Proof. The operators \(\mathcal{F}_{\omega^N}\) and \(\mathcal{G}\) of 8 –9 map step fields to step fields. If \(\theta\in \mathfrak{S}_N^\theta\) and \(a\in \mathfrak{S}_N^a\), choose step representatives such that \(\theta=\theta_i\) a.e. on \(I_i^N\) and \(a=a_{ij}\) a.e. on \(I_i^N\times I_j^N\). Then, for a.e. \(x\in I_i^N\), \[\mathcal{F}_{\omega^N}(\theta,a)(x) = \omega_i^N+\sum_{j=1}^N\int_{I_j^N}a_{ij}\,\Gamma(\theta_j-\theta_i)\,\,\mathrm{d}y = \omega_i^N+\frac{1}{N}\sum_{j=1}^N a_{ij}\,\Gamma(\theta_j-\theta_i),\] and \(\mathcal{G}(\theta,a)(x,y)=-a_{ij}+H(\theta_i,\theta_j)\) for \((x,y)\in I_i^N\times I_j^N\). Thus the vector field of 10 preserves \(\mathfrak{S}_N\), and its restriction to \(\mathfrak{S}_N\simeq\mathbb{R}^N\times\mathbb{R}^{N\times N}\) coincides with 4 . Picard–Lindelöf on this finite-dimensional subspace identifies the restricted flow with the finite-\(N\) solution. ◻

Proof. For \(\mathcal{F}_\omega\), add and subtract \(a_2(x,y)\Gamma(\theta_1(y)-\theta_1(x))\): \[\begin{align} &|\mathcal{F}_\omega(\theta_1,a_1)(x)-\mathcal{F}_\omega(\theta_2,a_2)(x)| \\ &\le \int_0^1 |a_1-a_2|(x,y)\,|\Gamma(\theta_1(y)-\theta_1(x))|\,\,\mathrm{d}y \\ &\quad + \int_0^1 |a_2(x,y)|\, |\Gamma(\theta_1(y)-\theta_1(x)) - \Gamma(\theta_2(y)-\theta_2(x))|\,\,\mathrm{d}y \\ &\le M_\Gamma\|a_1-a_2\|_{L^{\infty}([0,1]^2)} + R_aL_\Gamma\int_0^1 \big(|\theta_1(y)-\theta_2(y)|+|\theta_1(x)-\theta_2(x)|\big)\,\,\mathrm{d}y \\ &\le M_\Gamma\|a_1-a_2\|_{L^{\infty}([0,1]^2)} +2R_aL_\Gamma\|\theta_1-\theta_2\|_{L^{\infty}([0,1])}. \end{align}\] Taking the essential supremum in \(x\) gives 11 .

For \(\mathcal{G}\), we use \[\begin{align} &|\mathcal{G}(\theta_1,a_1)(x,y)-\mathcal{G}(\theta_2,a_2)(x,y)| \\ &\le |a_1-a_2|(x,y) + |H(\theta_1(x),\theta_1(y)) - H(\theta_2(x),\theta_2(y))| \\ &\le \|a_1-a_2\|_{L^{\infty}([0,1]^2)} + {M_{DH}}\big(|\theta_1(x)-\theta_2(x)|+|\theta_1(y)-\theta_2(y)|\big) \\ &\le \|a_1-a_2\|_{L^{\infty}([0,1]^2)} +2 {M_{DH}}\|\theta_1-\theta_2\|_{L^{\infty}([0,1])}. \end{align}\] Taking the essential supremum in \((x,y)\) gives 12 . The local Lipschitz claim for \(\mathcal{V}_\varepsilon\) is immediate. ◻

Proof. Set \[f(t):=H\bigl(\theta(t,\cdot),\theta(t,\cdot)\bigr)\in X_a.\] More explicitly, \(f(t)(x,y)=H(\theta(t,x),\theta(t,y))\), and \[\|f(t)-f(s)\|_{X_a}\le 2 {M_{DH}}\|\theta(t)-\theta(s)\|_{X_\theta};\] hence \(t\mapsto f(t)\) is continuous as an \(X_a\)-valued function. The fast equation is the linear inhomogeneous equation \[\varepsilon a'(t)=-a(t)+f(t)\] in \(X_a\), or \(a'(t)=-\varepsilon^{-1}a(t)+\varepsilon^{-1}f(t)\), where \(-\varepsilon^{-1}I\) generates the strongly continuous semigroup \(e^{-t/\varepsilon}I\). Duhamel’s formula therefore gives \[a(t)=e^{-t/\varepsilon}a_0+\frac{1}{\varepsilon}\int_0^t e^{-(t-s)/\varepsilon} H\bigl(\theta(s,\cdot),\theta(s,\cdot)\bigr)\,\,\mathrm{d}s \quad\text{in }X_a,\] which is 13 . Taking \(\|\cdot\|_{X_a}\) and using \(\|H(\theta(s,\cdot),\theta(s,\cdot))\|_{X_a}\le M_H\) gives 14 . ◻

Proof. By Lemma 2, \(\mathcal{V}_\varepsilon\) is locally Lipschitz on the Banach space \(X\). Hence Picard–Lindelöf for ODEs in Banach spaces yields local existence and uniqueness. ◻

Proof. By Lemma 3, \(\|a^\ell(t)\|_{L^{\infty}([0,1]^2)}\le R_a\) for \(\ell=1,2\), hence Lemma 2 applies along both trajectories. Subtracting the two variation-of-constants formulas from Lemma 3 and using the Lipschitz bound for \(H\) gives \[\|\delta a(t)\|_{L^{\infty}} \le e^{-t/\varepsilon}\|\delta a(0)\|_{L^{\infty}} + \frac{2 {M_{DH}}}{\varepsilon}\int_0^t e^{-(t-s)/\varepsilon}\|\delta\theta(s)\|_{L^{\infty}}\,\,\mathrm{d}s. \label{eq:sec2-delta-a-voc}\tag{15}\] Integrating 15 in time and applying Fubini’s theorem yields \[\int_0^t \|\delta a(s)\|_{L^{\infty}}\,\,\mathrm{d}s \le \varepsilon\|\delta a(0)\|_{L^{\infty}} + 2 {M_{DH}}\int_0^t\|\delta\theta(s)\|_{L^{\infty}}\,\,\mathrm{d}s, \label{eq:sec2-delta-a-integral}\tag{16}\] because \(\int_0^t e^{-s/\varepsilon}\,\,\mathrm{d}s\le\varepsilon\) and \(\int_r^t \varepsilon^{-1}e^{-(s-r)/\varepsilon}\,\,\mathrm{d}s\le1\). Integrating only the slow equation and using 11 gives \[\begin{align} \|\delta\theta(t)\|_{L^{\infty}} &\le \|\delta\theta(0)\|_{L^{\infty}} +t\|\delta\omega\|_{L^{\infty}} +\int_0^t \Big(M_\Gamma\|\delta a(s)\|_{L^{\infty}} +2R_aL_\Gamma\|\delta\theta(s)\|_{L^{\infty}}\Big)\,\,\mathrm{d}s. \end{align}\] Substituting 16 into this bound gives \[\|\delta\theta(t)\|_{L^{\infty}} \le \|\delta\theta(0)\|_{L^{\infty}} +t\|\delta\omega\|_{L^{\infty}} +\varepsilon M_\Gamma\|\delta a(0)\|_{L^{\infty}} +\Lambda_a\int_0^t\|\delta\theta(s)\|_{L^{\infty}}\,\,\mathrm{d}s.\] The integral form of Grönwall’s inequality yields ?? . Inserting that bound into 15 and using the monotonicity of the right-hand side in time gives ?? ; summing the two estimates gives ?? . ◻

Proof of Theorem 1. Local existence and uniqueness follow from Proposition 3. To prove global existence, fix a local solution on \([0,T_{\mathrm{loc}}]\). Lemma 3 gives \[\|a(t)\|_{L^{\infty}([0,1]^2)}\le R_a^*:=\max\{\|a_0\|_{L^{\infty}([0,1]^2)},M_H\}\] for all times where the solution exists. Therefore \[\|\partial_t\theta(t)\|_{L^{\infty}([0,1])} \le \|\omega\|_{L^{\infty}([0,1])} + M_\Gamma R_a^*,\] and therefore \[\|\theta(t)\|_{L^{\infty}([0,1])} \le \|\theta_0\|_{L^{\infty}([0,1])} +t\bigl(\|\omega\|_{L^{\infty}([0,1])}+M_\Gamma R_a^*\bigr)\] on every interval on which the local solution exists. The Banach-space continuation criterion therefore rules out finite-time blow-up, and the maximal existence time is infinite.

Uniqueness follows from ?? with identical data. To obtain the Lipschitz estimate on a ball of radius \(R\), apply Proposition 4 to two solutions with data in that ball. Then \(R_a\le \max\{R,M_H\}\), so \[\Lambda_a \le 2L_\Gamma\max\{R,M_H\}+2M_\Gamma {M_{DH}}.\] For every \(\varepsilon_0>0\), the right-hand side of ?? is therefore bounded uniformly for \(\varepsilon\in(0,\varepsilon_0]\) by replacing \(\varepsilon M_\Gamma\) with \(\varepsilon_0M_\Gamma\). This gives the claimed finite-time Lipschitz bound into \(C([0,T];X)\), uniformly for \(\varepsilon\in(0,\varepsilon_0]\). This proves the theorem. ◻

Proof of Theorem 2. For each \(N\), let \((\theta^N,a^N)\) denote the embedded step-field solution of 4 –5 , and let \((\theta,a)\) solve 10 . By Lemma 1, \((\theta^N,a^N)\) is a strong solution of 10 with data \((\theta_0^N,a_0^N,\omega^N)\). Hence we may apply Proposition 4 with \((\theta^1,a^1,\omega^1)=(\theta^N,a^N,\omega^N)\) and \((\theta^2,a^2,\omega^2)=(\theta,a,\omega)\). Using \(R_a\le R_{a,*}\) in ?? and ?? gives the two componentwise bounds stated in the theorem, and hence \[\sup_{t\in[0,T]} \|(\theta^N,a^N)(t)-(\theta,a)(t)\|_X \le E_a^N + (1+2 {M_{DH}})e^{\Lambda_*T} \bigl(E_\theta^N+TE_\omega^N+\varepsilon M_\Gamma E_a^N\bigr).\] Assumption 2 [ass:sec2-A4] implies the right-hand side tends to zero, proving both the quantitative estimate and convergence. The final uniform bound for \(\varepsilon\in(0,\varepsilon_0]\) follows by replacing \(\varepsilon M_\Gamma\) with \(\varepsilon_0M_\Gamma\). ◻

Proof. On any cell \(I_i^N\) of length \(1/N\), the Lipschitz hypothesis on \(\omega\) gives \(|\omega(x)-\omega(x')|\le L_\omega/N\) for all \(x,x'\in I_i^N\); the same applies to \(\theta_0\) on \([0,1]\) and to \(a_0\) on the \(2\)-dimensional cell \(I_i^N\times I_j^N\) of diameter \(\sqrt{2}/N\). Cell-evaluation therefore gives the displayed step-approximation bound. Substituting into Theorem 2 yields the trajectory bound. ◻

Proof. For \(x\in I_i^N\), the operators \(K[\theta](x)\), \(P[\theta;\omega](x)\), and \(T[\theta](x)\) are constant on \(I_i^N\) whenever \(\theta,\omega\in \mathfrak{S}_N^\theta\). Indeed, their integrals over cells \(I_j^N\) and \(I_k^N\) reduce exactly to the \(1/N\)- and \(1/N^2\)-weighted sums in 17 . Hence the reduced continuum vector field is tangent to \(\mathfrak{S}_N^\theta\), and Picard–Lindelöf on the finite-dimensional subspace identifies the restricted flow with the finite-\(N\) reduced ODE. ◻

Proof. The bounds 27 –29 follow immediately from 20 –22 and \(|[0,1]|=1\).

Lipschitz bound for \(K\). Add and subtract \(H(\theta_2(x),\theta_2(y))\Gamma(\theta_1(y)-\theta_1(x))\) in the integrand: \[\begin{align} |K[\theta_1](x)-K[\theta_2](x)| &\le \int_0^1 |H(\theta_1(x),\theta_1(y))-H(\theta_2(x),\theta_2(y))|\, |\Gamma(\theta_1(y)-\theta_1(x))|\,\,\mathrm{d}y \\ &\quad + \int_0^1 |H(\theta_2(x),\theta_2(y))|\, |\Gamma(\theta_1(y)-\theta_1(x))-\Gamma(\theta_2(y)-\theta_2(x))|\,\,\mathrm{d}y. \end{align}\] The first integral is bounded by \(2 {M_{DH}}M_\Gamma\|\theta_1-\theta_2\|_{L^{\infty}}\) and the second by \(2M_HL_\Gamma\|\theta_1-\theta_2\|_{L^{\infty}}\), giving \(C_K=2( {M_{DH}}M_\Gamma+M_HL_\Gamma)\).

Lipschitz bound for \(P\). Split \[P[\theta_1;\omega_1]-P[\theta_2;\omega_2] =\big(P[\theta_1;\omega_1]-P[\theta_2;\omega_1]\big) +\big(P[\theta_2;\omega_1]-P[\theta_2;\omega_2]\big).\] The second term is linear in \(\omega\) with bounded kernel, giving the \(C_P^{\omega}\|\omega_1-\omega_2\|_{L^{\infty}}\) contribution with \(C_P^{\omega}=2M_\Gamma {M_{DH}}\). For the first term, each of the two summands in \(P\) contains the product \(\Gamma(\theta(y)-\theta(x))\partial_rH(\theta(x),\theta(y))\omega(\cdot)\). The \(\Gamma\)-factor contributes at most \(2L_\Gamma {M_{DH}}M_\omega \|\theta_1-\theta_2\|_{L^{\infty}}\), and the \(\partial_rH\)-factor contributes at most \(2M_\Gamma {M_{D^2H}}M_\omega\|\theta_1-\theta_2\|_{L^{\infty}}\). Summing over \(r=1,2\), we may take \[C_P^\theta=4M_\omega\bigl(L_\Gamma {M_{DH}}+M_\Gamma {M_{D^2H}}\bigr).\]

Lipschitz bound for \(T\). Each of the two integrands in 22 is a product of four bounded Lipschitz factors depending on \(\theta(x),\theta(y),\theta(z)\). Using the telescoping identity \[\textstyle\prod_{m=1}^4 a_m-\prod_{m=1}^4 b_m =\sum_{m=1}^4 (a_m-b_m)\prod_{n<m}b_n\prod_{n>m}a_n,\] one double-integral term is bounded by \[2\Bigl(2L_\Gamma {M_{DH}}M_HM_\Gamma+ {M_{D^2H}}M_HM_\Gamma^2+ {M_{DH}}^2M_\Gamma^2\Bigr) \|\theta_1-\theta_2\|_{L^{\infty}}.\] The second double-integral term has the same bound, so 26 holds with \[C_T^\theta = 4\Bigl(2L_\Gamma {M_{DH}}M_HM_\Gamma+ {M_{D^2H}}M_HM_\Gamma^2+ {M_{DH}}^2M_\Gamma^2\Bigr).\] ◻

Proof. Local existence/uniqueness follows from Lemma 5 and Picard–Lindelöf on \(X_{\mathrm{red}}\). Bound ?? is the direct estimate of the right-hand side in 23 using 27 –29 . Let \(M_{\partial}\) denote the right-hand side of ?? . Integrating the derivative bound gives \[\|\theta(t)\|_{L^{\infty}([0,1])} \le \|\theta_0\|_{L^{\infty}([0,1])}+tM_{\partial}\] on every interval on which the local solution exists. Hence the \(X_{\mathrm{red}}=L^{\infty}([0,1])\) norm stays finite on every finite time interval, and the standard Banach-space continuation criterion rules out finite-time blow-up. The local solution therefore extends globally. ◻

Proof. Subtract the two reduced equations and use 24 –26 : \[\|\theta^1(t)-\theta^2(t)\|_{L^{\infty}} \le \|\theta_0^1-\theta_0^2\|_{L^{\infty}} +tB_\varepsilon\|\omega^1-\omega^2\|_{L^{\infty}} +\Lambda_\varepsilon\int_0^t \|\theta^1(s)-\theta^2(s)\|_{L^{\infty}}\,\,\mathrm{d}s.\] Apply Grönwall. ◻

Proof of Theorem 3. Global existence and uniqueness are given by Proposition 9. Lipschitz dependence on \((\theta_0,\omega)\) on \(R\)-balls follows from Proposition 10 with \(M_\omega\le R\). ◻

Proof of Theorem 4. By Lemma 4, \(\theta^N\) is a strong solution of 23 with data \((\theta_0^N,\omega^N)\). Apply Proposition 10 with \((\theta^1,\omega^1)=(\theta^N,\omega^N)\) and \((\theta^2,\omega^2)=(\theta,\omega)\). By Remark 8, the constant \(M_\omega\) in Proposition 10 may be chosen uniformly in \(N\), namely \(M_\omega\le M_\omega^{\mathrm{step}}\). This yields \[\sup_{t\in[0,T]}\|\theta^N(t)-\theta(t)\|_{L^{\infty}([0,1])} \le e^{\Lambda_\varepsilon T}\max\{1,TB_\varepsilon\} \Big( \|\theta_0^N-\theta_0\|_{L^{\infty}([0,1])} +\|\omega^N-\omega\|_{L^{\infty}([0,1])} \Big).\] Set \(C_T^{\mathrm{red}}:=e^{\Lambda_\varepsilon T}\max\{1,TB_\varepsilon\}\). Assumption 7 [ass:sec3-B3] implies convergence to zero. ◻

Proof. On any cell of length \(1/N\), the Lipschitz hypothesis gives the cell-evaluation bound \(\|\omega^N-\omega\|_{L^{\infty}([0,1])}\le L_\omega/N\) and \(\|\theta_0^N-\theta_0\|_{L^{\infty}([0,1])}\le L_{\theta_0}/N\). Substituting into Theorem 4 yields the displayed estimate. ◻

Proof. Since \(\mathcal{G}(\theta,a)= -a+h_0(\theta)\), one has \(D_2\mathcal{G}(\theta,a)=-I_{X_a}\). For the full layer linearisation at \((h_0(\theta),\theta)\), \[\partial_\tau\delta a=-\delta a+\mathrm{D}h_0(\theta)\delta\theta,\qquad \partial_\tau\delta\theta=0.\] Writing \(\delta\xi=\delta a-\mathrm{D}h_0(\theta)\delta\theta\) gives \(\partial_\tau\delta\xi=-\delta\xi\), which is the claimed splitting. ◻

Proof. Fix \(\eta<0\), \(\theta_0\in X_\theta\), and \((\xi,\theta),(\tilde{\xi},\tilde{\theta})\in {\mathcal{X}_\eta}\). Using the estimates above, together with \[e^{-\eta t}\int_{-\infty}^t e^{-\varepsilon^{-1}(t-s)}e^{\eta s}\,\,\mathrm{d}s =\int_0^\infty e^{-(\varepsilon^{-1}+\eta)\tau}\,\,\mathrm{d}\tau =\frac{\varepsilon}{1+\varepsilon\eta}, \qquad e^{-\eta t}\int_t^0 e^{\eta s}\,\,\mathrm{d}s\le \frac{1}{|\eta|},\] we obtain \[\begin{align} \|\mathcal{L}^{(1)}_{\varepsilon,\theta_0,r}(\xi,\theta)\|_{C_{\eta,X_a}} &\le C_1(r)\frac{\varepsilon}{1+\varepsilon\eta},\\ \|\mathcal{L}^{(2)}_{\varepsilon,\theta_0,r}(\xi,\theta)\|_{C_{\eta,X_\theta}} &\le \|\theta_0\|_{C_{\eta,X_\theta}}+\frac{C_2(r)}{|\eta|}=\|\theta_0\|_{X_\theta}+\frac{C_2(r)}{|\eta|}. \end{align}\] Taking the difference between two solutions yields \[\begin{align} \|\mathcal{L}^{(1)}_{\varepsilon,\theta_0,r}(\xi,\theta)-\mathcal{L}^{(1)}_{\varepsilon,\theta_0,r}(\tilde{\xi},\tilde{\theta})\|_{C_{\eta,X_a}} &\le C_3(r)\frac{\varepsilon}{1+\varepsilon\eta}\,{\|(\xi,\theta)-(\tilde{\xi},\tilde{\theta})\|_{\mathcal{X}_\eta}},\\ \|\mathcal{L}^{(2)}_{\varepsilon,\theta_0,r}(\xi,\theta)-\mathcal{L}^{(2)}_{\varepsilon,\theta_0,r}(\tilde{\xi},\tilde{\theta})\|_{C_{\eta,X_\theta}} &\le \frac{C_4(r)}{|\eta|}\,{\|(\xi,\theta)-(\tilde{\xi},\tilde{\theta})\|_{\mathcal{X}_\eta}}. \end{align}\]

Set \[q_{\varepsilon,\eta,r} := C_3(r)\frac{\varepsilon}{1+\varepsilon\eta} +\frac{C_4(r)}{|\eta|}.\]

Choose \(\eta_r<0\) so that \(C_4(r)/|\eta_r|<1/2\). Then choose \(\varepsilon_{0,r}>0\) sufficiently small that \[1+\varepsilon\eta_r>0,\qquad q_{\varepsilon,\eta_r,r}<1,\] for every \(\varepsilon\in(0,\varepsilon_{0,r}]\). With these choices, \(\mathcal{L}_{\varepsilon,\theta_0,r}\) maps the Banach space \(\mathcal{X}_{\eta_r}\) into itself by the bounds above, and is a \(q_{\varepsilon,\eta_r,r}\)-contraction there by the difference estimates. Hence the Banach fixed-point theorem gives a unique fixed point in \(\mathcal{X}_{\eta_r}\). ◻

Proof. Fix the cut-off level \(r>M_H\), and let \(\eta_r<0\), \(\varepsilon_{0,r}>0\) be given by Lemma 6. For \(\theta_0\in X_\theta\) and \(\varepsilon\in(0,\varepsilon_{0,r}]\), denote by \[(\xi_{\varepsilon,r}^{\theta_0},\theta_{\varepsilon,r}^{\theta_0})\] the unique fixed point of \(\mathcal{L}_{\varepsilon,\theta_0,r}\) in \(\mathcal{X}_{\eta_r}\).

Property (i): Define \[\label{eq:he-def} h_{\varepsilon,r}(\theta_0):=h_0(\theta_0)+\xi_{\varepsilon,r}^{\theta_0}(0),\tag{46}\] and \[a_{\varepsilon,r}^{\theta_0}(t):=h_0(\theta_{\varepsilon,r}^{\theta_0}(t))+\xi_{\varepsilon,r}^{\theta_0}(t),\qquad t\le 0.\] By construction, \((a^{\theta_0}_{\varepsilon,r},\theta^{\theta_0}_{\varepsilon,r})\) is a mild solution of the truncated system on \((-\infty,0]\), with \[\theta^{\theta_0}_{\varepsilon,r}(0)=\theta_0,\qquad a^{\theta_0}_{\varepsilon,r}(0)=h_{\varepsilon,r}(\theta_0).\] This yields the graph representation.

Before claiming invariance for the original system, we first show that the cut-off is inactive on the Lyapunov–Perron orbit. Because \((\xi_{\varepsilon,r}^{\theta_0},\theta_{\varepsilon,r}^{\theta_0})\) is a fixed point, \[\begin{align} \left\lVert \xi_{\varepsilon,r}^{\theta_0}(t) \right\rVert_{X_a} &\leq \int_{-\infty}^t e^{-\varepsilon^{-1}(t-s)} \|\mathrm{D}h_0(\theta_{\varepsilon,r}^{\theta_0}(s))B_r(\xi_{\varepsilon,r}^{\theta_0}(s), \theta_{\varepsilon,r}^{\theta_0}(s))\|_{X_a}\,\,\mathrm{d}s\\ &\leq C_1(r)\int_{-\infty}^t e^{-\varepsilon^{-1}(t-s)}\,\,\mathrm{d}s =\varepsilon C_1(r) \end{align}\] for all \(t\le 0\). Hence, at \(t=0\), \[\label{eq:sec4-initial-graph-close} \left\lVert h_{\varepsilon,r}(\theta_0)-h_0(\theta_0) \right\rVert_{X_a} = \left\lVert \xi_{\varepsilon,r}^{\theta_0}(0) \right\rVert_{X_a}\le \varepsilon C_1(r),\tag{47}\] and for every \(t\le 0\), \[\|a_{\varepsilon,r}^{\theta_0}(t)\|_{X_a} \le \|h_0(\theta_{\varepsilon,r}^{\theta_0}(t))\|_{X_a} +\|\xi_{\varepsilon,r}^{\theta_0}(t)\|_{X_a} \le M_H+\varepsilon C_1(r)<r\] by 38 . Therefore \(\mathsf{N}_r(a_{\varepsilon,r}^{\theta_0}(t))=a_{\varepsilon,r}^{\theta_0}(t)\) for all \(t\le 0\), so the truncated orbit is also an orbit of the original transformed system 32 –33 . We set \[h_\varepsilon:=h_{\varepsilon,r},\qquad \theta_\varepsilon^{\theta_0}:=\theta_{\varepsilon,r}^{\theta_0},\qquad a_\varepsilon^{\theta_0}:=a_{\varepsilon,r}^{\theta_0}.\]

Next, we show the Lipschitz continuity of the map \(h_\varepsilon\). Let \(\theta_0,\widetilde{\theta}_0\in X_\theta\), and write \[(\xi,\theta):=(\xi_{\varepsilon,r}^{\theta_0},\theta_{\varepsilon,r}^{\theta_0}), \qquad (\widetilde{\xi},\widetilde{\theta}):=(\xi_{\varepsilon,r}^{\widetilde{\theta}_0},\theta_{\varepsilon,r}^{\widetilde{\theta}_0}).\] Using the fixed-point identities from Lemma 6 we obtain \[{\left\lVert (\xi,\theta)-(\widetilde{\xi},\widetilde{\theta}) \right\rVert_{\mathcal{X}_{\eta_r}}} \le q\,{\left\lVert (\xi,\theta)-(\widetilde{\xi},\widetilde{\theta}) \right\rVert_{\mathcal{X}_{\eta_r}}} + \left\lVert \theta_0-\widetilde{\theta}_0 \right\rVert_{X_\theta},\] where \(q:=q_{\varepsilon,\eta_r,r}<1\). Hence \[\label{eq:fp-lip} {\left\lVert (\xi,\theta)-(\widetilde{\xi},\widetilde{\theta}) \right\rVert_{\mathcal{X}_{\eta_r}}} \le \frac{1}{1-q}\left\lVert \theta_0-\widetilde{\theta}_0 \right\rVert_{X_\theta}.\tag{48}\] By 39 , \(q\le q_*<1\), so \(1/(1-q)\le 1/(1-q_*)\) uniformly in \(\varepsilon\). Using the Lipschitz estimate for \(h_0\), we conclude that \[\begin{align} \left\lVert h_{\varepsilon,r}(\theta_0)-h_{\varepsilon,r}(\widetilde{\theta}_0) \right\rVert_{X_a} &\le \left\lVert h_0(\theta_0)-h_0(\widetilde{\theta}_0) \right\rVert_{X_a}+\left\lVert \xi_{\varepsilon,r}^{\theta_0}(0)-\xi_{\varepsilon,r}^{\widetilde{\theta}_0}(0) \right\rVert_{X_a}\\ &\le \Bigl(2 {M_{DH}}+\frac{1}{1-q_*}\Bigr)\left\lVert \theta_0-\widetilde{\theta}_0 \right\rVert_{X_\theta}. \end{align}\] Thus \(h_{\varepsilon,r}\) is globally Lipschitz, with a constant uniform for \(\varepsilon\in(0,\varepsilon_{0,r}]\).

Concerning invariance, let \(\tau\le 0\) and define \[\xi_\tau(t):=\xi_{\varepsilon,r}^{\theta_0}(t+\tau), \qquad \theta_\tau(t):=\theta_{\varepsilon,r}^{\theta_0}(t+\tau).\] The shifted pair belongs to \(\mathcal{X}_{\eta_r}\). Indeed, if \(u\) denotes either \(\xi_{\varepsilon,r}^{\theta_0}\) or \(\theta_{\varepsilon,r}^{\theta_0}\), with target space \(Y=X_a\) or \(Y=X_\theta\), then with \(s=t+\tau\) \[e^{-\eta_r t}\|u(t+\tau)\|_Y = e^{\eta_r\tau}e^{-\eta_r s}\|u(s)\|_Y \le e^{\eta_r\tau}\|u\|_{C_{\eta_r,Y}}<\infty.\] A direct change of variables in the definition of \(\mathcal{L}_{\varepsilon,\theta_0,r}\) then shows that \((\xi_\tau,\theta_\tau)\) is a fixed point of \(\mathcal{L}_{\varepsilon,\theta_\varepsilon^{\theta_0}(\tau),r}\); the weighted bound above, together with \(1+\varepsilon\eta_r>0\), is the condition ensuring convergence of the improper fast integral at \(-\infty\). By Lemma 6, the fixed point in \(\mathcal{X}_{\eta_r}\) is unique. Hence \[h_\varepsilon(\theta_\varepsilon^{\theta_0}(\tau)) = h_0(\theta_\varepsilon^{\theta_0}(\tau))+\xi_\tau(0) = h_0(\theta_\varepsilon^{\theta_0}(\tau))+\xi_{\varepsilon,r}^{\theta_0}(\tau) = a_\varepsilon^{\theta_0}(\tau).\] Thus every Lyapunov–Perron backward orbit lies on the graph. Forward invariance for the truncated system is the content of Lemma 7 below: applied with anchor \(\theta_\ast\), it states that the unique forward solution \((\widetilde{a},\widetilde{\theta})\) of the truncated system on \([0,T]\) starting at \((h_\varepsilon(\theta_\ast),\theta_\ast)\) satisfies \(\widetilde{a}(\tau)=h_\varepsilon(\widetilde{\theta}(\tau))\) for every \(\tau\in[0,T]\). The proof, recorded separately below, is the standard six-step concatenation argument: concatenate the backward Lyapunov–Perron orbit anchored at \(\theta_\ast\) with the forward solution into a single curve, observe that translating in time produces another fixed point of the Lyapunov–Perron operator, and apply uniqueness of the fixed point. The estimate proved above, \[\|h_\varepsilon(\theta)\|_{X_a}\le M_H+C_1(r)\varepsilon<r,\] shows that the cut-off is inactive on the resulting orbit, so it is also a solution of the original (uncut-off) system. Uniqueness of the original Cauchy problem then gives forward invariance of \(S_\varepsilon\) for the original dynamics on \([0,T]\); since \(T\) was arbitrary, this holds for all \(t\ge 0\).

Property (ii). The estimate already obtained at \(t=0\), namely 47 , is 41 with \(C_r=C_1(r)\).

Property (iii).

By definition, \[h_\varepsilon(\theta_0)=h_0(\theta_0)+\Theta_\varepsilon(\theta_0).\] Lemma 16 shows that \(\Theta_\varepsilon\) is \(C^1\) and that \[\mathrm{D}h_\varepsilon(\theta_0)\phi = \mathrm{D}h_0(\theta_0)\phi+U_{\theta_0}^{\phi}(0).\] Therefore \(h_\varepsilon\in C^1(X_\theta,X_a)\). Lemma 17 gives \[\sup_{\theta_0\in X_\theta} \|\mathrm{D}h_\varepsilon(\theta_0)-\mathrm{D}h_0(\theta_0)\|_{\mathcal{B}(X_\theta,X_a)} \le \frac{2C_3(r)}{1-q_*}\,\varepsilon.\] Thus property (iii) holds with \[C_{D,r}:=\frac{2C_3(r)}{1-q_*},\] and \(S_\varepsilon\) is a \(C^1\)-manifold.

Property (iv). By the cut-off inactivity estimate proved in Property (i), \(\mathcal{F}_r(\theta,h_\varepsilon(\theta)) =\mathcal{F}(\theta,h_\varepsilon(\theta))\); hence the slow-manifold identities used below are the same for the truncated and original systems. Let \[L:=2 {M_{DH}}+\frac{1}{1-q_*}.\] By the previous properties \[L_{h,\varepsilon,r}:=\sup\|\mathrm{D}h_\varepsilon(\vartheta)\|_{\mathcal{B}(X_\theta,X_a)}\le 2 {M_{DH}}+\frac{1}{1-q_*}=:L_r<\infty,\] for all \(\varepsilon\in(0,\varepsilon_{0,r}]\). After possibly shrinking \(\varepsilon_{0,r}\) once more, define \[\mu_r:=1-\varepsilon_{0,r}M_\Gamma L_r>0.\] Hence, for every \(\varepsilon\in(0,\varepsilon_{0,r}]\), \[\varepsilon^{-1}-M_\Gamma L_{h,\varepsilon,r} \ge \varepsilon^{-1}-M_\Gamma L_r = \frac{1-\varepsilon M_\Gamma L_r}{\varepsilon} \ge \frac{\mu_r}{\varepsilon}.\]

Now let \((a,\theta)\) be a solution of the original system on \([0,T]\), and set \[d(t):=a(t)-h_\varepsilon(\theta(t)).\] Invariance of the graph gives \[\mathrm{D}h_\varepsilon(\theta)\,\mathcal{F}(\theta,h_\varepsilon(\theta))=\varepsilon^{-1}\bigl(-h_\varepsilon(\theta)+h_0(\theta)\bigr),\] so subtracting the equation for \(h_\varepsilon(\theta(t))\) from the equation for \(a(t)\) yields \[\partial_t d = -\varepsilon^{-1}d -\mathrm{D}h_\varepsilon(\theta)\Bigl(\mathcal{F}(\theta,h_\varepsilon(\theta)+d)-\mathcal{F}(\theta,h_\varepsilon(\theta))\Bigr).\] Since \(\mathcal{F}\) is globally Lipschitz in its second variable with constant \(M_\Gamma\), \[\|\mathcal{F}(\theta,h_\varepsilon(\theta)+d)-\mathcal{F}(\theta,h_\varepsilon(\theta))\|_{X_\theta}\le M_\Gamma \|d\|_{X_a}.\] Therefore the upper Dini derivative of \(\|d(t)\|_{X_a}\) satisfies \[D^+\|d(t)\|_{X_a} \le -\Bigl(\varepsilon^{-1}-M_\Gamma L_{h,\varepsilon,r}\Bigr)\|d(t)\|_{X_a} \le -\frac{\mu_r}{\varepsilon}\|d(t)\|_{X_a},\] and Gronwall’s inequality gives 42 .

Next, let \((\bar a,\bar\theta)\) be the solution on \(S_{\varepsilon}\) with \(\bar\theta(0)=\theta(0)\), and define \[\delta\theta(t):=\theta(t)-\bar\theta(t).\] Then \[\partial_t\delta\theta = \mathcal{F}(\theta,h_\varepsilon(\theta)+d)-\mathcal{F}(\bar\theta,h_\varepsilon(\bar\theta)).\] Let \[R_a:=\max\{\|a_0\|_{X_a},M_H,r\}.\] By the a priori bound from Section 2, \(\|a(t)\|_{X_a}\le \max\{\|a_0\|_{X_a},M_H\}\), while \(\|\bar a(t)\|_{X_a}\le r\). Hence \[\|\partial_t\delta\theta(t)\|_{X_\theta} \le M_\Gamma\|d(t)\|_{X_a} + (M_\Gamma L_{h,\varepsilon,r}+2R_aL_\Gamma)\|\delta\theta(t)\|_{X_\theta} .\] Since \(R_a\le \max\{\rho,M_H,r\}\) under the standing hypothesis \(\|a_0\|_{X_a}\le \rho\), the Gronwall coefficient \(M_\Gamma L_{h,\varepsilon,r}+2R_a L_\Gamma\) is bounded by a constant that depends only on \(r\), \(\rho\), and the data of the problem. Consequently the constant \(C_{r,T,\rho}\) appearing below depends on \(r\), \(T\), and \(\rho\), but is otherwise independent of the particular initial datum \((a_0,\theta_0)\). Integrating in time, using \(\delta\theta(0)=0\), and applying 42 and Gronwall’s inequality, we obtain \[\sup_{0\le t\le T}\|\delta\theta(t)\|_{X_\theta} \le C_{r,T,\rho}\int_0^T e^{-\mu_r s/\varepsilon}\,\,\mathrm{d}s\,\|d(0)\|_{X_a} \le C_{r,T,\rho}\,\varepsilon\,\|d(0)\|_{X_a},\] which is 43 . Finally, \[a(t)-\bar a(t)=d(t)+h_\varepsilon(\theta(t))-h_\varepsilon(\bar\theta(t)),\] so the Lipschitz bound for \(h_\varepsilon\), together with 42 and 43 , yields 44 . ◻

Proof. We concatenate the backward Lyapunov–Perron orbit anchored at \(\theta_\ast\) with the forward solution and apply uniqueness of the Lyapunov–Perron fixed point.

1. Let \((\xi^{\theta_\ast}_{\varepsilon,r},\theta^{\theta_\ast}_{\varepsilon,r})\) denote the backward Lyapunov–Perron fixed point anchored at \(\theta_\ast\), so that \[a^{\theta_\ast}_{\varepsilon,r}(t):=h_0(\theta^{\theta_\ast}_{\varepsilon,r}(t))+\xi^{\theta_\ast}_{\varepsilon,r}(t),\qquad t\le 0,\] is a backward solution of the truncated system with \((a^{\theta_\ast}_{\varepsilon,r}(0),\theta^{\theta_\ast}_{\varepsilon,r}(0))=(h_\varepsilon(\theta_\ast),\theta_\ast)\).

2. Existence and uniqueness of the forward solution \((\widetilde{a},\widetilde{\theta})\) on \([0,T]\) are guaranteed by the global Lipschitz bounds for the truncated rectified fields \(B_r\) and \(A_r=\mathrm{D}h_0\,B_r\) imposed by the cut-off \(\mathsf N_r\).

3. Concatenate the two pieces into the single curve \[\gamma\colon(-\infty,T]\to X_a\times X_\theta,\qquad \gamma(t):= \begin{cases} (a^{\theta_\ast}_{\varepsilon,r}(t),\theta^{\theta_\ast}_{\varepsilon,r}(t)), & t\le 0,\\ (\widetilde{a}(t),\widetilde{\theta}(t)), & 0\le t\le T, \end{cases}\] which agrees on \(t=0\) and is continuous; it is a mild solution of the truncated system on \((-\infty,T]\).

4. Define \(\widetilde{\xi}(t):=\widetilde{a}(t)-h_0(\widetilde{\theta}(t))\) on \([0,T]\). For fixed \(\tau\in[0,T]\), set, for \(t\le0\), \[(\xi_\tau(t),\theta_\tau(t)):= \begin{cases} (\xi^{\theta_\ast}_{\varepsilon,r}(t+\tau),\theta^{\theta_\ast}_{\varepsilon,r}(t+\tau)), & t+\tau\le 0,\\ (\widetilde{\xi}(t+\tau),\widetilde{\theta}(t+\tau)), & 0\le t+\tau\le T. \end{cases}\] This is a backward mild solution of the transformed truncated system 34 –35 , anchored at \(\widetilde{\theta}(\tau)\).

5. The pair \((\xi_\tau,\theta_\tau)\) lies in \(\mathcal{X}_{\eta_r}\). On the part \(t+\tau\le0\), this follows from the same weighted translation estimate used above. On the finite interval \(-\tau\le t\le0\), the concatenated forward solution is continuous and the factor \(e^{-\eta_r t}\) is bounded, so the weighted norm is finite.

6. Since \((\xi_\tau,\theta_\tau)\) is a backward mild solution and belongs to \(\mathcal{X}_{\eta_r}\), the improper fast integral at \(-\infty\) is legitimate; equivalently, the homogeneous boundary term there vanishes. Hence \((\xi_\tau,\theta_\tau)\) satisfies the Lyapunov–Perron fixed-point equations with anchor \(\widetilde{\theta}(\tau)\). By the uniqueness statement in Lemma 6, it is the Lyapunov–Perron fixed point anchored at \(\widetilde{\theta}(\tau)\). Evaluating at \(t=0\) gives \[\widetilde{a}(\tau)=h_0(\widetilde{\theta}(\tau))+\xi_\tau(0)=h_\varepsilon(\widetilde{\theta}(\tau)),\] so the forward orbit lies on the graph for every \(\tau\in[0,T]\).

 ◻

Proof. By 41 , \[\|h_\varepsilon(\theta)\|_{X_a} \le \|h_0(\theta)\|_{X_a}+\|h_\varepsilon(\theta)-h_0(\theta)\|_{X_a} \le M_H+C_1(r)\varepsilon \le M_H+C_1(r)\varepsilon_{0,r} <r\] by 38 . Since the \(L^\infty\)-norm is the essential supremum, this implies \(|h_\varepsilon(\theta)(x,y)|<r\) for almost every \((x,y)\in[0,1]^2\), hence \(\mathsf{N}_r(h_\varepsilon(\theta))=h_\varepsilon(\theta)\). The identity for \(\mathcal{F}_r\) follows immediately from its definition. ◻

Proof. Let \((\bar a,\bar\theta)\) be the trajectory on \(S_{\varepsilon}\) with \(\bar\theta(0)=\theta\). Since \(\bar a(t)=h_\varepsilon(\bar\theta(t))\), differentiating gives \[\partial_t \bar a = \mathrm{D}h_\varepsilon(\bar\theta)\,\partial_t\bar\theta = \mathrm{D}h_\varepsilon(\bar\theta)\,\mathcal{F}(\bar\theta,h_\varepsilon(\bar\theta)).\] On the other hand, the fast equation reads \[\partial_t \bar a = \varepsilon^{-1}\mathcal{G}(\bar\theta,h_\varepsilon(\bar\theta)) = \varepsilon^{-1}\big({-h_\varepsilon(\bar\theta)+h_0(\bar\theta)}\big).\] Equating at \(t=0\) and multiplying by \(\varepsilon\) yields ?? . Since \(\theta\in X_\theta\) is arbitrary and the slow manifold is invariant, one can always choose the trajectory \((\bar a, \bar\theta)\) with \(\bar\theta(0) = \theta\), so ?? holds for every \(\theta \in X_\theta\). ◻

Proof. Insert \(h_\varepsilon=h_0+\varepsilon h_1 + r_\varepsilon\) into ?? . The left-hand side becomes \[-h_0(\theta)-\varepsilon h_1(\theta) - r_\varepsilon(\theta)+h_0(\theta) = -\varepsilon h_1(\theta) - r_\varepsilon(\theta).\] By the definition ?? , the left-hand side is \(\varepsilon\,\mathrm{D}h_0(\theta)\mathcal{F}(\theta,h_0(\theta))-r_\varepsilon(\theta)\). Rearranging the exact invariance equation gives \[\begin{align} \label{eq:sec5-r-split} r_\varepsilon(\theta) &= \varepsilon\,\mathrm{D}h_0(\theta)\,\mathcal{F}(\theta,h_0(\theta)) -\varepsilon\,\mathrm{D}h_\varepsilon(\theta)\,\mathcal{F}(\theta,h_\varepsilon(\theta))\nonumber\\ &= \varepsilon\bigl(\mathrm{D}h_0(\theta)-\mathrm{D}h_\varepsilon(\theta)\bigr) \mathcal{F}(\theta,h_\varepsilon(\theta)) +\varepsilon\,\mathrm{D}h_0(\theta)\bigl(\mathcal{F}(\theta,h_0(\theta)) -\mathcal{F}(\theta,h_\varepsilon(\theta))\bigr). \end{align}\tag{50}\] Thus the two possible sources of error are derivative-closeness and graph-closeness.

Term involving derivative closeness. By Theorem 5(iii), \[\|\mathrm{D}h_\varepsilon(\theta)-\mathrm{D}h_0(\theta)\|_{\mathcal{B}(X_\theta,X_a)} \le C_{D,r}\varepsilon\] uniformly in \(\theta\). The slow vector field is uniformly bounded along the slow manifold by \[\|\mathcal{F}(\theta,h_\varepsilon(\theta))\|_{X_\theta} \le M_\omega+M_\Gamma(M_H+C_1(r)\varepsilon_{0,r});\] call this bound \(\overline{M}_{\mathcal{F},r}\). Hence \[\bigl\|\bigl(\mathrm{D}h_0(\theta)-\mathrm{D}h_\varepsilon(\theta)\bigr)\mathcal{F}(\theta,h_\varepsilon(\theta))\bigr\|_{X_a} \le C_{D,r}\,\overline{M}_{\mathcal{F},r}\,\varepsilon, \qquad\text{uniformly in }\theta\in X_\theta.\]

Term involving graph closeness. The Lipschitz bound for \(\mathcal{F}\) in its second argument and 41 give \[\|\mathcal{F}(\theta,h_\varepsilon(\theta))-\mathcal{F}(\theta,h_0(\theta))\|_{X_\theta} \le M_\Gamma\|h_\varepsilon(\theta)-h_0(\theta)\|_{X_a} \le M_\Gamma C_1(r)\varepsilon,\] together with \(\|\mathrm{D}h_0(\theta)\|_{\mathcal{B}(X_\theta,X_a)}\le 2 {M_{DH}}\), yielding \[\bigl\|\mathrm{D}h_0(\theta)\bigl(\mathcal{F}(\theta,h_0(\theta))-\mathcal{F}(\theta,h_\varepsilon(\theta))\bigr)\bigr\|_{X_a} \le 2 {M_{DH}}M_\Gamma C_1(r)\,\varepsilon, \qquad\text{uniformly in }\theta\in X_\theta.\]

Combining the two estimates in 50 gives \[\|r_\varepsilon(\theta)\|_{X_a} \le \bigl(C_{D,r}\overline{M}_{\mathcal{F},r}+2 {M_{DH}}M_\Gamma C_1(r)\bigr)\varepsilon^2, \qquad\text{uniformly in }\theta\in X_\theta.\]

This proves the claimed \(O(\varepsilon^2)\) bound.

For the explicit formula, recall from ?? that \[\big(\mathrm{D}h_0(\theta)\,\varphi\big)(x,y) = \partial_1 H\!\big(\theta(x),\theta(y)\big)\,\varphi(x) +\partial_2 H\!\big(\theta(x),\theta(y)\big)\,\varphi(y),\] and on the critical manifold \[\mathcal{F}(\theta,h_0(\theta))(x) = \omega(x)+\int_0^1 H\!\big(\theta(x),\theta(z)\big)\,\Gamma\!\big(\theta(z)-\theta(x)\big)\,\,\mathrm{d}z.\] Substituting \(\varphi=\mathcal{F}(\theta,h_0(\theta))\) and applying the minus sign gives ?? . ◻

Proof. The critical-manifold graph \(h_0\), the Nemytskii cut-off \(\mathsf N_r\), and the operators \(\mathcal{F}_{\omega^N}\) and \(\mathcal{G}\) all preserve \(\mathfrak{S}_N\), by Lemma 1 and pointwise inspection of their definitions. The Lyapunov–Perron operator \(\mathcal{L}_{\varepsilon,\theta_0,r}\) of Lemma 6 therefore maps the closed subspace \(C_{\eta_r}(\mathfrak{S}_N^a)\times C_{\eta_r}(\mathfrak{S}_N^\theta)\) into itself whenever \(\theta_0\in \mathfrak{S}_N^\theta\); its contractivity on the ambient weighted product space \(\mathcal{X}_{\eta_r}\) localises to this subspace, so the fixed point \((\xi^{\theta_0},\theta^{\theta_0})\) lies there. Evaluating at \(t=0\) and using \(h_{\varepsilon,\omega^N}(\theta_0)=h_0(\theta_0)+\xi^{\theta_0}(0)\) yields \(h_{\varepsilon,\omega^N}(\theta_0)\in \mathfrak{S}_N^a\).

If \(\widetilde{\vartheta}=\vartheta+2\pi m\) with \(m\in\mathbb{Z}^N\), then the associated step fields \(\theta^{N,\vartheta}\) and \(\theta^{N,\widetilde{\vartheta}}\) induce the same restricted finite-dimensional vector field because \(\Gamma(\widetilde{\vartheta}_j-\widetilde{\vartheta}_i)=\Gamma(\vartheta_j-\vartheta_i)\) and \(H(\widetilde{\vartheta}_i,\widetilde{\vartheta}_j)=H(\vartheta_i,\vartheta_j)\). Thus the restricted fixed point depends only on the torus class \([\vartheta]\).

Read as a Lyapunov–Perron problem on \(\mathbb{R}^N\times\mathbb{R}^{N\times N}\) for the finite-\(N\) unreduced ODE (Lemma 1), the restricted fixed point is an exact slow-manifold graph for that ODE, lying in the cut-off ball on which the cut-off is inactive (Lemma 8). The identification now uses only uniqueness of this selected Lyapunov–Perron fixed point, not generic uniqueness of slow manifolds. We do not claim that slow manifolds near \(S_0^N\) are unique as invariant manifolds. The slow directions are parametrised by the anchor \(\theta_0\); once the anchor, the cut-off level \(r\), the weight \(\eta_r\), and the Lyapunov–Perron boundary condition on \((-\infty,0]\) are fixed, Lemma 6 gives a unique fixed point in \(\mathcal{X}_{\eta_r}\). Concretely, the same Lyapunov–Perron operator \(\mathcal{L}_{\varepsilon,\theta_0,r}\) used in Lemma 6 restricts, when \(\theta_0\in \mathfrak{S}_N^\theta\), to a contraction on the closed subspace \[C_{\eta_r}(\mathfrak{S}_N^a)\times C_{\eta_r}(\mathfrak{S}_N^\theta)\simeq C_{\eta_r}(\mathbb{R}^{N\times N})\times C_{\eta_r}(\mathbb{R}^N),\] and produces a unique fixed point there. Since the closed step-field subspace is invariant under the same operator, this ambient fixed point with step-field anchor coincides with the fixed point of the restricted finite-dimensional contraction. By definition, the coefficient graph of this unique fixed point is \(h_{\varepsilon,\omega^N}^N\). Since \(h_{\varepsilon,\omega^N}(\theta^{N,\vartheta})\) is obtained by the same fixed point, restricted to the same finite-dimensional subspace, its coefficients are exactly \(h_{\varepsilon,\omega^N}^N([\vartheta])\). Passage from \(\vartheta\in\mathbb{R}^N\) to its torus class \([\vartheta]\in\mathbb{T}^N\) uses the periodicity argument established just above. ◻

Proof. Let \(\vartheta\in\mathbb{R}^N\) and let \(\theta^{N,\vartheta}\in \mathfrak{S}_N^\theta\) be the associated step field. Define the continuum remainder at the step-field anchor by \[r_\varepsilon(\theta^{N,\vartheta}) := h_{\varepsilon,\omega^N}(\theta^{N,\vartheta})-h_0(\theta^{N,\vartheta})-\varepsilon h_1(\theta^{N,\vartheta}).\] The algebraic formulas \(h_0(\theta)(x,y)=H(\theta(x),\theta(y))\) and ?? send step-field inputs to step-field outputs, and Lemma 9 does the same for \(h_{\varepsilon,\omega^N}\); hence \(r_\varepsilon(\theta^{N,\vartheta})\in \mathfrak{S}_N^a\), and its step-coefficients coincide with \(r_\varepsilon^N([\vartheta])\). Since \(\omega=\omega^N\) is a step field with \(\|\omega^N\|_{L^{\infty}([0,1])}\le M_\omega^{\mathrm{step}}\), Proposition 25 therefore gives a constant independent of \(N\): \[\|r_\varepsilon^N([\vartheta])\|_{\ell^\infty} = \|r_\varepsilon(\theta^{N,\vartheta})\|_{X_a} \le C(r)\,\varepsilon^2,\] with \(C(r)\) the constant tracked in the proof of Proposition 25, depending on the continuum data listed above and on \(M_\omega^{\mathrm{step}}\), but not on \(N\). ◻

Proof. By Lemma 8 the cut-off is inactive on the slow manifold, so \(\mathcal{F}(\theta,h_\varepsilon(\theta))=\mathcal{F}_r(\theta,h_\varepsilon(\theta))\). The reduced dynamics on the slow manifold are \[\partial_t\theta(x) = \mathcal{F}(\theta,h_\varepsilon(\theta))(x) = \omega(x) + \int_0^1 h_\varepsilon[\theta](x,y)\,\Gamma\!\big(\theta(y)-\theta(x)\big)\,\,\mathrm{d}y.\] Substituting \(h_\varepsilon=h_0+\varepsilon h_1+r_\varepsilon\) gives \[\begin{align} \partial_t\theta(x) &= \omega(x) +\int_0^1 H\!\big(\theta(x),\theta(y)\big)\,\Gamma\!\big(\theta(y)-\theta(x)\big)\,\,\mathrm{d}y \\ &\quad +\varepsilon\int_0^1 h_1[\theta](x,y)\,\Gamma\!\big(\theta(y)-\theta(x)\big)\,\,\mathrm{d}y + R_\varepsilon^{\mathrm{CF}}[\theta](x), \end{align}\] where \[R_\varepsilon^{\mathrm{CF}}[\theta](x):= \int_0^1 r_\varepsilon[\theta](x,y)\, \Gamma\!\big(\theta(y)-\theta(x)\big)\,\,\mathrm{d}y .\] The first line is \(\omega(x)+K[\theta](x)\).

It remains to show that \(\int_0^1 h_1[\theta](x,y)\,\Gamma(\theta(y)-\theta(x))\,\,\mathrm{d}y = P[\theta;\omega](x)+T[\theta](x)\).

Substituting the explicit formula ?? for \(h_1\) yields \[\begin{align} &\int_0^1 h_1[\theta](x,y)\,\Gamma\!\big(\theta(y)-\theta(x)\big)\,\,\mathrm{d}y \\ &= -\int_0^1 \Gamma\!\big(\theta(y)-\theta(x)\big)\,\partial_1 H\!\big(\theta(x),\theta(y)\big) \bigg[\omega(x)+\int_0^1 H\!\big(\theta(x),\theta(z)\big)\,\Gamma\!\big(\theta(z)-\theta(x)\big)\,\,\mathrm{d}z\bigg]\,\,\mathrm{d}y\\ &\quad -\int_0^1 \Gamma\!\big(\theta(y)-\theta(x)\big)\,\partial_2 H\!\big(\theta(x),\theta(y)\big) \bigg[\omega(y)+\int_0^1 H\!\big(\theta(y),\theta(z)\big)\,\Gamma\!\big(\theta(z)-\theta(y)\big)\,\,\mathrm{d}z\bigg]\,\,\mathrm{d}y. \end{align}\] We separate the \(\omega\)-dependent and the integral terms.

\(\omega\)-dependent terms. Collecting the terms involving \(\omega\) gives \[\begin{align} &-\int_0^1 \Gamma\!\big(\theta(y)-\theta(x)\big) \Big(\partial_1 H\!\big(\theta(x),\theta(y)\big)\,\omega(x) +\partial_2 H\!\big(\theta(x),\theta(y)\big)\,\omega(y)\Big)\,\,\mathrm{d}y = P[\theta;\omega](x), \end{align}\] by the definition 21 .

Integral terms. The remaining double integrals are \[\begin{align} &-\int_0^1 \Gamma\!\big(\theta(y)-\theta(x)\big)\,\partial_1 H\!\big(\theta(x),\theta(y)\big) \int_0^1 H\!\big(\theta(x),\theta(z)\big)\,\Gamma\!\big(\theta(z)-\theta(x)\big)\,\,\mathrm{d}z\,\,\mathrm{d}y\\ &-\int_0^1 \Gamma\!\big(\theta(y)-\theta(x)\big)\,\partial_2 H\!\big(\theta(x),\theta(y)\big) \int_0^1 H\!\big(\theta(y),\theta(z)\big)\,\Gamma\!\big(\theta(z)-\theta(y)\big)\,\,\mathrm{d}z\,\,\mathrm{d}y. \end{align}\] Since \(\Gamma\), \(H\), \(\partial_1 H\), \(\partial_2 H\) are bounded and measurable, each integrand is in \(L^\infty([0,1]^2)\). By Fubini’s theorem we may write each term as a double integral over \((y,z)\in[0,1]^2\): \[\begin{align} &-\int_0^1\!\int_0^1 \Gamma\!\big(\theta(y)-\theta(x)\big)\,\partial_1 H\!\big(\theta(x),\theta(y)\big)\, H\!\big(\theta(x),\theta(z)\big)\,\Gamma\!\big(\theta(z)-\theta(x)\big)\,\,\mathrm{d}z\,\,\mathrm{d}y\\ &-\int_0^1\!\int_0^1 \Gamma\!\big(\theta(y)-\theta(x)\big)\,\partial_2 H\!\big(\theta(x),\theta(y)\big)\, H\!\big(\theta(y),\theta(z)\big)\,\Gamma\!\big(\theta(z)-\theta(y)\big)\,\,\mathrm{d}z\,\,\mathrm{d}y = T[\theta](x), \end{align}\] by the definition 22 . Combining gives 51 .

The remainder in 51 comes entirely from the manifold remainder \(r_\varepsilon\). Since \(\|r_\varepsilon(\theta)\|_{X_a}\le C_r\varepsilon^2\) uniformly in \(X_\theta\) and \(\Gamma\) is bounded, \[\|R_\varepsilon^{\mathrm{CF}}[\theta]\|_{X_\theta} \le M_\Gamma C_r\varepsilon^2,\] which gives the stated bound. ◻

Proof. For part (i), write the finite-\(N\) graph expansion as \[h_\varepsilon^N=h_0^N+\varepsilon h_1^N+r_\varepsilon^N.\] Substituting this into the finite-\(N\) phase vector field shows that the graph remainder contributes exactly \[R_{\varepsilon,i}^{N,\mathrm{RF}}([\vartheta]) = \frac{1}{N}\sum_{j=1}^N r_{\varepsilon,ij}^N([\vartheta])\Gamma(\vartheta_j-\vartheta_i).\] Therefore \[\|R_\varepsilon^{N,\mathrm{RF}}[\theta^{N,\vartheta}]\|_{X_\theta} \le M_\Gamma\|r_\varepsilon^N([\vartheta])\|_{\ell^\infty} \le M_\Gamma C(r)\varepsilon^2\] by Corollary 3. The remaining zeroth- and first-order terms are the embedded forms of \(K\), \(P[\cdot;\omega^N]\), and \(T\), as in Section 3.

Part (ii) is Theorem 6, applied with the frequency profile \(\Omega\). The constants are uniform over the stated frequency class because the estimates in Sections 4–5 depend on the frequency only through the common \(L^\infty\)-bound.

For 52 , apply Lemma 9 to the continuum system with frequency profile \(\omega^N\). The continuum graph \(h_{\varepsilon,\omega^N}(\theta^{N,\vartheta})\) then lies in \(\mathfrak{S}_N^a\) and has step coefficients \(h_{\varepsilon,\omega^N}^N([\vartheta])\). Since \(\mathcal{F}_{\omega^N}\) restricted to \(\mathfrak{S}_N^\theta\times \mathfrak{S}_N^a\) is precisely the embedded finite-\(N\) phase vector field, the exact vector fields agree.

Finally, subtract the expansions in (i) and (ii) with \(\Omega=\omega\). The operators \(K\) and \(T\) cancel, while \[\|P[\theta;\omega^N]-P[\theta;\omega]\|_{X_\theta} \le 2M_\Gamma {M_{DH}}\,\|\omega^N-\omega\|_{X_\theta}\] by the explicit formula for \(P\). The two named remainders are bounded by constants times \(\varepsilon^2\), giving 53 after renaming the constant. ◻

Proof. Write \(V\) in the form 54 . Since \(\kappa\) has bounded derivatives up to order two, differentiation under the integral sign is justified by dominated convergence. Hence \[\begin{align} DV[\theta]\phi(x) &= \int_0^1 \Big( \partial_1\kappa\bigl(\theta(x),\theta(y);x,y\bigr)\phi(x) +\partial_2\kappa\bigl(\theta(x),\theta(y);x,y\bigr)\phi(y) \Big)\,\,\mathrm{d}y, \end{align}\] and a second differentiation gives \[\begin{align} D^2V[\theta](\phi,\psi)(x) &= \int_0^1 \partial_{11}\kappa\bigl(\theta(x),\theta(y);x,y\bigr)\phi(x)\psi(x)\,\,\mathrm{d}y\\ &\quad +\int_0^1 \partial_{12}\kappa\bigl(\theta(x),\theta(y);x,y\bigr) \bigl(\phi(x)\psi(y)+\psi(x)\phi(y)\bigr)\,\,\mathrm{d}y\\ &\quad +\int_0^1 \partial_{22}\kappa\bigl(\theta(x),\theta(y);x,y\bigr)\phi(y)\psi(y)\,\,\mathrm{d}y. \end{align}\] If \(\phi(x)=\psi(x)=0\) and \(\operatorname{supp}\phi\cap\operatorname{supp}\psi=\varnothing\), then \(\phi(y)\psi(y)=0\) for every \(y\in [0,1]\), so all three terms vanish. This proves ?? . ◻

Proof. If such a pairwise kernel representation were valid for all \(\theta\in L^{\infty}([0,1])\), then the same formula, with the same smooth bounded-kernel assumptions, would hold after restricting to the subspace \(C([0,1])\subset L^{\infty}([0,1])\). This would make \(V|_{C([0,1])}\) pairwise-representable, contradicting the assumed genuine nonpairwiseness on \(C([0,1])\). ◻

Proof. This is the contrapositive of Proposition 31. ◻

Proof. Under the stated regularity assumptions, \(F\in C_b^2(\mathbb{R}^3)\). Standard Nemytskii calculus on \(C([0,1])\) shows that \(T:C([0,1])\to C([0,1])\) is \(C^2\), and differentiation under the double integral yields \[\begin{align} D^2T[\theta](\phi,\psi)(x) &= \int_0^1\!\int_0^1 \partial_{11}F(\Theta)\,\phi(x)\psi(x)\,\,\mathrm{d}z\,\,\mathrm{d}y \nonumber\\ &\quad +\int_0^1\!\int_0^1 \partial_{12}F(\Theta)\bigl(\phi(x)\psi(y)+\psi(x)\phi(y)\bigr)\,\,\mathrm{d}z\,\,\mathrm{d}y \nonumber\\ &\quad +\int_0^1\!\int_0^1 \partial_{13}F(\Theta)\bigl(\phi(x)\psi(z)+\psi(x)\phi(z)\bigr)\,\,\mathrm{d}z\,\,\mathrm{d}y \nonumber\\ &\quad +\int_0^1\!\int_0^1 \partial_{22}F(\Theta)\,\phi(y)\psi(y)\,\,\mathrm{d}z\,\,\mathrm{d}y \nonumber\\ &\quad +\int_0^1\!\int_0^1 \partial_{33}F(\Theta)\,\phi(z)\psi(z)\,\,\mathrm{d}z\,\,\mathrm{d}y \nonumber\\ &\quad +\int_0^1\!\int_0^1 \partial_{23}F(\Theta)\bigl(\phi(y)\psi(z)+\psi(y)\phi(z)\bigr)\,\,\mathrm{d}z\,\,\mathrm{d}y, \label{eq:sec5-second-variation-T} \end{align}\tag{57}\] where \(\Theta=(\theta(x),\theta(y),\theta(z))\).

Choose \(x_\ast\in [0,1]\) and two disjoint closed intervals \(I_y,I_z\subset [0,1]\setminus\{x_\ast\}\) with nonempty interiors. Let \(\theta\in C([0,1])\) satisfy \[\theta(x_\ast)=u_\ast, \qquad \theta\equiv v_\ast \text{ on }I_y, \qquad \theta\equiv w_\ast \text{ on }I_z,\] and choose nonnegative functions \(\phi,\psi\in C([0,1])\) such that \[\operatorname{supp}\phi\subset I_y, \qquad \operatorname{supp}\psi\subset I_z, \qquad \int_0^1 \phi(y)\,\,\mathrm{d}y>0, \qquad \int_0^1 \psi(z)\,\,\mathrm{d}z>0.\] Then \(\phi(x_\ast)=\psi(x_\ast)=0\) and \(\operatorname{supp}\phi\cap\operatorname{supp}\psi=\varnothing\). Hence every term in 57 vanishes except the \(\partial_{23}F\) contribution, and because \(\theta\) is constant on \(\operatorname{supp}\phi\) and \(\operatorname{supp}\psi\), we obtain \[\begin{align} D^2T[\theta](\phi,\psi)(x_\ast) &= \int_0^1\!\int_0^1 \partial_{23}F\bigl(u_\ast,v_\ast,w_\ast\bigr)\phi(y)\psi(z)\,\,\mathrm{d}z\,\,\mathrm{d}y\\ &\quad +\int_0^1\!\int_0^1 \partial_{23}F\bigl(u_\ast,w_\ast,v_\ast\bigr)\psi(y)\phi(z)\,\,\mathrm{d}z\,\,\mathrm{d}y\\ &= M(u_\ast,v_\ast,w_\ast) \Bigl(\int_0^1\phi(y)\,\,\mathrm{d}y\Bigr) \Bigl(\int_0^1\psi(z)\,\,\mathrm{d}z\Bigr). \end{align}\] By 56 , this quantity is nonzero. Proposition 32 therefore implies that \(T\) is genuinely nonpairwise on \(C([0,1])\). By Lemma 10, \(T\) is also genuinely nonpairwise in the original \(L^\infty\)-based continuum setting, within the same smooth bounded-kernel class. ◻

Proof. For this choice of \(\Gamma\) and \(H\), the integrand 55 becomes \[F(u,v,w) = -\sin^2(v-u)\bigl(\alpha+\cos(u-w)\bigr)\sin(w-u) +\sin^2(v-u)\bigl(\alpha+\cos(v-w)\bigr)\sin(w-v).\] Set \[P(y):=\sin^2 y, \qquad Q(x):=(\alpha+\cos x)\sin x.\] Then \[F(u,v,w)=-P(v-u)Q(w-u)+P(v-u)Q(w-v).\] Hence \[\partial_v\partial_wF(u,v,w) = -P'(v-u)Q'(w-u)+P'(v-u)Q'(w-v)-P(v-u)Q''(w-v).\] We evaluate at \((u_\ast,v_\ast,w_\ast)=(0,\pi/2,0)\). The point is chosen because \(P'(\pi/2)=0\) annihilates the two pairwise-type cross terms \(P'(v-u)Q'(\cdot)\) in \(\partial_v\partial_wF\), isolating the single genuinely-triplet contribution \(-P(v-u)Q''(w-v)\), which is exactly the mixed second derivative that no pairwise kernel can reproduce. Any triple with \(P'(v-u)=0\), \(P(v-u)\neq0\), and \(Q''(w-v)\neq0\) would serve equally well. Concretely, using \[P'(\pi/2)=0, \qquad P(\pi/2)=1, \qquad Q''(-\pi/2)=\alpha,\] gives \[\partial_v\partial_wF(0,\pi/2,0)=-\alpha.\] For the swapped arguments, \[F(u,w,v)=-P(w-u)Q(v-u)+P(w-u)Q(v-w),\] so \[\partial_v\partial_wF(u,w,v) = -P'(w-u)Q'(v-u)+P'(w-u)Q'(v-w)-P(w-u)Q''(v-w).\] Evaluating \(\partial_v\partial_w[F(u,w,v)]\) at \((u,v,w)=(0,\pi/2,0)\) gives \[\partial_v\partial_wF(0,0,\pi/2)=0,\] because \(P(0)=P'(0)=0\). Therefore \[M(0,\pi/2,0) = \partial_v\partial_wF(0,\pi/2,0) +\partial_v\partial_wF(0,0,\pi/2) = -\alpha \neq 0.\] Theorem 8 now applies and shows that \(T\) is genuinely nonpairwise. ◻

Proof. For \(u\in L^\infty(\Omega;\mathbb{R}^m)\), the essential range of \(u\) lies in a compact ball of \(\mathbb{R}^m\). The pointwise Taylor formula for \(f\), together with the uniform continuity of the relevant derivatives on a slightly larger compact ball, gives the Frechet derivative in \(L^\infty\) after taking essential suprema. Iterating the argument gives 58 ; the displayed operator bound follows immediately from the essential-supremum norm. Bounded linear pre- and postcomposition preserve \(C^k\)-regularity. For the finite-window assertion, the set \(z([-T,0])\) is compact in \(E\). The continuity of \(\mathrm{D}\Phi\), a finite-cover argument on this compact set, and the smallness of \(\|w\|_{C([-T,0];E)}\) give the uniform Taylor remainder. ◻

Proof. The \(C^1\)-regularity follows from Lemma 11, the smoothness of the cut-off \(\sigma_r\), the bounded periodic coefficient hypotheses, and boundedness of the averaging operator \(J:L^\infty([0,1]^2)\to L^\infty([0,1])\). To make the estimate explicit, write \[\rho(x,y):=H(\theta(x),\theta(y))+\xi(x,y), \qquad \Delta(x,y):=\theta(y)-\theta(x).\] For \((u,v)\in X_a\times X_\theta\), \[\delta\rho_{u,v}(x,y):= u(x,y)+\partial_1H(\theta(x),\theta(y))v(x) +\partial_2H(\theta(x),\theta(y))v(y).\] \[\begin{align} \bigl(\mathrm{D}B_r(\xi,\theta)[u,v]\bigr)(x) &= \int_0^1 \sigma_r'(\rho(x,y))\delta\rho_{u,v}(x,y)\Gamma(\Delta(x,y))\,\,\mathrm{d}y \nonumber\\ &\quad+ \int_0^1 \sigma_r(\rho(x,y))\Gamma'(\Delta(x,y))(v(y)-v(x))\,\,\mathrm{d}y . \label{eq:sec4-DBr-formula} \end{align}\tag{59}\] Using \(|\sigma_r'|\le1\), \(|\sigma_r|\le2r\), and the bounds on \(H\) and \(\Gamma\), this gives \[\|\mathrm{D}B_r(\xi,\theta)[u,v]\|_{X_\theta} \le M_\Gamma\|u\|_{X_a} +\bigl(2M_\Gamma {M_{DH}}+4rL_\Gamma\bigr)\|v\|_{X_\theta} \le C_4(r)\|(u,v)\|.\] Since \(A_r(\xi,\theta)=\mathrm{D}h_0(\theta)B_r(\xi,\theta)\), the product rule gives \[\label{eq:sec4-DAr-formula} \mathrm{D}A_r(\xi,\theta)[u,v] = \mathrm{D}^2h_0(\theta)[v,B_r(\xi,\theta)] + \mathrm{D}h_0(\theta)\mathrm{D}B_r(\xi,\theta)[u,v].\tag{60}\] Together with \[\|B_r(\xi,\theta)\|_{X_\theta}\le C_2(r),\qquad \|\mathrm{D}h_0(\theta)\|\le2 {M_{DH}},\] and \[\|\mathrm{D}^2h_0(\theta)[v,\phi]\|_{X_a} \le4 {M_{D^2H}}\|v\|_{X_\theta}\|\phi\|_{X_\theta},\] this yields the stated \(C_3(r)\)-bound. ◻

Proof. The estimate 61 is the contraction estimate for two fixed points with different anchors: the second component of \(\mathcal{L}_{\varepsilon,\theta_0,r}\) changes by exactly \(\theta_0-\widetilde{\theta}_0\), while the remaining terms are \(q_*\)-Lipschitz. For 62 –63 , the linear part has the same contraction constant because Lemma 12 gives the bounds \(C_3(r)\) and \(C_4(r)\) for \(\mathrm{D}A_r\) and \(\mathrm{D}B_r\). The affine term is only the anchor direction \(\phi\) in the second component. Banach’s fixed-point theorem therefore gives existence, uniqueness, linearity in \(\phi\), and 64 . ◻

Proof. By 61 , \[\sup_{-T\le s\le0}\|z^\psi(s)-z^0(s)\|_{X_a\times X_\theta} \le \frac{e^{-\eta_rT}}{1-q_*}\|\psi\|_{X_\theta}.\] Thus \(z^\psi\to z^0\) uniformly on every fixed finite window. Applying the finite-window part of Lemma 11 to the \(C^1\)-maps \(A_r\) and \(B_r\) gives Taylor remainders which are \(o(\|z^\psi-z^0\|_{C([-T,0])})\), and the displayed estimate follows from the preceding Lipschitz bound. ◻

Proof. The Lipschitz bound for \(A_r\), the derivative bound for \(A_r\), and 61 –64 give, for \(s\le0\), \[\begin{align} &\|A_r(z^\psi(s))-A_r(z^0(s)) -\mathrm{D}A_r(z^0(s))[U_{\theta_0}^{\psi}(s),V_{\theta_0}^{\psi}(s)]\|_{X_a}\\ &\qquad\le \frac{2C_3(r)}{1-q_*}\|\psi\|_{X_\theta}e^{\eta_r s}. \end{align}\] Since \(1+\varepsilon\eta_r\ge1/2\), \[\int_{-\infty}^{-T} e^{(\varepsilon^{-1}+\eta_r)s}\,\,\mathrm{d}s = \frac{\varepsilon}{1+\varepsilon\eta_r}e^{-(\varepsilon^{-1}+\eta_r)T} \le 2\varepsilon_{0,r}e^{-(\varepsilon^{-1}+\eta_r)T}.\] The result follows, for instance with \(K_r=4C_3(r)\varepsilon_{0,r}/(1-q_*)\). ◻

Proof. The point is to differentiate the evaluated fast component at \(t=0\), not the full weighted half-line superposition map \(\xi\mapsto\bigl(t\mapsto\int_{-\infty}^t e^{-\varepsilon^{-1}(t-s)}A_r(\xi(s),\theta(s))\,\,\mathrm{d}s\bigr)\). The latter need not be Fréchet differentiable as a map from \(C_{\eta_r,X_a}\times C_{\eta_r,X_\theta}\) into itself: a perturbation \(w\in C_{\eta_r,X_\theta}\) only satisfies \(\|w(s)\|_{X_\theta}\lesssim e^{\eta_r s}\) as \(s\to-\infty\), which is small in the weighted norm but unbounded pointwise; the candidate linearisation \(\mathrm{D}A_r(z^0(s))[U(s),V(s)]\) can therefore have, on the same scale, growth that the kernel \(e^{-\varepsilon^{-1}(t-s)}\) does not damp uniformly in \(t\le 0\). The Taylor remainder of the half-line superposition then fails to be \(o(\|w\|)\) in operator norm.

Evaluation at \(t=0\) escapes this obstruction because the kernel \(s\mapsto e^{s/\varepsilon}\) decays so rapidly as \(s\to-\infty\) that, when paired with the \(e^{\eta_r s}\)-controlled growth of the perturbation, the contribution of the remote past is absorbed into a quantitative tail bound (Lemma 15). The differentiability proof therefore proceeds by combining a finite-window Taylor remainder on \([-T,0]\) (Lemma 14) with the fast-kernel tail estimate on \((-\infty,-T)\), letting \(\|\psi\|_{X_\theta}\to 0\) at fixed \(T\) and then \(T\to\infty\); the two limits commute precisely because the fast kernel decay is independent of the weighted-space modulus of \(\psi\).

Let \(R^\psi(t):=z^\psi(t)-z^0(t)- (U_{\theta_0}^{\psi}(t),V_{\theta_0}^{\psi}(t))\). Subtracting the fixed-point equations for \(z^\psi\) and \(z^0\), and then subtracting the variational equations, gives on every finite window \([-T,0]\) a linear Volterra system for \(R^\psi\). Its forcing terms are the Taylor remainders \(\mathcal{E}_A^\psi,\mathcal{E}_B^\psi\), plus the fast-component tail from \((-\infty,-T)\). The same kernel estimates as in Lemma 6 give \[\begin{align} \|R^\psi_1(0)\|_{X_a} &\le \frac{1}{1-q_*} \left[ \frac{\varepsilon}{1+\varepsilon\eta_r} \sup_{-T\le s\le0}\|\mathcal{E}_A^\psi(s)\|_{X_a} \right.\\ &\qquad\left. +\frac{1}{|\eta_r|} \sup_{-T\le s\le0}\|\mathcal{E}_B^\psi(s)\|_{X_\theta} +K_r\|\psi\|_{X_\theta}e^{-(\varepsilon^{-1}+\eta_r)T} \right], \end{align}\] where \(R^\psi_1\) denotes the \(X_a\)-component. Dividing by \(\|\psi\|_{X_\theta}\), applying Lemma 14, first letting \(\|\psi\|_{X_\theta}\to0\) for fixed \(T\), and then letting \(T\to\infty\), yields \[\frac{\|\Theta_\varepsilon(\theta_0+\psi)-\Theta_\varepsilon(\theta_0) -U_{\theta_0}^{\psi}(0)\|_{X_a}}{\|\psi\|_{X_\theta}} \longrightarrow0 .\] Thus \(\Theta_\varepsilon\) is Frechet differentiable with the stated derivative.

It remains to record continuity of the derivative. Let \(\theta_n\to\theta_0\) in \(X_\theta\). The anchor estimate 61 gives uniform convergence \(z^{\theta_n}\to z^{\theta_0}\) on each finite window. By Lemma 12, the coefficient maps \(\mathrm{D}A_r(z^{\theta_n}(\cdot))\) and \(\mathrm{D}B_r(z^{\theta_n}(\cdot))\) converge uniformly on that window in operator norm. Subtracting the two variational equations and using the same contraction estimate gives convergence of \(\phi\mapsto U_{\theta_n}^{\phi}(0)\) to \(\phi\mapsto U_{\theta_0}^{\phi}(0)\) in \(\mathcal{B}(X_\theta,X_a)\); the remote fast tail is again controlled by Lemma 15, uniformly for \(\|\phi\|_{X_\theta}\le1\). Hence \(\Theta_\varepsilon\in C^1(X_\theta,X_a)\). ◻

Proof. Evaluating 62 at \(t=0\) and using 64 gives \[\|U_{\theta_0}^{\phi}(0)\|_{X_a} \le C_3(r)\int_{-\infty}^{0}e^{s/\varepsilon}e^{\eta_r s}\,\,\mathrm{d}s\, \frac{\|\phi\|_{X_\theta}}{1-q_*} = C_3(r)\frac{\varepsilon}{1+\varepsilon\eta_r} \frac{\|\phi\|_{X_\theta}}{1-q_*}.\] The assumption \(1+\varepsilon\eta_r\ge1/2\) gives the displayed estimate. ◻