the difference of the two leading logarithmic magnitudes has derivative at least \(2c_1B-O(1)\). The set on which that difference has modulus at most \(\sqrt B\) has length \(O(B^{-1/2})\). Outside it one term dominates the other by at least \(e^{\sqrt B+O(1)}\). Each individual leading logarithmic magnitude is strictly monotone with slope of absolute value at least \(c_1B-O(1)\), so it spends at most \(O((K^{\alpha/2}+1)/B)=o(1)\) length within \(K^{\alpha/2}+1\) of zero. The same conclusion holds when one coefficient vanishes. Thus the fraction violating (356) on every fixed scaled interval tends to zero.

The countable product of continuous-function spaces used above is Polish. By the Skorokhod representation theorem (Pollard 2002, chap. 10, Theorem 4, p. 239), any weakly convergent subsequence of profile laws admits an almost-sure coupling. The deterministic criterion therefore applies almost surely. The local bad fraction is bounded by one, so its expectation tends to zero. Translation averaging identifies this expectation with the full-period bad fraction on the selected row. Every subsequence has the same conclusion, proving the lemma. ◻

Proof. Scale by \(d/8\), writing \[F(\zeta)=f(z_0+(d/8)\zeta),\qquad Q(\zeta)=(d/8)^2q(z_0+(d/8)\zeta).\] These functions are holomorphic on \(|\zeta|<4\), and \(|Q|^{1/2}\le H_b/4\). The nonvanishing vector \(Y=(F,F'/H_b)\) satisfies \[Y'=\begin{pmatrix}0&H_b\\ Q/H_b&0\end{pmatrix}Y.\] The matrix norm is at most \(CH_b\). Gronwall along a straight segment, used in both directions, bounds the change of \(\log\|Y\|\) by \(CH_b\) on fixed interior disks. Since physical depth there is comparable to \(d\), replacing \(\log\|Y\|\) by \(\mathcal A\) changes the oscillation by at most \(C\log(2+H_b)\). In particular, \[ |\mathcal A(z)-\mathcal A(z_0)|\le CH_b \qquad (|z-z_0|\le3d/8). \tag{361}\]

Set \(A_0=\mathcal A(z_0)\) and \(G=e^{-A_0}F\). Cauchy’s estimate on a sufficiently small fixed circle shows that there is a point \(|\zeta_*|\le1/16\) with \(\log|G(\zeta_*)|\ge-C\): indeed a fixed multiple of \(\sup_{|\zeta|\le1/16}|F(\zeta)|\) bounds \(|F(0)|+8|F'(0)|=e^{A_0}\). On \(|\zeta|\le3\), (361) gives \(\log|G|\le CH_b\). Choose \(M= C_0H_b\) large enough and set \(U=M-\log|G|\ge0\). Then \(U(\zeta_*)\le CH_b\).

Jensen’s formula on a circle centered at \(\zeta_*\), of radius in \([2,9/4]\) avoiding boundary zeros, bounds by \(CH_b\) the number of zeros in \(B(\zeta_*,7/4)\), with multiplicities. Choose a further radius in \([3/2,7/4]\) avoiding boundary zeros. The boundary mean of \(U\) is at most \(U(\zeta_*)\), by submean for \(\log|G|\). Poisson–Jensen expresses \(U\) inside this latter circle as the harmonic extension of that nonnegative boundary data plus the positive disk Green functions of the zeros. On \(|\zeta|\le1\), the harmonic term is bounded by \(CH_b\). Each Green function is bounded there by \[C+\log^-|\zeta-\zeta_j|,\] which has uniformly bounded \(L^2\) norm on the unit disk and on its horizontal diameter. For the segment assertion use \(\log^-|t+\mathrm ib-\zeta_j|\le\log^-|t-\Re\zeta_j|\) and the integrability of \(|\log|t||^2\). There are at most \(CH_b\) zeros, so the triangle inequality bounds the required \(L^2\) norms of \(U\) by \(CH_b\). Finally, \[\mathcal A-a=(\mathcal A-A_0-M)+U,\] and (361) bounds the other summand. Fixed smaller disks and comparable segments follow by the same fixed ratios, or by a finite covering. ◻

Proof. Start on the upper row of depth \(d=e^{-v}\) and stop on first hitting either row of depth \(d'=de^{-s}\). Let \(U\) be the unwrapped horizontal displacement. The distance to the upper target line is \(d-d'\asymp d\). Ignoring the lower target line gives half-plane exit with Cauchy density \[\frac{d-d'}{\pi((d-d')^2+u^2)}.\] Consequently \[ \mathbb P(\text{same side},\ |U|>Rd)\le C/R\quad(R\ge1), \qquad \mathbb P(\text{opposite side})\le Cd. \tag{364}\] The second estimate is also the elementary linear hitting probability between two horizontal lines. Conditional on displacement and on both sides, the starting horizontal point is still uniform modulo \(2\pi\); the terminal point is its translate, hence uniform as well. This follows directly by translating the Brownian path and its start.

For a same-side displacement \(u\) with \(R=|u|/d\le e^{K/20}\), join the endpoints by a prescribed path going first to depth \(D=d+|u|\), then horizontally by \(u\), and then to depth \(d'\). Subdivide the vertical pieces in fixed small relative steps. The horizontal piece requires only a bounded number of depth-sized tiles. The whole chain has at most \(C(1+\log(1+R))\) tiles. All their fixed enlargements lie in the low-moment band for large \(K\). At fixed \(u\), their centers are fixed translates of the uniform starting point. By 83, (362), and Minkowski, \[\|\mathcal A(\text{end})-\mathcal A(\text{start})\|_{L^2(dx/(2\pi))} \le CK^\alpha(1+\log(1+R)).\] The square of the last factor has bounded expectation by (364).

For all remaining transitions, compare amplitudes by a chain from each endpoint to a fixed real collar, connecting there modulo period. The global bound (352) makes the sum of the tile costs at most \(C(1+K)^3\): there are \(O(K)\) levels and the cost at level \(j\) is \(O((1+j)^2)\). Thus the amplitude difference is pointwise polynomially bounded even when an endpoint is near a zero of \(f\). The remaining transitions have probability at most \(C(e^{-K/20}+e^{-v})\), by (364). They contribute \(o(1)\) to its second moment. We have proved \[\mathbb E|\mathcal A(\text{end})-\mathcal A(\text{start})|^2 \le CK^{2\alpha}.\] Finally subtract \(\mathcal A-a\) at both endpoints. Their uniform horizontal marginal laws and (362) bound each squared error by \(CK^{2\alpha}\). This proves (363). Only the amplitude difference was bounded pointwise on rare transitions; no pointwise bound for the logarithmic singularities is needed. ◻

Proof. Choose the aperture and all fixed enlargements large enough for 38, with the polynomial exponent in (352). That proposition supplies fixed constants \(L_0,C_1,C_2,c_*>0\). At every target \((x,k)\), there is a witness \(z\) with \[ |\Re z-x|\le L_0\delta(z),\qquad k-C_1\log k\le-\log\delta(z)\le k+C_2, \qquad \delta(z)(|f(z)|+\sqrt{|q(z)|})\ge c_*. \tag{367}\] Increasing the fixed margins allows all disks and paths below to stay inside this enlarged cone window.

For a uniformly chosen target \((x,k)\) in the stated central band, (362) and Markov’s inequality show, with probability tending to one, that \[ (\mathcal A-a)(x+\mathrm i(H-e^{-k}))\le K^{2\alpha}, \qquad \delta\sqrt{|q|}\le K^{2\alpha} \tag{368}\] throughout every required enlarged conic tile. For the second assertion, cover the window by a bounded number of tiles in each unit log-depth band and use the union bound over \(O(\log K)\) bands. The failure probability is \(O(K^{-2\alpha}\log K)=o(1)\).

Call such a target typical. If it is also moderate, meaning \(|a|\le h_0/4\), then 83 along tile chains of length \(O(1+\log K)\) gives \[ |\mathcal A|\le h_0/3 \tag{369}\] on the needed cone cells, for all large \(K\). Indeed the initial error and total variation are \(O(K^{2\alpha}(1+\log K))=o(h_0)\). On this entire window, \(\delta|f|\le\exp(-1.1K+h_0/3)\) for large \(K\), so the first field cannot provide (367). At a witness point \(z_1\) we therefore have \[ \delta_1\sqrt{|q(z_1)|}\ge c_*/2, \qquad \delta_1=\delta(z_1). \tag{370}\]

Let \(\mathcal D_1\) be a disk centered at \(z_1\) of a sufficiently small fixed radius times \(\delta_1\). There is a fixed \(c_2>0\) such that, for every real number \(z\), \[ \frac1{|\mathcal D_1|}\int_{\mathcal D_1} |a-z|\,\,\mathrm d\mathrm{area}\ge c_2. \tag{371}\] Here \(|\mathcal D_1|\) denotes its area. To prove uniformity, suppose the assertion failed along a sequence. Scale the disks to a fixed disk and multiply \(f\) by \(e^{-z}\). The resulting holomorphic functions \(g_n\) satisfy \(\log|g_n|\to0\) in \(L^1\). Submean bounds \(|g_n|\) uniformly on compact subdisks, so a subsequence converges holomorphically. The limit is not zero, since uniform convergence to zero on a subdisk would contradict the \(L^1\) convergence of the logarithms. Off its zeros its modulus is one, again by that convergence. Thus it is a nonzero constant. At the center, \(g_n''/g_n\to0\), contradicting (370) in scaled coordinates. This proves (371) without an upper bound for the witness curvature independent of \(K\).

For \(|z|\le h_0/2\), the lower bound persists after restricting to \(|a|\le h_0\) and truncating \(|a-z|\) at \(h_0\). Indeed, (369) and 83 give \[\frac1{|\mathcal D_1|}\int_{\mathcal D_1}(\mathcal A-a)^2 \,\,\mathrm d\mathrm{area}\le CK^{4\alpha}.\] On \(|a|>h_0\), the upper bound \(a\le\mathcal A\le h_0/3\) forces \(a<-h_0\), so \(\mathcal A-a\ge2h_0/3\). The discarded \(L^1\) mass of \(|a-z|\) is at most \(CK^{4\alpha}/h_0=o(1)\), by Cauchy–Schwarz or the elementary tail estimate \(\mathbb E[e\mathbf 1_{e>t}]\le\mathbb Ee^2/t\). On the retained set, \(|a-z|\le3h_0/2\), and truncation at \(h_0\) retains at least two thirds of it. Cauchy–Schwarz now yields \[ \frac1{|\mathcal D_1|}\int_{\mathcal D_1}\mathbf 1_{\{|a|\le h_0\}} \min\{|a-z|^2,h_0^2\}\,\,\mathrm d\mathrm{area} \ge c_3>0\qquad (|z|\le h_0/2). \tag{372}\]

Take a second disk \(\mathcal D_2\), with the same horizontal center, centered at depth \(e^{-3}\delta_1\), and with radius a small fixed multiple of that depth. It lies in the same enlarged cone window. By (369) and the disk deficit estimate, all but \(o(1)\) of its normalized area has \(|a|\le h_0/2\). For every starting point of \(\mathcal D_1\), the transitions to the upper row at an additional log depth \(s\in[2,4]\), integrated in \(s\), dominate a fixed positive multiple of normalized area on \(\mathcal D_2\). Here the radii were chosen small enough that all height increments lie in \([2,4]\) with room. The upper-exit strip Poisson kernel has density at least \(c/\delta_1\) for these displacements, since both horizontal displacement and vertical distance to the target row are \(O(\delta_1)\), and the latter is comparable to \(\delta_1\). Also normalized area at depth comparable to \(\delta_1\) is comparable to \(\,\mathrm dx\,\,\mathrm dv/\delta_1\), with only fixed factors for the second disk.

For a starting point \(x'+\mathrm i(H-e^{-v})\), let \[C(x',v)=\int_2^4\mathbb E_{x',v}\left[ \mathbf 1_{\{|a(x',v)|\le h_0\}} \min\{|a(v+s)-a(x',v)|^2,h_0^2\}\right]\,\,\mathrm ds,\] where the conditional expectation denotes ordinary shape Brownian exit to the two next rows, and \(a(x',v)\) abbreviates the starting value. Integrate the kernel lower bound over \(\mathcal D_1\), then over the terminal points of \(\mathcal D_2\) with \(|a|\le h_0/2\). Formula (372), applied with each such terminal value as \(z\), shows that every typical moderate target forces \[ 1\le C\int_{\mathcal W(x,k)}C(x',v) \frac{\,\mathrm dx'\,\,\mathrm dv}{e^{-v}}, \tag{373}\] where \(\mathcal W(x,k)\) is the full enlarged cone window. Replacing the witness disk by this window is legitimate because the integrand is nonnegative; it also avoids any measurable-choice issue for the witness.

Integrate (373) over all typical moderate targets. For a fixed source \((x',v)\), the horizontal measure of target centers whose window contains it is \(O(e^{-v})\); this cancels the factor \(e^v\). Their allowed \(k\)-range has length \(O(\log K)\). Fubini therefore bounds their unnormalized measure by \[C\log K\int_{1.1K}^{1.9K}\int_0^{2\pi}C(x',v)\,\,\mathrm dx'\,\,\mathrm dv.\] Uniform horizontal law and the upper-side probability \(1/2\) bound the last integral by a fixed multiple of (366). The result is \(o(K)\). Atypical targets already have fraction \(o(1)\), proving the lemma. ◻

Proof. If \(\mu(j)\ge K^\alpha\) for some \(j\in[0.9K,2.1K]\), use 82. Otherwise (360) holds, and 85 provides a deterministic row in \([1.2K,1.8K]\) with exceptional fraction tending to zero for the stronger threshold \(h_0/4\). Since \(K^{\alpha/2}=o(h_0)\), it also gives (374). Real symmetry gives the lower-row conclusion from the upper one. The sequential proofs imply the stated conclusion as \(K\to\infty\), without restricting to either alternative along a fixed subsequence. ◻

Proof. Write \(d=e^{-k}\). By (352), \(|q|\le C d^{-2}K^4\) throughout a fixed depth-sized neighborhood of either tested row. Zeros of \(f\) in this neighborhood are separated by at least \(c dK^{-2}\). Indeed, scale around a zero by \(r=c dK^{-2}\), choosing \(c\) small. The scaled potential has arbitrarily small supremum on a fixed disk, and the solution normalized to have derivative one and value zero at the center satisfies \[F(\zeta)=\zeta+ \int_0^\zeta(\zeta-u)Q(u)F(u)\,\,\mathrm du.\] The integral equation gives \(|F(\zeta)-\zeta|\le C\|Q\|_\infty |\zeta|^3\) on a smaller fixed disk. Thus the zero is the only one there. This also reproves simplicity in this setting.

Packing disks of radius comparable to \(dK^{-2}\) in a periodic band of width \(O(d)\) bounds its number of zeros by \(C d^{-1}K^4\). Delete from the tested row the shadows of disks of radius \[r_{\mathrm{exc}}=d e^{-h_1/10}\] about all these zeros. Their total relative length is at most \(CK^4e^{-h_1/10}=o(1)\). Zeros outside this band are too far away to affect the following smaller disks.

At a remaining point, and throughout its disk of radius \(r_{\mathrm{rob}}\), \[ \left|\frac{f'}f\right|\le C/r_{\mathrm{exc}}. \tag{376}\] Here is the local zero test proving this bound. If \(\lambda=f'(z)/f(z)\) has modulus much larger than \(r_{\mathrm{exc}}^{-1}\), rescale by \(1/\lambda\) and divide by \(f(z)\). On \(|u|\le2\) the resulting equation has arbitrarily small potential, because \(\sup|q|/|\lambda|^2\le CK^4e^{-h_1/5}\), and its initial data are \(F(0)=F'(0)=1\). The integral equation makes \(F\) uniformly close to \(1+u\). Rouche’s theorem gives a zero near \(u=-1\), hence within \(2/|\lambda|\) of \(z\). That contradicts the distance to the deleted zero disks. Since \(r_{\mathrm{rob}}/r_{\mathrm{exc}}=e^{-3h_1/20}\), the same argument applies throughout the robustness disk.

The disk is zero-free, so integrating (376) along segments bounds the change of \(\log|f|\) by \(Ce^{-3h_1/20}=o(1)\). The original test has threshold \(2h_1\), which proves the assertion after discarding its exceptional set. For the Brownian statement, 75 dominates the unconditional first-hit law of each site by a fixed multiple of ordinary shape Brownian motion from uniform horizontal start. The latter has uniform horizontal distribution on each tested row. The bad first-hit probability is therefore \(o(1)\), including paths on which the site is stopped before reaching that row. ◻

Proof. Choose a fixed integer \(M>10N\). Choose positive exponents \(\gamma_1<\cdots<\gamma_M<1\), small enough and separated so that \(\gamma_m<\gamma_{m+1}\alpha/8\). Then the intervals \[(L^{\gamma_m\alpha/8},3L^{\gamma_m}),\qquad 1\le m\le M,\] are pairwise disjoint and lie below \(L\), for all large \(L\). Each path has at most \(N\) failed menu choices: a single terminal history belongs to at most one of the intervals. Averaging failure probabilities over the menu gives a deterministic choice with failure probability at most \(N/M<0.1\). One history equals \(L\) at \(\tau_L\), so the large class is nonempty. Because the menu is fixed and finite, all subsequent \(o(1)\) estimates can be made uniform over its choices. ◻

Proof. Choose the arbitrarily small fixed exponent in 80 using the smallest exponent in the menu. Its geometry cost is then \(o(u_0)\), uniformly through \(\tau_L\). This controls logarithms of the sines and lengths, and hence all entries of \(M_0\), at cost \(o(h_2)\). The identities (324) and (325) read \[ w_i=c\sin d_i^0\,f_i^{-2}s_i',\qquad 2f_i s_i'=(M_0w)_i. \tag{380}\] If \(a_i<0\), the first gives \(|s_i'|\le e^{o(h_2)+2a_i}w_{\max}\), which implies (378). If \(a_i\ge0\), the second gives that inequality directly.

We spell out the uniform inverse estimate needed for the second bound. Fix \(I\) and a subset \(F\subset J\). Retain the sites \(I\) with their actual curvatures, retain \(F\) with curvature zero, and delete all other sites. This is one of the generically invertible modified matrices of 72; it has at least one deletion or flattening, since \(J\ne\varnothing\). Stop at the first shallow history \(2u_0\), and at the geometry cuts just selected. The upper bound for \(a_i\) on truthful sites is \(O(u_0)\) by (352). Every entry and the determinant of this fixed matrix thus have upper logarithmic bound \(Cu_0\). At a real starting point its determinant is nonzero, with a uniform lower bound on the compact real period. Apply the stopped logarithmic-deficit estimate from 76, with lower threshold \(u_0^{3/2}\). Except with probability \(O(u_0^{-1/2})\), the inverse determinant costs at most \(e^{u_0^{3/2}}\). The adjugate costs \(e^{Cu_0}\), so \[ \|B_{I,F}^{-1}\| \le \exp(O(u_0^{3/2}))=e^{o(h_2)}. \tag{381}\] Empty determinants are one. This is also the specialization of 77 with upper threshold \(O(u_0)\) and lower threshold \(u_0^{3/2}\). Take the finite union over all fixed pairs \((I,F)\) before using any time-dependent set.