Rigidity of Smooth Billiards with a Continuous Caustic Collar

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 Equation (2.2), 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

exists, is independent of x, and satisfies

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 Equation (2.3).□

Proof. We give the composition estimates, including their width dependence. For a trial mesh in the ball Equation (2.10), set

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

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}):

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

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 Equation (2.11) gives, for 1\le i\le3,

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,

Differentiating the same equations therefore gives, for 1\le i\le3,

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 Equation (2.7) is

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')}\,\dd v' and \operatorname{sinc}z=\sin z/z. Thus Equations (2.12) and (2.13) apply to this expression and its first variations. The calculation following Equation (2.8) gives

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

Taylor’s integral remainder gives

where the evaluated functions a(X), b(X,T) have bounded C^p norms. Their first variations are bounded by

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

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

Using Equation (2.14) and factoring t^2-s^2 rewrites it as

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

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 (2.15) 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 Equation (2.5) and reversal of the two integrations give the kernel

integrated against v(z)\,\dd z. On the right half, the corresponding formula is

The moving-midpoint terms cancel because the two original integrands agree at \mu.

Parametrize the two halves, always from left to right, by

and set \tau_-=\alpha/2, \tau_+=(1+\alpha)/2. Uniformly in j,\alpha, the mesh and midpoint estimates give

where

Each remainder in Equation (2.21) 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

The constant exponential factors cancel. The reflected expression gives the right half. Since

and similarly from the right endpoint, these formulas give Equation (2.22). An integrated difference gains one factor q^{-1} also after p derivatives. Applying the composition-difference identity used in Lemma 2.3 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

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 (2.17) follows with

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 Equation (2.24) has leading value

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 Equation (2.24) thus gives

The integral equals -3. Indeed symmetry reduces it to

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 Equation (2.18). Since the leading term is independent of u, the same computation with differences proves Equation (2.19). 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 Lemma 2.2, 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 (2.14) first gives s\asymp t, and then, after factoring t^2-s^2, gives t-s=O(s^4). Consequently

Summing along any part of a circuit costs at most

All steps are therefore uniformly comparable. Their sum implies each is O(q^{-1}), so adjacent differences are O(q^{-4}). It follows that

These are exactly the real position and step bounds in a sufficiently large fixed uniqueness ball of Lemma 2.3. 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 Equation (2.25).□

Proof. Contour displacement and two integrations by parts give

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\Z and |n|\ge N,

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

Its column sum is at most

This proves Equation (2.26). 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 Theorem 2.1. Let u be the actual smooth function in Equation (2.4). For the remainder of the proof set m=2 and r_0=m+15. Choose

and choose N large enough for Equation (2.25) and Lemma 2.6, 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,

We first prove that its residual is arbitrarily small in powers of P:

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 Lemma 2.6 at width zero. The sum of these weighted coefficients is bounded by

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

Use Proposition 2.4 here with one fixed integer m>A+2. By Equation (2.29), 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

and sum the resulting power (1+|n-k|)^{A-m}. It follows that

Dividing by (1+P)^A proves the tail bound and hence Equation (2.30). 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

At step j, let

The correction has only high modes and satisfies

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

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 Equation (2.31). Although its inverse was computed at width \sigma_j, it is the same coefficient-defined derivative on the smaller strip, by the compatibility following Lemma 2.6. Therefore

Equivalently,

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 \eps_0\le P^{-2r_0}, as permitted by Equation (2.30). Increase it further so that the fixed constant in Equation (2.33) satisfies CP^{-r_0}2^{r_0+1}<1. Induction gives

Indeed the ratio of the right hand side of Equation (2.33), with this bound substituted, to the proposed bound at j+1, is at most CP^{-r_0}2^{r_0+1}2^{-j}. Moreover, Equations (2.32) and (2.34) give

For sufficiently large P this is smaller than M/4. By Equation (2.29), 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 Equation (2.33) at step j, the already known bound on \eps_j and Equation (2.32) 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 \eps_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,

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 Equation (2.4), \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 Equations (2.6) and (2.8); the final assertion is Lemma 2.2.□

Proof. By antisymmetry,

Put v_+=T_tU-U=tA_+ and v_-=T_{-t}U-U=-tA_-, where

The leading part of t^{-3}\mathcal R is

The last quotient is holomorphic, because

The remaining part, using (3.1), is

This proves regularity and the formula for E(y,0). Differentiating (3.1) 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

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 (3.8), subtract from the off-diagonal Hessian terms those in which v is evaluated at y. The remaining terms are

where

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 (3.8). This proves (3.9).□

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

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:

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

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,

Consequently

Cauchy’s estimate gives \|V\|_{Y_1}\le CQ\eps. A Neumann perturbation therefore gives the same inverse bound for A_0+V if CQ^3\eps 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

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\eps and contracts there if CQ^7\eps is small. This gives holomorphic corrections on Y_1, with \|\xi\|\le CQ^2\eps. 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 (3.16) 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 Lemma 2.2, 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

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

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

is an exact solution at t_0, and U_*-U(\,\cdot\,,t_0)=O(P\eps). 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\eps) on Y_6. Indeed E_t=O(P\eps) by Cauchy’s estimate, and replacing U by U_*, with the same assigned jet, changes the explicit differentiated formula for E by O(P\eps). Only finitely many derivatives occur, and every denominator t_0 is controlled by (3.16). More explicitly, for any assigned jet V, the raw derivative is

This formula makes both the fixed derivative count and the worst explicit denominator t^{-4} transparent.

Let

be the orbit-average projection. Equation (3.9) 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

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

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

Correcting \dot U_* by U_*'z, with z\in\ker\Pi, now makes \dot E_*=0, at cost O(P\eps).

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\eps). Taylor expansion and (3.8) give

Differentiating this Taylor identity once in the assigned parameter jets gives the second assertion in (3.13). The term (Ew)' and its jet are quadratic, because E,E_t,w,\dot w are all bounded by a polynomial times \eps. 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}\eps, and its correction costs at most CQ^{12}\eps. 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\eps^2 and its first jet by CQ^{18}\eps^2: a correction and its first y-derivative cost Q^2\eps,Q^3\eps, whereas the assigned correction jet and its first y-derivative cost Q^{12}\eps,Q^{13}\eps. 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}\eps^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 (3.20) 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

gives

Outside disks centered at the divisor zeros with radius c_1\min(|n|^{-1},\delta), elementary estimates for the sine give

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+\ii v and use |\sin(u+\ii v)|^2=\sin^2u+\sinh^2v. Outside the nonzero-root disks, |\sin(u+\ii v)| 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

After division its imaginary-parameter loss is (\nu-1/2)|\Im t|. The interpolant contribution after division has no such loss. Hence

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 Proposition 3.3. First divide f=-E. Its mean is zero by Lemma 3.2. 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 (3.13) gives

Apply Lemma 3.5 with \nu=1, j_0=0, a=10, b=3, and J=11. Call its output h. Then

The term involving the interpolation values is absorbed by the smallness condition. Define

The denominator is bounded away from zero because a^{-1} is close to four. Thus b has mean zero and is O(P\eps) 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 Lemma 3.4. For retained nonzero modes the exact first-division identity is

Its value and its first parameter derivative at t_0 are bounded by P\eps^2e^{-|n|(s-10\delta)}, by the two compatibility conditions and (3.23). If d_n(t_0)\ne0, reducedness of l/q gives

If d_n(t_0)=0, then

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\eps on Y_{10}. Its tail on Y_{13} is bounded by P\eps e^{-3N\delta}, hence by P\eps^2. Summing retained and omitted modes gives

Since \langle a^{-1}g\rangle=\langle\partial w\rangle=0, the mean choice in (3.25) gives \lambda=g_0+O(P\eps^2). Consequently

At a nonzero divisor zero for a retained mode, (\partial w)_n=0. Thus (3.26) supplies the values needed for the second division, with bound P\eps^2e^{-|n|(s-13\delta)}.

Apply Lemma 3.5 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 (3.25). Call the output v. The remaining half of the imaginary-parameter loss is canceled, giving

Both divisions, multiplication by a^{-1}, and the mean adjustment preserve real symmetry and evenness.

Because ab=h+\lambda(t) and \partial\lambda=0, (3.24) and (3.27) imply

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 (3.8) gives a quadratic linearized residual, since (Ev)'=O(P\eps^2). The normalized integral formulas in the proof of Lemma 3.2 give

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 Lemma 3.4. 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 (3.11)–(3.12), 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 (3.7); 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 Theorem 3.1. Choose a fixed sufficiently small y-width s_0. The constants in Proposition 3.3 are now fixed. Start with Lemma 3.6, choosing its order below, and let the initial parameter radius be d_0<s_0/4. At step i, use losses

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 \eps_i>0, choose

If it is zero, the construction has already finished. Enlarge one fixed integer B if necessary. Since every power of 1+\log(1/\eps_i) is bounded by a constant times \eps_i^{-1/4}, the step estimates imply

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 (3.28) shows inductively that

Indeed the ratio of the resulting right-hand side of (3.28) to the next bound in (3.30) is at most C d_0^{2B+3}2^{4B+4-(2B+3)i}, which is less than one for small d_0. Equation (3.29) then bounds the corrections by

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 (3.2). At t=0, (3.7) 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 Theorem 3.1.□

Completion of the proof of Theorem 3.1. The change from \mathcal S to D subtracted a telescoping endpoint integral and changed the boundary coordinate. Thus (3.2) is equivalent to

The momentum formula in (3.3) 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

Hence both oriented line coordinates are jointly analytic and even. Taylor expansion of the support-point formula for the momentum gives

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 (3.31) 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, (3.31) gives the coefficient of dy\wedge dt as

The coefficient is odd because the line coordinates are even. This proves (3.4) and completes the theorem.□

Proof. Intersection in (4.1) gives

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 (4.3).□

Proof. Locally choose a coordinate direction transverse to all the finitely many foliations; write it as the a-direction. Put

The A_i are nonzero and the \rho_i are distinct. Differentiate (4.4) by \partial_a^{m-r}\partial_\delta^r, 0\le r<m. The terms containing F_{i,m} have coefficient matrix

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

The chain rules

then give a full first-order system

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 (4.5) 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 (4.5) 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,

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.

The critical value and its endpoint data. Use the initial angle a and the total angular increment \delta as local parameters. This is a nonsingular change from (y,t): at t=0, the two nonzero diagonal factors are \phi'(y) and k\phi'(y). Write

The intermediate angles are stationary for

Its derivatives with respect to the two endpoint angles are -p_0,p_k. Thus all its derivatives of positive order are known from endpoint jets. This statement uses only algebraic extensions and algebraic operations on those jets. Indeed, oriented unit normals and signed distances are obtained from the unoriented projective lines by square roots near the real branch; half-angle exponentials require only further radicals. Differentiating angles and changing the two derivations then involve rational expressions in these algebraic data and their derivatives.

On a small rectangle a\in I, 0<\delta<\delta_*, analyticity of the glancing family gives

Consequently the gradients of the \psi_i have distinct directions and \partial_a\psi_i\ne0. The stationary Hessian is invertible. To see its sign and scale, subtract the telescoping integral \int_a^{a+\delta}h. The leading summands become

At equal steps their Hessian is negative definite on Dirichlet variations of the interior angles. The differentiated analytic remainders preserve invertibility for positive sufficiently small \delta, uniformly on a smaller a-interval.

Finite algebraic core. Fix a generic interior basepoint. Allow the k occurrences of the support in (4.6) to be different germs h_i. The finite core consists of the interior half-angle exponentials and the jets of every h_i through order

at its sampled angle. Impose base stationarity and equality of every positive-order critical-value derivative with the given endpoint data. Formal critical-point Taylor coefficients are rational functions of the support jets: at each order one inverts the same finite stationary Hessian and performs rational operations.

For an order m>m_0, the new support jets h_i^{(m)} enter the m-th critical-value derivative tensor with coefficients

Indeed a profile change beginning in order m moves the formal stationary point only in order m-1. Its critical-value correction beyond direct substitution has order at least 2m-2>m. The direct substitution gives (4.8). Choose tensor components with 0,\ldots,k-1 \delta-derivatives. After dividing nonzero column factors, their matrix is Vandermonde in (\partial_\delta\psi_i)/(\partial_a\psi_i). It is invertible. The gradients themselves depend only on second-order core jets, by differentiated stationarity. Thus every higher support jet is rationally expressible in the core and the endpoint data.

The denominators are products of powers of a fixed finite collection of nonzero quantities: the stationary Hessian, the coordinate normalizations, the weights \sin d_i, the chosen coordinate derivatives, and the differences of gradient ratios. Work in the open locus where they are nonzero. After elimination, the remaining requirements are polynomial equations in the finite core, with coefficients in the endpoint data field. Noetherianity reduces this possibly infinite list to finitely many equations on that open locus.

Analytic continuation of a hypothetical deformation. Suppose the actual core were not isolated. Slicing a positive-dimensional algebraic component to a curve and parametrizing a branch gives a nonconstant complex analytic arc of admissible cores, with parameter \eps. Every eliminated jet is holomorphic along this arc near zero. Move the sampled Taylor centers back to their original locations. Since the center changes are O(\eps), formal translation is well-defined in the two formal variables: each coefficient uses finitely many shift powers. The resulting support germs have the form

with at least one nonzero f_{i,p}. If all support variations vanished, formal implicit uniqueness for the nondegenerate stationary point would also keep the angle core fixed, contradicting nonconstancy of the arc. Matching all positive-order derivatives of the critical value means that its change is

where each K_j is constant in the endpoint parameters.

We next show that every coefficient f_{i,j}, initially formal, is analytic and extends to the coalescing edge. Induct on j. First variation at the j-th order gives

The term B_j depends only on lower perturbation orders and on the original stationary branch. By the induction hypothesis and Hessian inversion it is analytic throughout the positive rectangle. Apply Lemma 4.3 with the weights 2\sin d_i and the foliations \psi_i. It proves convergence, continuation, and extension to \delta=0 near a smaller interior a-interval. Notice that this is coefficientwise in j; no assertion of convergence of the full \eps-series is needed.

The coalescing stationary limit. Fix an interior value of a. Put

Subtract the unperturbed integral of h and divide the action by \delta^3. Coefficientwise in e, its stationary value tends to

Here direct Taylor expansion gives the limit of the action and of its derivatives near the equal-step point. A perturbation of order j>p carries the factor \delta^{2j/p-2} and therefore disappears. The zeroth-order scaled Hessian tends to an invertible one. Recursion for the formal stationary coefficients thus gives coefficientwise convergence of the stationary point and its value.

The coefficient of e^j contributed by the matched critical value is K_j\delta^{2j/p-3}. The limit in (4.11) depends only on x=e^p. At j=p, boundedness forces K_p=0; at j=mp, m\ge2, the exponent 2m-3 is positive. Therefore every positive x-coefficient of G vanishes, and G is constant.

Write f_i=f_{i,p}(a) and r=r(a) in the remainder of this paragraph. Differentiating the critical value gives \sum_i f_i z_i=0. If \Lambda is its Lagrange multiplier, stationarity and weighted summation give

Taking the roots with constant term z_i(0)=1/k and summing them yields

Every coefficient of positive degree in the square-root series is nonzero. Hence \sum_i f_i^m=0 for every m\ge1. Newton’s identities make all elementary symmetric functions of the f_i vanish; each f_i is zero. Varying a contradicts the first nonzero coefficient in (4.9). The actual core is therefore isolated.

From isolated values to algebraic germs. An isolated point of a complex algebraic set defined over a field K has coordinates algebraic over K. For example, pass to an algebraic closure of K. Its zero-dimensional components remain zero-dimensional after further extension of algebraically closed fields, and an isolated point must lie in such a component.

To apply this fact to functions, do not specialize at an arbitrary point. Begin over a countable field containing \Q(\ii) and the fixed numerical constants used in the coordinates. Adjoin the endpoint coordinate functions, all their derivatives, the actual local radical branches, and also the actual core functions. The resulting function field is countable. Choose the real basepoint outside the poles and the zero sets of every nonzero member of this field. This is possible: a nonzero analytic function cannot vanish on an open real rectangle, and the excluded countable union has measure zero. Evaluation is then injective on the field.

The algebraic equations used above are defined over the evaluated endpoint field, with its stated algebraic extensions. Isolation makes the evaluated core algebraic over that field. Lift the coefficients of such a polynomial relation back to functions. Injectivity of evaluation, also after adjoining the actual core, makes the lifted polynomial identity hold as a germ. The contact momentum formulas recover the offsets of the intermediate lines from the core. Eliminating the endpoint radical extensions proves the asserted algebraicity over the original unoriented endpoint differential field. Only finitely many endpoint derivatives occur in the resulting polynomial. These identities persist under continuation.

Finally, in characteristic zero an algebraic element is separable. Differentiating its minimal polynomial expresses each derivative algebraically over the differential field. This proves the assertion about jets, and translation of (y,t) proves the assertion for other index intervals.□

Proof.

The length functional and its mixed derivatives. Near a real glancing configuration choose orthogonal coordinates in which the boundary is Y=F(x), with F''\ne0, and orient the arc so that x increases along the chain. Use the endpoint abscissas

as parameters. This change of coordinates is nonsingular near glancing. Write x_i=x(P_i). On a small rectangle a\in I, 0<\delta<\delta_*,

Allow the interior occurrences of the graph to be independent functions F_i, while keeping F_0=F_k=F fixed. Consider the critical value

The roots are the positive real lengths at the original configuration and are continued with that choice.

The endpoint graph jets are determined by the point jets, since x is a regular coordinate. All mixed derivatives of \mathcal A_{\rm point} are determined as well. Indeed, if e_{\theta_0} is the first unit chord velocity, endpoint differentiation gives

The endpoint momentum identity then gives

The first edge is L(y+t/2,t). Its half-shift in y preserves \dd y\wedge\dd t, so the area coefficient is exactly the same A(t). Equation (4.15), the Jacobian of the endpoint coordinate change, and differentiation recover every derivative having at least one a- and one b-derivative.

Elimination of higher graph jets. At the original configuration subtract the telescoping graph arc lengths between successive endpoints. Taylor expansion of chord length minus arc length gives the leading summands

For completeness, if v=F'(a), r=F''(a), and g=1+v^2, the cubic coefficient of a chord over an interval of length s, after subtraction of the integral of \sqrt{1+F'^2}, is -r^2/(24g^{3/2}). This follows by expanding the square root of s^2+(vs+rs^2/2+F'''(a)s^3/6)^2 through degree three; the F'''(a)-terms cancel against the arc integral. At equal positive steps the Hessian of the sum in (4.16), on interior variations with fixed endpoints, is negative definite. Thus the stationary Hessian is invertible for small positive \delta.

Take as finite core the x_i, the jets of F_i through order m_0=\max(2,k), 0<i<k, and the nonzero length roots. Impose base stationarity, the algebraic equations for the roots, and equality of every mixed critical-value derivative with the given endpoint data, including the low-order derivatives. In order m>m_0, the new graph jets enter the critical-value derivative tensor through

where w_i is the vertical component of the incoming unit velocity minus the outgoing unit velocity. The same stationary-value differentiation as in (4.8) proves this formula: movement of the critical point contributes only in orders greater than m. The weights are nonzero because the tangent of our graph is nonvertical and the real incidence is nonzero.

Only mixed tensor components are known here, but they suffice. In (a,b)-coordinates,

Both derivatives are nonzero and their ratios are distinct. Use the components with 1,\ldots,k-1 derivatives in b and the remaining derivatives in a. Since m>k, all are mixed components. Their coefficient matrix is an invertible Vandermonde matrix after nonzero column factors have been removed. Hence each higher graph jet is a rational function of the finite core and the given mixed value derivatives. The gradients of the x_i are obtained by inverting the second-order stationary Hessian, so they too are already rational in this data.

Restrict to the finite open locus where the length roots, Hessian, weights, coordinate derivatives and Vandermonde differences are nonzero. All denominators in the recursive computation are products of these fixed quantities. Elimination and Noetherianity leave an algebraic set of possible cores defined by finitely many equations on this open locus. We prove that the actual core is isolated.

Continuation of the variation coefficients. A nonisolated core would admit a nonconstant analytic arc. Translate the moving Taylor centers as in the line proof. The resulting formal graph variations have the form

with at least one nonzero first coefficient f_{i,p}. Equality of all mixed value derivatives says precisely that the positive-order change of critical value is

initially as a formal identity. At order j, first variation yields

where the right side is determined by lower orders. There are k+1 one-variable unknowns in this web. The foliations x_1,\ldots,x_{k-1},a,b have pairwise distinct gradient directions by (4.13); their weights w_i,-1,-1 are nonzero. A generic coordinate direction is transverse to all of them, so the full Vandermonde argument of Lemma 4.3 applies. Induction on j proves that all f_{i,j},U_j,V_j are analytic on the positive rectangle and extend to the coalescing edge near an interior a-interval. For the edge assertion use (a,\delta): all the foliation coordinates, including b=a+\delta, are a+O(\delta) and have a-derivative 1+O(\delta). No bound on the web system at that edge is needed.

The scaled critical length. Fix an interior a, and abbreviate

Thus f_0=f_k=0, \sum_i z_i=1, and r\ne0. Subtract the unperturbed graph arc from the length, multiply by g^{3/2}/\delta^3, and substitute \eps=e\delta^{2/p}. The coefficientwise stationary limit is

Here are the individual terms in this limit. The cubic term is (4.16). The constant vertical first-variation coefficient of an edge is F'(a)/\sqrt g. Summed against the graph offsets it telescopes to zero because the endpoints are fixed. The next coefficient is

Summation by parts gives

which produces the displayed linear term. The second vertical derivative of ordinary edge length gives the inverse-z_i quadratic term. Higher powers of the vertical perturbation vanish in the limit. Terms of perturbation order j>p have, after the constant telescoping part has canceled, a factor \delta^{2j/p-2} in their linear contribution and vanish as well. These expansions hold with derivatives in the stationary variables near z_i=1/k. The limiting Hessian at x=0 is invertible, so formal stationary recursion justifies the stated coefficientwise limit.

For each multiple j=mp\ge2p, the separated matched value contributes

which tends to zero. The limit depends only on x=e^p. It follows that

Let \lambda be the stationary multiplier. Differentiating the functional in (4.20) gives

Multiply by z_i, sum, and compare with the derivative of the critical value in x. The weighted homogeneity of its three terms gives

Consequently

Take each square root with constant term r/(2k). Adding both roots for each edge and then all edges gives r\sum_i z_i=r. Equivalently,

Every positive power sum of the multiset

therefore vanishes. Newton’s identities make every member zero. The fixed endpoints give g_1=0, and then f_i=0 for every i. Varying a contradicts the first nonzero term of (4.18). The core is isolated.

Return to the endpoint function field. Use a countable field containing the endpoint functions, their derivatives, the derivatives of A(t), the actual length roots, and the actual core. Choose the real basepoint outside the zeros and poles of all its nonzero members. Evaluation is injective there. Isolation over the evaluated endpoint field gives polynomial algebraicity of the core values, and injectivity lifts those polynomial identities to germs, exactly as in the last part of Theorem 4.4. Eliminating the finitely many radical extensions leaves the original endpoint differential field. The intermediate graph values give all points; joining consecutive points gives every edge by rational projective operations. Differentiation of separable polynomials gives the assertion for jets. This completes the point-endpoint determination.□

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

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 Lemma 4.6; 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

and let g be a holomorphic polynomial whose boundary imaginary part is q. Thus

For real x_0, 0\le\lambda\le1, and a small complex variable w, consider

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

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

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.

Endpoint equations across the full band. Fix a nearby real t_1 with t_1/(2\pi)\notin\Q. Forward and backward translates of any real phase meet the good arc. On a small phase box choose integers m<0<n so that both L(y+mt,t) and L(y+nt,t) are supplied by the patch throughout the narrower vertical band. Compactness modulo 2\pi gives finitely many boxes covering the whole band. Only finitely many shifts occur, so these assertions hold for t in a complex neighborhood of t_1, possibly depending on t_1. Shrink this neighborhood to retain a common bottom strip on which every shifted field is the primary one.

Apply Theorem 4.4 on that bottom strip in each box. It gives nonzero polynomial equations for the coordinates of L(y,t), with coefficients meromorphic across the full band. Use squarefree equations with nonzero leading coefficients and discriminants. These remain nonzero as germs: their coefficients are continued actual endpoint jets, so a nonzero identity cannot become identically zero during continuation. The primary branch below any first wall in the band satisfies the equations, by vertical continuation from the bottom strip.

A first obstruction has nontrivial monodromy. The exceptional sets are the coefficient poles and the zeros of the leading coefficients and discriminants. They are locally analytic curves. Outside countably many parameter values, a slice meets their reduced union at isolated points where it is a smooth graph over t. Indeed, use a countable cover by analytic boxes. Vertical components give countably many constant parameter values; on nonvertical components, singularities and critical points of projection are discrete. Intersections of distinct components are also discrete. Exclude all their parameter images. Relatively compact boxes leave only finitely many exceptional points in a periodic row.

Choose a real irrational t_2 outside those exceptions. Suppose h_L(t_2)\le H_1+\eta, so its first wall is strictly inside the band. Away from exceptional points, following simple roots gives joint passage through the wall. Near an exceptional point the only exceptional curve is y=z(t). Off it the roots have finite continuation on a punctured disk times a disk, whose fundamental group is generated by a small y-loop about z(t_2). If the actual branch has trivial monodromy on that loop, it is single-valued on the product. After denominators are cleared, the leading coefficient is a unit times a power of y-z(t). A polynomial root bound therefore gives

for a finite N. Multiplication by that power and removal of a bounded singularity gives joint meromorphy. If every wall point had trivial monodromy, these extensions would glue on overlaps just below the wall. A finite cover of the periodic wall would then continue the tube to a strictly greater height on a parameter neighborhood, contrary to the definition of h_L(t_2). Some small loop must act nontrivially.

A defect affects only finitely many shifted indices. Put L_j=L(y+jt,t). At a fixed point of the row, a small loop can change L_j only if translation by jt_2 aligns its center with an exceptional point. There are finitely many such points modulo period and j\mapsto jt_2\pmod{2\pi} is injective. Only finitely many indices can be aligned. Precisely, for each prescribed finite set of indices one can choose the loop sufficiently small to fix all its unaligned members. No single loop fixing an infinite collection is asserted.

First choose the loop sufficiently small for the whole finite set of aligned indices. Further shrinking preserves its monodromy on those fields. Among the aligned indices with nontrivial monodromy, take the smallest and translate the center along the orbit so that this index is 1. Choose K so that every aligned index now has absolute value at most K. Choose b>K, b\ge2, and any further finite collection of far indices, including both signs when needed below. Shrink the loop to fix the unaligned members of that collection and the unaligned members of \{-1,0\}. Its action on aligned fields is unchanged, so L_{-1},L_0 return unchanged. It produces

while fixing L_{-1},L_0,L_b and all selected far fields.

Descent to glancing. Below the wall at the real parameter t_2, all selected primary fields are single-valued jointly near every strict subheight. Descend there and then move along real phase and parameter to the original collar. A connected open region around this connection carries all selected primary data meromorphically, so the outward path is available. The small loop avoids every shifted exceptional point relevant to the finite selection.

On return M_1 is algebraic over L_0,L_b and their jets, by transporting Theorem 4.4 around the loop. In the connected lower region these endpoints are back on their single-valued primary branches. Continue the alternative root down along paths avoiding the proper analytic exceptional sets of its squarefree equation. Complex detours, including an initial move off such a set within the starting germ, are allowed. They do not change the selected primary data.

Near a generic point (y_0,0), only t=0 can be an exceptional divisor for that polynomial. Other analytic curves meet this divisor discretely unless they contain it. On a sufficiently small bidisk, a power cover t=\tau^d kills the finite monodromy. The polynomial root bound makes the root meromorphic in (y,\tau). Thus M_1 has a joint meromorphic Puiseux germ at glancing. Its inequality from L_1 persists, since the primary L_1 is single-valued in the lower region.

The alternate germ is even. Put Z=L_0\cap M_1. Continued incidence, rank and reflection identities show that Z has rank at most one and that reflection in its tangent relates the two lines. It differs as a germ from the unchanged preceding intersection L_{-1}\cap L_0: continuing back to the original real table would otherwise identify the preceding and following intersections.

On the Puiseux cover there is a projective limit

holomorphic at generic y. It is nonconstant, because it lies on L(y,0) and the tangents of the original strictly convex arc have no common projective point. At a regular point of Z^0, rank one implies that the nearby image of Z lies on the same fixed analytic arc traced by Z^0. This arc cannot lie at infinity or have identically isotropic tangent: in either case it would be a fixed projective line, and continuation would force the original real table arc into that line. Indeed, the line equation would hold on this rank-one impact branch and continue back along the reverse path to the original impact L_0\cap L_1, contradicting strict convexity.

If L(y,0) were generically tangent to the fixed arc at Z^0(y), differentiated incidence and tangency would identify Z^0 with the envelope of those lines, namely \Gamma(y). The fixed arc would be the primary boundary germ. Near its simple quadratic tangency, L_0(y,t) has exactly the two primary nearby intersections with that germ, distinct for generic small t\ne0. The preceding intersection has been excluded; reflection at the following one gives L_1, contrary to (4.25).

The fixed arc is therefore transverse to L(y,0) at a generic point. Intersect it with the even line L_0(y,t) by the ordinary holomorphic implicit function theorem, then reflect in its nonisotropic tangent. Both operations are holomorphic and uniquely determined near that point. They produce M_1, and prove

A continuation operation reflecting indices. The incoming leg can now end at a regular (y_0,0) itself; take the outgoing primary leg from the same point. The resulting closed continuation path is denoted A. Let I denote pullback by t\mapsto-t. Perform, in order,

The distinguished field follows the succession

This operation fixes L_0 and every selected far field, including its jets, provided both signs of each far index were selected. Indeed, evenness of L makes the middle pullback send L_j to L_{-j}; the reversed path fixes that selected primary germ, and the last pullback restores L_j. The total base pullback is the identity, so the derivative chain rules cancel. The operation is depicted in Figure 2.

Paths and generic basepoints can be chosen anew for each finite selection. The identities before and after are identities of actual primary germs near real glancing, and hence transport between those basepoints.

Apply (4.27) to algebraicity of L_1 over the jets of L_0,L_b. It proves algebraicity of L_{-1} over the same jets. Translate this assertion. From the endpoints of any interval [B,B+b], the line immediately to its left is now algebraic, as are all interior lines by Theorem 4.4. Differentiation preserves algebraicity over a characteristic-zero differential field. Shift the interval one place left and repeat; transitivity proves algebraicity of every line to the left over the original endpoint jets.

Choose B>K+2. There is one fixed nonzero polynomial over jets of L_B,L_{B+b} for a generic direction coordinate of L_1. For each n\ge0, first shear an identity by (y,t)\mapsto(y+2nt,t), apply (4.27) with far indices including \pm(B+2n),\pm(B+b+2n), and shear back. It sends

and fixes the original coefficient jets: the shear sends L_{1-2n} to L_1 and the coefficient fields to the selected far indices, while the inverse shear returns those fields exactly. For S_qF(y,t)=F(y+qt,t), S_q(\partial_tF)=\partial_t(S_qF)-q\partial_y(S_qF); thus the shear chain rules cancel with those of its inverse. Inductively this same polynomial has the direction coordinates of

as roots in the field of primary germs. They are distinct: since \Theta(y,t)=\phi(y)+O(t^2), the first t-derivative of a generic slope coordinate of L_j is j\phi'(y) times a nonzero chart factor. A nonzero polynomial over a field cannot have infinitely many distinct roots. This contradiction excludes a first wall at or below H_1+\eta for generic real irrational parameters in the neighborhood under consideration.

Transfer to the original real parameter. For every nearby irrational t_1, the preceding construction supplies a parameter neighborhood on which

for real t outside a countable exceptional set. The increment \eta depends only on the initial patch. A countable subcover covers all nearby irrational real parameters. Discarding their exceptional sets and the countably many rational shifts yields the displayed bound almost everywhere on an interval about t_0.

Put M=H_1+\eta. On an upper half disk with diameter in that interval, approximate the nonnegative lower semicontinuous boundary values of h_L from below by continuous functions supported away from the diameter endpoints. Superharmonic comparison and the almost-everywhere bound give

where \omega_+ is the harmonic measure of the diameter. The lower half disk gives h_L\ge M\omega_-. Both harmonic measures tend to one near the interior diameter point t_0. The small-circle supermean inequality centered at t_0, followed by shrinking the circle, gives h_L(t_0)\ge M. This contradicts h_L(t_0)=H_1.

The same conclusion for impact patches. Suppose now that X has a joint patch through its finite first wall at a real parameter. At real parameters (4.3) gives h_X=h_L. For nearby irrational shifts, forward and backward translates

of the patch supply enclosing impacts for every edge. Choose a finite covering by phase boxes, enclosing indices with at least two steps between them, and a common narrower vertical band exactly as above. Theorem 4.5, with the analytic coefficient A(t), supplies full-band joint polynomial equations for the edge L(y,t), attached to the primary branch on a common bottom strip.

These equations are exactly the input to the first part of the argument. Generic slices, root bounds and monodromy turn a first obstruction into a finite defect. Its exclusion still uses the line-endpoint identities of Theorem 4.4 and the evenness operation (4.27); no additional point identity is needed in that exclusion. Thus h_X(t)=h_L(t)\ge H_1+\eta almost everywhere on a real interval. Apply the same superharmonic transfer to h_X, using Proposition 4.8, to contradict its value at the original parameter. Real symmetry supplies the lower-wall assertion.□

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

These equations prove (4.28) 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

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 (4.28) 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.

A regular starting segment. Choose a row slightly below H, sufficiently close that all short slanted segments with normalized vertical component of absolute value at least 1/3, ending near s_{\rm edge}, remain inside the regular patch. Their starting points lie in a compact horizontal segment. Choose the exact row height to avoid joint poles and indeterminacies of L,X on t=0 over this segment. This is possible because in affine charts finite along the segment there is no polar component equal to the zero slice; joint exceptions intersect it discretely. Compactness then gives a neighborhood of the segment on which L,X are analytic for small complex t. Evenness gives the line angle expansion

uniformly there.

The rank-one identity identifies the nearby impact image with the fixed regular curve arc traced on t=0. The reflection identity is its ordinary local reflection law. Thus the successive line angles, expressed in the normalized coordinate

satisfy the stationary recurrence of the local action D. The support and all root branches are chosen locally; the line angles are close enough that their mean is the normal angle at the shared impact.

Cubic accuracy of the local recurrence. The same local Taylor calculation used for the normalized action, now using constancy of r(\phi)(\phi')^3, gives

There is no cubic term. Antisymmetry of D turns stationarity at a node into equality of this expression for the forward and backward increments. Its analytic square root is s\sqrt{\kappa+O(s^2)}. Choosing the opposite root from the repeated previous node gives the nonrepeated successor analytically and uniformly:

The offsets of the corresponding lines are also analytic, being the endpoint momenta of the local generating function \mathcal S.

Propagation in steep directions. Assume |\Im t|\ge |t|/3. Replace t by -t if necessary, using evenness, so that the step is upward. For a target y near the edge choose N with y-Nt in the starting neighborhood. There is a uniform bound N=O(1/|t|). Initialize the recurrence with

using (4.29). If the step sizes stay at most 2|t|, each recurrence error is O(|t|^3). Summing the errors once gives

and summing a second time gives

These estimates close the assumed step bound for small |t|. Choose the geometric paths with fixed slack in the regular patch before shrinking |t|; then every iterate remains in that patch.

For locally constant N, this construction is holomorphic in y,t. Choices of different N’s agree: take their starting points in a thin slab of known analytic data, where the exact recurrence relates all intervening indices, then apply uniqueness of the nonrepeated successor. It follows that the resulting germs glue near the target and along a vertical corridor down to the starting slab, uniformly for all steep directions. They are attached to the primary data on a fixed lower part of that corridor. At any chosen strict subheight below H, make |t| smaller if needed so that uniqueness identifies them with the already known primary branch. The line construction also gives X by the neighboring intersections. All choices have been made in order: first the regular patch and geometric slack, then the vertical travel distance, and finally the allowed size of |t|.

Quadratic disks fill the shallow directions. Choose \eta>0 small compared with both the corridor size and the depth of the starting row. Next choose a sufficiently small parameter radius r_0>0. Consider the analytic disks

where w is a small complex transverse parameter. At |\zeta|=1, write \zeta=e^{\ii\theta}. On the near-real arcs |\sin\theta|<1/2,

These arcs therefore have joint primary data for small r_0. On the remaining arcs, |\Im t|/|t|\ge1/2, so the steep continuation just constructed applies with a strict margin over 1/3. The continued germs glue to the primary ones on the overlaps by their attachment below H. Compactness supplies a common annular collar in \zeta and a common small disk in w on which the pullbacks are meromorphic, throughout the indicated interval of \sigma.

At \sigma=H-2\eta, the full disks have height at most H-\eta for w=0. Hence they, and a small neighborhood of them, already lie in the primary joint domain when r_0 is sufficiently small. Increase \sigma in small steps. The displacement of one center is absorbed by the transverse w-disk, so the full disks from the previous step give a nonempty open seed at the next step. Apply Lemma 4.6 and Corollary 4.7 at each step. Uniqueness on the collars and on the seed overlaps attaches the resulting extensions to the lower branch throughout the ascent.

At \sigma=H, the disk center \zeta=0 gives a joint meromorphic patch through (y,t)=(x_*+\ii H,0); varying w and \zeta are local coordinates there because r_0\ne0. This contradicts Theorem 4.9, applied to L at the real parameter 0. The supposed shape patch cannot exist. Real symmetry excludes a lower-edge patch as well.□

Proof. Write z=s-p. By (5.2), f=n/(\ii z)+O(1). Integration of (5.1) therefore gives

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

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 (5.6) 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

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:

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\ii\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

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 (5.6). 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

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 (5.7) 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 \ord_p v_j''\geq-1-2j, which proves (5.7) 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\Z and leading coefficient c_P\ne0. Composition with (5.6) and (5.7) show that its coefficient of t^l has order at least b-l. Consequently

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 \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 [Hör90, 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

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,

locally uniformly, with derivatives, on |w|>R_0; the limit is rational. Applying this to the two coordinates in (5.5) 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 (5.10) 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 (5.4), 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

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

After cancellation of the common powers of R(w), (5.4) becomes

We use the Riemann–Hurwitz formula: a rational map of degree s has total ramification 2s-2; see [For81].

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 (5.12) 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

has a pole of order \alpha at infinity. The function on the right of (5.12), 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

At the preceding point w_0-1, which is not a pole of \rho, \rho must take a nontrivial \alpha-th root with multiplicity

The preceding points of different runs are distinct. Their ramification contributions are their displayed multiplicities minus one. If m\geq2, (5.13) 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 (5.13), 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 (5.12) reads

Equivalently, 2/(\rho-1) increases by one under translation. A rational one-periodic function is constant, so

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 (5.10), \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 (5.3), \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 (5.11); 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 (5.11) 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 Corollary 4.10. 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 Lemma 5.2. The degree of a on \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

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

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,

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

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

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

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 Corollary 4.10. 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

It consequently induces the symmetry

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 (5.19) would give an edge patch. Thus only \omega=\pm1, b\in\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 (5.19). 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 (5.2). 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 (5.19) and makes a real translation an integer multiple of 2\pi. We conclude that

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 \dd\mathfrak s is already single-valued, because all its monodromy is translational. It extends meromorphically by (5.18). 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 (5.18) for the other component.

Write \Gamma(s)=G(e^{\ii s}), where G is meromorphic on \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\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 (5.2) then make a rational in w=e^{\ii s}. A rational function with no zeros or poles on \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 \dd\mathfrak s=\dd v/(\ii v) 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 \dd\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 \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 \dd\mathfrak s. Normalize by the original germ. This gives the required single-valued lift. The composition and all symmetries in (5.17), 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 (5.1) 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 Theorem 5.4, 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

For its signed offset p_{\rm chord}, differentiation of p_{\rm chord}=n_\theta\cdot\Gamma(\mathfrak s) gives

The final identity is the continued area identity (3.4); 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 \dd\mathfrak s\wedge\dd d =z^{m+1}\dd\mathfrak s\wedge\dd b. The left side of (5.21) 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'=\ii fa, 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 \ii c 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 (5.8) put g=f^{-2}, g_\pm=g e^{\pm\ii\phi}. Midpoint integration through the fourth derivative gives

For clarity, define

Multiplying the two midpoint integral expansions and taking the glancing square root gives

Differentiating g_\pm explicitly yields

The two (f')^2 contributions cancel, since 15/1920=9/(2\cdot24^2). Therefore

At the middle impact s=\mathfrak s(y,t), the previous and next shape increments are

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

Substitute (5.26), integrate once, and use periodicity of V to determine the integration constant. This proves (5.24). The zero mean of V follows from (5.22).□

Proof. The integral equation

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

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 (6.5) 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,

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 Figure 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

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

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

Thus Lemma 6.1 applies uniformly. If a(s,J;t)=p(s,J;t)-J^3/24, then

for the finitely many derivatives used below. The s-derivative bounds follow by Cauchy on the same thickenings.

Evaluate (6.6) 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

For \beta>1, comparison with an integral on the depth mesh gives

The possible final summand is controlled because |t|/x is bounded. Summing (6.15) first, and then summing the position increments t(J-1), gives

The second sum uses 2\alpha>1. Initial errors from the fixed slab are absorbed because B\ge1. Moreover

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

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

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 (6.17). 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

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

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 Lemma 4.6 and Corollary 4.7, 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 (6.16) 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 (6.13).

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 Theorem 6.3 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 (6.20). 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 (6.21); on the tail D\ge R, (6.19) 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 Theorem 6.3 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 (6.22).

For (iii), boundedness of b_n turns the vanishing weighted bound into the same statement with normalization d_nM. Use Theorem 6.3 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

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^{\ii\theta}, the nearer of the two selected impacts has depth

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 (6.23) up to \sigma=H. At their centers it gives a joint patch of L through (x_n+\ii H,0), attached to the primary zero-parameter data, contrary to Theorem 4.9.□

Proof. Choose \alpha with \max\{p_0,1/2\}<\alpha<1. On any cone with \delta_s\ge d,

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 Corollary 6.5(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,

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

The remaining fixed lower-strip portion has bounded M and is smaller still. This is the vanishing cone bound forbidden by Corollary 6.5(iii), with b_n=1.□

Proof. Write the limsup in (6.25) as \rho, and fix

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

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 (6.26). 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

Then b_n\to\infty, and in fact

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-\ii H)/d_n and fields

Their domains exhaust the lower half-plane. The envelopes give, with D=-\Im S,

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

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 (6.30).

We claim that some sequence of centers, after passage to a subsequence, has F=Q=0. Suppose otherwise. Fix a small closed disk about -\ii 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 -\ii, 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

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, Lemma 5.5 gives f''(z)=0, and hence

Deforming C_n to the real period, with residues included, therefore gives

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 (6.32). This is a contradiction.

Suppose next that the common type is the Q-type. There are centers x_n with

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

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 -\ii, 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

by (6.28), contradicting (6.33). 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 Corollary 6.5(ii). The bound (6.30) 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

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

Fix a real T\ne0. Define, now as a function of the rescaled phase coordinate Y,

By (6.34), its phase domains exhaust \Im Y<0. By the shape constraint in Theorem 5.4, \Im S_n<0. For any 1/2<\alpha<1, the coarse cone estimate anchors these half-plane-valued functions on deep compact sets:

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 (6.35) gives the same anchor for S.

The limiting energy equation. Set h_n=T/b_n, which tends to zero. Then

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 Lemma 6.2; no additional loop is used. Let p_n(A,J) denote the normalized action with rescaled shape basepoint A. Its fast fields are

They converge analytically to the frozen fields TF(A), T^2Q(A). By Lemma 6.2, the actions and their derivatives converge analytically to

The second expression depends analytically on \lambda^2 near zero, so no choice of a square root is needed there.

In these coordinates (6.6) reads

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

and analytic convergence gives convergence of G_n'. The limit is consequently the Euler–Lagrange equation

For E(A,J)=Jp_J(A,J)-p(A,J), the chain rule now gives

The deep decay of the fields and the anchor, with Cauchy estimates, give S_Y\to1 as D\to\infty. Hence the constant is

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 (6.9)–(6.10). The analytic implicit function theorem applied to (6.36), together with (6.30), yields

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

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 \eps<D<R, its top boundary is bounded below by -\eps, and its bottom boundary is bounded below by -\eta_R, where \eta_R=\sup_x|S(x-\ii R)-(x-\ii R)|\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

First let \eps\downarrow0, then R\to\infty, to get v\ge0. Harnack’s inequality in the half-plane gives, for R\ge1,

For example, this is the ordinary disk Harnack inequality after mapping -\ii to the origin, since -\ii R has disk modulus (R-1)/(R+1). By (6.37) its left side is o(R^{-1}). Hence v(-\ii)=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 (6.36). Deep down the field parameter is small, and the energy expansions are

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 -\ii, proving the Theorem.□

Proof. We first make a uniformity observation. With D=\delta/\ell_n, weakening the exponent in (7.3) gives

For each fixed L, this is bounded by an absolute constant beyond a finite depth threshold depending on L. On the remaining finite cone, (7.2) 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 Theorem 6.3 and its reference-height formulation in Remark 6.4 with reference height h_n, rather than H, and with

The cone hypothesis is exactly rM\le\eps(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

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 (7.6). 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 Theorem 6.3, 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 (7.4) implies

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 Theorem 6.3 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

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

by (7.7). 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

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,

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, (7.8) 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 (7.7). 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 Theorem 6.3 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 (7.6). 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,

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 (7.9). The action p_n(S,J;T) and its derivatives converge on these neighborhoods by Lemma 6.2. In particular the case F_n\to0 uses the analytic division in Lemma 6.1, rather than division of two limiting zeros.

The exact recurrence takes the form

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 (7.11) by a_n. Both terms are evaluations of the same analytic function p_{n,J} at neighboring states. Analytic convergence therefore gives

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^{-\ii Y} and z=\lambda J/2. The energy equation is

At (\lambda,J)=(0,1), \partial_J E=1/4; thus it determines J analytically as a function of \lambda near that point. Moreover

Integrating this differential relation gives

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 (7.12) put v=z and define

In the second case put v=\tanh z and define

All branches at zero are normalized by g(0)=P(0)=1. Direct differentiation of (7.12) gives, in both cases,

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 (7.14), 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

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 [McM17, Theorem 3.7, p. 43] therefore gives

Comparison with the sine product yields

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 (7.15),

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

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

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 (7.16) is analytic at zero:

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

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 (7.16) 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\ii\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 (7.13) defines the impact model there. Its regularity tests can be made explicit. In the first case,

In the second case, with z=\operatorname{arctanh}v,

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,

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,

Integration of w_v=w/(vg(v)^{2/3}) gives

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,

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, \ii\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 Lemma 6.2 applies on these charts, following the branches initialized in the known data. Given the old state (S_-,J_-), set S=S_-+a_nJ_-. Equation (7.11) 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

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

and has local Lipschitz bound 1+C|a_n|.

The model solves the ordinary differential equation in (7.18): its second component follows by expanding \dd_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 Theorem 7.1. Let

so that T_* is real and |T_*c_0|=1. Take the parameter bound in Lemma 7.2 larger than all the parameter disks below. We first reach, vertically and with joint convergence, the inner regular region

for any fixed \nu>0 and every needed compact horizontal range.

Fix a center T_c in this parameter disk and use

The parameter disk is bounded away from zero, so this logarithm is single-valued. Its sign is important:

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^{-\ii Y}| 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 Lemma 7.2 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 (7.19). By the maximum principle for \Im g, the full model disks in (7.20) stay within radius Re^{-\nu/2}. Initially the full disks lie in known data. The Laurent-coefficient argument used in the proof of Lemma 7.2 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 (7.19). 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

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 (7.20). This time choose a polynomial with

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 (7.21). 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

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 Theorem 4.9. The same argument excludes equality at the endpoint, because the passage has a neighborhood there. Thus, for all sufficiently large n,

This is the one place where a local passage is promoted to an entire horizontal band, and the parameter here is real. By Theorem 5.4, 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 (7.21). 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 Lemma 7.4. 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 (7.22). 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

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

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 (7.23). 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 (7.25) 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

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

We claim that

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 (7.26).

Choose R\to\infty along values with \mu(R)=o(R), and fix a lower base value R_b. The positive measure \dd\mu on [R_b,R] has mass o(R). The weak maximal-interval inequality gives a number \eps_R\to0 and a continuity point v\in[R/3,2R/3] such that

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 \eps_R has length at most a constant times the total mass divided by \eps_R, by the disjoint-interval covering argument. Choose, for example, a constant multiple of \sqrt{(1+\mu(R)-\mu(R_b))/R} for \eps_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 (7.27).

Set

For bounded u, integrate the inverse derivative using (7.27), then invert the result. It gives, uniformly on each fixed bounded real range,

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

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 \C.

Choose x_n realizing the row maximum at h_n. By (7.29), the entire subsequential limits of

satisfy |F(S)|\le e^{\Im S} and |F(0)|=1. The maximum principle applied to e^{\ii S}F(S) therefore gives

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 (7.29) give the global log bound K=v+O_{C_1}(1) on their heights. Equations (7.24)–(7.25) and (7.30) imply

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 (7.27) give

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 (7.33). The result is at most

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

by (7.30). This absorbs that compact range and, using (7.32), 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

uniformly in horizontal position. In the finite-width case retain any fixed smaller range \delta\le\delta_b<H/2. In the entire case (7.34) is exactly the additional condition (7.4).

All hypotheses of Theorem 7.1 now hold, with the first profile in (7.2), by (7.31) and (7.32). Its contradiction proves both assertions.□

Proof. At a fixed center the cone theorem gives a joint holomorphic vertical passage at the row point for

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

The fixed factors in c are included in the constant here.

By (8.12) there is a center with b(x,k)\le C. The real no-patch transfer from Theorem 4.9 then implies

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, (8.7) gives

For small z, choose h\ge0 so that e^{-h}=C'z^{1/(1+a)}, where the fixed constant C' is large enough for (8.17) to apply at level j. This proves (8.13). 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 (8.16). Combining that estimate with (8.6) gives

The two powers of b coincide and equal the exponent in (8.14). 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

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 \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 (8.18).□

Proof of the quadratic and higher-moment assertions. Let C_* =\sup_{j\ge k_0}\mu(j)/N(j)<\infty. By (8.12), \|b(\cdot,k)\|_2\le C(1+C_*). Hence (8.15) gives

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

Here (8.5) 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 (8.19), 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 (8.15) give, on all rows in any fixed range of depth ratios,

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

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

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 (8.19). The convolution is an L^p contraction, so Holder’s inequality and (8.22) bound the second normalized term by C(R\sigma)^\vartheta. The negative mass directly on E has the same bound. Consequently

The interior harmonic estimates used above, followed by Lemma 8.3, now give

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 (8.25) 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 (8.26) 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 (8.26). Selecting the cells for which A_k(I)>b_0\ge1 gives

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 (8.20), including its fixed-tile versions.□

Proof of the global assertions in Theorem 8.4. Suppose N is nondecreasing. We first prove that, for a fixed small c_2>0,

Choose a fixed \eps_0>0 so small that (8.13) bounds \Delta by d_k/16 on the real interval |u|\le u_0:=\eps_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

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 (8.28) is at most C_{\eps_0}(1+k)N(k). For earlier levels j\le k, monotonicity gives N(j)\le N(k), so (8.17) 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_{\eps_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 (8.27).

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+\ii(\mathfrak s-y), gives

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 (8.5), 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 (8.21).□

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 (8.30) on the coarser part. This gives

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,

The laws \kappa_j and \nu are comparable. Moreover the absolute value of the right side, in fine-row normalization, is at most

The negative-part estimate (8.15), integrated using (8.33), 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

The strict inequality 2\beta<1 gives uniform decay in (8.34).

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 Lemma 8.3 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 (8.32).□

Proof. Suppose first that (8.35) holds. For any fixed c,T_0>0, apply Theorem 6.3 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 (8.35) supplies its small parameter. The real no-patch transfer gives

for all sufficiently large n. Since c>0 is arbitrary,

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 (8.12), the cone costs b(x,j) have uniformly bounded second moments. Also (8.15) gives

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 (8.38), the relevant rows lie in the actual strip for large n. Combining (8.6) and (8.16), now in this normalization, gives

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 (8.39) 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

The disk bound and B_n\to\infty now imply

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, (8.37) bounds the expression in (8.35) 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 (8.35), which was just excluded. In fact compact convergence within this fixed cone is enough for the argument.

Finally the scaled differential identity is

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

Here is an explicit selection. Failure of (8.45) 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 (8.2). It is still a record among all earlier levels. Its value satisfies

where we used \log(\kappa_n/u_n)\le(\kappa_n-u_n)/u_n. The maximizing property gives the last inequality in (8.46).

For each fixed \eta>0, and all sufficiently large n, set

This nondecreasing envelope controls the first moment uniformly in n. For |f| use (8.46); for |q|^{1/2} use (8.3) 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. Theorem 8.4 consequently gives

Fix one positive \eta to use the higher-moment assertion, and let p>2 be the resulting exponent. On the row \kappa_n put

Holder’s inequality gives

Therefore

We next select a normal level for the full moment. For any fixed small \eps>0, maximize \mu(j)e^{-\eps|j-\kappa_n|} over j\ge k_0. Let k be a maximizer and B=\mu(k). The triangle inequality gives

Using (8.47) with rate \eps/2 also gives

For fixed \eps, the right side is o(\log\kappa_n), by (8.46). A slow diagonal choice thus gives \eps_n\downarrow0 and maximizing levels k_n such that

while (8.49) holds with \eps=\eps_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 (8.48) and (8.50); 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

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 Theorem 8.4 to the envelope

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 (8.41) and (8.42), with constants independent of n for every fixed choice.

It remains to prove (8.43). 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+\ii H+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, (8.47) implies

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 (8.44) bounds its average on every target disk by \kappa_n^{o(1)}, uniformly in the indicated centers. Lemma 7.6, with fixed inner margins, therefore gives

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 (8.43). We have constructed all the data of Definition 8.7, a contradiction.□

Proof. By Proposition 8.8 and (8.3), \E d_jM\le C(1+j). At a target level K, use the nondecreasing envelope

Its envelope and first-moment constants are independent of K, and its exponential tail makes the bootstrap ratio finite. Apply Theorem 8.4 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

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^{-\eps_n|j-\kappa_n|}, where \eps_n\downarrow0 sufficiently slowly, and denote the maximizer by k_n, with B_n=\mu(k_n)\ge b_n. The maximizing argument in (8.49) gives primary normality. The polynomial bound \mu(j)\le C(1+j) shows that

Indeed, on the future side use (1+j)/(1+\kappa_n)\le e^{(j-\kappa_n)/(1+\kappa_n)} and choose \eps_n\ge2/(1+\kappa_n); the past side is immediate. We can therefore arrange

Here both ratios denoted by \ll tend to zero. For example, after a further slow choice of C_n, one may use \eps_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 (8.51). 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

At the chosen ancestor point, monotonicity and (8.53) give

Both levels in this difference lie inside the interval in (8.53), 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

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 Lemma 7.6 on a finite covering of the target compact set by disks with fixed inner margins. It yields

Since B_n\to\infty, Markov’s inequality proves (8.43). 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 Lemma 8.10,

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

Multiplying by the bound from Proposition 8.8 gives

On the complementary set the integral is at most \int_{K_0}^K k^{2\alpha}\,\dd k=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 (8.55).□

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 (9.1) gives

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 (9.3) pass to the limit.□

Proof. Since \rho(v)-cv\ge(1-c)v, the infimum in (9.6) is attained. The same inequality gives

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 (9.5),

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 (9.5) 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

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,

Taking |h|\le g(x_j)=P_j^{-1/2} shows that

Integration, with \log(1-u/P_j)=-u/P_j+O(P_j^{-2}), proves (9.7).

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

and \alpha_0<c, the increase is at least \alpha_0\log(1+v)-O(1), uniformly for all v\ge1. Exponentiating proves (9.8).

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

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

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:

and use \rho_c(x)\ge x. Birkhoff’s pointwise ergodic theorem [Dur19, 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

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, (9.13) 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 (9.5),

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 (9.13) 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 (9.10).

Finally, the lower increment bound for \rho_c, its uniform Lipschitz bound, and the bounded difference at x_j give (9.11) and (9.12) 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

The diagonal can be chosen so that

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 (9.7) and (9.8) 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 \eps_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+\eps_n} is below a fixed multiple of the right side of (9.8), with the normalizing factor m_n(D_j), whenever D\ge D_{\rm cut}. Such a cutoff exists because 1-\eps_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

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

let

There is an exact cancellation in the expectation:

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 (9.15) 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

for 1\le-\Im S\le\delta_b/\ell_n and |\Re S|\le L(-\Im S). Also

Since m_0\ell_n\to\infty, it follows that m_x\ell_n\to\infty in probability.

Now consider the nonnegative subharmonic function

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 (9.16) 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

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

At S=0 the normalized sum is exactly 1, so the limit is nonzero. Moreover

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 Theorem 7.1. 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

The identity

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

which tends, after the diagonal selection, to (9.18).

We spell out the global envelope because its constants must remain uniform after diagonalization. With v=k-l,

On a finite v-range, the limiting estimates (9.11) and (9.12) 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 (9.21) by

If \eps_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 (9.19) with one constant C.

Let k_a=k-s_n and \omega,\nu be the ancestors and measures in Definition 8.7. To apply Proposition 8.5, 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

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 (9.19) 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 (9.20). 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 Theorem 8.6; 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 Theorem 6.3 to (9.20) 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

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

On the fast segment based at a scaled shape coordinate S, the fields entering the normalized chord action are

They converge analytically, including on small thickenings, to the constant fast pair (TF(S),0). The kernel division in Lemma 6.1 and the frozen first-field formula (6.7) therefore give the local action limit

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

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

In particular no prescribed rate of convergence relative to h is needed. Differentiating E=Jp_J-p along the limiting solution gives

The deep anchor implies J\to1, while F(S)\to0, so E=1/12. The frozen energy identity

is exactly (9.26).

Near zero field, (9.26) 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

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,

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

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

For D/C_0\le v\le1, first use m_*(v)\le C_\eps v^{-\lambda-\eps/2}, then take the original index sufficiently large at this fixed D. For v\ge1, use (9.19), whose field exponent is 1-\beta>c_1. These give (9.28) 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 Theorem 6.3 at that center with

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 Theorem 4.9 as after Theorem 6.3, consequently gives the global width bound

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 (9.29).

The full-width domains in (9.29) exhaust (9.27), 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 (9.27); 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 (9.26) first shows that J\ne0. At fixed w=TF(S), its J-derivative is

It can vanish only when z=n\pi, n\in\Z\setminus\{0\}. The equivalent entire equation

then has a nonzero right side. Differentiating in Y gives

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, (9.31) 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 (9.30) 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

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

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 \eps>0, use Proposition 9.6 with r=1/(\lambda+\eps)\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

The fractional power is the branch on the right half-plane; \Re(\eta_0-\ii T)>0 in the upper half-disk. The auxiliary \eta_0 is distinct from the fixed growth exponent \eps.

On the real diameter,

Choose C_* large compared with the constant in (9.27), 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,

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 Proposition 9.6(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 Lemma 9.7 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)=-\ii C_*(\eta_0+\ii T)^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

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

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 (9.35), or the Caratheodory coefficient estimate, gives

This also holds on the depth-sized Y-neighborhood, so Cauchy’s estimate yields

The energy equation determines the derivative:

Its coefficient of T^2 is 3A'(Y)-F(Y)^2/40=0. Hence

Combining these bounds gives |F(Y)|\le C D^{-1/r}=C D^{-\lambda-\eps}. Arbitrariness of \eps>0 gives (9.33).□

Proof. The coarse part of (9.19) 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 (9.39). 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

The epochs obtained this way are within o(w) of every sufficiently large w.□

Proof. Fix

and choose L_0 large enough for the quiet-cone form of Theorem 8.6 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

for every

Pass to a subsequence with u_j\ge j^2.

Return to the actual data of Lemma 9.5, taking their sequence index sufficiently large after each fixed u=u_j. Set

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

locally uniformly. The old envelope says

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

For all v\le0, (9.43) 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:

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

Indeed the new depth ratio is e^{u-v}, and

where M_{\rm shape}=\max(|f|,\sqrt{|q|}).

Increase the actual index further so that

throughout

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, (9.46) is at least

The errors in (9.47) 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 (9.41) bounds the logarithmic ratio by

For 0\le v<u_j-\eta u_j, use (9.33) 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

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

The inherited bound (9.20) gives

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 Theorem 8.6. 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 (9.40).□

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

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

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 (9.52). Summing their lower bounds gives (9.51). 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

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

List these zeros with multiplicity as a_1,\ldots,a_N. The function

extends harmonically across the zeros on B(c,20\rho). Each logarithm has bounded absolute integral in scaled coordinates, so (9.54) gives \int_{B(c,20\rho)}|H|\,\dd A\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\C, outside disks of sum of radii at most 3h,

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 (9.55).

Apply this estimate to b_\nu=(a_\nu-c)/\rho. Outside the resulting disks,

This implies (9.53).

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 Theorem 9.11. We work in fixed horizontal intervals contained in [-5,5]. All errors below are uniform on these intervals. Fix the aperture from (9.50) for this compact interval.

Separated bands with neighboring logarithmic depths. Choose a nonincreasing function \theta(v)\to0, with \theta(v)\ge v^{-1/4}, whose square bounds, eventually, the relative errors in (9.49) and (9.50), uniformly for all arguments at least v. Such a majorant is obtained by taking the decreasing tail supremum of these errors, its square root, and the maximum with v^{-1/4}. Enlarging it slightly accommodates approximate nearest epochs if E is not closed.

Starting at a sufficiently large u_1\in E, set s_j=\theta(u_j)u_j, and choose

Indeed the error in finding an epoch near u_j+8s_j is at most C\theta(u_j)^2u_j=o(s_j). The monotonicity of \theta implies s_{j+1}\le(1+o(1))s_j. Consequently the bands [u_j-s_j,u_j+s_j] are disjoint, their gaps tend to infinity, s_j\to\infty, and

Every witness at epoch u_j has logarithmic depth u_j+o(s_j). These statements survive any fixed additive changes to the ends of the logarithmic bands.

Choose dyadic depths a_j,b_j with

The horizontal rectangle is R_j=[-3,3]\times[a_j,b_j] in the coordinates (x,\delta). The two vertical rectangles have horizontal intervals [-2,-3/2] and [3/2,2], and depth interval [a_{j+2}/8,8b_j]; dyadic rounding can be incorporated into these fixed factors. The factors eight leave room at the ends for all subsequent local enlargements. A witness at epoch u_{j-1} is deeper than the entire vertical rectangle by a factor tending to infinity. In fact, the logarithmic gap from that witness to its bottom is at least (7-o(1))s_{j-1}.

Take a slightly larger open region containing these three rectangles, the witnesses at epochs u_j,u_{j-1} needed for centers in [-3,3], and the fixed-factor boxes that will be used about them. Its horizontal coordinates lie in [-5,5]; all its logarithmic depths are u_j+o(u_j), by (9.57). By (9.48) we can choose

throughout this region. The function U_j is superharmonic and may have logarithmic poles at the zeros of F. Its value at every witness just specified is o(u_j), uniformly in its center.

Tiles and the exclusion of bad barriers. Fix a sufficiently large dyadic integer N_0. In the layer 2^{-l-1}\le\delta\le2^{-l}, tile by squares of side \rho=2^{-l-1}/N_0, using the common dyadic horizontal grid. The sides of a tile are thus a small fixed fraction of its depth. Take N_0 large enough that the disks of radius 100\rho required in Lemma 9.13 lie within the fixed-factor enlargements already allowed. The rectangles have dyadic boundaries, so they are finite unions of these tiles. Tiles that touch, even at a corner, have side ratios between one half and two.

Choose \eps_j\to0 so slowly that the uniform witness deficits in (9.58) are o(\eps_j u_j), and so that M_j-\lambda u_j=o(\eps_j u_j). Color a tile good when

and bad otherwise. On every bad tile, U_j\ge\eps_j u_j. For any witness w at one of the two indicated epochs, Lemma 9.12 therefore implies that the union of the projections of bad tiles with center depth at most \delta_w/2 and horizontal center distance at most K\delta_w from w has length

Here K is any fixed constant needed below. For this application put the fictitious top of the Brownian rectangle a fixed factor shallower than the shallowest tile under consideration. The rectangle then lies in the enlarged region of (9.58).

There is no bad chain in R_j from its shallow edge to its deep edge, even when corner contacts are allowed. Suppose such a chain exists and let x_0 be a point where it meets the shallow edge. Choose the epoch-u_j witness p-\ii\delta_w for the center x_0. The shallow edge is much shallower than \delta_w, and the deep edge is much deeper. Follow the chain from its shallow end until it first reaches depth \delta_w/4 or first leaves the horizontal interval [x_0-\delta_w,x_0+\delta_w]. Taking N_0 large guarantees that all tiles up to and including this first event have center depths at most \delta_w/2 and center distances from p at most (A_I+2)\delta_w. In the first case the last tile has side at least c\delta_w/N_0, so its shadow alone has length at least a fixed positive multiple of \delta_w. In the second case the projections of this connected chain prefix cover an interval of length at least c\delta_w. Both conclusions contradict (9.60).

Neither vertical rectangle has a bad chain joining its two lateral sides. Choose an epoch-u_{j-1} witness for the midpoint of its fixed horizontal interval. This witness has depth \delta_w, and every tile of the rectangle has center depth less than \delta_w/2 for large j. Its horizontal coordinate p lies in the interior of the strip, since its distance from the midpoint is at most A_I\delta_w\to0. A lateral bad crossing projects onto the whole strip. The tiles of that crossing with centers in [p-2\delta_w,p+2\delta_w] therefore have projections covering [p-\delta_w,p+\delta_w], once N_0 is large. This again contradicts (9.60).

Planar duality and routing the good chains. We include the finite grid argument to specify the corner convention. Subdivide every tile of a rectangle to its finest dyadic square grid, retaining the parent tile’s color. On a finite square grid, either there is a good left-to-right path using shared edges, or there is a bad top-to-bottom path using shared edges or corners. To prove this, mark all good cells reachable by shared edges from the left side and adjoin an exterior collar along the left. If this set reaches the right, the first alternative holds. Otherwise follow its right-hand boundary from the top of the rectangle to the bottom, with the marked set on the left. At vertices with several possible continuations, keep the marked set on the left, equivalently resolve the vertex by a small rounded arc. Closed boundary components can be ignored. The cells immediately on the right of the resulting boundary are bad: an edge-adjacent good cell would have been marked. Consecutive such cells share an edge or a corner, and they join the top to the bottom. If no cell was marked, the first column itself is such a bad chain. This proves the alternative. Interchanging the two coordinate directions gives its vertical version. Mapping a chain back to its parent tiles preserves corner adjacency; removing loops and repeated tiles gives a simple parent chain.

The bad barriers have already been excluded. Thus R_j has a good horizontal chain, and each vertical rectangle has a good vertical chain. On a good tile the infimum of U_j is at most M_j-\lambda u_j+\eps_j u_j=o(u_j), uniformly. Apply Lemma 9.13 to route each chain. We obtain a horizontal path \gamma_j, extending from near -3 to near 3, and vertical paths \sigma_j^-, \sigma_j^+ in slight enlargements of the two side strips, with

The horizontal path stays in a fixed-factor enlargement of its band. The vertical paths start deeper than all of \gamma_j and end shallower than all of \gamma_{j+2}. They remain to the left or right, respectively, of the entire interval [-1,1].

The length estimate in the routing lemma is subexponential in u_j. Indeed, in any of these rectangles the smallest tile side is e^{-u_j+o(u_j)}, the largest side is e^{-u_j+o(u_j)}, its height is e^{-u_j+o(u_j)}, and its width is bounded. Even its entire finest grid contains at most e^{u_j+o(u_j)} cells. Hence the number of parent tiles times their largest side, and therefore each routed path’s length, is at most

All local constants, including those in the Cartan covering, have been fixed before j tends to infinity.

Each vertical path intersects each of \gamma_j,\gamma_{j+1},\gamma_{j+2}. One elementary way to verify this is to erase loops from a horizontal polygonal path and trim it at two vertical lines outside the side strips. It is then a simple arc joining the opposite sides of a rectangle, confined to its horizontal band. Closing this arc along three sides of the rectangle and applying the Jordan curve theorem separates points above the band from points below it. A vertical path in either side strip has endpoints on these different sides and must meet the arc. The fixed-factor margins ensure that local routing has not changed this separation.

A primitive with no logarithmic monodromy. Set q=F''/F, holomorphic by hypothesis. Every zero z_0 of F is simple. Otherwise F(z_0)=F'(z_0)=0, and uniqueness for the holomorphic equation F''=qF would imply F\equiv0. At a simple zero this equation also gives F''(z_0)=0, and hence

In particular all residues of F^{-2} vanish. Integration around any closed curve avoiding its poles is zero: the curve is compact in the simply connected half-plane and encloses only finitely many poles, so this is the residue theorem. Thus there is a single-valued meromorphic function C_0 in \mathbb H_- with

Its singularity at a zero z_0 is -a^{-2}(z-z_0)^{-1}.

The paths have no zeros, by (9.61). Equations (9.61)–(9.62) imply that the oscillation of C_0 along any one of the three paths with index j is at most

Choose an arbitrary point z_j\in\gamma_j. The connections just proved join z_j to z_{j+1} by pieces of \gamma_j,\sigma_j^-,\gamma_{j+1}. Thus |C_0(z_{j+1})-C_0(z_j)|\le e^{-2\lambda u_j+o(u_j)}. The gaps u_{j+1}-u_j eventually exceed one. For any fixed \eta\in(0,2\lambda) and all sufficiently large j, the telescoping tail is therefore bounded by

It follows that C_0(z_j) tends to a constant c_*. Letting \eta tend to zero, and using (9.64) to reach every point of the three paths, gives the uniform estimate

This is one common constant for the whole mesh, not a separate constant for each band or each side strip.

The function

extends holomorphically through every zero of F: its apparent pole cancels, with limiting value -a^{-1} at the zero above. The upper bound on the enlarged regions, together with (9.65), gives

Enclosing every shallow point and reflecting. The arrangement of the paths is illustrated in Figure 4.

Write [\alpha_j,\beta_j] for a logarithmic-depth interval containing the entire routed horizontal path \gamma_j. The fixed-factor enlargements give \alpha_j=u_j-s_j-O(1) and \beta_j=u_j+s_j+O(1), and \beta_j<\alpha_{j+1} for large j. Given z=x-\ii e^{-v} with x\in[-1,1] and v large, choose j so that

Then \gamma_j lies strictly below z, whereas \gamma_{j+2} lies strictly above z. The two paths \sigma_j^- and \sigma_j^+ connect across both bands and lie strictly to the left and right of z.

Choose an intersection of each side path with the lower and upper horizontal paths, and use the corresponding path segments to form a closed walk, in the order lower, right, upper, left. More explicitly, along either side path take its last lower-path hit before its first subsequent upper-path hit; its intervening segment then joins those two paths. Join these endpoints along the horizontal paths. Every piece lies in the indicated open half-plane relative to z, and the four corner points lie in the corresponding quadrants. This walk has winding number one about z: replace each of its pieces by a straight segment within its open half-plane, which is convex and avoids z. The resulting quadrilateral traverses successively the four quadrants and has total change of argument 2\pi. Equivalently, the arguments on the lower, right, upper, and left pieces can be chosen in their respective half-plane intervals and then joined in this order. Self-intersections of a path or of the walk do not affect this winding calculation.

The complementary component containing z is therefore bounded and has boundary in this walk. It lies compactly in the half-plane. The maximum principle for the holomorphic function G gives its bound there from the walk. The upper horizontal piece has index j+2, and u_{j+2}=u_j+o(u_j), so (9.67) yields

The same estimate is immediate if the point lies on a mesh path. As v\to\infty, the selected j tends to infinity uniformly for x\in[-1,1]. Hence G has continuous boundary value zero on (-1,1), uniformly on compact subintervals.

Schwarz reflection across this interval extends G holomorphically by \overline{G(\overline z)}. The extension vanishes on the interval, so the identity theorem gives G\equiv0. On any disk where F\ne0, this says C_0\equiv c_*, contradicting C_0'=F^{-2}\ne0. The contradiction proves the Theorem.□

Proof of Theorem 9.1. Assume that a sequence in Definition 8.7 exists. Its primary moments have a limit by Lemma 9.2. Apply Lemma 9.3 to its logarithmic law. The steep-contact alternative contradicts Theorem 7.1, as proved in Lemma 9.4. The bounded-contact alternative gives the fields in Lemma 9.5. Their nonzero first field has holomorphic quotient F''/F. The limiting impact equation and parameter continuation, Propositions 9.6 and 9.8, give its uniform upper growth. Lemma 9.9 and Proposition 9.10 give common relatively dense epochs and uniform witnesses at every horizontal center. These are precisely the hypotheses contradicted by Theorem 9.11. Both alternatives are impossible, proving the Theorem.□

Proof. The sole forward assumption in the critical reduction was the nonexistence of the sequences in Definition 8.7. This is Theorem 9.1. 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 Equation (10.10). 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=\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

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

Hence the full matrix is invertible. If one of the two blocks is empty, one of the already proved definite-matrix statements applies.□