Proof. Let \(R\) be the circumradius, let \(r\) be the inradius, and choose centers \(o,a\) so that \[B_r(a)\subset \Omega\subset B_R(o).\] Domain monotonicity of indexed Dirichlet eigenvalues gives \[\lambda_2(\Omega)\ge \lambda_2(B_R(o)),\qquad \lambda_1(\Omega)\le \lambda_1(B_r(a)).\] Therefore \[\lambda_2(\Omega)-\lambda_1(\Omega) \ge \lambda_2(B_R^{\mathbb{H}^n})-\lambda_1(B_r^{\mathbb{H}^n}).\] Set \(\alpha_R=R-r\). The inradius/circumradius estimate gives \(0\le\alpha_R\le\log 2\). Kristály’s large-radius first-eigenvalue expansion [17], together with the \(n\mapsto n+2\) dimension shift for the first \(l=1\) branch and the Benguria–Linde identification of the second ball eigenvalue [27], gives the ball-gap asymptotic. Using also \((r+\alpha)^{-2}=r^{-2}-2\alpha r^{-3}+O(r^{-4})\) uniformly for bounded \(\alpha\), we obtain, uniformly for \(0\le\alpha\le\log2\), \[\lambda_2(B_{r+\alpha}^{\mathbb{H}^n})-\lambda_1(B_r^{\mathbb{H}^n}) = \left({4\pi^2\over n-1}-2\alpha\pi^2\right)r^{-3}+o(r^{-3}).\] The little-\(o(r^{-3})\) term is uniform for \(\alpha\in[0,\log2]\), since the expansion is evaluated at \(r+\alpha\in[r,r+\log2]\). For \(n=2,3\), \[\gamma_n:=\pi^2\left({4\over n-1}-2\log2\right)>0.\] After increasing \(R_n^{\rm ball}\), the preceding expansion with \(\alpha=\alpha_R\) gives \[\lambda_2(B_R^{\mathbb{H}^n})-\lambda_1(B_r^{\mathbb{H}^n}) \ge {\gamma_n\over 2}r^{-3}.\] Since \(r\le R\), this is at least \((\gamma_n/2)R^{-3}\). ◻

Proof. Every supporting horoball in the ACW representation contains \(\Omega\), hence contains \(B_{R-C}(o)\). In particular it contains \(o\) in its interior. Since horoballs are geodesically convex, each supporting horoball meets a geodesic ray from \(o\) in an interval containing the initial point. The endpoint may be included or excluded depending on the open-domain convention; only the radial endpoint matters here. Intersecting these intervals over all supporting horoballs shows that \(\Omega\) meets each ray in an interval \([0,\rho(\theta))\). The outer inclusion \(\Omega\subset B_R(o)\) makes \(\rho\) finite, and the inner inclusion gives \(\rho(\theta)\ge R-C\). The intersection formula for \(\Omega\) then gives the infimum formula for \(\rho\). Subtracting from \(R\) gives the supremum formula for \(h\), and the positive part may be inserted because \(\Omega\subset B_R(o)\) implies \(h\ge0\). ◻

Proof. The \(L^\infty\) bound follows from the annulus and Lemma 1. Use the ACW support convention [28] \[u_{\Omega}(e) = \sup_{x\in \Omega} \log\bigl(\cosh\rho(x)-\sinh\rho(x)\langle\theta(x),e\rangle\bigr).\] The centered ball \(B_a(o)\) has support value \(a\), and monotonicity gives \[R-C\leq u_{\Omega}(e)\leq R.\] Write \(u_{\Omega}(e)=R-\sigma(e)\), with \(0\leq \sigma(e)\leq C\). Since \(\Omega\) is the intersection of its supporting horoballs, it is enough to estimate one supporting wall and then take suprema. Here the centered geodesic balls \(B_a(o)\) are used only to normalize the support range; the single-wall kernel below is the boundary of a horoball \[\mathcal{H}_e(u) = \left\{ \log(\cosh\rho-\sinh\rho\,\langle\theta,e\rangle)\leq u \right\}.\] This is exactly ACW’s hyperboloid convention: if \(X=(\sinh\rho\,\theta,\cosh\rho)\), then \(-X\cdot(e,1)=\cosh\rho-\sinh\rho\langle\theta,e\rangle\).

For a single supporting horoball with support value \(R-\sigma\), put \(c=\langle\theta,e\rangle\). Its radial boundary is determined by \[\cosh\rho-\sinh\rho\,c=e^{R-\sigma}.\] Solving the resulting quadratic in \(e^\rho\) gives, for \(c<1\), \[\rho_{R,\sigma}(\theta,e) = \log \frac{e^{R-\sigma}+\sqrt{e^{2(R-\sigma)}-(1-c^2)}}{1-c}.\] Thus the height relative to \(B_R(o)\) is \[H_{R,\sigma}(\theta,e) = \sigma+\log(1-c) -\log\!\left(1+\sqrt{1-e^{-2(R-\sigma)}(1-c^2)}\right).\] Let \[q=\sqrt{1-e^{-2(R-\sigma)}(1-c^2)}.\] On the active set \(H_{R,\sigma}\geq0\), \[1-c\geq e^{-\sigma}(1+q)\geq e^{-C}.\] Indeed, \(H_{R,\sigma}\ge0\) is equivalent to \[e^{\rho_{R,\sigma}}\le e^R,\] and the quadratic solution \[e^{\rho_{R,\sigma}} = \frac{e^{R-\sigma}(1+q)}{1-c}\] then gives the displayed lower bound for \(1-c\). Moreover \(R\geq C+1\) implies \(q\geq\sqrt{1-e^{-2}}\). Using \[|\nabla_{\mathbb{S}^{n-1}}c|^2=1-c^2,\] one obtains the exact identity \[\frac{dH_{R,\sigma}}{dc} = -\frac{1}{1-c} -\frac{e^{-2(R-\sigma)}c}{q(1+q)} = -\frac{q+c}{(1-c^2)q}, \qquad |\nabla_{\mathbb{S}^{n-1}}H_{R,\sigma}| = \frac{|q+c|}{q\sqrt{1-c^2}}.\] The apparent singularity at \(c=-1\) is removable because \[q^2-c^2=(1-c^2)(1-e^{-2(R-\sigma)}),\] or, equivalently, \[\frac{dH_{R,\sigma}}{dc} = -\frac{1-e^{-2(R-\sigma)}}{(q-c)q}\] where the right side has a finite limit as \(c\to-1\), while the active-set lower bound on \(1-c\) keeps the kernel away from the singular direction \(c=1\). Equivalently, the differentiated formula gives the crude estimate \[|\nabla_{\mathbb{S}^{n-1}}H_{R,\sigma}| \leq \sqrt{\frac{1+c}{1-c}} +\frac{e^{-2(R-\sigma)}|c|\sqrt{1-c^2}}{q(1+q)} \leq \sqrt{\frac{1+c}{1-c}} +\frac{e^{-2(R-\sigma)}}{q(1+q)} \leq \sqrt{2}\,e^{C/2}+1.\] In the last step, the first term is bounded on the active set by \(1-c\ge e^{-C}\), which gives \((1+c)/(1-c)\le2e^C\). The second term is bounded by \(1\) because \(R-\sigma\ge1\) and \(q\ge\sqrt{1-e^{-2}}\). This estimate is only needed on the active side of the truncation. Indeed, the contrapositive of the active-set bound gives \[1-c<e^{-C}\quad\Longrightarrow\quad H_{R,\sigma}<0.\] Thus the possible large-gradient region near \(c=1\) is killed by the positive-part truncation. On the open set \(\{H_{R,\sigma}>0\}\) the displayed gradient bound holds, while on \(\{H_{R,\sigma}<0\}\) the positive part is identically zero. The preceding implication gives an open neighborhood of the only singular direction \(c=1\) on which the positive part is identically zero; on the complement \(H_{R,\sigma}\) is smooth away from the removable point \(c=-1\). Hence the weak spherical gradient of \(H_{R,\sigma,+}\) is \[\nabla_{\mathbb{S}^{n-1}}H_{R,\sigma}\, \mathbf{1}_{\{H_{R,\sigma}>0\}}\] and is bounded in \(L^\infty\) by \(L(C)\). Equivalently, \(H_{R,\sigma,+}\in W^{1,\infty}(\mathbb{S}^{n-1})\) with weak gradient bounded by \(L(C)\), and integration along minimizing spherical geodesics gives the same global Lipschitz bound. Equivalently, the standard absolutely-continuous-on-lines property for \(W^{1,\infty}\) functions on the sphere integrates the weak-gradient bound along geodesic arcs, and continuity fixes the representative. Therefore \(H_{R,\sigma,+}=\max\{0,H_{R,\sigma}\}\) is globally \(L(C)\)-Lipschitz.

By Lemma 1, the radial boundary is the infimum of the supporting horoball boundaries: \[\rho(\theta)=\inf_e \rho_{R,\sigma(e)}(\theta,e).\] Consequently \[h(\theta)=R-\rho(\theta) = \sup_e H_{R,\sigma(e)}(\theta,e) = \sup_e H_{R,\sigma(e),+}(\theta,e),\] where the last equality uses \(\Omega\subset B_R(o)\). A supremum of functions with a common Lipschitz constant has the same Lipschitz constant; no regularity of the index map \(e\mapsto\sigma(e)\), nor uniqueness of a maximizing supporting wall, is used. This proves the lemma. ◻

Proof. The chain rule gives \[\partial_r z=F_s^{-1}\partial_s\widetilde{z},\qquad \nabla_\theta z=\nabla_\theta\widetilde{z} +\frac{\beta_R}{F_s}\partial_s\widetilde{z}\,\nabla h .\] Multiplying the radial and angular gradient terms by the pulled-back volume element \(F_s\sinh^{n-1}(F_h)\,ds\,d\theta\) gives the displayed identity. Since \(\beta_R=0\) near the singular endpoint and \(\beta_R\le s/R\) elsewhere, while \(F_h\ge s/2\) after increasing \(R\), \[\frac{\beta_R|\nabla h|}{\sinh(F_h)}\le \frac{C_{C,L}}{R}.\] Thus the mixed integral is bounded by \[\frac{C_{C,L}}{R} \left(\int F_s^{-1}\sinh^{n-1}(F_h)|\partial_s\widetilde{z}|^2\right)^{1/2} \left(\int F_s\sinh^{n-3}(F_h)|\nabla_\theta\widetilde{z}|^2\right)^{1/2},\] and Young’s inequality with parameter \(t=\eta R/C_{C,L}\), followed by renaming the constant, proves the stated asymmetric estimate. ◻

Proof. Integrate by parts on the sphere. Since \(\Delta_{\mathbb{S}^{n-1}}h_j=\operatorname{div}\nabla h_j\), \[I_j[w]= -\int \nabla h_j\cdot\nabla_\theta(b_jw^2)\,ds\,d\theta .\] The derivative falling on \(w^2\) is bounded by \[C_L\int \beta_R\coth(F_j)\sinh^{-2}(F_j) |w|\,|\nabla_\theta w|\,ds\,d\theta .\] Because \(\beta_R=0\) near \(s=0\), \(\beta_R\le s/R\), and \(F_j\ge s/2\) after increasing \(R\), the factor \((\beta_R\coth(F_j))^2\) is bounded by \(\alpha_j\). Young’s inequality gives the stated angular kinetic loss and the \(\alpha_j\sinh^{-2}(F_j)|w|^2\) scalar remainder.

When the derivative falls on \(b_j\), the identity \[\nabla_\theta\{\coth(F_j)\sinh^{-2}(F_j)\} = O\!\left((\operatorname{csch}^2(F_j)+\coth^2(F_j))\sinh^{-2}(F_j)\right) \beta_R\nabla h_j\] and \(|\nabla h_j|\le L\) give \[|\nabla_\theta b_j| \le C_{C,L}\alpha_j\sinh^{-2}(F_j),\] again using \(\alpha_j=\beta_R^2(\coth^2(F_j)+\operatorname{csch}^2(F_j))\). This proves the scalar bound. The version with a bounded smooth coefficient has the same proof: when the derivative falls on the coefficient, it supplies one factor \(\beta_R|\nabla h_j|\), while \(b_j\) already supplies the factor \(\beta_R\coth(F_j)\sinh^{-2}(F_j)\), and the product is again controlled by \(\alpha_j\sinh^{-2}(F_j)\). Finally, \(\alpha_j\le C_C\) everywhere, while on \(s\le1\) the estimate \(\beta_R\le s/R\) and \(\coth^2(F_j)+\operatorname{csch}^2(F_j)\le C/s^2\) give \(\alpha_j\le C_CR^{-2}\). ◻

Proof. The scalar \(b_n\) does not affect gaps, so write \[A_h=T_n+2\pi^2h .\] We first record that the first eigenvalue of \(A_h\) is simple for every measurable \(h\) with \(0\le h\le C\). We use the functional-calculus definition of \(T_n\) from Subsection 6.2: if \(D_S\) is the spherical degree operator, \(D_SY_l=lY_l\), equivalently \[D_S=\sqrt{-\Delta_{\mathbb{S}^{n-1}}+\left(\frac{n-2}{2}\right)^2} -\frac{n-2}{2}.\] Again, this is an angular operator on \(\mathbb{S}^{n-1}\), not the radial derivative \(D_s\) used in the one-dimensional estimates below. The operator \(T_n\) is defined from \(D_S\) by spectral calculus, and the dimension-shift asymptotic identifies the scalar function: \[T_n=\phi_n(D_S),\qquad \phi_n(x)=2\pi^2 \left[ \psi\!\left(\frac{n-1}{2}+x\right) -\psi\!\left(\frac{n-1}{2}\right) \right],\] so \(T_nY_l=(b_{n+2l}-b_n)Y_l\). For \(a=(n-1)/2\), the integral representation \[\psi(a+x)-\psi(a) = \int_0^\infty \frac{e^{-at}(1-e^{-xt})}{1-e^{-t}}\,dt\] shows that \(\phi_n\) is a Bernstein function. Hence \(e^{-\sigma T_n}\) is a Bochner subordinate of the spherical Poisson semigroup \(e^{-tD_S}\) [29]. For \(r=e^{-t}\), this semigroup agrees with integration against the spherical Poisson kernel \[P_r(\theta,\eta) = |\mathbb{S}^{n-1}|^{-1} \frac{1-r^2}{(1-2r\,\theta\cdot\eta+r^2)^{n/2}},\] with respect to surface measure \(d\eta\). Indeed, by the standard Poisson-kernel expansion, equivalently by Funk–Hecke, the degree \(l\) zonal component of \(P_r\) is \(r^l\), so the integral operator acts on degree \(l\) harmonics by the multiplier \(r^l=e^{-tl}\). Let \(\mu_\sigma\) be the subordinator law. Since \(\phi_n(0)=0\) and \(\phi_n(\lambda)\to\infty\), \[\mu_\sigma(\{0\}) = \lim_{\lambda\to\infty}\int_0^\infty e^{-\lambda t}\,d\mu_\sigma(t) = \lim_{\lambda\to\infty}e^{-\sigma\phi_n(\lambda)} =0 .\] Thus there is a compact interval \([t_0,t_1]\subset(0,\infty)\) with \(\mu_\sigma([t_0,t_1])>0\). On this interval the Poisson kernel is bounded below by a positive constant; for instance \[P_{e^{-t}}(\theta,\eta) \ge |\mathbb{S}^{n-1}|^{-1}\frac{1-e^{-2t_0}}{2^n} =:p_* .\] Therefore, for \(f\ge0\), \[(e^{-\sigma T_n}f)(\theta) = \int_0^\infty (e^{-tD_S}f)(\theta)\,d\mu_\sigma(t) \ge m_\sigma\int_{\mathbb{S}^{n-1}}f(\eta)\,d\eta,\qquad m_\sigma:=\mu_\sigma([t_0,t_1])p_*>0 .\] Let \(V_h=2\pi^2h\). Since \(T_n\) is self-adjoint and nonnegative and \(V_h\) is a bounded symmetric multiplication operator, \(A_h=T_n+V_h\) is semibounded self-adjoint. The semigroup Trotter product formula [30], together with the inequalities \[e^{-2\pi^2C\sigma/k}I\le e^{-(\sigma/k)V_h}\le I\] give, first for the positive Trotter products and then by strong \(L^2\) convergence, \[e^{-\sigma A_h}f \ge e^{-2\pi^2C\sigma}e^{-\sigma T_n}f \ge e^{-2\pi^2C\sigma}m_\sigma \int_{\mathbb{S}^{n-1}}f\] for every nonzero \(f\ge0\). Since \(T_n\) has compact resolvent and \(V_h\) is bounded, \(e^{-\sigma A_h}\) is compact. The last display makes it positivity improving. By the Perron–Frobenius theorem for compact positivity-improving operators [31], the spectral radius of \(e^{-\sigma A_h}\) is a simple eigenvalue with a strictly positive eigenvector. Equivalently, the first eigenvalue of \(A_h\) is simple for every such \(h\). Since \(A_h\) is real and self-adjoint, this real simplicity is the same as complex simplicity: the real and imaginary parts of any complex first eigenfunction are real first eigenfunctions.

We now prove uniformity by compactness. Let \[\mathcal{H}_C=\{h\in L^\infty(\mathbb{S}^{n-1}):0\le h\le C\}.\] The unit ball of \(L^\infty\) is weak-\(*\) compact, and here it is sequentially compact on bounded sets because \(L^1(\mathbb{S}^{n-1})\) is separable. Suppose, to the contrary, that the gaps of \(A_h\) are not uniformly bounded below on \(\mathcal{H}_C\). Then there are \(h_j\in \mathcal{H}_C\) such that \[\mu_2(A_{h_j})-\mu_1(A_{h_j})\to0 .\] After passing to a subsequence, \(h_j\to h\) weak-\(*\) in \(L^\infty\) for some \(h\in\mathcal{H}_C\).

Let \(u_j,v_j\) be \(L^2\)-orthonormal eigenfunctions for the first two eigenvalues of \(A_{h_j}\). The min-max principle and \(0\le V_{h_j}\le 2\pi^2C\) give \[\mu_2(A_{h_j})\le \mu_2(T_n)+2\pi^2C,\] so \[\langle u_j,T_nu_j\rangle,\;\langle v_j,T_nv_j\rangle \le \mu_2(T_n)+2\pi^2C .\] Thus \(u_j\) and \(v_j\) are bounded in the form domain \[\mathcal{Q}=\operatorname{Dom}(T_n^{1/2}),\qquad \|w\|_{\mathcal{Q}}^2=\|w\|_2^2+\langle w,T_nw\rangle .\] Since \(T_n\) has eigenvalues tending to infinity and each spherical-harmonic degree has finite multiplicity, the eigenvalue list with multiplicity tends to infinity. Hence the embedding \(\mathcal{Q}\hookrightarrow L^2(\mathbb{S}^{n-1})\) is compact. Passing to a further subsequence, \[u_j\to u,\qquad v_j\to v \quad\text{strongly in }L^2,\] and weakly in \(\mathcal{Q}\). The limits are still orthonormal. Passing to a further subsequence, the two eigenvalues converge to a common value \(\lambda\), because their difference tends to zero.

For each fixed \(\varphi\in\mathcal{Q}\), the products \(u_j\overline{\varphi}\) and \(v_j\overline{\varphi}\) converge in \(L^1\). For instance, \[\begin{align} &\left|\int h_j u_j\overline{\varphi} -\int h u\overline{\varphi}\right| \\ &\qquad\le C\|(u_j-u)\overline{\varphi}\|_{L^1} +\left|\int (h_j-h)u\overline{\varphi}\right| \to0 . \end{align}\] The same argument applies to \(v_j\). Together with weak convergence in \(\mathcal{Q}\), this allows passage to the limit in the weak eigenvalue equations. More explicitly, the equations are used in form-pairing form: for every \(\varphi\in\mathcal{Q}\), \[\langle T_n^{1/2}u_j,T_n^{1/2}\varphi\rangle +2\pi^2\int h_ju_j\overline{\varphi} = \mu_1(A_{h_j})\langle u_j,\varphi\rangle,\] and similarly for \(v_j\). Weak convergence in \(\mathcal{Q}\) gives convergence of the kinetic half-power pairing, while the displayed \(L^1\) estimate gives convergence of the potential term. Hence \[A_hu=\lambda u,\qquad A_hv=\lambda v .\] It remains to identify \(\lambda\) as the first eigenvalue of \(A_h\). For any normalized \(\varphi\in\mathcal{Q}\), \[\mu_1(A_{h_j})\le \langle\varphi,T_n\varphi\rangle +2\pi^2\int_{\mathbb{S}^{n-1}}h_j|\varphi|^2 \to \langle\varphi,T_n\varphi\rangle +2\pi^2\int_{\mathbb{S}^{n-1}}h|\varphi|^2 .\] Taking the infimum over \(\varphi\) gives \(\lambda\le\mu_1(A_h)\). Since \(\lambda\) is an eigenvalue of \(A_h\), it is not below the bottom of the spectrum, so \(\mu_1(A_h)\le\lambda\). Thus \(\lambda=\mu_1(A_h)\), and \(u,v\) are two orthonormal first eigenfunctions of \(A_h\). This contradicts the simplicity proved above. Therefore \[\inf_{h\in\mathcal{H}_C}\bigl(\mu_2(A_h)-\mu_1(A_h)\bigr)>0 .\] This infimum is the required \(\delta_{\rm eff}(n,C)\), and adding the scalar \(b_n\) does not change the gap. ◻

Proof. Write \[c_l=l(l+n-2)+\frac{(n-1)(n-3)}{4}.\] Except in the critical channel \((n,l)=(2,0)\), one has \(c_l\ge0\), and the only zero case is \((n,l)=(3,0)\). For \(c_l\ge0\), the potential \(c_l\sinh^{-2}s\) is convex on \((0,\infty)\), since \[\frac{d^2}{ds^2}\sinh^{-2}s = 4\coth^2s\,\sinh^{-2}s+2\sinh^{-4}s>0.\] This includes the zero potential in the \((n,l)=(3,0)\) channel. Truncate the endpoint to \((\epsilon,R)\) and impose Dirichlet conditions at both endpoints. The one-dimensional convex-potential gap theorem [2], [32] then gives \[\lambda_2(H_{l,R}^{\epsilon})-\lambda_1(H_{l,R}^{\epsilon}) \ge 3\pi^2(R-\epsilon)^{-2}.\] Extend \(H_0^1(\epsilon,R)\) functions by zero to \((0,R)\). As \(\epsilon\downarrow0\) these closed form domains increase monotonically, and their union is dense in the Friedrichs form domain, the completion of \(C_c^\infty(0,R)\) under the form norm. The corresponding eigenvalues therefore converge to the singular-endpoint Friedrichs eigenvalues by monotone form convergence. For each \(\epsilon\), \[\lambda_2(H_{l,R}^{\epsilon}) \geq \lambda_1(H_{l,R}^{\epsilon})+3\pi^2(R-\epsilon)^{-2}.\] Taking limits in this inequality gives \(\lambda_{2,l,R}\geq\lambda_{1,l,R}+3\pi^2R^{-2}\). The right side of the truncated gap bound depends only on the interval length, not on the coefficient \(c_l\). Consequently the same limiting inequality holds for every noncritical angular degree with the same constant \(3\pi^2\). No uniform-in-\(l\) convergence rate as \(\epsilon\downarrow0\) is being used; the limit is taken degree by degree after a bound whose right side is already independent of \(l\).

For the exceptional critical channel, where \((n,l)=(2,0)\), first subtract the harmless additive constant \(a_2^2=1/4\): \[\widehat H_R=H_R-\frac{1}{4} =-\frac{d^2}{ds^2}-\frac{1}{4}\sinh^{-2}s.\] This does not change the branch gap. Compare \(\widehat H_R\) with the exact critical Bessel operator \[B_R=-\frac{d^2}{ds^2}-\frac{1}{4}s^{-2}\] with the same Friedrichs endpoint at \(0\) and Dirichlet endpoint at \(R\). Because \[\sinh^{-2}s=s^{-2}-\frac{1}{3}+O(s^2)\qquad(s\downarrow0),\] the difference \(\frac{1}{4}(s^{-2}-\sinh^{-2}s)\) extends to a bounded potential at the singular endpoint. Thus \(\widehat H_R\) and \(B_R\) are closures of the same critical Friedrichs core, and have the same form domain: the bounded potential perturbation is continuous with respect to the closed critical Friedrichs form and does not change its closure. Since \(\sinh s\ge s\), one has \(\widehat H_R\ge B_R\) in form order on that common domain, and the Courant–Fischer min-max monotonicity for closed semibounded forms therefore gives \(\lambda_k(\widehat H_R)\ge\lambda_k(B_R)\) for every indexed eigenvalue \(k\). In particular, the critical Bessel spectrum gives \[\lambda_2(\widehat H_R)\ge j_{0,2}^2R^{-2}.\] Here the Friedrichs eigenfunctions of \(B_R\) are \(\sqrt{s}\,J_0(j_{0,k}s/R)\); the companion \(\sqrt{s}\,Y_0(j_{0,k}s/R)\sim\sqrt{s}\log s\) branch is excluded by the critical Friedrichs endpoint condition. For the first eigenvalue, the simpler Dirichlet trial \[u_R(s)=\sin(\pi s/R)\] is admissible in the same critical Friedrichs form domain: it vanishes at \(R\), has \(u_R(s)=O(s)\) at the singular endpoint, and has finite critical form energy. Since the potential in \(\widehat H_R\) is nonpositive, \[\begin{align} \lambda_1(\widehat H_R) &\le \frac{\int_0^R |u_R'(s)|^2\,ds -\frac{1}{4}\int_0^R\sinh^{-2}s\,|u_R(s)|^2\,ds}{\int_0^R |u_R(s)|^2\,ds} \\ &\le \frac{\int_0^R |u_R'(s)|^2\,ds}{\int_0^R |u_R(s)|^2\,ds} = \pi^2R^{-2}. \end{align}\] The second positive zero of \(J_0\) satisfies \(j_{0,2}>5>\pi\) [33], so \[j_{0,2}^2-\pi^2>0 .\] Thus \(\widehat H_R\), and hence \(H_R\), has a critical branch gap \(\ge c_{\rm crit}R^{-2}\) with, for instance, \[c_{\rm crit}=\frac{1}{2}\bigl(j_{0,2}^2-\pi^2\bigr)>0 .\] Combining this with the noncritical channels fixes the dimension-only constants \(c_{\rm rad}^0(n)>0\) and \(R_{\rm rad}(n)<\infty\) used in the lower form. This uses the Bessel zero reference cited above, together with the standard Friedrichs-extension and Sturm–Liouville endpoint framework [30]. ◻

Proof. This lemma is proved before the projected Green estimate below; it uses only the regular Sturm–Liouville construction, Bessel comparison, and Volterra estimates for the finite-box radial equation. No later projected endpoint estimate is used here. We separate the standard regular-solution construction from the two local normalization calculations. For each fixed \(l\le N\), subtract the common constant \(a_n^2\) and write the finite-box equation in the form \[\bigl(-\partial_s^2+V_l(s)\bigr)u=\rho^2u,\qquad u(s)\sim s^{\nu_l+1/2}\quad(s\downarrow0).\] Here \[V_l(s)=\bigl(l(l+n-2)+c_n^0\bigr)\sinh^{-2}s.\] The choice order in this lemma is: fix \(N\), choose \(S_N\), then choose \(\rho_0\), and finally choose \(R_N\). On the regular tail \(s\ge S_N\), the potentials \(V_l\) are exponentially decaying, uniformly over the finite set \(l\le N\). The zero-energy critical regular solution is nonconstant, so on the regular tail the zeros of its derivative are isolated. Choose \(S_N\) away from those zeros and large enough that the Volterra operator built from the free sine kernel is a small contraction, for example by making \[\sup_{l\le N}\int_{S_N}^\infty (1+t)|V_l(t)|\,dt\] sufficiently small. The bound \[|\rho^{-1}\sin\rho(y-x)|\le y-x\] also covers the limiting case \(\rho=0\), so \(S_N\) is fixed before any positive-frequency threshold. After \(S_N\) is fixed, choose \(\rho_0>0\) by continuity of the Cauchy data on \([0,S_N]\), and then choose \(R_N\) large enough for the phase averages on \([S_N,R]\). The Volterra construction and spectral-parameter continuity are the standard Sturm–Liouville construction [30]; the Bessel comparisons at the critical endpoint use the large- and small-argument estimates for \(J_0,Y_0\) recorded in [33].

For the first branch, the regular solution on the tail is a perturbation of a single sine mode with frequency \(\pi/R+O_N(R^{-2})\). Its \(L^2(0,R)\)-normalization has amplitude \(\sqrt{2/R}+O_N(R^{-3/2})\). More explicitly, if \(U_R\) is the unnormalized regular solution and \(A_R\) is its sine-tail amplitude on \([S_N,R]\), then \(A_R\asymp_N R\) and tail averaging gives \[\|U_R\|_{L^2(0,R)}^2 = \frac{1}{2}A_R^2R+O_N(A_R^2)+O_N(R^2) = \frac{1}{2}A_R^2R\bigl(1+O_N(R^{-1})\bigr).\] Here the fixed endpoint interval contributes lower order mass, and the Volterra dressing on the exponentially small tail contributes the displayed lower-order error. Thus the normalized tail amplitude is \(\sqrt{2/R}(1+O_N(R^{-1}))\). Since the Dirichlet endpoint phase is \(\sin(\rho_{1,l,R}(R-s))\) up to the same Volterra dressing and \(\rho_{1,l,R}=\pi/R+O_N(R^{-2})\), differentiating at \(s=R\) gives \[\partial_s y_{l,R}(R) = -\sqrt{2/R}\,\pi/R+O_N(R^{-5/2}),\] uniformly for the finite set \(l\le N\). This includes the critical channel \((n,l)=(2,0)\): after \(S_N\) is fixed, the critical Friedrichs endpoint only determines the fixed Cauchy data at \(S_N\), while the long interval \([S_N,R]\) is governed by the same low-frequency Volterra tail and the same linear zero-frequency growth that sets the \(R^{3/2}\) normalization scale.

It remains to record the critical-channel estimates. In this paragraph \((n,l)=(2,0)\), \(V_0(s)=-\frac{1}{4}\sinh^{-2}s\), and we use the \(\rho_k,\mu_k,\psi_k\) notation from the statement. The lower root growth is a comparison with the exact critical Bessel operator \[B_R=-\frac{d^2}{ds^2}-\frac{1}{4}s^{-2}\] with the same Friedrichs endpoint at \(0\) and Dirichlet endpoint at \(R\). Since \(\sinh s\ge s\), \(V_0(s)\ge -\frac{1}{4}s^{-2}\), hence \[\rho_k^2\ge j_{0,k}^2R^{-2}.\] The first critical eigenvalue estimate proved in Lemma 6 gives \(\rho_1^2\le \pi^2R^{-2}\). Since \(j_{0,k}=(k-\frac{1}{4})\pi+O(k^{-1})\), there are \(k_0\) and \(c>0\) such that \(j_{0,k}^2-\pi^2\ge ck^2\) for \(k\ge k_0\). For the finitely many \(2\le k<k_0\), the strict inequality \(\pi^2<j_{0,2}^2\le j_{0,k}^2\) gives the same bound after decreasing \(c\). Therefore, after enlarging \(R_N\), \[\mu_kR^2=\rho_k^2R^2-\rho_1^2R^2 \ge j_{0,k}^2-\pi^2\ge ck^2,\qquad k\ge2.\] The separate lower root bound follows from \(j_{0,k}\ge ck\): \[\rho_k\ge c\,k/R,\qquad k\ge2 .\] The upper root growth is the min-max half of the argument. On \((S_N,R)\), take the \(k\)-dimensional space spanned by the first \(k\) Dirichlet sine modes, vanishing at \(S_N\) and \(R\), and extend it by zero to \((0,R)\). These functions lie in the Friedrichs form domain, and on this space \[\frac{\int |\phi'|^2+V_0|\phi|^2}{\int |\phi|^2} \le C_N k^2R^{-2}\] for \(R\ge R_N\), because \(S_N\) is fixed and \(V_0\le0\) on the tail. Thus \(\rho_k\le Ck/R\).

The endpoint amplitudes use two normalizations of the same Friedrichs-regular solution. If \(\rho_k<\rho_0/2\), the bounded-frequency Volterra expansion and the zero-energy critical solution give \[|u_0(s,\rho_k)|\le Cs^{1/2},\qquad 0<s\le1 .\] The normalization needs the small-frequency Cauchy amplitude at the fixed matching point \(S_N\). Let \[\mathcal{A}_0(\rho)^2 = u_0(S_N,\rho)^2+\rho^{-2}\partial_su_0(S_N,\rho)^2 .\] As \(\rho\downarrow0\), the regular solution converges in Cauchy data on \([0,S_N]\) to the zero-energy Friedrichs solution. The matching point \(S_N\) is chosen on the tail where this zero-energy solution has nonzero slope. Hence, after decreasing \(\rho_0\) if needed, \[\rho^2\mathcal{A}_0(\rho)^2 \longrightarrow |\partial_su_0(S_N,0)|^2>0, \qquad \mathcal{A}_0(\rho)\ge c_N\rho^{-1}, \qquad 0<\rho<\rho_0/2 .\] Equivalently, after using the Dirichlet condition to write the tail phase from \(R\), the sine tail has amplitude comparable to \(\rho_k^{-1}\). Write \(\zeta_k=\rho_kR\). The lower root bound gives \(\zeta_k\ge c k\), hence \(\zeta_k\ge c_*>0\) for \(k\ge2\). If \(\zeta_k\) is bounded, the rescaled sine-tail integral has a positive lower bound by compactness on \(\zeta_k\in[c_*,K]\); if \(\zeta_k>K\), the same lower bound follows from averaging over many half-periods. Equivalently, the tail Volterra solution is a perturbed sinusoid of amplitude comparable to \(\mathcal{A}_0(\rho_k)\), and the same reverse-triangle estimate used below absorbs the relative sup-norm Volterra error into the principal sinusoidal term. The preceding compactness/phase-averaging argument gives \[\int_{S_N}^{R}|u_0(s,\rho_k)|^2\,ds \ge c(R-S_N)\mathcal{A}_0(\rho_k)^2 \ge c\rho_k^{-2}R .\] Enlarging \(K\) and then \(R_N\) gives, uniformly over all low roots, \[\|u_0(\cdot,\rho_k)\|_{L^2(0,R)}\ge c\rho_k^{-1}R^{1/2}.\] Therefore the normalized eigenfunction satisfies \[|\psi_k(s)|\le C\rho_kR^{-1/2}s^{1/2} \le CkR^{-3/2}s^{1/2}.\] For \(\rho_k\ge\rho_0/2\), write \[V_0(s)=-\frac{1}{4}s^{-2}+W(s),\] where \(W\) is bounded on \((0,S_N]\). We first record the a priori critical Bessel-Volterra bound, before using it in the Cauchy-data lower estimate. Let \(u_B(s,\rho)=s^{1/2}J_0(\rho s)\), and write the regular solution on \((0,S_N]\) as \[u_0(s,\rho)=u_B(s,\rho)+\int_0^sK_\rho(s,t)W(t)u_0(t,\rho)\,dt,\] where \(K_\rho\) is the Green kernel built from \(s^{1/2}J_0(\rho s)\) and \(s^{1/2}Y_0(\rho s)\). The small- and large-argument Bessel bounds in DLMF Sections 10.8 and 10.17, together with the Wronskian normalization in DLMF Section 10.5, give \[|K_\rho(s,t)| \le C_N \bigl(s^{1/2}\wedge\rho^{-1/2}\bigr) \bigl(t^{1/2}\wedge\rho^{-1/2}\bigr) \bigl(1+\mathbf{1}_{\{\rho t\le1\}}|\log(\rho t)|\bigr), \qquad 0<t\le s\le S_N .\] With the weighted norm \[\|u\|_\rho = \sup_{0<s\le S_N} {|u(s)|\over s^{1/2}\wedge\rho^{-1/2}},\] and \(m_\rho(t)=t^{1/2}\wedge\rho^{-1/2}\), the kernel estimate gives \[\int_0^{S_N} m_\rho(t)^2 \bigl(1+\mathbf{1}_{\{\rho t\le1\}}|\log(\rho t)|\bigr)\,dt \le C(S_N,\rho_0),\qquad \rho\ge\rho_0/2.\] Indeed, the part \(0<t\le\rho^{-1}\) is \(O(\rho^{-2})\) after the change of variables \(y=\rho t\), while the remaining part is bounded by \(S_N\rho^{-1}\). The Volterra operator is triangular, so its iterated norms are bounded by the usual factorial estimate with constants depending only on \(S_N\), \(\rho_0\), and \(\|W\|_\infty\). Picard iteration gives, uniformly in \(\rho\ge\rho_0/2\), \[|u_0(s,\rho)|\le \begin{cases} Cs^{1/2}, & \rho s\le1,\\ C\rho^{-1/2}, & \rho s\ge1 . \end{cases}\] This a priori bound for the unnormalized regular solution is independent of the normalization estimate below. We next make the Cauchy-data lower bound explicit. At the fixed matching point \(S_N\), set \[\mathcal{A}(\rho)^2 = u_0(S_N,\rho)^2+\rho^{-2}\partial_su_0(S_N,\rho)^2 .\] For the model solution \(u_B(s,\rho)=s^{1/2}J_0(\rho s)\), \[\rho^{-1}\partial_su_B(S_N,\rho) = \frac{1}{2\rho\sqrt{S_N}}J_0(\rho S_N) -\sqrt{S_N}\,J_1(\rho S_N).\] The large-argument expansions of \(J_0\) and \(J_1\), with \(x=\rho S_N\) and \(S_N\) fixed, give \[\begin{align} S_NJ_0(x)^2 &+ \left( \frac{1}{2\rho\sqrt{S_N}}J_0(x) -\sqrt{S_N}\,J_1(x) \right)^2 \\ &=\frac{2}{\pi\rho}+O_N(\rho^{-2}). \end{align}\] Indeed \(J_0(x)=\sqrt{2/(\pi x)}\cos(x-\pi/4)+O(x^{-3/2})\) and \(J_1(x)=\sqrt{2/(\pi x)}\sin(x-\pi/4)+O(x^{-3/2})\), so the two leading Cauchy components are phase-shifted and their squares add to \(2/(\pi\rho)\); in particular they cannot vanish simultaneously. The Bessel Volterra equation for the bounded residual \(W\) perturbs this scaled Cauchy vector by only \(O_N(\rho^{-3/2})\): splitting the integral at \(t=\rho^{-1}\), the collar \(0<t<\rho^{-1}\) uses \[|K_\rho(S_N,t)|+\rho^{-1}|\partial_sK_\rho(S_N,t)| \le C_N\rho^{-1/2}t^{1/2}(1+|\log(\rho t)|),\] so the logarithmic small-argument \(Y_0\) bound and the regular estimate \(u_0(t,\rho)=O(t^{1/2})\) give an \(O_N(\rho^{-5/2})\) contribution, while \(\rho^{-1}\le t\le S_N\) contributes \(O_N(\rho^{-3/2})\) by the large-argument Bessel bounds and the compact-collar high-window bound \(u_0(t,\rho)=O_N(\rho^{-1/2})\). Therefore \[\mathcal{A}(\rho)\ge c\rho^{-1/2}\] for all sufficiently large \(\rho\). On the remaining compact interval \(\rho_0/2\le\rho\le\rho_1\), the same lower bound follows after decreasing \(c\), because the regular solution depends continuously on \(\rho\) and vanishing of both Cauchy data at \(S_N\) would force the solution to be identically zero.

The free-region Volterra construction on the fixed tail transfers this amplitude to \[u_0(s,\rho) = \mathcal{A}(\rho)\cos(\rho(s-S_N)-\theta(\rho))+E(s,\rho), \qquad \sup_{S_N\le s\le R}|E(s,\rho)| \le\epsilon\,\mathcal{A}(\rho),\] where \(\epsilon\in(0,1/8)\) is fixed by the original choice of \(S_N\); the tail Volterra operator built from the free sine kernel is an \(\epsilon\)-contraction in the sup norm, uniformly for \(\rho\ge\rho_0/2\). Hence \[\int_{S_N}^{R}|E(s,\rho)|^2\,ds \le \epsilon^2\mathcal{A}(\rho)^2(R-S_N).\] The reverse triangle inequality in \(L^2(S_N,R)\) gives \[\begin{align} \|u_0(\cdot,\rho)\|_{L^2(S_N,R)} &\ge \mathcal{A}(\rho) \left(\int_{S_N}^{R} \cos^2(\rho(s-S_N)-\theta(\rho))\,ds \right)^{1/2} \\ &\qquad -\left(\int_{S_N}^{R}|E(s,\rho)|^2\,ds\right)^{1/2}. \end{align}\] For \(\rho=\rho_k\ge\rho_0/2\) and \(R\ge R_N\), \[\int_{S_N}^R \cos^2(\rho_k(s-S_N)-\theta(\rho_k))\,ds \ge \frac{R-S_N}{2}-\frac{1}{2\rho_k} \ge \frac{R}{8},\] after enlarging \(R_N\) so that \(R-S_N\ge R/2\) and \((2\rho_k)^{-1}\le R/8\). Since \(R-S_N\le R\), the preceding reverse triangle inequality gives \[\|u_0(\cdot,\rho_k)\|_{L^2(S_N,R)} \ge \mathcal{A}(\rho_k)R^{1/2} \left(\frac{1}{\sqrt8}-\epsilon\right) \ge c\mathcal{A}(\rho_k)R^{1/2}.\] Here \(c>0\) because \(\epsilon<1/8<1/\sqrt8\). Since \(\mathcal{A}(\rho_k)^2\ge c/\rho_k\), this gives \[\|u_0(\cdot,\rho_k)\|_{L^2(0,R)}^2\ge cR/\rho_k .\] Normalizing and using the upper root bound \(\rho_k\le Ck/R\) gives \[|\psi_k(s)|\le C\rho_k^{1/2}R^{-1/2}s^{1/2} \le Ck^{1/2}R^{-1}s^{1/2} \quad(\rho_ks\le1),\] while the large-argument Bessel estimate gives \[|\psi_k(s)|\le CR^{-1/2}\quad(\rho_ks\ge1).\] Taking the maximum of the finitely many thresholds and the minimum of the positive constants gives the stated uniform constants. ◻

Proof. We begin with the finite-box input, then derive the projected Green estimates and the endpoint collar bounds.

Finite-box Jost supply. In the critical channel write \[\rho_k=\rho_{k,0,R},\qquad \psi_k=\psi_{k,0,R},\qquad \mu_k=\lambda_{k,0,R}-\lambda_{1,0,R}.\] Here \(\lambda_{k,l,R}\) and \(\psi_{k,l,R}\) denote the eigenvalues and normalized eigenfunctions of the separated radial operator \(H_{l,R}\) on \((0,R)\), so that \(\lambda_{1,l,R}=e_{l,R}\) and \(\psi_{1,l,R}=y_{l,R}\). Let \(u_l(s,\rho)\) be the Friedrichs-regular solution of \[\bigl(-\partial_s^2+V_l(s)\bigr)u=\rho^2u,\qquad u_l(s,\rho)\sim s^{\nu_l+1/2}\quad (s\downarrow0),\] and write \(u_0\) for the critical channel. After subtracting the common constant \(a_n^2\), the finite-box equation has potential \[V_l(s)=\bigl(l(l+n-2)+c_n^0\bigr)\sinh^{-2}s ;\] in the critical channel \((n,l)=(2,0)\), this is \(V_0(s)=-\frac{1}{4}\sinh^{-2}s\). For each fixed \(l\le N\) the same construction is run with \(V_l\) in place of \(V_0\). Since only finitely many degrees \(l\le N\) occur, the common constants below are obtained by taking the maximum of the resulting thresholds and the minimum of the resulting positive lower constants. The displayed argument is written in the critical channel, where the logarithmic endpoint companion is the binding case. Lemma 7 supplies constants \(\rho_0>0\), \(R_N<\infty\), and \(0<c_0<C_0<\infty\), depending only on the fixed angular cutoff, such that for \(R\ge R_N\) and \(k\ge2\), \[\mu_k\ge c_0k^2R^{-2},\qquad c_0\,\frac{k}{R}\le \rho_k\le C_0\,\frac{k}{R},\] together with the three endpoint amplitude regimes \[\begin{array}{ll} |\psi_k(s)|\le CkR^{-3/2}s^{1/2}, & \rho_k<\rho_0/2,\\[3pt] |\psi_k(s)|\le Ck^{1/2}R^{-1}s^{1/2}, & \rho_k\ge\rho_0/2,\;\rho_k s\le1,\\[3pt] |\psi_k(s)|\le CR^{-1/2}, & \rho_k s\ge1, \end{array} \qquad 0<s\le1 .\] These finite-box estimates are the only Jost input used in the rest of this lemma. The choice order \(S_N\to\rho_0\to R_N\), the high-window Cauchy-data normalization, and the first-branch boundary slope have all been packaged in Lemma 7; the projected Green argument below uses only the displayed root-growth and endpoint-amplitude consequences.

Projected Green diagonal. If \(Q_{0,R}\) denotes projection away from the first critical radial branch, set \[k[q]=\langle q,Q_{0,R}(H_{0,R}-\lambda_{1,0,R})Q_{0,R}q\rangle .\] The projection is essential: without removing the first critical branch, the right side could vanish while the endpoint mass is positive. For \[G_R(s,s)=\sum_{k\ge2}\frac{|\psi_k(s)|^2}{\mu_k},\] Cauchy–Schwarz in the projected spectral expansion gives \(|q(s)|^2\le k[q]G_R(s,s)\): indeed, if \(q=\sum_{k\ge2}a_k\psi_k\), then \[|q(s)|^2 \le \left(\sum_{k\ge2}\mu_k|a_k|^2\right) \left(\sum_{k\ge2}\frac{|\psi_k(s)|^2}{\mu_k}\right) =k[q]G_R(s,s).\] The diagonal bound \(\sup_{\varepsilon_R\le s\le1}G_R(s,s)\le C\) is proved by splitting the sum according to the global frequency threshold \(\rho_0/2\) and the local collar product \(\rho_k s\): \[G_R(s,s)=\Sigma_{\rm low}(s)+\Sigma_{\rm seam}(s)+\Sigma_{\rm high}(s), \qquad \varepsilon_R\le s\le1 .\] The index enlargements used below are \[\begin{array}{c|c} \text{regime} & \text{consequence of the root bounds} \\ \hline \rho_k<\rho_0/2 & k\le C R,\\ \rho_k\ge\rho_0/2,\;\rho_k s\le1 & cR\le k\le C R/s,\\ \rho_k s\ge1 & k\ge c R/s . \end{array}\] Let \(k_\ast\) be the first index with \(\rho_{k_\ast}\ge\rho_0/2\). The root bounds give \(c_0R\le k_\ast\le K_{\rm root}R\), and the upper root-growth bound \(\rho_k\le Ck/R\) is used only for \(k\ge k_\ast\). For genuinely low roots \(\rho_k<\rho_0/2\), the lower root-growth bound \(\rho_k\ge ck/R\) gives \(k\le \rho_0R/(2c)\), hence \(O(R)\) such indices, and \[\Sigma_{\rm low}(s) \le \sum_{\rho_k<\rho_0/2} \frac{Ck^2R^{-3}s}{ck^2R^{-2}} \le \sum_{\rho_k<\rho_0/2}CR^{-1}s \le Cs .\] For seam roots \(\rho_k\ge\rho_0/2\) and \(\rho_k s\le1\), the lower root-growth bound gives \(k\le C_1R/s\). The lower endpoint \(k\ge c_0R\) follows from \(\rho_k\ge\rho_0/2\) and the upper root-growth bound \(\rho_k\le Ck/R\). Thus the actual seam index set is contained in \(c_0R\le k\le C_1R/s\). The following sum is therefore over a superset of the actual seam indices; indices in this range with \(\rho_k s>1\) are not in \(\Sigma_{\rm seam}\) and are handled by \(\Sigma_{\rm high}\). Hence \[\Sigma_{\rm seam}(s) \le \sum_{k_\ast\le k\le C_1R/s} \frac{CkR^{-2}s}{ck^2R^{-2}} \le C_2s\sum_{c_0R\le k\le C_1R/s}\frac{1}{k} \le C_3s\bigl(1+\log(C_4/s)\bigr)\le C_5 .\] The second inequality only enlarges the index range, using \(k_\ast\ge c_0R\). The last inequality uses the elementary bound \(\sup_{0<s\le1}s\log(1/s)=e^{-1}\). For high local-Bessel roots \(\rho_k s\ge1\), the upper root-growth bound \(\rho_k\le Ck/R\) gives \(k\ge cR/s\), so the actual high index set is contained in that tail. Therefore \[\Sigma_{\rm high}(s) \le \sum_{k\ge cR/s} \frac{CR^{-1}}{ck^2R^{-2}} \le CR\sum_{k\ge cR/s}\frac{1}{k^2} \le Cs .\] In fact these three estimates give the sharper bound \[G_R(s,s)\le Cs\bigl(1+\log(C/s)\bigr),\qquad 0<s\le1,\] and in particular the projected diagonal is uniformly bounded on \([\varepsilon_R,1]\). The point of the genuinely low/seam/high local-Bessel split is to keep distinct the global frequency dichotomy controlling box normalization from the local Bessel dichotomy controlling endpoint size. On the transition layer \(I_R=[\varepsilon_R,2\varepsilon_R]\), the same bound gives \[\int_{I_R}|q|^2\,ds \le C\varepsilon_R^2\log(C/\varepsilon_R)\,k[q]\] for every projected vector \(q\). This is the estimate used when \(G_s'=O(R^{-1}\varepsilon_R^{-2})\) is routed in the transition coefficient class.

Endpoint form controls. For the collar scale \(\varepsilon_R=R^{-\delta_{\rm cut}}\), \(1/2<\delta_{\rm cut}<2/3\), the cruder uniform diagonal bound gives the weighted collar mass estimate \[R^{-1}\int_{\varepsilon_R}^{1}s^{-2}|q(s)|^2\,ds \le C R^{-1}\varepsilon_R^{-1}k[q] =O(R^{\delta_{\rm cut}-1})k[q] =o_R(1)k[q].\] Here the last equality uses \(\delta_{\rm cut}<1\). The derivative estimate is not obtained by differentiating a singular projected Green kernel. On the endpoint collar write \(q=yv\), where \(y\) is the positive zero-energy critical solution for the shifted operator \(\widehat H=H_{0,R}-a_2^2\); equivalently \(H_{0,R}y=a_2^2y\). The local ground-state identity needed here is \[\langle yv,(H_{0,R}-\lambda_{1,0,R})yv\rangle = \int y^2|v'|^2 -(\lambda_{1,0,R}-a_2^2)\|yv\|^2 .\] For \(v\in C_c^\infty((0,R))\), expand \(q=yv\): \[|q'|^2+V_0q^2 = y^2|v'|^2+y'^2v^2+2yy'vv'+V_0y^2v^2 .\] Integrating \(2yy'vv'=yy'(v^2)'\) by parts gives the bulk contribution \(-(yy')'v^2=-(y'^2+yy'')v^2\). Since \(-y''+V_0y=a_2^2y\), the remaining zero-order terms collapse to \(a_2^2y^2v^2\), and no endpoint contribution appears because \(v\) is compactly supported. Subtracting \(\lambda_{1,0,R}\|yv\|^2\) gives the displayed identity. For finite spectral sums and then for a general projected complement vector, this identity is closed in the critical Friedrichs form sense, not by evaluating a classical endpoint trace. Insert the standard logarithmic cutoff at the critical endpoint \(s=0\), use the Dirichlet trace at \(R\), and let the cutoff tend to one. The cutoff cost tends to zero because \(y^2\sim s\) and the Friedrichs endpoint excludes the \(s^{1/2}\log s\) branch. Thus no singular-endpoint boundary term is produced in the closed-form identity. Keeping the closure inside the projected spectral subspace, write \[q=\sum_{k\ge2}a_k\psi_k,\qquad q^{(M)}=\sum_{2\le k\le M}a_k\psi_k .\] The finite sums \(q^{(M)}\) lie in the Friedrichs operator domain and in \(\operatorname{Ran}Q_{0,R}\). The compact-core identity, closed in the operator graph norm on these finite sums, gives \[\int y^2\left|\left(q^{(M)}/y\right)'\right|^2 = k[q^{(M)}]+(\lambda_{1,0,R}-a_2^2)\|q^{(M)}\|^2 .\] Since \(q^{(M)}=Q_{0,R}q^{(M)}\) is projected off the first critical branch, the branch gap \(\lambda_{2,0,R}-\lambda_{1,0,R}\ge c_{\rm rad}^0R^{-2}\) from Lemma 6 gives the same inequality with the smaller post-reservation constant \(c_{\rm rad}\le c_{\rm rad}^0\). Thus, through the spectral expansion, \[k[q^{(M)}]\ge c_{\rm rad}R^{-2}\|q^{(M)}\|^2, \qquad \|q^{(M)}\|^2\le c_{\rm rad}^{-1}R^2k[q^{(M)}].\] Together with \(k[q^{(M)}]\to k[q]\), \(q^{(M)}\to q\) in \(L^2\), and the critical first-eigenvalue estimate \(\lambda_{1,0,R}-a_2^2=O(R^{-2})\), this makes the right side bounded by \(Ck[q]\) and Cauchy. This defines the closed \(h\)-transform derivative for \(q=yv\) and yields \[\int y^2|v'|^2\le Ck[q].\] Thus no pointwise singular endpoint trace is evaluated for the limiting function, and the closure uses the projected form norm for which \(k[\cdot]\) is the exact energy. The shifted critical endpoint expansion gives \(y(s)=c_0s^{1/2}+O(s^{5/2})\) and \(|y'/y|\le Cs^{-1}\), while the conjugated radial derivative \(D_s\) has its Jacobian/log-derivative coefficient absorbed into the same \(s^{-2}|q|^2\) term. Hence \[|D_sq|^2\le C y^2|v'|^2+C s^{-2}|q|^2\] on the fixed collar, and therefore \[R^{-2}\int_{\varepsilon_R}^{1}|D_sq|^2\,ds \le CR^{-2}k[q] +CR^{-1} \left(R^{-1}\int_{\varepsilon_R}^{1}s^{-2}|q|^2\,ds\right) =o_R(1)k[q].\] On the regular tail \(s\ge1\), the coefficient-error terms carry an exponentially decaying factor, bounded by \(Ce^{-2s}(1+s)^2\). The same projected Jost split, now in the bounded-potential region, gives the corresponding exponentially weighted tail estimates beyond the endpoint collar. Thus this local estimate uses a projected Green diagonal and a closed \(h\)-transform, not a critical Hardy inequality. ◻

Proof. Stage 1: Hadamard identity and sign convention. The matrix is computed for the physical hyperbolic Dirichlet form before comparing the separated \(l\)-rows. Let \(L=-\Delta_{\mathbb{H}^n}\) on \(B_R\), and let \(u_{l,R}\) and \(u_{l',R}\) be exact ball eigenfunctions with eigenvalues \(e_{l,R}\) and \(e_{l',R}\). Introduce the path \[F_t(s,\theta)=s-t\beta_R(s)h(\theta),\qquad 0\le t\le1,\] and let \(X_h=-\beta_Rh\,\partial_s\) be its velocity field at \(t=0\). On \(\partial B_R\), \(X_h\cdot\nu=-h\). Let \(B_h^{E_R}\) denote the derivative at \(t=0\) of the \(L^2\)-unitarily pulled-back shifted physical form \(q_{E_R}=q-E_R\|\cdot\|^2\). We use the local-height sign convention: positive \(h\) is inward motion of the boundary. Thus the outward normal velocity is \(V=X_h\cdot\nu=-h\), and the Dirichlet Hadamard sign is \(-V\). In polarized form, for Dirichlet eigenfunctions \(u\) and \(v\) with eigenvalues \(e_u\) and \(e_v\), the exact Hadamard–Rellich identity is \[B_h^{E_R}(u,v) = -\int_{\partial B_R}(X_h\cdot\nu)\, \partial_\nu u\,\partial_\nu v\,d\sigma_R +\mathcal{C}_X^{E_R}(u,v).\] Here \[A_X=-X\cdot\nabla-\frac{1}{2}\operatorname{div}_gX,\] denotes the skew-adjoint operator associated with the unitary pullback. With this orientation of the unitary, \[\mathcal{C}_X^{E_R}(u,v) = (e_u-E_R)\langle A_Xu,v\rangle +(e_v-E_R)\langle u,A_Xv\rangle = (e_u-e_v)\langle A_Xu,v\rangle .\] The inverse unitary convention changes only the sign of the final commutator. This sign is immaterial below; the proof uses only the exact diagonal cancellation and the absolute value of the off-diagonal term. The mass part has no boundary term because \(u=v=0\) on \(\partial B_R\). The boundary part of the identity is therefore \[\int_{\partial B_R}h\,\partial_\nu u\,\partial_\nu v\,d\sigma_R,\] so an inward deformation gives the positive first variation expected from domain monotonicity. The flux is evaluated on the unperturbed sphere \(s=R\), because it is the first-variation coefficient at \(t=0\); the remaining effects of moving to the graph boundary \(s=R-h(\theta)\) are absorbed into the finite-block remainders \(\Delta_{ll',R}\) controlled in Stage 5. Here is the calculation behind the identity. Before the half-density unitary is inserted, differentiating the pulled-back Dirichlet energy gives the integrand \[(\operatorname{div}X)\langle\nabla u,\nabla v\rangle -\langle\nabla_{\nabla u}X,\nabla v\rangle -\langle\nabla_{\nabla v}X,\nabla u\rangle -E_R(\operatorname{div}X)uv .\] Writing \(Xu=X\cdot\nabla u\), define the Rellich vector field \[\mathcal{R}_X(u,v) = \langle\nabla u,\nabla v\rangle X -(Xu)\nabla v-(Xv)\nabla u .\] Then \[\begin{align} \operatorname{div}\mathcal{R}_X(u,v) &= (\operatorname{div}X)\langle\nabla u,\nabla v\rangle -\langle\nabla_{\nabla u}X,\nabla v\rangle -\langle\nabla_{\nabla v}X,\nabla u\rangle \\ &\qquad -(Xu) \Delta_{\mathbb{H}^n}v-(Xv)\Delta_{\mathbb{H}^n}u . \end{align}\] Because \(\beta_R=0\) on a neighborhood of the origin, \(X_h\) and \(\mathcal{R}_{X_h}(u,v)\) vanish there. Thus the divergence theorem produces no inner-boundary contribution from the singular polar endpoint. On \(\partial B_R\), \(u=v=0\), their tangential derivatives vanish there, and \[\mathcal{R}_X(u,v)\cdot\nu =-(X\cdot\nu)\partial_\nu u\,\partial_\nu v .\] Integrating the divergence identity therefore gives the boundary term above. The two \(J_t^{1/2}\) factors in the \(L^2\)-unitary pullback add the corresponding \(-\frac{1}{2}(\operatorname{div}X)\) contributions in the two slots. Thus the remaining bulk terms are exactly the skew-adjoint commutator terms encoded by \(A_X=-X\cdot\nabla-\frac{1}{2}\operatorname{div}X\). Pairing these terms with \((-\Delta_{\mathbb{H}^n}-E_R)u=(e_u-E_R)u\) and \((-\Delta_{\mathbb{H}^n}-E_R)v=(e_v-E_R)v\) gives the displayed \(\mathcal{C}_X^{E_R}(u,v)\), up to the harmless global sign determined by which of the two inverse unitary conventions is used. For the fixed finite first-branch block, the off-diagonal commutator part is lower order: \[(e_{l,R}-e_{l',R})\langle A_Xu_{l,R},u_{l',R}\rangle =O_{N,C,L}(R^{-4}),\] as verified below from the branch asymptotics and the \(O_N(R^{-1})\) size of the radial generator. Thus the leading \(R^{-3}\) term in the first variation comes only from the Rellich boundary flux.

Stage 2: finite-height reduction. The displayed identity supplies the first derivative at the round ball. To use it for the actual finite graph displacement, let \(\mathcal{B}_t^{E_R}\) be the shifted pulled-back finite-sector form obtained from the path \(F_t(s,\theta)=s-t\beta_R(s)h(\theta)\), evaluated on the fixed first-branch inputs. Then \[\mathcal{B}_1^{E_R}-\mathcal{B}_0^{E_R} = \dot{\mathcal{B}}_0^{E_R} +\int_0^1(1-t)\ddot{\mathcal{B}}_t^{E_R}\,dt .\] The estimates below are uniform in \(t\), because \(0\le th\le C\) and \(\operatorname{Lip}(th)\le L\). Each second \(t\)-derivative term contains either a second height/Jacobian factor, an extra \(R^{-1}\), a transition coefficient on \([\varepsilon_R,2\varepsilon_R]\), or a tangential correction carrying \(\sinh^{-2}(F_t)\). More explicitly, after writing \(a_t=\partial_sF_t=1-t\beta_R'h\) and expanding the Schrodinger-flat pullback, \(\ddot{\mathcal{B}}_t^{E_R}\) is a finite sum of bilinear integrals of the following five types: \[\begin{array}{ll} \text{endpoint:} & R^{-1}s^{-2}\beta_Rh \text{ or its differentiated transition version},\\ \text{transition:} & \beta_R'',\,(\beta_R')^2,\text{ or }G_s' \text{ on } [\varepsilon_R,2\varepsilon_R],\\ \text{shear:} & \beta_R^2|\nabla h|^2\sinh^{-2}(F_t)D_sy\,D_sy',\\ \text{Jacobian:} & (a_t-1)^2,\;\partial_s\log a_t,\text{ and angular half-density factors},\\ \text{extra height:} & \beta_R^2h^2 \text{ multiplying a regular-bulk coefficient.} \end{array}\] These are exactly the term families later routed through the endpoint, transition, shear, Jacobian, and extra-height-factor estimates; no second derivative introduces a new order-\(R^{-3}\) coefficient family. The endpoint, transition, and regular-bulk bounds displayed below therefore give, uniformly for \(l,l'\le N\), \[|\ddot{\mathcal{B}}_t^{E_R}(u_{l,R},u_{l',R})| \le C_{N,C,L}R^{-4}.\] Thus the finite-height displacement has the same leading coefficient as the round-sphere first variation, and the integral remainder is included in \(\Delta_{ll',R}\).

Applying this to \(u_{l,R}\) and \(u_{l',R}\), the shifted spectral part is \[(e_{l,R}-E_R)\langle A_Xu_{l,R},u_{l',R}\rangle +(e_{l',R}-E_R)\langle u_{l,R},A_Xu_{l',R}\rangle .\] For Dirichlet eigenfunctions \(u\) and \(v\), integration by parts gives the skew-adjointness identity \[\langle A_Xu,v\rangle+\langle u,A_Xv\rangle = -\int_{\partial B_R}(X\cdot\nu)u\overline{v}\,d\sigma_R=0,\] because \(u=v=0\) on \(\partial B_R\), with no inner-boundary contribution since \(\beta_R=0\) near the origin. Taking \(v=u\) gives the diagonal vanishing, and taking \(u=u_{l,R}\), \(v=u_{l',R}\) gives the off-diagonal skew relation on the finite first-branch span. This skew-adjointness identity is distinct from the Hadamard–Rellich boundary flux identity, which involves the normal derivatives \(\partial_\nu u\,\partial_\nu v\). The off-diagonal commutator term \[(e_u-E_R)\langle A_Xu,v\rangle +(e_v-E_R)\langle u,A_Xv\rangle\] therefore simplifies to \((e_u-e_v)\langle A_Xu,v\rangle\) by this skew relation, not by a boundary flux calculation. The normal-derivative flux \(-(X\cdot\nu)\partial_\nu u\,\partial_\nu v\) was produced above by the Rellich vector field \(\mathcal{R}_X(u,v)\). Thus for \(l\ne l'\), the shifted spectral part rewrites as \[(e_{l,R}-e_{l',R})\langle A_Xu_{l,R},u_{l',R}\rangle,\] which is an off-diagonal error absorbed into the Stage 5 remainder estimates. Therefore the leading coefficient is the boundary flux; row-dependent separated Schrodinger centrifugal terms have already canceled in the physical Hadamard identity before the \(l\)-rows are read off.

Stage 3: boundary-flux coefficient. If \(u_{l,R}=(\sinh s)^{-(n-1)/2}y_{l,R}Y_l\), then \(y_{l,R}(R)=0\). Hence the hyperbolic boundary measure cancels the Schrodinger half-density, and the boundary-flux scalar, after the angular factor is separated, is \[\partial_s y_{l,R}(R)\,\partial_s y_{l',R}(R)\] with the sign convention fixed above. Indeed, \[\begin{align} \partial_\nu u_{l,R}|_{s=R} &= (\sinh R)^{-(n-1)/2} \partial_s y_{l,R}(R)Y_l(\theta), \\ &\int_{\partial B_R} h\,\partial_\nu u_{l,R}\,\partial_\nu u_{l',R}\,d\sigma_R \\ &\qquad = \partial_s y_{l,R}(R)\partial_s y_{l',R}(R) \int_{\mathbb{S}^{n-1}} h\,Y_l\overline{Y_{l'}}\,d\theta , \end{align}\] because the derivative of the half-density term is multiplied by \(y_{l,R}(R)=0\), and the two factors \((\sinh R)^{-(n-1)/2}\) cancel the boundary measure \(d\sigma_R=\sinh^{n-1}R\,d\theta\) exactly. The Hadamard derivative is taken at the round sphere, where the normal is radial. Since \(y_{l,R}(R)=0\), the tangential derivatives of \((\sinh s)^{-(n-1)/2}y_{l,R}(s)Y_l(\theta)\) vanish on the unperturbed boundary. If one instead expands the graph normal for \(r=R-h(\theta)\), the remaining tangential correction carries a factor \(\sinh^{-2}R\) and bounded finite-sector angular derivatives, hence contributes only \(O_{N,L}(e^{-2R}R^{-3})\).

Stage 4: spectral-defect commutator and cutoff flow. For \(l\ne l'\), the first branches are not exactly \(E_R\)-eigenvectors. The fixed-degree ball asymptotics give \[e_{l,R}-E_R=b_{n+2l}R^{-3}+o_l(R^{-3})=O_N(R^{-3})\] uniformly for \(l\le N\). Here we use Kristály’s fixed-dimension expansion with a genuine \(o_m(R^{-3})\) remainder after the dimension shift \(m=n+2l\), as encoded by the first-four-terms statement in [17]; the set \(\{b_{n+2l}:l\le N\}\) is finite, and the finitely many fixed-degree remainders have maximum \(o_N(R^{-3})\). Hence \(e_{l,R}-e_{l',R}=O_N(R^{-3})\). The resulting shifted-form generator terms are lower order. More precisely, after the angular factor has been separated, the radial part of the Schrodinger-conjugated generator for the reference scaling field is, up to the harmless global sign determined by the height-variation convention, \[G_R=-(s/R)\partial_s-\frac{1}{2R}.\] The fixed-sector endpoint and bulk estimates from Lemma 7 give \[\|G_Ry_{l,R}\|_2 \le \|(s/R)\partial_s y_{l,R}\|_2+\frac{1}{2R}\|y_{l,R}\|_2 \le C_NR^{-1}.\] This is an \(L^2\)-operator-vector bound, so by Cauchy–Schwarz it applies to off-diagonal matrix elements. Hence the off-diagonal generator term is \(O_N(R^{-3})O_N(R^{-1})=O_N(R^{-4})\). In the diagonal case the Dirichlet boundary cancellation above gives zero before any estimate is needed. Here the Schrodinger-conjugated flat generator is computed as follows. If \(X=X^s\partial_s\), then \[\operatorname{div}_gX=\partial_sX^s+(n-1)\coth(s)X^s .\] Conjugating \(-X^s\partial_s-\frac{1}{2}\operatorname{div}_g X\) by \((\sinh s)^{(n-1)/2}\) gives \[-X^s\partial_s-\frac{1}{2}\partial_s X^s .\] The \((n-1)\coth(s)X^s\) term cancels exactly against the derivative of the half-density, so a field \(X^s=\sigma(s)h(\theta)\) has flat generator \[-\sigma(s)h\partial_s-\frac{1}{2}\sigma'(s)h .\] It contains no angular derivative. The actual cutoff flattening is compared with the pure normal flow at the level of the physical shifted form, before the separated rows are read off. The generator estimate keeps the actual weighted derivative: \[\|(s/R)\partial_s y_{l,R}\|_2\le C_NR^{-1}.\] On \([S_N,R]\) the common finite-box profile gives \[|\partial_s y_{l,R}(s)|\le C_NR^{-3/2}, \qquad \int_{S_N}^R|\partial_s y_{l,R}|^2\,ds\le C_NR^{-2}.\] On the endpoint interval the factor \(s/R\) is essential in the critical channel; the endpoint bound below gives a contribution \(O_N(R^{-5/2})\) to the \(L^2\)-norm of \((s/R)\partial_s y_{l,R}\) there.

Thus the radial interior extension affects the finite first-branch block only through the displayed eigenvalue-defect commutators. If one instead writes a nonradial graph-normal extension, the tangential boundary polarization is a symmetric finite-dimensional angular bilinear form, with typical terms such as \(\sinh^{-2}R\langle\nabla_S Y_l,h\nabla_S Y_{l'}\rangle\). Hence it is bounded by \[C_{N,L}\sinh^{-2}R\,|\partial_s y_{l,R}(R)|\,|\partial_s y_{l',R}(R)| =O_{N,L}(e^{-2R}R^{-3}).\] The cutoff field \(-\beta_R h\partial_s\) and the reference scaling field \(-(s/R)h\partial_s\) are radial, have the same boundary normal component, and coincide on the regular bulk \(s\ge2\varepsilon_R\). Their difference is radial, vanishes at \(s=R\), has zero boundary normal trace, and has a unitary generator vector of \(L^2\)-size \(O_{N,L}(R^{-1})\) on first-branch inputs. Indeed, if \(\sigma_{\rm diff}=s/R-\beta_R\), then the flat generator contains \(-\sigma_{\rm diff}h\partial_s-\frac{1}{2}\sigma_{\rm diff}'h\), but no \(\sigma_{\rm diff}''\); the second-derivative transition coefficients are absorbed into the Stage 5 finite-block remainders. On the protected origin \(\sigma_{\rm diff}=s/R\), and on the transition layer \(\sigma_{\rm diff}=O(\varepsilon_R/R)\), \(\sigma_{\rm diff}'=O(R^{-1})\), so the same endpoint and transition estimates give the stated \(O_{N,L}(R^{-1})\) vector bound. The zero boundary trace means the difference field contributes no boundary flux. Its bulk contribution is instead the commutator \[(e_{l,R}-e_{l',R}) \langle A_{X_{\rm diff}}u_{l,R},u_{l',R}\rangle = O_N(R^{-3})O_{N,L}(R^{-1}) = O_{N,L}(R^{-4}),\] which is \(o_{N,C,L,\delta_{\rm cut}}(R^{-3})\).

Therefore the first-order cutoff-to-normal-flow difference is already \(O_{N,L}(R^{-4})\) globally. The remaining finite first-branch coefficient remainders after subtracting the first-order local-height model are only endpoint, transition, and higher-order terms.

Stage 5: endpoint, transition, and shear remainders. On the transition layer \(I_R=[\varepsilon_R,2\varepsilon_R]\), the fixed-sector endpoint estimates give, uniformly for \(l\le N\), \[|y_{l,R}(s)|\le C_NR^{-3/2}s^{1/2},\qquad |s\partial_s y_{l,R}(s)|\le C_NR^{-3/2}s^{1/2}, \qquad 0<s\le2\varepsilon_R,\] with stronger powers of \(s\) in noncritical channels. These are normalized first-branch bounds, not projected-complement estimates. For each fixed \(l\le N\), the regular finite-box solution \[u_l(s,\rho),\qquad \rho=\pi/R+O_N(R^{-2}),\] is a small-energy perturbation of the positive zero-energy regular solution on \(0<s\le2\varepsilon_R\). That zero-energy solution has endpoint behavior \(s^{\nu_l+1/2}\) and a nonzero linear coefficient at infinity; hence its \((0,R)\)-norm is of order \(R^{3/2}\). The finite-box Jost normalization from Lemma 7, the same input used for the boundary-slope asymptotic, transfers this uniformly for \(l\le N\) to the actual normalized first eigenfunctions: \[|y_{l,R}(s)|+|s\partial_s y_{l,R}(s)| \le C_NR^{-3/2}s^{\nu_l+1/2} \le C_NR^{-3/2}s^{1/2}, \qquad 0<s\le2\varepsilon_R .\] In the critical \((n,l)=(2,0)\) channel this says that the comparison solution is \(s^{1/2}\) at the endpoint and grows like \(s\) at infinity, so the normalized endpoint coefficient is \(O(R^{-3/2})\). The same fixed-sector normalization gives the boundary-slope size \(R^{-3/2}\). Together with the finite-box bulk profile \(|\partial_s y_{l,R}|\le C_NR^{-3/2}\) on \([S_N,R]\), this yields \[\|(s/R)\partial_s y_{l,R}\|_2\le C_NR^{-1}.\] Thus the worst critical transition scalar satisfies \[R^{-1}\varepsilon_R^{-2}\int_{I_R}|y_{l,R}y_{l',R}|\,ds \le C_NR^{-4},\] and the mixed transition term satisfies \[R^{-1}\varepsilon_R^{-1} \int_{I_R}\bigl(|\partial_s y_{l,R}|\,|y_{l',R}| +|y_{l,R}|\,|\partial_s y_{l',R}|\bigr)\,ds \le C_NR^{-4}.\] The \(G_s^2\), quadratic metric, and cutoff-square terms are smaller, since they contain an additional factor \(R^{-1}\), \(\varepsilon_R\), or a positive local majorant. There is no hidden \(O(R^{-2})\) finite-height bulk term here. The constant-height case is a consistency check on the linear coefficient and on the constant-direction quadratic scale: if \(h\equiv c\), the domain is exactly \(B_{R-c}(o)\), and \[\frac{\pi^2}{(R-c)^2} = \frac{\pi^2}{R^2} +2\pi^2cR^{-3} +O_C(R^{-4});\] hence the quadratic height contribution is \(O_C(R^{-4})\), not \(O(R^{-2})\). The genuinely variable-height quadratic pieces are controlled separately. In the exact coordinate pullback the radial metric quadratic remainder is bounded, on fixed first-branch inputs, by \[\int \frac{(\beta_R'h)^2}{1-\beta_R'h}\, |\partial_s y_{l,R}|^2\,ds \le C_{C}R^{-2}\int|\partial_s y_{l,R}|^2\,ds =O_{N,C}(R^{-4}).\] On the endpoint transition collar the displayed endpoint bounds give the same or smaller order. In the critical channel the displayed endpoint bound gives \(|\partial_s y_{l,R}|\le C_NR^{-3/2}s^{-1/2}\), so on \([\varepsilon_R,2\varepsilon_R]\) \[\int_{I_R}(\beta_R')^2|\partial_s y_{l,R}|^2\,ds \le C_NR^{-5}\int_{\varepsilon_R}^{2\varepsilon_R}s^{-1}\,ds =O_N(R^{-5})=o_N(R^{-4}).\] The noncritical channels have better endpoint powers. The shear-square term has schematic size \[\int \frac{\beta_R^2|\nabla h|^2}{\sinh^2(F_h)} |D_sy_{l,R}|\,|D_sy_{l',R}|\,ds .\] Splitting at a fixed \(S_N\), the endpoint/Jost bounds on \([\varepsilon_R,S_N]\) and the exponential factor \(\sinh^{-2}(F_h)\) on \([S_N,R]\) give \[\int \frac{\beta_R^2|\nabla h|^2}{\sinh^2(F_h)} |D_sy_{l,R}|\,|D_sy_{l',R}|\,ds \le C_{N,C,L}R^{-4}.\] Here \(D_s\) is the conjugated radial derivative appearing in the Schrodinger pullback; the protected origin has \(\beta_R=0\), and the transition collar is covered by the same endpoint estimates. In the critical endpoint channel this estimate is borderline but smaller than required. On \([\varepsilon_R,S_N]\) one has \(\beta_R=O(s/R)\), \(\sinh(F_h)\simeq s\), and \(|D_sy_{l,R}|\le C_NR^{-3/2}s^{-1/2}\), hence the endpoint integrand is bounded by \(C_{N,C,L}R^{-5}s^{-1}\). Its integral is \[C_{N,C,L}R^{-5}\log(S_N/\varepsilon_R)=O_{N,C,L}(R^{-5}\log R)=o(R^{-4}).\] On \([S_N,R]\) the same derivative factors are \(O_N(R^{-3})\) and \(\sinh^{-2}(F_h)\) is exponentially small after enlarging \(S_N\), giving an even smaller contribution. The pulled-back mass term after the physical radial Schrodinger conjugation has coefficient \(1-\beta_R'h\) and is strictly linear in \(h\). Although the unitary generator \(A_Xu_{l,R}\) has \(L^2\)-size \(O_N(R^{-1})\), the exact Hadamard–Rellich identity uses only the skew-adjoint spectral-defect commutator. The identity may be written in physical \(L^2(dV_{\mathbb{H}^n})\); after the radial Schrodinger conjugation, \(A_X\) is unitarily equivalent to the flat radial generator used in the size estimate. No angular derivative is hidden here: \(X_h=-\beta_Rh\,\partial_s\) has only a radial component, so \(X_h\cdot\nabla\) differentiates only in \(s\), while \(\operatorname{div}X_h\) contains \(h(\theta)\) as a multiplier and no \(\nabla_\theta h\) term. Thus the relevant term is \[(e_{l,R}-e_{l',R}) \langle A_Xu_{l,R},u_{l',R}\rangle =O_N(R^{-3})O_{N,C,L}(R^{-1}) =O_{N,C,L}(R^{-4}),\] not an independent \(\|A_Xu_{l,R}\|^2\) term.

The estimate uses only the \(C^0\) and Lipschitz bounds on the smooth height. If angular second derivatives of \(h\) appear in a coordinate expansion of this finite block, one integrates by parts on the sphere at this stage; the derivatives then fall on the fixed finite set of spherical harmonics or on the weak gradient of \(h\), giving constants depending only on \(N\) and \(L\). Thus no \(C^2\) norm of a smooth approximating height enters \(\Delta_{ll',R}\). Let \(\Delta_{ll',R}\) denote the sum of these finite-block remainders after the boundary flux and the skew-adjoint commutator have been subtracted. Consequently \[|\Delta_{ll',R}|=o_{N,C,L,\delta_{\rm cut}}(R^{-3})\] uniformly for \(l,l'\le N\). Hence the cutoff-flattened matrix has the same leading flux coefficient as the normal-flow derivative.

Stage 6: assembly of the finite first-branch block. The common finite-sector boundary-slope asymptotic is the fixed-\(l\) first-branch part of Lemma 7. It does not rely on the projected Green estimates in Lemma 8: \[\partial_s y_{l,R}(R) = -\sqrt{2/R}\,\pi/R+O_N(R^{-5/2}),\] and therefore, uniformly for \(l,l'\le N\), \[\partial_s y_{l,R}(R)\partial_s y_{l',R}(R) = \left(\sqrt{2/R}\,\pi/R\right)^2+O_N(R^{-4}) = 2\pi^2R^{-3}+O_N(R^{-4}).\] Thus, in particular, \[B_h^{E_R}(u_{l,R},u_{l,R}) = 2\pi^2R^{-3}\int_{\mathbb{S}^{n-1}}h|Y_l|^2\,d\theta +o_{N,C,L,\delta_{\rm cut}}(R^{-3}),\] and the polarized version gives the same \(2\pi^2h\) boundary matrix coefficient on the whole finite first-branch block. This gives \(M_{ll',R}=2\pi^2R^{-3}+o_{N,C,L,\delta_{\rm cut}}(R^{-3})\), uniformly for \(l,l'\le N\). The finite-dimensional angular form identity follows by summing the displayed matrix identity against \(\int hY_\alpha\overline{Y_\beta}\,d\theta\); the error is uniform on the finite \(P_N\) space and over \(0\le h\le C\). The exact finite-block form is obtained by adding this first-variation matrix and the finite-block remainders to the unperturbed separated first-branch energies. Since \[e_{l,R}-E_R=b_{n+2l}R^{-3}+o_l(R^{-3}) = R^{-3}(b_n+\tau_{n,l})+o_l(R^{-3}),\] the unperturbed finite block contributes \[R^{-3}\langle f_{\le},P_N(T_n+b_n)P_Nf_{\le}\rangle +o_N(R^{-3})\|f_{\le}\|^2 .\] Combining this with the boundary-flux matrix identity and the bound on \(\Delta_{ll',R}\) gives the exact shifted finite-block form asserted in the statement. ◻

Proof. Before the bounded coefficient-error classes are routed, the raw high separated form contains the nonnegative shifted radial kinetic part and the unperturbed centrifugal term \[\sum_{l>N}\int\left(|\partial_sw_l|^2-\pi^2R^{-2}|w_l|^2\right) + \sum_{l>N}c_l\int\sinh^{-2}(F_h)|w_l|^2 .\] The first sum is nonnegative by the one-dimensional Dirichlet–Friedrichs Poincaré inequality on \((0,R)\): high angular modes have the zero-trace Friedrichs behavior at \(s=0\) and Dirichlet trace at \(s=R\). More explicitly, for \(l>N_A(n)\) the inverse-square endpoint exponent satisfies \(\nu_l>1/2\). This holds for all \(l\ge1\), hence for \(l>N_A(n)\), because the coefficient satisfies \(c_l>0\) and \(c_l=\nu_l^2-1/4\). Functions in the Friedrichs form domain therefore have ordinary \(H^1\) trace \(w_l(0)=0\); the graph boundary gives \(w_l(R)=0\). Thus \[\int_0^R|\partial_sw_l|^2\,ds \ge \pi^2R^{-2}\int_0^R|w_l|^2\,ds .\] This is the Dirichlet–Dirichlet constant, not the mixed Neumann–Dirichlet constant. The coefficient \(F_h\) occurs only in the centrifugal weight above; the Jacobian perturbations of the radial kinetic term are not part of the raw reserve and are charged in classes (3)–(6). Since \(c_l\ge\lambda_l/2\) for \(l>N_A(n)\), this part alone gives \[\operatorname{Sep}_>[w_>]\ge c_A(n)\operatorname{Ang}[w_>], \qquad c_A(n)=1/2.\] ◻

Proof. The high-mode classes fix the accounting. The symbol \(\operatorname{Err}_{\rm hi}\) denotes a finite sum of the high-mode pieces left after the six coefficient classes have been expanded once and each piece has been assigned to exactly one route: favorable sign, scalar high-angular conversion, \(R\)-vanishing shear reserve, transition reserve, or exponentially weighted tail. This classification organizes the perturbative part of the pulled-back form; it is not a second decomposition of the clean separated form \(\operatorname{Sep}_>\). By definition, \(\operatorname{Err}_{\rm hi}\) excludes the pure high-mode singular-potential displacement. This exclusion is legitimate because class (2) gives \[c_l\bigl(\sinh^{-2}(F_h)-\sinh^{-2}s\bigr)|w_l|^2\ge0\] for \(N\ge N_A(n)\), because \(c_l\ge0\), \(0\le\beta_Rh\), and \(F_h=s-\beta_Rh\le s\). Thus this term is removed before the reserve estimate is applied and is not used as an additional positive term in the absorption inequality. In contrast, the lower-order scalar residuals from the angular Jacobian and half-density coefficients have no angular derivative, so the scalar-to-angular conversion is legitimate and gives \[C_{C,L}\lambda_{N+1}^{-1}\operatorname{Ang}[w_{>}].\] Here the factor \(C_{C,L}\lambda_{N+1}^{-1}\) is one contribution to \(\sigma_N\), and it tends to zero when the high cutoff \(N\) is increased. The principal shear contribution is instead a kinetic error controlled by Lemma 3, before angular projection; it contributes vanishing multiples of \(\operatorname{Ang}[w_>]\) and \(\operatorname{Sep}_>[w_>]\) to the reserve. The mollifier-sensitive angular Jacobian scalar is resolved by Lemma 4, uniformly in the smooth approximation to \(h\). The remaining high-mode errors come only from multiplicative, transition, and Jacobian errors, and are absorbed by the cutoff-small and \(R\)-vanishing reserve before the remaining high block is compressed. Equivalently, after \(N\) is fixed the high-mode error satisfies \[|\operatorname{Err}_{\rm hi}[w_>]| \leq \tau_{N,R}\operatorname{Sep}_>[w_>] +(\sigma_N+\sigma_{N,R})\operatorname{Ang}[w_>] +o_{N,C,L}(R^{-3})\|w_>\|^2 .\] Here \(\sigma_N\to0\) as \(N\to\infty\), while \(\sigma_{N,R},\tau_{N,R}\to0\) as \(R\to\infty\) for fixed \(N\). Since \(\operatorname{Ang}[w_>]\le c_A(n)^{-1}\operatorname{Sep}_>[w_>]\) by Lemma 10, \(N\) is chosen so large that \(\sigma_N/c_A(n)\le\rho_{\rm res}/2\), and then the final \(R\)-threshold is enlarged so that \[\tau_{N,R}+\sigma_{N,R}/c_A(n)\le \rho_{\rm res}/2.\] Then \[\rho_{\rm res}\operatorname{Sep}_>[w_>] +\operatorname{Err}_{\rm hi}[w_>] \ge -o_{N,C,L}(R^{-3})\|w_>\|^2.\] Only the clean remainder \((1-\rho_{\rm res})\operatorname{Sep}_>[w_>]\) is passed to the block-form compression step. This is a scalar decomposition of the nonnegative separated high-mode form: \[\operatorname{Sep}_> = \rho_{\rm res}\operatorname{Sep}_> +(1-\rho_{\rm res})\operatorname{Sep}_>.\] The first summand is spent only in this reserve absorption estimate. The finite-block scalar compression below uses only the second summand, so the high-mode reserve is not reopened or counted twice. If the low/high physical cross terms in Lemma 13 contribute an additional \(\tau'_{N,R}\operatorname{Sep}_>\), this is treated exactly like \(\tau_{N,R}\operatorname{Sep}_>\): for fixed \(N\) the coefficient \(\tau'_{N,R}\) tends to zero as \(R\to\infty\), and the last threshold is replaced by \[\tau_{N,R}+\tau'_{N,R}+\sigma_{N,R}/c_A(n) \le \rho_{\rm res}/2.\] ◻

Proof. In the exact coordinate-unitary flat representation the mass form is the reference \(L^2(ds\,d\theta)\) inner product. Hence the mass cross term between \(J_Rf_{\leq}\) and \(q_{\rm rad}\) vanishes exactly. The unperturbed separated shifted form has no first-branch/radial-complement cross term either, because \(J_R\) uses the first eigenfunction of each fixed degree and \(q_{\rm rad}\) is projected off that branch in the same flat fiber.

All remaining mixed terms are perturbative coefficient classes. For the regular coefficient, angular-drift, and shear classes, the first branch satisfies, uniformly on the fixed finite angular sector, \[\int |D_sp|^2\le C_NR^{-2}\|f_{\leq}\|^2,\qquad \int\sinh^{-2}(F_h)|p|^2\le C_NR^{-3}\|f_{\leq}\|^2,\] and its angular form is \(O_N(R^{-3})\|f_{\leq}\|^2\). The shear coefficient has \[{\beta_R|\nabla h|\over \sinh(F_h)}\le C_{C,L}R^{-1}.\] Cauchy–Schwarz and Young’s inequality therefore route the derivative factor on \(q_{\rm rad}\) into \(\eta{\mathcal{Q}}_{\rm rad}[q_{\rm rad}]\), while the first-branch side costs at most \[C_{\eta,N,C,L}R^{-2}\int |D_sp|^2 +C_{\eta,N,C,L}R^{-2}\int\sinh^{-2}(F_h)|p|^2 = o_{N,C,L,\eta}(R^{-3})\|f_{\leq}\|^2 .\] The purely angular first-order drift terms are even smaller on the finite first branch, because their coefficients are \(O_L(R^{-1})\) and the weighted first-branch scalar or angular forms are \(O_N(R^{-3})\).

It remains to check the endpoint coefficient classes, where pointwise smallness is false. In the critical channel the finite first-branch Jost bounds give, on \(0<s\le2\varepsilon_R\) and then on the fixed collar, \[|p(s,\theta)|+|sD_sp(s,\theta)| \le C_NR^{-3/2}s^{1/2}\|f_{\leq}\|.\] The projected Green estimates of Lemma 8 give \[R^{-1}\int_{\varepsilon_R}^{1}s^{-2}|q_{\rm rad}|^2=o_R(1){\mathcal{Q}}_{\rm rad}[q_{\rm rad}], \qquad R^{-2}\int_{\varepsilon_R}^{1}|D_sq_{\rm rad}|^2=o_R(1){\mathcal{Q}}_{\rm rad}[q_{\rm rad}]\] in the critical projected complement, and stronger estimates hold in the noncritical finite set of degrees. Thus a typical scalar cross term with coefficient \(O(R^{-1}s^{-2})\) is bounded by \[\begin{align} R^{-1}\int_{\varepsilon_R}^{1}s^{-2}|p\,q_{\rm rad}| &\le \left(R^{-1}\int_{\varepsilon_R}^{1}s^{-2}|q_{\rm rad}|^2\right)^{1/2} \left(R^{-1}\int_{\varepsilon_R}^{1}s^{-2}|p|^2\right)^{1/2} \\ &\le o_R(1)^{1/2}{\mathcal{Q}}_{\rm rad}[q_{\rm rad}]^{1/2} \,O_N(R^{-2}\sqrt{\log R})\|f_{\leq}\|. \end{align}\] Young’s inequality makes this \[\eta{\mathcal{Q}}_{\rm rad}[q_{\rm rad}] +o_{N,C,L,\eta}(R^{-3})\|f_{\leq}\|^2 .\] The transition terms are the same calculation on \([\varepsilon_R,2\varepsilon_R]\), using \(\int_{I_R}|q_{\rm rad}|^2\le C\varepsilon_R{\mathcal{Q}}_{\rm rad}[q_{\rm rad}]\) for scalar transition coefficients. For mixed transition terms with a derivative on \(q_{\rm rad}\), no integration by parts is needed. Since \(|G_s|\le C_CR^{-1}\varepsilon_R^{-1}\) on \(I_R\), \[\begin{align} \int_{I_R}|G_s p\,D_sq_{\rm rad}| &\le C_CR^{-1}\varepsilon_R^{-1} \left(\int_{I_R}|p|^2\right)^{1/2} \left(\int_{I_R}|D_sq_{\rm rad}|^2\right)^{1/2} \\ &\le o_R(1)^{1/2}R^{-3/2} \|f_{\leq}\|\,{\mathcal{Q}}_{\rm rad}[q_{\rm rad}]^{1/2}. \end{align}\] Here \(\int_{I_R}|p|^2\le C_NR^{-3}\varepsilon_R^2\|f_{\leq}\|^2\), while Lemma 8 gives \(\int_{I_R}|D_sq_{\rm rad}|^2\le o_R(1)R^2{\mathcal{Q}}_{\rm rad}[q_{\rm rad}]\) in the critical channel, with stronger estimates in the noncritical finite set. Young’s inequality again gives \[\eta{\mathcal{Q}}_{\rm rad}[q_{\rm rad}] +o_{N,C,L,\eta}(R^{-3})\|f_{\leq}\|^2.\] The conjugated difference \(D_s-\partial_s=O(s^{-1})\) is absorbed into the same scalar transition estimates above. The regular tail \(s\ge1\) carries an exponentially decaying weight and is controlled by the same Young splitting with an \(o(R^{-3})\) first-branch cost.

Choosing \(\eta\) as part of the fixed radial-complement reserve, absorbing the finite first-branch \(o(R^{-3})\) cost into the trailing remainder, and then enlarging \(R_0\) proves the stated mixed estimate. ◻

Proof. Start from the exact physical coordinate pullback \(r=F_h(s,\theta)=s-\beta_R(s)h(\theta)\), followed by the fixed radial Schrodinger half-density conjugation. The radial and angular gradient principal parts are given by Lemma 3; the remaining metric, Jacobian, half-density, and singular-potential differences are precisely the six coefficient classes in Definition 1. The finite first-radial block \(J_RP_Nf\) is treated as a physical finite-dimensional Hadamard problem, not by termwise separated-coordinate estimates. The exact-form conclusion of Lemma 9 gives the finite low block \[R^{-3}\langle f_{\leq},P_N(T_n+2\pi^2h+b_n)P_Nf_{\leq}\rangle +o_{N,C,L}(R^{-3})\|f_{\leq}\|^2 .\]

Terms with one factor in the finite first branch \(J_RP_Nf\) and one factor in the high block \(w_>\) are not identified with the angular multiplication matrix \(2\pi^2h\). The finite Hadamard lemma gives that coefficient only for pairs of degrees \(\le N\). Instead these low/high physical cross terms are estimated in the pulled-back form before compression. The genuinely order-\(R^{-3}\) part is the bounded multiplication coupling generated by the boundary-height coefficient on the first-radial part of the high angular sector. Writing, only for this estimate, the high block as \(w_>=J_R^{>}g_>+q_>\), where \(J_R^{>}\) is the degree-adapted first-radial lift in degrees \(>N\), this leading part has the form \[2\pi^2R^{-3}\langle f_{\leq},h\,g_>\rangle_{L^2(\mathbb{S}^{n-1})},\] up to finite-\(R\) normalization errors. Those errors are only the mismatch between the exact degree-dependent first-branch boundary slopes and the common \(2\pi^2R^{-3}\) flux coefficient, together with the difference between the flattened \(L^2\) normalization of \(J_R^{>}g_>\) and the angular norm of \(g_>\). The fixed-degree boundary-slope asymptotic makes the former \(o_{N,C,L}(R^{-3})\|f_{\leq}\|\,\|g_>\|\) for the finite side, and the latter is an \(R\)-vanishing perturbation of the high separated form after Cauchy–Schwarz. Hence both are assigned to the same finite-low \(o_{N,C,L}(R^{-3})\) cost and high-side \(\tau'_{N,R}\operatorname{Sep}_>\) reserve as the other non-leading low/high classes. The high radial-complement part \(q_>\), transition pieces, and the non-leading boundary-trace corrections are likewise among the remaining low/high classes routed below into \(\tau'_{N,R}\operatorname{Sep}_>\) and the finite-low \(o_{N,C,L}(R^{-3})\) remainder. Since \(\|M_h\|_{2\to2}\le C\), the choice \(M_C=4\pi^2C+1\) in Subsection 6.2 dominates the displayed bounded multiplication coupling after the final enlargement of \(R_0\). Hence it is bounded below by \[-M_CR^{-3}\|f_{\leq}\|\,\|w_>\|.\] Young’s inequality with parameter \(\eta_{\rm gap}\) gives \[-M_CR^{-3}\|f_{\leq}\|\,\|w_>\| \ge -\eta_{\rm gap}R^{-3}\|f_{\leq}\|^2 -\frac{M_C^2}{4\eta_{\rm gap}}R^{-3}\|w_>\|^2 .\] This is the fixed low/high Schur cost recorded in \(B_N^h\); it is not an \(o(R^{-3})\) term.

The remaining regular, shear, Jacobian, and transition low/high classes have an additional small coefficient after the leading bounded multiplication coupling has been separated. The finite-low first branch has \[\int |D_sJ_RP_Nf|^2\le C_NR^{-2}\|f_{\leq}\|^2,\qquad \int\sinh^{-2}(F_h)|J_RP_Nf|^2\le C_NR^{-3}\|f_{\leq}\|^2,\] and finite angular form \(O_NR^{-3}\|f_{\leq}\|^2\). Applying Cauchy–Schwarz and Young’s inequality to the regular, shear, Jacobian, and transition coefficient classes after the fixed bounded-multiplication part has been removed gives a high-side contribution of the form \[\tau'_{N,R}\operatorname{Sep}_>[w_>], \qquad \tau'_{N,R}\to0\quad(R\to\infty,\;N\text{ fixed}),\] and a low-side contribution \[o_{N,C,L}(R^{-3})\|f_{\leq}\|^2 .\] The high-side term is folded into the reserve absorption by using the strengthened threshold \[\tau_{N,R}+\tau'_{N,R}+\sigma_{N,R}/c_A(n) \le \rho_{\rm res}/2\] from Lemma 11; this is allowed after enlarging \(R_0\) because all three quantities tend to zero for fixed \(N\). The low-side term is charged to the trailing \(o_{N,C,L}(R^{-3})\) remainder, not to an additional fixed fraction of \(\delta_{\rm eff}\). The only fixed low-side loss from the low/high coupling is the Young loss \(\eta_{\rm gap}R^{-3}\|f_{\leq}\|^2 =(\delta_{\rm eff}/32)R^{-3}\|f_{\leq}\|^2\) already displayed in the statement of the lemma. The endpoint pieces use the same first-branch endpoint bounds and projected endpoint controls as Lemma 12; the high angular factor is controlled by Lemma 10.

On the pure high block, the nonnegative singular-potential displacement is discarded, and the remaining high errors are absorbed by Lemma 11. This spends only the reserved \(\rho_{\rm res}\operatorname{Sep}_>\) summand and leaves the clean \((1-\rho_{\rm res})\operatorname{Sep}_>\) term appearing in \(B_N^h\). On finite radial complements and first-branch/radial-complement mixed terms, Lemma 8 and Lemma 12 route the critical endpoint mass and derivative contributions into a fixed radial-complement reserve; after this reserve is removed the remaining radial-complement form has the post-reservation lower bound \({\mathcal{Q}}_{\rm rad}\ge c_{\rm rad}R^{-2}\|q_{\rm rad,\leq}\|^2\).

Summing the six routed classes, the finite Hadamard low block, the high reserve absorption, and the radial-complement mixed estimate gives the displayed lower bound. All constants have already been fixed before the final enlargement of \(R_0\), so the remaining fixed-\(N\) and fixed-parameter remainders are \(o_{N,C,L}(R^{-3})\). ◻

Proof. After the physical low/high cross terms have already been Young-routed in Lemma 13, the remaining high block is compressed by a scalar lower bound. Let \(\widetilde{B}_N^h\) be the block-diagonal form obtained from \(B_N^h\) after this scalar compression. Set \[\Gamma_N = (1-\rho_{\rm res})(b_n+\tau_{n,N+1}) -\frac{M_C^2}{4\eta_{\rm gap}}.\] After the high-mode compression estimate below, \(\widetilde{B}_N^h\) is the block-diagonal form on \[P_NL^2(\mathbb{S}^{n-1})\oplus {\mathcal{H}}_{>,R},\] where \({\mathcal{H}}_{>,R}\) denotes the flattened fixed-cylinder Hilbert subspace of angular degrees \(>N\), with the same Dirichlet–Friedrichs endpoint conditions as in the high separated form. It is given by \[\widetilde{B}_N^h[g_{\leq}+z_{>}] = \langle g_{\leq},P_NL_hP_Ng_{\leq}\rangle +\Gamma_N\|z_{>}\|^2.\] Thus the full high cylinder block has been replaced by the scalar level \(\Gamma_N\), repeated with the multiplicity of \({\mathcal{H}}_{>,R}\); no identification of \(z_>\) with a purely angular function is being made. This step does not reopen \(\operatorname{Sep}_>\), does not reabsorb class-2, class-4, class-6, or low/high physical cross errors, and uses only the already reduced scalar high coefficient \((1-\rho_{\rm res})(b_n+\tau_{n,N+1})\), minus the fixed Young penalty \(M_C^2/(4\eta_{\rm gap})\) from the bounded low/high first-branch coupling. The positive coefficient comes from the single clean summand \((1-\rho_{\rm res})\operatorname{Sep}_>\), not from a second use of the reserve summand. Indeed, after the raw high form has been split as \[\operatorname{Sep}_> = \rho_{\rm res}\operatorname{Sep}_> +(1-\rho_{\rm res})\operatorname{Sep}_>,\] the first summand is used only in Lemma 11. This is a scalar split of the full nonnegative quadratic-form value, not a decomposition into separate kinetic and centrifugal subenergies. Consequently the reserve inequality and the compression floor may use different lower bounds for \(\operatorname{Sep}_>\), because they are applied to the disjoint scalar multiples \(\rho_{\rm res}\operatorname{Sep}_>\) and \((1-\rho_{\rm res})\operatorname{Sep}_>\), respectively. For the second summand, monotonicity in angular degree and the favorable inequality \(\sinh^{-2}(F_h)\ge\sinh^{-2}s\) give, for \(l>N\), \[\begin{align} \operatorname{Sep}_{l}[w_l] &= \int\left(|\partial_sw_l|^2-\pi^2R^{-2}|w_l|^2\right) +c_l\int\sinh^{-2}(F_h)|w_l|^2 \\ &\ge \int\left(|\partial_sw_l|^2-\pi^2R^{-2}|w_l|^2\right) +c_{N+1}\int\sinh^{-2}(s)|w_l|^2 \\ &= \langle w_l,(H_{N+1,R}-E_R)w_l\rangle \\ &\ge \bigl((b_n+\tau_{n,N+1})R^{-3}+o_N(R^{-3})\bigr)\|w_l\|^2 . \end{align}\] The last inequality is the Rayleigh–Ritz lower bound for the fixed-degree operator \(H_{N+1,R}-E_R\), together with the fixed-\(N\) first-eigenvalue asymptotic; no decomposition of \(w_l\) into radial eigenfunctions is needed. Here \(c_l\) is increasing for \(l\ge1\), \(N\ge N_A(n)\ge1\), and \(N\) is fixed before \(R_0\). The asymptotic in this last line is the fixed-degree \(N+1\) ball asymptotic. Since \(b_n+\tau_{n,N+1}=b_{n+2(N+1)}\), this is exactly the degree-\(N+1\) shifted coefficient. Thus compression uses only \[(1-\rho_{\rm res}) \bigl((b_n+\tau_{n,N+1})R^{-3}+o_N(R^{-3})\bigr)\|w_>\|^2 .\] All radial-bottom and centrifugal contributions in this scalar floor have already been discounted by the factor \(1-\rho_{\rm res}\). Here the compression estimate is fixed by the same choice of \(N\). Let \[U_{n,C}=b_n+\tau_{n,1}+2\pi^2C .\] The two-dimensional space spanned by a constant spherical harmonic and any degree-one harmonic gives, uniformly for \(0\le h\le C\), \[\mu_2(L_h)\le U_{n,C}.\] Since \(\tau_{n,N+1}=b_{n+2(N+1)}-b_n\to\infty\), the cutoff \(N\) is chosen large enough, after \(\eta_{\rm gap}\) and \(\rho_{\rm res}\) are fixed, that \[(1-\rho_{\rm res})(b_n+\tau_{n,N+1}) -\frac{M_C^2}{4\eta_{\rm gap}} \ge U_{n,C}+\delta_{\rm eff}/8 .\] This is a simultaneous finite list of requirements on \(N\): we enlarge the same cutoff so that \(N\ge N_A(n)\), \(\sigma_N/c_A(n)\le\rho_{\rm res}/2\), the strengthened displayed compression inequality holds, and the finite-block ground-state energy approximation recorded in Subsection 6.2 holds. The later threshold \(R_0\) then handles only the \(R\)-dependent remainders \(\sigma_{N,R}\), \(\tau_{N,R}\), and the fixed-\(N\) asymptotic errors. For this fixed \(N\), the scalar compression estimate has an additional \(o_N(1)\) contribution after multiplication by \(R^3\). The norm \(\|w_>\|\) is the flattened Schrodinger \(L^2(ds\,d\theta)\) norm used in the high separated form. After one further enlargement of \(R_0\), the fixed-\(N\) scalar error is bounded below by \(-\delta_{\rm eff}/16\). Hence the actual compressed high block is above \(U_{n,C}+\delta_{\rm eff}/16\), and therefore above \(\mu_2(L_h)+\delta_{\rm eff}/16\). On the low block, min-max for the restricted form gives \[\mu_2(P_NL_hP_N|_{P_NL^2})\ge \mu_2(L_h),\] because the form domain has been restricted. The form \(\widetilde{B}_N^h\) is block diagonal on \[P_NL^2(\mathbb{S}^{n-1})\oplus {\mathcal{H}}_{>,R},\] so its eigenvalue list is the union of the eigenvalues of the low block \(P_NL_hP_N|_{P_NL^2}\) and the scalar high level \(\Gamma_N\), with \(\Gamma_N\) repeated with the multiplicity of \({\mathcal{H}}_{>,R}\). The compression-floor choice above gives \[\Gamma_N\ge U_{n,C}+\delta_{\rm eff}/16 \ge \mu_2(L_h)+\delta_{\rm eff}/16 .\] Therefore \[\nu_2(\widetilde{B}_N^h) \ge \min\{\mu_2(P_NL_hP_N|_{P_NL^2}),\Gamma_N\} \ge \mu_2(L_h).\] The low-block Young loss is not included in \(\widetilde{B}_N^h\); it remains the single external \(\eta_{\rm gap}\) loss in Lemma 13. The compression margin \(\delta_{\rm eff}/16\) is used only to keep the high scalar block away from the low spectral window. ◻

Proof. Let \(\varepsilon_j=\|h_j-h\|_\infty\). Define the offset radial domains \[\Omega^-(\varepsilon)=\{0\leq \rho<R-h(\theta)-\varepsilon\}, \qquad \Omega^+(\varepsilon)=\{0\leq \rho<R-h(\theta)+\varepsilon\}.\] Then \(\varepsilon_j\to0\). Discarding finitely many indices, assume \(\varepsilon_j<\min\{R-C,1\}\), so the inner offsets are nonempty and all offset domains lie in \(B_{R+1}(o)\). Daners is applied to the monotone one-parameter offset families as \(\varepsilon\downarrow0\), and the resulting convergence holds along the sequence \(\varepsilon=\varepsilon_j\). The monotonicity here is in the continuous parameter \(\varepsilon\); after Daners gives convergence as \(\varepsilon\downarrow0\), we evaluate that convergence on the particular sequence \(\varepsilon_j=\|h_j-h\|_\infty\). Because \(h-\varepsilon_j\leq h_j\leq h+\varepsilon_j\), \[\Omega^-(\varepsilon_j)\subset \Omega_j\subset \Omega^+(\varepsilon_j).\] To invoke Daners’ Euclidean varying-domain results, freeze this value of \(R\) and use the Poincaré ball model with \(o\) mapped to the Euclidean origin. Then \(B_{R+1}(o)\) is the Euclidean ball \[|x|<\tanh((R+1)/2)<1,\] and the hyperbolic metric is conformal to the Euclidean metric with conformal factor \(4(1-|x|^2)^{-2}\). On the compact closure of this fixed Euclidean ball the coefficients are smooth, bounded, and uniformly elliptic; there is no polar-coordinate singularity at \(o\). The ellipticity constants may depend on this frozen \(R\), which is harmless because only the limit \(j\to\infty\) is being taken in the domain squeeze. If \(w=\sqrt{\det g}\) is the hyperbolic volume density in this chart, the map \(U\psi=w^{1/2}\psi\) is a unitary identification from \(L^2(dV_{\mathbb{H}^n})\) to Euclidean \(L^2(dx)\). Since \(w^{1/2}\) is smooth, bounded, and bounded away from zero on the fixed chart, this unitary identifies the Dirichlet form domain with the Euclidean \(H_0^1\) domain and preserves the Dirichlet spectrum. It conjugates the Dirichlet hyperbolic Laplacian to a symmetric uniformly elliptic operator \[-\operatorname{div}(A\nabla\cdot)+V\] with \(A\) smooth and uniformly elliptic and \(V\) smooth and bounded on the fixed chart. The domain-theoretic point is the Mosco convergence of the Dirichlet spaces \(H_0^1(\Omega^\pm(\varepsilon))\) for this one fixed uniformly elliptic form. The coefficients do not vary with \(\varepsilon\), and the graph norm of the principal part is equivalent to the standard \(H_0^1\) norm on every offset domain. Daners’ monotone convergence and stability hypotheses therefore give uniform-resolvent convergence for the fixed principal form; the bounded symmetric potential term \(\int V|u|^2\) is form-continuous on \(L^2\) and is carried through the same resolvent convergence after shifting by a constant if needed. The eigenvalue convergence used below is then the standard finite-eigenvalue consequence of this resolvent convergence, via Daners’ spectral-continuity corollaries. The value of \(R\) is fixed throughout this domain-convergence step. In the Poincaré chart the radial graph \(\rho=R-g(\theta)\) has Euclidean radius \[\varrho_g(\theta)=\tanh((R-g(\theta))/2),\] and hence \[|\nabla_{\mathbb{S}^{n-1}}\varrho_g| \le \frac{1}{2} \operatorname{sech}^2((R-C-\varepsilon_0)/2)\, \operatorname{Lip}(g)\] whenever \(-\varepsilon_0\le g\le C+\varepsilon_0\). Thus the inner and outer offset radial graphs, corresponding to \(g=h+\varepsilon\) and \(g=h-\varepsilon\), have Euclidean Lipschitz constants bounded uniformly for \(0<\varepsilon<\varepsilon_0\). In particular, both offset families are bounded Lipschitz domains in the fixed chart with a uniform Lipschitz bound; this is the regularity input used in the decreasing-domain convergence step below.

The inner domains \(\Omega^-(\varepsilon)\) increase to \(\Omega\), so Daners’ monotone-inside criterion [34], followed by uniform resolvent convergence and finite-eigenvalue continuity [34], gives convergence of the indexed eigenvalues \[\lambda_k(\Omega^-(\varepsilon))\to\lambda_k(\Omega)\] for every fixed \(k\). For the decreasing outer family, the domains \(\Omega^+(\varepsilon)\) decrease in the fixed chart and have uniformly bounded Lipschitz constants as noted above. Since \(h\) is Lipschitz and \(R>C\), the open limit domain \[\Omega=\{0\le\rho<R-h(\theta)\}\] is a bounded Lipschitz domain in this chart. Hence its closure \(\overline{\Omega}\) has Lipschitz boundary, satisfies the segment property, and is stable in Daners’ sense by the Lipschitz-domain stability criterion [34]. Moreover \[\operatorname{int}\left(\bigcap_{\varepsilon>0}\Omega^+(\varepsilon)\right)=\Omega\] because \(h\) is continuous. Therefore Daners’ outside convergence criterion, applied to the decreasing outer family with open limit \(\Omega\), stable closed limit \(\overline{\Omega}\), the displayed interior-intersection identity, the bounded-domain uniform-resolvent corollary, and the finite-eigenvalue continuity result [34] Proposition 7.4, Proposition 7.3(1), Corollaries 4.7, 4.2 and Remark 4.3 give \[\lambda_k(\Omega^+(\varepsilon))\to\lambda_k(\Omega)\] for every fixed \(k\). The last step is indexed eigenvalue convergence counting multiplicity: Corollary 4.2 gives persistence of the corresponding finite spectral projections under uniform resolvent convergence, and Remark 4.3 states the resulting continuity of each finite eigenvalue system. Thus the assertion applies to \(k=2\) even if eigenvalue multiplicities cross. In particular, along \(\varepsilon=\varepsilon_j\), \[\lambda_k(\Omega^-(\varepsilon_j))\to\lambda_k(\Omega), \qquad \lambda_k(\Omega^+(\varepsilon_j))\to\lambda_k(\Omega).\] Domain monotonicity of indexed Dirichlet eigenvalues in the inclusions above gives \[\lambda_k(\Omega^+(\varepsilon_j)) \leq \lambda_k(\Omega_j) \leq \lambda_k(\Omega^-(\varepsilon_j)),\] and hence \(\lambda_k(\Omega_j)\to\lambda_k(\Omega)\). ◻