Proof. The pair plane and pair strip are simply connected, so the nonvanishing vector has a continuous real argument. For a single leaf normalize it to \(\pi/4\) on \(\beta=\alpha+2\pi\). For distinct leaves, both entries equal \(\beta-\alpha+O(1)\), uniformly in the pair variables, so normalize the argument to \((0,\pi/2)\) on the connected region where \(\beta-\alpha\) is sufficiently large. Either normalization implies simultaneous periodicity: the difference after a simultaneous translation is a continuous integer multiple of \(2\pi\) and vanishes in the normalizing region.

Consecutive vectors have the form \((u,v),(v,w)\). A positive quarter-turn relation \((v,w)=c(-v,u)\), \(c>0\), would force \(v=0\) and \(u,w\) to have the same sign, contrary to (11). Thus \(A_1-A_0\) avoids \(\pi/2\) modulo \(2\pi\). Connectedness, together with the edge normalization or the large-separation normalization, puts it in \((-3\pi/2,\pi/2)\). Applying this to the iterated pair proves (12).

If \(A_j\le a_m\le A_{j+1}\), that bound gives \[a_m-\pi/2<A_j\le a_m\le A_{j+1}<a_m+\pi/2.\] On these two intervals \(\sin A_j\) is nonzero with sign \((-1)^m\), whereas \(\cos A_{j+1}\) is either zero or has the opposite sign. This is impossible: both have the sign, and the same zero condition, of \(x_{j+1}\). This proves (13).

Finally, if \(x_i\ne0\), then \(A_i\) lies strictly between two consecutive barriers. All subsequent arguments stay below the upper one. If \(x_k\) has the opposite sign, \(A_k\) cannot remain in the interval between those barriers, so lies strictly below its lower endpoint. All later arguments remain below that endpoint, a positive distance from \(A_i\). ◻

Proof. Periodic leaves are graphs. Suppose first that \(G_b^q=\mathrm{id}+2\pi l\) on a whole leaf. For its pair strip, simultaneous periodicity gives \(A_q=A_0\). If \(x_0=0\), then \(A_0\) is a barrier, and (13) makes \(A_q<A_0\), a contradiction. Thus \(x_0\) never vanishes on the connected strip. It equals \(2\pi\) on its upper edge, so is everywhere positive. This says precisely that \(\theta_b\) is strictly increasing.

Constant-translation intervals also consist of graphs. Let \(b\) be interior to an interval on which \(\tau\) is constant. If \(\theta_b\) were not strictly increasing, its degree-one property and continuity would give an ordered same-leaf pair with \(x_0=0\). By (11), \(x_{-1}\) and \(x_1\) have opposite strict signs. Start instead with that preceding pair and perturb it slightly. We obtain a nonempty open set \(O\) of pairs on distinct leaves for which \(x_0x_2<0\). We can take the two leaf parameters in disjoint relatively compact intervals \(I_1,I_2\) inside the constant-translation interval, and take \(|\beta-\alpha|\) bounded on \(O\).

Consider the cover of \(H(\mathbb{T}\times I_1)\times H(\mathbb{T}\times I_2)\) obtained from \((\mathbb{R}\times I_1)\times(\mathbb{R}\times I_2)\) by identifying only \[(\alpha,b_1,\beta,b_2) \sim(\alpha+2\pi,b_1,\beta+2\pi,b_2).\] The pair dynamics \(K\) induced by \((G_{b_1},G_{b_2})\) preserves the measure lifted from product billiard area. This measure is defined by integrating counting measure on the covering fibers against product area; \(K\) bijects the fibers over the area-preserving base dynamics. It has full support, and any region of bounded \(|\beta-\alpha|\) has finite measure, since it meets uniformly finitely many sheets over each base point. This construction requires no differentiability or absolute continuity of \(H\).

Equality of the two translation numbers and (7) imply, for every \(j\in\mathbb Z\), \[ \left|(G_{b_2}^j(\beta)-G_{b_1}^j(\alpha)) -(\beta-\alpha)\right|\le4\pi. \tag{15}\] Consequently the saturation \(\bigcup_{j\in\mathbb Z}K^j(O)\) is an invariant finite-measure set. Poincaré recurrence gives recurrent points in \(O\), indeed almost every point there is recurrent. One may obtain recurrence to a point from the usual measurable-set version by using a countable neighborhood base; the latter version follows because the forward images of a set of nonreturners are disjoint in finite measure.

For any such recurrent pair, the two leaf parameters remain fixed. Thus continuity of the argument for this fixed pair of leaves and simultaneous periodicity give \(A_{j_k}\to A_0\) along a sequence \(j_k\to\infty\). This contradicts Lemma 4, because \(x_0x_2<0\). Hence \(\theta_b\) is strictly increasing throughout every constant-translation interval.

Graph leaves are dense, and then every leaf is a graph. Take any open parameter interval \(J\). If it contains a subinterval where \(\tau\) is constant, the preceding argument supplies graph leaves. Otherwise choose a rational multiple \(r_*=2\pi l/q\) strictly between two values of the continuous function \(\tau\) on \(J\). Travel along the parameter segment from the smaller value to the larger one, and let \(c\) be the first position approached by values \(\tau>r_*\). Continuity gives \(\tau(c)=r_*\). Values strictly below \(r_*\) also approach \(c\): otherwise \(\tau=r_*\) on an interval immediately before \(c\), contrary to our case assumption. Taking limits in (8) from these two strict sides gives \(G_c^q=\mathrm{id}+2\pi l\). The periodic-leaf argument therefore supplies a graph in \(J\). Notice that no monotonicity of \(\tau\) has been used.

Graph parameters are thus dense. Continuity of the lifted foliation makes every \(\theta_b\) nondecreasing. If a same-leaf ordered pair had \(x_0=0\), its preceding and following pairs would still be ordered, because \(G_b\) is increasing. Nondecrease would give \(x_{-1},x_1\ge0\), contradicting (11). Every \(\theta_b\) is therefore strictly increasing, proving the graph property.

Ordering and the grazing end. Inverting the increasing angular parametrizations shows that \(g_b\) is jointly continuous. Distinct graphs are disjoint, so their order cannot change with \(\theta\). For each fixed angle, the continuous injective function \(b\mapsto g_b(\theta)\) is strictly monotone, in the same direction at every angle. Reverse \(b\) to make this direction decreasing. The upper-strip inclusion (6) then forces \(g_b(\theta)\to h(\theta)\) as \(b\downarrow0\) at every angle. Monotonicity and compactness upgrade this to uniform convergence. Formula (2) now shows that \(F_b(\theta)-\theta\) is positive and tends uniformly to zero at that end. Its positive minimum on each fixed graph gives \(\tau(b)>0\), and \(\tau(b)\to0\) follows.

The forward twist and decreasing graph order give \(F_{b_1}<F_{b_2}\) when \(b_1<b_2\). Since the maps are increasing, all their positive iterates have the same strict order. Passing to translation numbers shows that \(\tau(b)\) is nondecreasing.

Rational plateaus are impossible. If \(\tau(b)=2\pi l/q\) on an open parameter interval \(I\), almost every point in its finite-area invariant annulus is recurrent for \(T^q\). On each leaf, \(F_b^q-2\pi l\) has zero translation number and hence has fixed points: without a fixed point its periodic displacement would have a uniform strict sign. Every nonfixed point moves monotonically in a component of the complement of the fixed set towards an endpoint, so is not recurrent on the circle. Thus \(T^q\) is the identity almost everywhere in this annulus, and therefore everywhere, by continuity and full support of area. Its lift on each graph is necessarily \(F_b^q=\mathrm{id}+2\pi l\), with the integer shift determined by the translation number. This contradicts strict ordering of the iterates on two distinct graphs.

The continuity, monotonicity, positivity, and zero limiting value of \(\tau\) now show that every sufficiently small positive rational \(l/q\) occurs. Its parameter is unique, since two parameters with that value would give a plateau. Values on either side are strictly below and above \(2\pi l/q\); taking limits in (8) proves (14) on the entire graph. ◻

Proof. Choose \(\kappa<2\pi\) small enough that (2) and (6) place every line with angular step in \((0,\kappa)\) in \(U\). Decrease it further so that Proposition 5 applies to every positive rational smaller than \(\kappa/(2\pi)\). The generating relation and (16) give every stated transition; for the last transition use precisely the definition of \(p_q\) in (17). No endpoint stationarity has been assumed.

The initial state belongs to an individually invariant graph in \(U\), so every state of the chain belongs to that same graph. The equality \(\theta_q=\theta_0+2\pi l\) therefore forces \(p_q=p_0\). Repeating the orbit now shows that its translation number is \(2\pi l/q\). Since \(0<l/q<\kappa/(2\pi)\), uniqueness of the rational graph applies. In particular, all chains with the same \(q,l\) lie on the same graph, regardless of their starting angles.

Finally, differentiating the action along a \(C^1\) family cancels all interior terms by (16). The remaining terms are \(-p_0\dot\theta_0+p_q\dot\theta_q=0\), because the endpoint advance is fixed and the momenta agree. ◻

Proof. Apply Proposition 8 on the real axis. Its steps are \(2\pi/q+O(q^{-3})\), hence positive and uniformly small. Applying \(\phi\) gives a stationary chain for \(\mathcal{S}\), with positive uniformly small angular increments and total advance \(2\pi\). Proposition 5 and Lemma 6 show that its final momentum equals its initial momentum. Thus the chain is stationary also at the identified endpoints. More explicitly, differentiation of its critical action cancels every interior term, and the two endpoint terms are \[W_q'(x;u)=\frac{\phi'(x)}{c_*}(p_q-p_0)=0.\] Here \(\phi'(x+2\pi)=\phi'(x)\), and periodicity of \(h\) cancels the two endpoint contributions from the subtracted integral. The action is constant in \(x\), so its nonzero Fourier coefficients vanish. ◻

Proof. Contour displacement and \(m\) integrations by parts in (24) give \[ |[b_n(f)-1]_j|\le \frac{C}{|n|}(1+|j|)^{-m} e^{-(1-K/n^2)\sigma|j|}. \tag{26}\] For positive \(\sigma\), displace first to an interior closed strip and then take its width to the indicated limit; for \(\sigma=0\), only integration by parts is needed. Coefficient differences satisfy the same bound multiplied by \(\|f-\widehat f\|_{r_0,\sigma}\).

Increase \(N\) so that \(N^2\ge2K\), and set \(\alpha=K/n^2\le1/2\). The triangle inequality yields the useful identity and bound \[\begin{align*} |n|-|k|-(1-\alpha)|n-k| &=(1-\alpha)(|n|-|k|-|n-k|) +\alpha(|n|-|k|)\\ &\le \alpha|n|=K/|n|. \tag{27}\end{align*}\] Consequently the weighted matrix entry from \(k\) to \(n\) of \(R(f)\) is at most \[\frac{C}{|n|}(1+|n-k|)^{-m}e^{K\sigma/|n|}.\] At \(m=2\) its column sums are \(O(N^{-1})\), proving the stated operator bounds on weighted \(\ell^1\). A Neumann series gives the inverse. The equality \(C_{-q}(f)=\overline{C_q(f)}\) for real inputs gives the real symmetry. Finally, \[\frac{(1+|n|)^a}{(1+|k|)^a}\le(1+|n-k|)^a\] reduces the polynomially weighted estimate to the convergent sum of \((1+|n-k|)^{a-m}\). ◻

Proof of Theorem 7. Fix \(m=2\), \(r_0=17\), and \[M>4e(1+|u|_{0,r_0}).\] Choose \(N\) large enough for Lemmas 10 and 11, with \(\|R\|\le1/2\) throughout this input ball. These choices precede all choices of approximation parameters.

Let \(f_0=\sum_{|k|\le P}[u]_ke^{\mathrm{i}kx}\), where \(P>N\), and put \(\sigma_0=P^{-1}\). For every fixed \(a\ge0\), \[ |f_0|_{\sigma_0,a}\le e|u|_{0,a}. \tag{28}\] We claim that, for every fixed positive integer \(a\), \[ |C(f_0)|_{\sigma_0}=O(P^{-a}). \tag{29}\] For \(N\le|n|\le P\), the exponential weights are at most \(e\). Integrate the real derivative along the segment from \(u\) to \(f_0\), use \(C(u)=0\), and apply Lemma 11 at width zero. This part is bounded by \(C|f_0-u|_0\), which decays faster than any fixed power of \(P\) because \(u\) is smooth.

For \(|n|>P\), instead integrate along \(\lambda f_0\), \(0\le\lambda\le1\). The zero input has a uniform stationary mesh and a constant critical action, so \(C(0)=0\). For the chosen \(a\), use Proposition 9 with one fixed integer \(m>a+1\). Formula (28) bounds all its required input derivatives uniformly in \(P\). Once \(P\) exceeds this higher-order row threshold, real mesh uniqueness identifies these rows with the original ones. The final assertion of Lemma 11 gives \[\sum_{|n|>P}(1+|n|)^a|C_n(f_0)|e^{\sigma_0|n|} \le C_a|f_0|_{\sigma_0,a}\le C'_a.\] Division by \((1+P)^a\) proves the tail bound. This establishes (29); its constants and threshold may depend on \(a\), and no uniformity over all smoothness orders is asserted.

Keep the modes \(|k|<N\) fixed and set \[\sigma_j=\frac{1+2^{-j}}{2P},\qquad \sigma_\infty=\frac1{2P},\qquad \varepsilon_j=|C(f_j)|_{\sigma_j}.\] At the \(j\)-th step define \[ \eta_j=-A_H(f_j)^{-1}C(f_j),\qquad f_{j+1}=f_j+\eta_j. \tag{30}\] The correction is real symmetric, has only high modes, and obeys \(|\eta_j|_{\sigma_j}\le2\varepsilon_j\). Since \(\sigma_j-\sigma_{j+1}=2^{-j-2}/P\), the elementary Fourier smoothing estimate gives \[ \|\eta_j\|_{r_0,\sigma_{j+1}} \le C P^{r_0}2^{r_0j}\varepsilon_j. \tag{31}\] If the correcting segment is in the fixed input ball on this smaller strip, integration of the derivative difference gives \[ \varepsilon_{j+1}\le C\|\eta_j\|_{r_0,\sigma_{j+1}}|\eta_j|_{\sigma_{j+1}} \le C P^{r_0}2^{r_0j}\varepsilon_j^2. \tag{32}\] The linear term cancels by (30): compatibility of the real row derivatives makes the matrix computed at width \(\sigma_j\) the same matrix on the smaller strip. Equivalently, one can integrate each smooth real row first and then take the smaller-strip norm, avoiding any derivative assertion for rough variations at the original strip boundary.

Choose \(P\) so large that \(\varepsilon_0\le P^{-2r_0}\), possible by (29), and enlarge it further as needed below. The simultaneous induction is \[ \varepsilon_j\le P^{-2r_0}2^{-(r_0+1)j}, \qquad \|\eta_j\|_{r_0,\sigma_{j+1}} \le C P^{-r_0}2^{-j}. \tag{33}\] Indeed, before using (32), the known bound for \(\varepsilon_j\) and (31) put the entire current segment in the ball: the initial norm is less than \(M/4\), and the cumulative corrections are at most \(CP^{-r_0}\sum_{i\ge0}2^{-i}<M/4\). Thus the quadratic estimate is applicable. Its ratio to the proposed bound for \(\varepsilon_{j+1}\) is at most \(CP^{-r_0}2^{r_0+1-j}\), which is less than one for one sufficiently large fixed \(P\). This proves both the error bound and the input bound at every step.

The summable corrections converge on the positive-width strip \(\mathcal A_{\sigma_\infty}\), with all derivatives through \(r_0\), to a real-symmetric holomorphic function \(f_*\). Its low Fourier modes agree with those of \(u\), and continuity of each real mesh gives \(C_n(f_*)=0\) for every \(|n|\ge N\). The real segment between \(u\) and \(f_*\) lies in the same input ball. Put \(v=f_*-u=\Pi_Hv\). Integrating the real derivative, \[0=C(f_*)-C(u) =v+\int_0^1R(u+\lambda v)v\,\mathop{}\!\mathrm{d}\lambda.\] The bound \(\|R\|\le1/2\) implies \(|v|_0\le|v|_0/2\), hence \(v=0\). The original smooth \(u\) is therefore analytic.

Finally \(\phi'=e^u>0\), so \(\phi\) and its inverse are analytic. Equation (18) makes \(r\) analytic in the normal angle, and the equation \(h''+h=r\) makes \(h\) analytic. Thus \(h n_\theta+h'e_\theta\) is an analytic regular boundary parametrization, proving Theorem 7. ◻

Proof. The midpoint implicit function theorem, positivity of \(\phi'\) on the real axis, and compactness modulo the period give a uniform holomorphic domain for (19). Reversing its two endpoints gives antisymmetry. The calculation of (20) applies on this holomorphic domain. Differentiating at fixed second endpoint is the operator \(\partial_x-\partial_b\), which gives the second expansion. ◻

Proof. Antisymmetry gives \(\mathcal R=D_1(V,T_tV)-D_1(V,T_{-t}V)\). Put \[A_\pm=\frac{T_{\pm t}V-V}{\pm t} =\int_0^1 V'(y\pm ut,t)\,\mathop{}\!\mathrm{d}u.\] By (36), \[ \frac{E}{V'}= \frac{A_++A_-}{8}\frac{A_+-A_-}{t} +t\bigl[A_+^4B(V,tA_+)-A_-^4B(V,-tA_-)\bigr]. \tag{42}\] The remaining divided difference has the nonsingular expression \[\frac{A_+-A_-}{t} =\int_0^1\int_{-u}^{u}V''(y+vt,t)\,\mathop{}\!\mathrm{d}v\,\mathop{}\!\mathrm{d}u.\] These formulas prove regularity and the first identity in (39). Since \(D_{12}(x,x+b)=b/4+O(b^3)\), the second follows as well. They also show that the first two variations of \(E\), on domains with room for the shifts, have bounds in terms of a fixed number of derivatives through order two in \(y\). No negative power of \(t\) is needed in these bounds. Antisymmetry gives the stated parity.

For real \(t\), the mean action \[\frac1{2\pi}\int_0^{2\pi} D\bigl(V(y+\eta,t),V(y+\eta+t,t)\bigr)\,\mathop{}\!\mathrm{d}y\] is independent of \(\eta\). Differentiating at zero and shifting one integral gives \(\langle V'\mathcal R\rangle=0\). The mean-zero identity for complex \(t\) follows by holomorphic continuation.

For (40), separate each off-diagonal variation into its value at \(y\) and its difference from that value. With \[c_+=D_{12}(V,T_tV)V'T_tV',\qquad c_-=D_{12}(T_{-t}V,V)T_{-t}V'V',\] the difference terms are \[t^{-3}\bigl[c_+(T_tw-w)+c_-(T_{-t}w-w)\bigr] =\partial(a\partial w),\] since \(T_{\pm t/2}a=c_\pm/t\). The common-value terms, including the variation of \(V'\), are \((Ew)'\).

At an exact solution, differentiating the factor \(t^{-3}\) contributes zero. Differentiation of the explicit shifts contributes \((c_+-c_-)/t_0^3=\partial a/t_0\), whereas the assigned jet contributes \(\partial(a\partial z)\). This proves (41). ◻

Proof. We begin with \(t_0>0\). Write \(U_j(y)=V(y+jt_0,t_0)\) and seek interior-stationary nodes \[X_j=U_j+U'_j\xi_j,\qquad 0\le j\le q, \qquad \xi_0=\xi_q=0.\] The endpoint increment is \(2\pi l\). Multiply the interior stationarity equation at node \(j\) by the fixed factor \(t_0^{-3}U'_j\). At \(\xi=0\) the residual is \(E(y+jt_0,t_0)\). By (40), its derivative in \(\xi\) is the Dirichlet flux matrix \[ (A_0\xi)_j=t_0^{-2} \bigl[a_{j+1/2}(\xi_{j+1}-\xi_j) -a_{j-1/2}(\xi_j-\xi_{j-1})\bigr], \quad a_{j+1/2}=a(y+(j+1/2)t_0,t_0), \tag{48}\] plus the diagonal potential \[W_j=T_{jt_0}\left(E'-\frac{V''}{V'}E\right).\] The prefactor in the finite-dimensional equations is frozen; subtracting its variation explains the second term in this potential.

The inverse of \(A_0\) has a uniform complex bound. For \(A_0\xi=f\), let \(J_{j+1/2}=a_{j+1/2}(\xi_{j+1}-\xi_j)\). Then \[J_{j+1/2}=J_{1/2}+t_0^2\sum_{i=1}^j f_i.\] The condition \(\sum_{j=0}^{q-1}(\xi_{j+1}-\xi_j)=0\) determines \(J_{1/2}\). Since each \(a_{j+1/2}^{-1}\) is close to four, the modulus of their sum is at least a fixed positive multiple of \(q\). Summing the resulting increments gives \[ \|A_0^{-1}\|_{\infty\to\infty}\le Cq^2t_0^2\le CQ^2. \tag{49}\] On \(Y_1\), the potential is bounded by \(CQ\varepsilon\). Thus a Neumann series gives the same bound for the full inverse if \(CQ^3\varepsilon\) is small.

Reducedness supplies the useful lower bound \[ |t_0|^{-1}\le q/(2\pi)\le N/(2\pi). \tag{50}\] The nonlinear equations involve only three neighboring nodes, bounded fixed derivatives of \(D\), and the factor \(t_0^{-3}\). Their Hessian in the supremum norm therefore has size at most \(CQ^3\), independently of the number of nodes. Applying the linearized inverse to the nonlinear remainder defines a contraction on a ball of radius \(CQ^2\varepsilon\) when \(CQ^7\varepsilon\) is small. Hence \(\xi=O_{\mathrm p}(\varepsilon)\) on \(Y_1\), holomorphically in \(y\). Uniqueness preserves periodicity and real symmetry. Fixed derivatives are controlled on smaller strips by Cauchy’s estimate.

For real \(y\), the uncorrected increments are bounded below by a fixed multiple of \(t_0\). The correction, using (50) and a stronger fixed polynomial smallness condition, preserves positivity and keeps every step below \(3d\). After applying \(\phi\), these are actual stationary billiard chains: subtracting the endpoint primitive in the definition of \(D\) telescopes, and the coordinate change has nonzero derivative. Their initial states lie in the graph collar. Lemma 6 therefore gives momentum return as well as angle return. Each chain is periodic with rotation \(l/q\), and Proposition 5 identifies the same unique invariant graph for every starting \(y\).

Invert \(V(\,\cdot\,,t_0)\) locally around the uncorrected nodes to write \[X_j(y)=V(\widetilde y_j(y),t_0),\qquad \widetilde y_j=y+jt_0+O_{\mathrm p}(\varepsilon)\] on \(Y_2\). This inverse is obtained by contraction for the small displacement from \(y+jt_0\); it requires no assertion about a global complex inverse. On the real axis, uniqueness of the rational graph implies \(\widetilde y_j=\widetilde y_1^{\,j}\) and \(\widetilde y_q=y+2\pi l\). Consequently \[G(y)=\frac1q\sum_{j=0}^{q-1}\bigl(\widetilde y_j(y)-jt_0\bigr) \quad\hbox{satisfies}\quad G(\widetilde y_1(y))=G(y)+t_0.\] Its displacement and, on \(Y_3\), its derivative displacement are \(O_{\mathrm p}(\varepsilon)\). Contraction gives an inverse on \(Y_4\). Thus \[ V_*(y)=V(G^{-1}(y),t_0) \tag{51}\] is an exact solution at \(t_0\), with \(V_*-V(\,\cdot\,,t_0)=O_{\mathrm p}(\varepsilon)\). The stationarity identity follows first on the real axis from the graph dynamics and then throughout the strip by holomorphy.

We must also give \(V_*\) an appropriate first parameter jet. Initially assign \(\dot V_*=V_t(\,\cdot\,,t_0)\). Its residual jet is \(O_{\mathrm p}(\varepsilon)\) on \(Y_6\). Indeed Cauchy’s estimate gives \(E_t=O_{\mathrm p}(\varepsilon)\) there: the distance of \(t_0\) from the parameter boundary exceeds \(2\delta\). A polydisk of radius \(\delta/4\) in each variable about \((y,t_0)\), with \(y\in Y_6\), lies in \(\mathcal D_0^1\), so this estimate costs only \(C\delta^{-1}\). Replacing \(V\) by \(V_*\), with the same assigned jet, changes that derivative by \(O_{\mathrm p}(\varepsilon)\). To make its polynomial character explicit, if \(J\) is any assigned jet, direct differentiation gives \[\begin{align*} \dot E={}&-3t^{-4}V'\mathcal R+t^{-3}J'\mathcal R\\ &+t^{-3}V'\bigl[ D_{11}(V,T_tV)J+D_{12}(V,T_tV)(T_tJ+T_tV')\\ &\hspace{29mm} +D_{21}(T_{-t}V,V)(T_{-t}J-T_{-t}V') +D_{22}(T_{-t}V,V)J\bigr]. \tag{52}\end{align*}\] Only fixed derivatives occur, \(V_t\) and its needed \(y\)-derivatives have Cauchy bounds, and (50) controls every denominator.

Let \(\Pi=q^{-1}\sum_{j=0}^{q-1}T_{jt_0}\). This is a projection on periodic functions, and \(\Pi\partial=\partial\Pi=0\). Equation (41) at the exact solution shows that \(\Pi\dot E_*=0\). Put \(a_*=a(V_*)\); after the fixed derivative losses, it is still close to \(1/4\). On \(\ker\Pi\), the operator \(\partial(a_*\partial)\) has an inverse with polynomial norm. For an explicit verification, put \(T=T_{t_0}\) and \[R_q=\frac1q\sum_{j=0}^{q-1}jT^j.\] The identity \((T-1)R_q=1-\Pi\) gives \(\partial^{-1}=t_0R_qT_{t_0/2}\) on \(\ker\Pi\), of norm at most \(Cq|t_0|\). Projected multiplication \((1-\Pi)a_*:\ker\Pi\to\ker\Pi\) differs from multiplication by \(1/4\) by at most \(2\|a_*-1/4\|\), so is invertible by a Neumann series. Composing these three inverses yields \[\|\{\partial(a_*\partial)\}^{-1}\|_{\ker\Pi} \le Cq^2t_0^2.\] The projection commutes with all translations; no trivial action of the half-shift on its range is needed. We may therefore correct \(\dot V_*\) by \(V_*'z\), where \(z=O_{\mathrm p}(\varepsilon)\), to make its total residual jet zero.

Finally set \(w=(V_*-V)/V'\) and differentiate this quotient using the corrected jet of \(V_*\). Both \(w\) and \(\dot w\), with the fixed number of derivatives needed below, are \(O_{\mathrm p}(\varepsilon)\) after the losses up to \(Y_{10}\). Taylor’s formula and (40) give \[0=E(V_*)=E+\partial(a\partial w)+(Ew)' +O_{\mathrm p}(\varepsilon^2).\] The same formula differentiated in the assigned jets has a quadratic remainder. To see this directly, write the remainder as the integral of the second variation along the segment from \(V\) to \(V_*\). A first parameter derivative either replaces one of its two increments by its assigned jet, or differentiates a coefficient. The first case still contains two \(O_{\mathrm p}(\varepsilon)\) factors. The second costs only fixed derivatives in (52), with denominators at worst \(t_0^{-4}\), and also retains both factors. Cauchy estimates on the remaining fixed buffers absorb the required derivatives. Moreover \((Ew)'\) and its jet are quadratic since \(E,E_t,w,\dot w=O_{\mathrm p}(\varepsilon)\). This proves both (46) and (47). All costs are fixed powers of \(Q\): the mesh inverse costs \(Q^2\), its Hessian \(Q^3\), the cyclic inverse at most \(Q^2\), and all other steps use fixed derivatives and fixed Cauchy losses. Negative \(t_0\) follow by evenness, with the sign of the first jet reversed. ◻

Proof of Proposition 15. The first division takes \(F=-E\). Its mean vanishes by Lemma 14. Every nonzero divisor zero for a mode \(|n|\le N\) has reduced denominator at most \(N\). At those centers with \(|t_j|<d-3\delta\), Equation (46) and \(d_n(t_j)=0\) give \[|E_n(t_j)|\le CQ^B\varepsilon^2e^{-|n|(s-10\delta)}.\] Apply Lemma 17 with \[(j_0,\nu,a_{\mathrm{buf}},b_{\mathrm{buf}},J)=(0,1,10,3,11).\] Denote the result by \(H\). Then \[ H=O_{\mathrm p}(\varepsilon)\quad\hbox{on }\mathcal D_{11}^{1/2}, \qquad \partial H+E=O_{\mathrm p}(\varepsilon^2) \quad\hbox{on }\mathcal D_{11}^{1}. \tag{58}\] The extra polynomial factor multiplying \(\varepsilon^2\) in the bound for \(H\) is absorbed by the smallness requirement. Define \[ b=a^{-1}(H+\lambda),\qquad \lambda(t)=-\frac{\langle a^{-1}H\rangle}{\langle a^{-1}\rangle}. \tag{59}\] The denominator stays away from zero, so \(b\) has mean zero and is \(O_{\mathrm p}(\varepsilon)\) on \(\mathcal D_{12}^{1/2}\).

Before dividing again, we need the compatibility of \(b\) at its resonances. Fix a retained rational parameter \(t_0\) with \(|t_0|<d-14\delta\), and let \(g=a\partial w\) use the value and assigned jet from Lemma 16. For every retained nonzero mode, the exact first-division identity gives, as an identity of values and first jets at \(t_0\), \[ d_n(H_n-g_n)=-(E+\partial g)_n-R_n. \tag{60}\] The right-hand side and its first jet are bounded by \(CQ^B\varepsilon^2e^{-|n|(s-10\delta)}\), using both compatibility equations and (57).

If \(d_n(t_0)\ne0\), reducedness of \(t_0/(2\pi)=l/q\) gives \[|\sin(\pi nl/q)|\ge c/q,\qquad |d_n(t_0)|^{-1}\le Cq|t_0|.\] If \(d_n(t_0)=0\), then \(|d_n'(t_0)|=|n|/|t_0|\), and differentiating (60) bounds the same difference \(H_n-g_n\). Thus in both cases \[|H_n(t_0)-g_n(t_0)| \le CQ^B\varepsilon^2 e^{-|n|(s-10\delta)}.\] This is the reason for the first-jet compatibility: the value equation alone would give no estimate for a resonant coefficient of \(g\).

The function \(g\) has norm \(O_{\mathrm p}(\varepsilon)\) on \(Y_{10}\). Its omitted Fourier tail on \(Y_{13}\) is bounded by \(CQ^B\varepsilon e^{-3N\delta}=O_{\mathrm p}(\varepsilon^2)\). Summing the retained estimates on that smaller strip therefore gives \[H-(g-g_0)=O_{\mathrm p}(\varepsilon^2)\quad\hbox{on }Y_{13}.\] Since \(\langle a^{-1}g\rangle=\langle\partial w\rangle=0\), the mean correction in (59) satisfies \(\lambda=g_0+O_{\mathrm p}(\varepsilon^2)\). Hence \[ b-\partial w=O_{\mathrm p}(\varepsilon^2)\quad\hbox{on }Y_{13}. \tag{61}\] At a zero of \(d_n\), the coefficient \((\partial w)_n\) vanishes. Equation (61) supplies the second division with its required Fourier values.

Apply Lemma 17 to \(F=b\), with \[(j_0,\nu,a_{\mathrm{buf}},b_{\mathrm{buf}},J)=(12,1/2,13,15,17).\] All its centers satisfy the comparison range \(|t_0|<d-14\delta\). Its output \(v\) satisfies \[ v=O_{\mathrm p}(\varepsilon)\quad\hbox{on }\mathcal D_{17}^{0}, \qquad \partial v-b=O_{\mathrm p}(\varepsilon^2) \quad\hbox{on }\mathcal D_{17}^{1/2}. \tag{62}\] Both divisions, multiplication by \(a^{-1}\), and the mean adjustment preserve parity and real symmetry.

Since \(ab=H+\lambda(t)\) and \(\partial\lambda=0\), (58) and (62) yield \[E+\partial(a\partial v)=O_{\mathrm p}(\varepsilon^2) \quad\hbox{on }\mathcal D_{18}^{1}.\] Here the remaining difference costs only a derivative and a domain loss, by \(\partial f=\int_{-1/2}^{1/2}f'(y+ut,t)\,\mathop{}\!\mathrm{d}u\). Take \(\Delta V=V'v\). Cauchy estimates between indices 17 and 20 bound its first three \(y\)-derivatives by \(O_{\mathrm p}(\varepsilon)\). Equation (40) makes its linearized residual quadratic, because \((Ev)'=O_{\mathrm p}(\varepsilon^2)\). The nonsingular Formula (42) gives \[E(V+\Delta V)-E(V)-E_V(\Delta V)=O_{\mathrm p}(\varepsilon^2) \quad\hbox{on }\mathcal D_{20}^{1},\] including \(t=0\).

The construction uses fixed numbers of matrix inversions, contractions, products and derivatives, two applications of Lemma 17, and fixed Cauchy losses. All dimension and resonance costs are bounded by powers of \(N\), and all shrinking-domain costs by powers of \(\delta^{-1}\). The action bounds are fixed before \(d\) is shrunk. A single exponent \(B_0\) and constant \(C_0\) thus give (44)–(45) and absorb every smallness condition. This proves the Proposition with the asserted uniformity. ◻

Proof of Theorem 13. Fix a sufficiently small \(y\)-width \(s_0\); all constants in Proposition 15 are now fixed. Begin with (63) on radius \(d_0<s_0/4\), choosing \(m\) and then \(d_0\) below. At step \(i\), take \[\delta_i=2^{-i}d_0/200\] and reduce both current radii by \(20\delta_i\). Their limits satisfy \(d_\infty\ge4d_0/5\) and \(s_\infty\ge s_0-d_0/5\). If the current residual norm is \(\varepsilon_i>0\), choose \[N_i=\left\lceil\delta_i^{-1}\log(1/\varepsilon_i)\right\rceil.\] If it vanishes, an exact solution has already been obtained. Every fixed power of \(1+\log(1/\varepsilon_i)\) is bounded by a constant times \(\varepsilon_i^{-1/4}\). Thus, with one fixed enlarged exponent \(B\), Proposition 15 gives \[\begin{align*} \varepsilon_{i+1}&\le C d_0^{-B}2^{Bi}\varepsilon_i^{7/4}, \tag{64}\\ \|\Delta V_i\|_{C_y^3} &\le C d_0^{-B}2^{Bi}\varepsilon_i^{3/4} \tag{65}\end{align*}\] on the next domains, provided the polynomial smallness requirement holds.

Choose \(m\) so that \(2m+2>4B+4\), and only then shrink \(d_0\). The finite constant \(C_m\) is thereby absorbed, giving the initial bound in the induction \[ \varepsilon_i\le d_0^{4B+4}2^{-(4B+4)i}. \tag{66}\] Indeed substituting this bound in (64) and dividing by the desired next bound gives at most \[C d_0^{2B+3}2^{4B+4-(2B+3)i},\] which is at most one for small \(d_0\). The correction bound becomes \[\|\Delta V_i\|_{C_y^3} \le C d_0^{2B+3}2^{-(2B+3)i}.\] These bounds both imply the polynomial smallness condition at every step and keep the candidates in their fixed smallness class: the initializer has slack and the total correction is arbitrarily small as \(d_0\to0\). The induction is therefore justified before each application of the Newton estimate. The order of choices is \(B\), then \(m\), then \(d_0\).

The corrections converge uniformly, with their first three \(y\)-derivatives, on the limiting domains. Their limit is holomorphic, real-symmetric, even in \(t\), and periodic modulo \(y\), with \(V_y\) close to one. The residual tends to zero, so (35) holds wherever the two shifts remain in the limit domain. At \(t=0\), Equation (39) gives \(V''=0\); periodicity gives \(V(y,0)=y+c\), with real \(c\). Replacing \(y\) by \(y-c\) removes the constant and preserves the equation and all symmetries. Finally choose a smaller product neighborhood with enough \(y\)-margin for both shifts. This gives precisely the domain and conclusions stated in the Theorem. ◻