Proof. Let \(\varphi(x)=\pi^{-1/2}e^{-|x|}\) be the hydrogenic ground state. We recall the two hydrogenic bounds needed for localization. After spherical-harmonic decomposition, the radial operators and first-order factors are \[H_\ell=-\frac12\frac{d^2}{d\rho^2} +\frac{\ell(\ell+1)}{2\rho^2}-\frac1\rho, \qquad A_\ell=\frac d{d\rho}-\frac{\ell+1}{\rho}+\frac1{\ell+1}.\] Here \(H_\ell\) has its Friedrichs realization in \(L^2((0,\infty),d\rho)\), and \(A_\ell\) is closed from \(C_c^\infty(0,\infty)\). On the corresponding Friedrichs form domains, integration by parts gives \[\frac12A_\ell^*A_\ell=H_\ell+\frac1{2(\ell+1)^2}, \qquad \frac12A_\ell A_\ell^*=H_{\ell+1}+\frac1{2(\ell+1)^2}.\] The first identity and centrifugal comparison give \(H_\ell\ge-1/8\) for \(\ell\ge1\). Also \(\ker A_0=\operatorname{span}\{\rho e^{-\rho}\}\), and the second identity gives \(A_0A_0^*\ge3/4\). The polar decomposition transfers this last bound to \(A_0^*A_0\) on \((\ker A_0)^\perp\). Thus the hydrogenic operator has ground energy \(-1/2\) and is bounded below by \(-1/8\) on \(\varphi^\perp\); the radial \(\ell=1\) function \(\rho^2e^{-\rho/2}\) attains \(-1/8\).

Choose a Lipschitz angle \(\tau(\rho)\) that is zero for \(\rho\le D/4\), is \(\pi/2\) for \(\rho\ge D/2\), and is linear between these radii. Its slope is \(2\pi/D\). Around the two nuclei use cutoffs \(\chi_\pm(x)=\cos\tau(|x\mp De_z|)\), set equal to zero outside their respective balls of radius \(D/2\), and complete them by the nonnegative cutoff \(\chi_{\rm out}\) to a quadratic partition: \[\chi_+^2+\chi_-^2+\chi_{\rm out}^2=1.\] The two balls are disjoint. On either transition annulus the nonconstant pair is a cosine and a sine, so \[ \frac12\sum_{\nu\in\{+,-,{\rm out}\}}|\nabla\chi_\nu|^2 \le \frac{2\pi^2}{D^2}<\frac{20}{D^2}. \tag{5}\] Lipschitz cutoffs suffice for the form identity; alternatively one may smooth the ramp with arbitrarily small increases in these constants.

On either inner ball, the attraction of the other nucleus has magnitude at most \(2/(3D)<1/D\). On the support of \(\chi_{\rm out}\), both nuclear distances are at least \(D/4\); there the total attraction is at most \(8/D\). Applying the IMS identity and the hydrogenic ground-energy bound gives \[ h_D\ge-\frac12-\frac1D-\frac{20}{D^2} \quad\text{as a quadratic form}. \tag{6}\] Here and below the exterior lower bound \(-8/D\) is larger than the displayed inner lower bound for \(D\ge100\).

If a test function \(\psi\) is orthogonal to both \(\chi_\pm\varphi(\,\cdot\,\mp De_z)\), then \(\chi_\pm\psi\) is orthogonal to the ground state of the corresponding hydrogenic operator. The next hydrogenic energy is \(-1/8\). Consequently, \[ \langle\psi,h_D\psi\rangle \ge\left(-\frac18-\frac1D-\frac{20}{D^2}\right)\|\psi\|_2^2 \tag{7}\] on a subspace of codimension two.

For the upper bound, use the two disjointly supported trial functions \(\chi_\pm\varphi(\,\cdot\,\mp De_z)\). Their squared norms exceed \(1/2\), since the hydrogenic mass outside radius \(D/4\) is \[e^{-D/2}\left(1+\frac D2+\frac{D^2}{8}\right)<\frac12.\] The ground-state identity for the hydrogenic operator and the bound \(|\nabla\chi_\pm|\le2\pi/D\) show that their Rayleigh quotients for \(h_D\) are at most \[ -\frac12+\frac{4\pi^2}{D^2} <-\frac12+\frac{40}{D^2}. \tag{8}\] Their even and odd combinations give the same bound in the two parity subspaces. The Coulomb potential is relatively compact with respect to the free resolvent: one first truncates at infinity and near the nuclei, and uses local compactness for the bounded remainder. The omitted nuclear piece has operator norm from \(H^2\) to \(L^2\) of order \(\sqrt{\varepsilon}\) when cut off to a ball of radius \(\varepsilon\), by \(H^2% \BeginAccSupp{pdfliteral=direct,method=hex,unicode,ActualText=21AA}% \KSOriginalHookrightarrow\EndAccSupp{pdfliteral=direct}L^\infty\) and \(\||x|^{-1}\mathbf 1_{|x|<\varepsilon}\|_2=O(\sqrt\varepsilon)\). The far-field remainder tends to zero in \(L^\infty\). Hence the essential spectrum is \([0,\infty)\). The min–max principle now gives two negative levels, one in each parity, and no third level below the lower bound in [eq:two-centre-third-lower].

A full-space ground state can be chosen nonnegative by the absolute-value inequality for its kinetic energy. It is strictly positive, including at the nuclei. One direct way to include those points is to use \[b=\left(-\frac12\Delta+c\right)^{-1} (c+\epsilon_b-u)b,\qquad c>|\epsilon_b|,\] whose resolvent kernel is strictly positive; the right-hand side is nonnegative and nonzero, and \(b\in H^2(\mathbf R^3)\) is continuous. The same argument with the Dirichlet resolvent on \(z>0\) gives strict positivity of its ground state in that half-space. Positivity and the usual ground-state simplicity argument identify the low even level with \(b\) and the low odd level with the odd extension of that Dirichlet ground state. Simplicity also follows from the two-level count above. The full-space ground state cannot be odd, so \(\epsilon_b<\epsilon_e\). Rotation about the \(z\)-axis preserves the unique normalized positive full-space and half-space ground states, proving axial symmetry.

Combining [eq:two-centre-bottom-lower,eq:two-centre-two-upper] gives \[d_*\le\frac1D+\frac{60}{D^2}<\frac{100}{D}.\] The same two bounds place \(\gamma\) and \(k_0\) within \(200/D\) of \(1\), with the stated bound on their sum. Finally, [eq:two-centre-third-lower,eq:two-centre-two-upper] give the gap estimate \[\frac38-\frac1D-\frac{60}{D^2}>0.2 .\] Since \(e\) is odd, the same lower bound holds above \(b\) in the even subspace. ◻

Proof. At \(\lambda=0\), the Pauli principle fills the spin orbitals in [eq:zero-coupling-states]. In the sector \(S_z=1/2\), moving the doubly occupied spatial orbital from \(b\) to \(e\) costs \(d_*>0\); using any further spatial orbital costs a positive gap by 3. The displayed determinant is therefore a simple isolated ground state in that sector. The ionic ground state is likewise simple and isolated. The noninteracting ionization thresholds are \(2\epsilon_b\) and \(\epsilon_b\), respectively, and are strictly above these ground energies.

The common-domain analytic perturbation argument applies to each isolated rank-one spectral projection; see [10]. To include the graph-norm conclusion explicitly, write \(V_{\rm ee}=\sum_{i<j}|x_i-x_j|^{-1}\) and \(R_0(\zeta)=(H_{m,0}-\zeta)^{-1}\). On a contour enclosing only the relevant zero-coupling level, \[(H_{m,\lambda}-\zeta)^{-1} =R_0(\zeta)\bigl(I+\lambda V_{\rm ee}R_0(\zeta)\bigr)^{-1}.\] Here \(R_0(\zeta):L^2\to H^2\) and \(V_{\rm ee}R_0(\zeta):L^2\to L^2\) are bounded uniformly on the contour. A Neumann series therefore makes the resolvent analytic as a map into \(H^2\). Integrating around the contour, applying the resulting projection to the unperturbed state, and normalizing gives a real-analytic state in that space. Choose its real phase by positive overlap with the zero-coupling state. Spectral separation, or the min–max principle with the nonnegative repulsion, keeps this branch the lowest level of its sector.

It remains to check the absolute three-electron ground property. In the sectors \(S_z=\pm3/2\), all three spatial orbitals must be distinct. Their zero-coupling energy is at least \(\epsilon_b+\epsilon_e+\epsilon_3\), where \(\epsilon_3=\inf\sigma(h_D|_{\{b,e\}^\perp})\); it exceeds \(2\epsilon_b+\epsilon_e\) by more than \(0.2\). Nonnegative repulsion preserves this lower bound, whereas the continued \(S_z=1/2\) energy converges to \(2\epsilon_b+\epsilon_e\). Spin flip identifies the two \(S_z=\pm1/2\) ground energies. These four sectors exhaust the three-electron spin space, proving the claim.

All the stated symmetries commute with the Hamiltonian and its spectral projection. Their eigenvalues at zero coupling persist on the simple branch. For axial rotations, one may equivalently note that the projection of the invariant zero-coupling state is still invariant. The ionic singlet subspace is also invariant, so its spin character persists. ◻

Proof. Fubini’s theorem applied to the derivatives in the distinguished coordinate gives the Hilbert-valued \(H^2\) assertion. In three dimensions the corresponding Sobolev estimates yield \[\|F_{\rm true}\|_{L^\infty_x(\mathcal K)} +\|\nabla_xF_{\rm true}\|_{L^4_x(\mathcal K)} \le C\|F_{\rm true}\|_{H^2_x(\mathcal K)} .\] The distributional product identities \[\begin{align*} \partial_j n_\lambda &=2\operatorname{Re} \langle F_{\rm true},\partial_jF_{\rm true}\rangle_{\mathcal K}, \\ \partial_i\partial_j n_\lambda &=2\operatorname{Re} \langle\partial_iF_{\rm true},\partial_jF_{\rm true}\rangle_{\mathcal K} +2\operatorname{Re} \langle F_{\rm true},\partial_i\partial_jF_{\rm true}\rangle_{\mathcal K} \tag{12}\end{align*}\] then prove \(n_\lambda\in H^2\). The same identities and Cauchy–Schwarz, now using \(L^2_x\) on both factors, give \(\Delta n_\lambda\in L^1\). The norm chain rule gives \[ \int_{\mathbf R^3}|\nabla\sqrt{n_\lambda}|^2 \le \int_{\mathbf R^3}\|\nabla_xF_{\rm true}\|_{\mathcal K}^2 . \tag{13}\]

The product estimates also show continuity of \(F% \BeginAccSupp{pdfliteral=direct,method=hex,unicode,ActualText=21A6}% \KSOriginalMapsto\EndAccSupp{pdfliteral=direct}\|F\|_{\mathcal K}^2\) from \(H^2_x(\mathcal K)\) to \(H^2_x\). Thus the graph-norm convergence in 4 gives the claimed density convergence. For the \(L^1\) convergence, use \[\|n_\lambda-n_0\|_1 \le \bigl(\|F_{{\rm true},\lambda}\|_2+\|F_{{\rm true},0}\|_2\bigr) \|F_{{\rm true},\lambda}-F_{{\rm true},0}\|_2 .\] It implies common tightness. Since \(b\) is continuous and strictly positive everywhere, \(n_0\ge2b^2\) has a positive minimum on each compact set. Uniform convergence gives the corresponding compact lower bound for \(n_\lambda\). The spatial symmetries follow by taking the squared fiber norm. ◻

Proof. Put \(\rho=(|x|^2+|y|^2)^{1/2}\). A configuration-space partition according to a particle with distance at least a fixed multiple of \(\rho\) from the origin gives the exterior form bound \[ H_{3,\lambda}\geq2\epsilon_b-o(1) \quad\hbox{on functions supported where }\rho\to\infty. \tag{28}\] The nuclear attraction of the selected particle is \(O(1/\rho)\); the other two one-particle operators are each bounded below by \(\epsilon_b\), and repulsion can be dropped. The partition has derivatives \(O(1/\rho)\), so its kinetic localization error tends to zero. This argument is uniform for \(0\leq\lambda\leq1\).

The exponent margin needed below already follows for \(D\ge100\) from [eq:two-centre-bottom-lower,eq:two-centre-two-upper]: \[\begin{gathered} k_0^2\ge1-\frac{80}{D^2},\qquad \gamma^2\le1+\frac2D+\frac{40}{D^2},\\ \frac{10}{9}k_0^2-\gamma^2 \ge\frac19-\frac2D-\frac{1160}{9D^2}>0. \end{gathered}\] We can therefore choose \[0<\tau_0<\tau_1<\tau_2<k_0,\qquad \tau_0\sqrt{10/9}>\gamma.\] After decreasing \(\lambda_D\), the exterior bound and \(E_3(0)=2\epsilon_b-k_0^2/2\) leave a positive margin above the weight cost \(\tau_2^2/2\). The exponential-weight identity, first with an exterior cutoff and a smooth truncated exponent whose gradient has norm at most one, gives \[ \|e^{\tau_1\rho}F_{\mathrm{true}}\|_{H^1(\mathbf R^9)} +\|e^{\tau_1\rho}F_0\|_{H^1(\mathbf R^9)}\leq C_D. \tag{29}\] At zero coupling the same argument also gives \(\|e^{\tau_2\rho}F_0\|_{H^1}\leq C_D\), since \(\tau_2<k_0\). The finite spin sum is understood. The estimate also applies to an inhomogeneous equation: if \((H_{3,\lambda}-E_3(\lambda))v=f\), then, for a fixed ball \(K\), \[ \|e^{\tau_1\rho}v\|_{H^1} \leq C_D\left(\|e^{\tau_1\rho}f\|_2+\|v\|_{H^1(K)}\right). \tag{30}\] Indeed the exterior spectral lower bound controls the weighted \(L^2\) norm by the weight identity and Cauchy–Schwarz. Combining a small multiple of the kinetic form estimate with that lower bound also controls the gradient; the strict margin between \(\tau_1\) and \(\tau_2\) absorbs its additional weight cost. Hardy’s inequality absorbs the infinitesimally form-bounded Coulomb terms. These bounds are independent of the exponent truncation, which can therefore be removed.

The equation for \(v=F_{\mathrm{true}}-F_0\) has source \[\bigl(E_3(\lambda)-E_3(0)-\lambda V_{\mathrm{ee}}\bigr)F_0.\] Hardy’s inequality, with the spare exponent \(\tau_2\) on \(F_0\), bounds it by \(C_D\lambda\) in weighted \(L^2\). The compact term is \(O(\lambda)\) by 4. Thus \[ \|e^{\tau_1\rho}(F_{\mathrm{true}}-F_0)\|_{H^1}\leq C_D\lambda. \tag{31}\]

Choose smooth functions of \(|y|/r\), with values in \([0,1]\), such that \(\chi=1\) on \(|y|\leq r/3\), \(\chi=0\) on \(|y|\geq r/2\), and \(\zeta=1\) on \(|y|\leq r/2\), \(\zeta=0\) on \(|y|\geq3r/4\). Define \[F=F_0+\chi(F_{\mathrm{true}}-F_0),\qquad W=\zeta\sum_{i=1}^2|x-y_i|^{-1}.\] On the support of \(\zeta\), \(|x-y_i|\geq r/4\), proving [eq:cutoff-interaction-symbol]. Denote the operator on the left of [eq:cutoff-channel-equation] by \(\mathcal L_\lambda\). Since \(W\) is exact on the support of \(\chi\), \[ R=(1-\chi)\mathcal L_\lambda F_0 +[\mathcal L_\lambda,\chi](F_{\mathrm{true}}-F_0), \tag{32}\] where \[\mathcal L_\lambda F_0 =\left\{\lambda\left(|y_1-y_2|^{-1}+W\right) -E_3(\lambda)+E_3(0)\right\}F_0.\] Thus every residual term is \(O(\lambda)\) and is supported where \(|y|\geq r/3\). There, \[ \rho\geq\sqrt{10/9}\,r. \tag{33}\]

We give the derivative details, since higher ordinary derivatives of the uncut wavefunction across a collision cannot simply be iterated. Local \(H^2\) estimates on nested fixed-size configuration boxes have uniform constants: the Laplacian estimate with cutoffs and the infinitesimal Coulomb bound absorb the potential terms; the local energy estimate controls cutoff commutators. Where derivatives of \(\chi\) occur, \(x\) is separated from both other electrons. Further \(x\)-derivatives of the equation differentiate only smooth bounded coefficients, while singular ionic coefficients are independent of \(x\). Difference quotients followed by these local estimates give any fixed number of \(x\)-derivatives of the difference, at size \(O(\lambda)\) in weighted local norms. In the other residual term, \(F_0\) has the product form [eq:unperturbed-fibers]; its \(x\)-derivatives have no electron-electron singularity, and the ionic Coulomb multiplier is controlled by Hardy’s inequality. Summing the box estimates and using Sobolev embedding in the three \(x\)-coordinates gives [eq:channel-residual]. A small loss from \(\tau_1\) toward \(\tau_0\) absorbs polynomial integration factors. By [eq:cutoff-geometric-margin], one still has \(\sigma>\gamma\).

Density derivatives in the discarded region require a different coordinate choice. Choose smooth pair cutoffs \(\chi_{ij}\in[0,1]\) equal to one at distances at most \(1\) and zero at distances at least \(2\), and expand the product of the three identities \(1=\chi_{ij}+(1-\chi_{ij})\). For each term use the connected components of its near-edge graph as clusters. They have diameter at most \(4\), and distinct clusters are separated by at least \(1\). Nested boxes in which each particle moves by less than \(1/8\) preserve a positive separation. The cluster containing \(x\) is far from the nuclei when \(r\) is large. Translate with \(x\) all other variables in that cluster. A derivative in the translated \(x\)-coordinate is the sum of the corresponding particle derivatives. It annihilates within-cluster Coulomb singularities and singularities involving only outside particles; every coefficient differentiated is smooth with uniformly bounded derivatives. Nested-box energy and elliptic estimates can therefore be iterated in these three translation directions. This is the parallel-derivative method of [5]; here the fixed separations make the constants uniform, while [eq:configuration-weight,eq:weighted-state-difference] preserve the requisite exponential weights.

The difference \(n-\|F\|^2\) is a sum of integrals of products of \(F_{\mathrm{true}}\) and \(F_0\), with coefficients supported where \(|y|\geq r/3\). After the coordinate changes just described, differentiate these integrals and apply Cauchy–Schwarz to the weighted local estimates. Each product has two exponentially decaying factors. Sobolev embedding in the three translation directions and [eq:cutoff-geometric-margin] give [eq:density-cutoff-error], after another arbitrarily small loss of exponent. To obtain smoothness on a compact set disjoint from the nuclei, replace the pair-cutoff radii \(1,2\) by \(\varepsilon,2\varepsilon\), with \(\varepsilon\) smaller than one sixteenth of the set’s distance from the nuclei. The cluster containing \(x\) has diameter at most \(4\varepsilon\); sufficiently small nested boxes therefore keep it away from the nuclei and preserve the intercluster separation. The same argument without discarding a tail then proves smoothness of the density there. On fixed exterior annuli the differentiated local estimates also give the claimed \(O(\lambda)\) convergence of \(F-F_0\). ◻

Proof. The ionic gap and multipole estimate imply \[PW\phi=(2/r+O(r^{-2}))\phi,\qquad QW\phi=O(r^{-2}).\] Splitting a vector into \(P+Q\) and applying Young’s inequality to the off-diagonal term, while retaining half the positive \(Q\)-gap, gives \[ B+\lambda W\geq 2\lambda/r-C_D/r^2. \tag{41}\] The norm \(N(x)=\|F(x)\|\) consequently satisfies \[\left(-\Delta_x/2+k^2/2-(2-2\lambda)/r-C_D/r^2\right)N\leq\|R\|.\] Take the real inner product of the vector equation with \(F/\|F\|\) and use \(|\nabla\|F\||\leq\|\nabla F\|\). Replacing the norm by \((\|F\|^2+\varepsilon^2)^{1/2}\) proves the weak inequality also at zeros. The unbounded ionic operator is handled through its quadratic form, or by bounded spectral truncations.

Applying this scalar operator to \(A(1-M/r)\), and dividing by \(A\), gives \(r^{-2}\) coefficient \[kM-\beta(\beta+1)/2-C_D.\] Choose \(M\) to make this positive, and start outside \(r>2M\). A sufficiently large multiple dominates the inner boundary value and the exponentially smaller source, yielding \(N=O(A)\) by exterior comparison. The latter can be proved by testing the positive part of the difference with cutoffs: the zeroth-order coefficient is positive sufficiently far out, both functions decay, and the cutoff error tends to zero. The \(Q\)-equation has source \(-\lambda QW\phi f+QR=O(A/r^2)\). Its norm inequality retains the positive constant \(c_D\) in addition to \(k^2/2+u\), so \(A/r^2\) is a supersolution at sufficiently large radius. This proves [eq:channel-leading-bounds].

Rotation fields \(\Omega\) acting on \(x\) commute with \(\Delta_x,B,P,Q\). Although \(\|\Omega^jW\|=O(r^{-1})\), one has \(\|\Omega^jW\phi\|=O(r^{-2})\) for \(j\geq1\), and \(\Omega^ju=O_D(r^{-3})\). Suppose the two leading bounds hold for fewer than \(s\) rotations. Every commutator source for \(\Omega^sF\) is \(O(A/r^2)\): its \(P\)-part uses the improved estimate on \(W\phi\), and its \(Q\)-part uses \(O(r^{-1})O(A/r^2)\). First the full norm comparison gives \(\Omega^sF=O(A)\); then the \(Q\)-equation and its gap give \(Q\Omega^sF=O(A/r^2)\). This closes the induction. The absolute decay needed for comparison follows from the cutoff construction, allowing a small exponential loss for the polynomial coefficients of rotation fields.

Set \(h_0=f/A\), and define \[ K_j=-\tfrac12\Delta_{\mathbb S^2} -\tfrac12(\beta-j)(\beta-j+1),\qquad c_m=U_m+\lambda\langle\phi,W_m\phi\rangle. \tag{42}\] Axial symmetry and reflection give \(c_2=0\). The projected equation conjugated by \(A\) is \[ -\tfrac12h_0''+ \left(k-\frac{\beta+1}{r}\right)h_0' +\frac{K_0+c_2}{r^2}h_0+\frac{c_3}{r^3}h_0+\cdots =O(r^{-4}), \tag{43}\] with any fixed number of angular derivatives. The off-diagonal source has this order because \(\|PWQ\|=\|QW\phi\|=O(r^{-2})\) and \(QF=O(A/r^2)\).

For the radial step, if a bounded angular-valued function \(H\) satisfies \[-\tfrac12H''+\left(k-\frac{\beta+1}{r}\right)H'=G(r), \qquad G(r)=O(r^{-q}),\quad q>1,\] then \[ H'(r)=2e^{2kr}r^{-2(\beta+1)} \int_r^\infty e^{-2ks}s^{2(\beta+1)}G(s)\,ds. \tag{44}\] The other homogeneous branch grows exponentially and is excluded by boundedness. The integral is \(O(r^{-q})\), uniformly for the present bounded \(\beta\) and positive lower bound on \(k\); thus \(H\) has a limit with error \(O(r^{1-q})\). Initially the right side of the radial equation is \(O(r^{-2})\), since all needed angular derivatives of \(h_0\) are bounded. Hence \(h_0\to a_0\), with error \(O(r^{-1})\). Set \[ a_1=\frac{(K_0+c_2)a_0}{k},\qquad a_2=\frac{(K_1+c_2)a_1+c_3a_0}{2k}. \tag{45}\] Subtracting \(a_0+a_1/r\) cancels order two; the integrating factor improves the remainder to \(O(r^{-2})\), with derivative \(O(r^{-3})\). Subtracting \(a_2/r^2\) cancels order three, giving \(\mathcal E=O(r^{-3})\) and \(\mathcal E'=O(r^{-4})\). The equation gives \(\mathcal E''=O(r^{-4})\). The same argument with finitely more angular derivatives at each step proves [eq:scalar-remainder-derivatives]. Reflection makes \(f,a_j\), and all radial remainders odd. The mean-value theorem in the transverse angular coordinate gives the additional factor at the equator.

On a fixed radius, the projected function and its angular derivatives converge by the local estimates in 7. Since \(\|f/A-a_0\|_{C^1}\leq C_D/r\) uniformly, first choosing the radius large and then sending \(\lambda\) to zero proves \(C^1\) convergence of \(a_0^\lambda\). At zero coupling the scalar equation for \(e\) on the upper half-space has a subsolution which is a positive multiple of \[e^{-k_0r}r^{2/k_0-1}\omega_z(1+M/r).\] After division by the radial factor and \(\omega_z\), the controlling \(r^{-2}\) coefficient contains \(-k_0M\); it dominates the bounded angular and scalar corrections for large \(M\). Positivity of \(e\) on an inner hemisphere and the boundary point lemma on its equator supply a strictly positive starting multiple, including the equatorial limit. Exterior comparison proves \(\inf a_0^0/\omega_z>0\). Oddness and \(C^1\) convergence give [eq:simple-angular-zero]. The identical scalar argument for \(b\), without an angular zero and with rate \(\gamma\), proves [eq:fast-orbital-expansion]; positivity on the compact inner sphere gives \(\min b_\infty>0\). ◻

Proof. On localized angular coefficients define \[C_m=U_m\,\mathrm{id}_{Q\mathcal K}+\lambda QW_mQ.\] The qualification is essential: \(W_m\), and hence \(C_m\), can be unbounded on arbitrary ionic vectors. Their action on the localized vectors in this proof is controlled by [eq:weighted-ionic-inverse,eq:ionic-multipoles]. On a term \(r^{-j}T(\omega)\), the conjugated \(Q\)-operator has expansion \[ r^{-j}B_QT-jk r^{-j-1}T +r^{-j-2}K_jT+\sum_{m\geq2}r^{-j-m}C_mT. \tag{48}\] The source from \(P\), after division by \(\lambda A\), is \(-QW\phi\,f/A\). Its coefficients of orders two through four are \[\begin{split} S_2&=-a_0QW_2\phi,\\ S_3&=-a_1QW_2\phi-a_0QW_3\phi,\\ S_4&=-a_2QW_2\phi-a_1QW_3\phi-a_0QW_4\phi. \end{split}\] We therefore define the coefficients by \[ \begin{split} B_QT_2&=S_2,\\ B_QT_3&=S_3+2kT_2,\\ B_QT_4&=S_4+3kT_3-(K_2+C_2)T_2. \end{split} \tag{49}\] Repeated use of [eq:weighted-ionic-inverse], decreasing a positive weight after polynomial multiplication when necessary, gives smooth localized coefficients with every finite angular derivative bound used here. The nuclear quadrupole has already entered \(a_2\) in [eq:scalar-channel-recursion]; the operator \(C_3\) acts directly on this complementary expansion first at order five.

Every \(S_j\) vanishes on the equator because its scalar factors \(a_i\) are odd. The first two recursion equations give \(T_2=T_3=0\) there. In the third equation all terms except the angular Laplacian of \(T_2\) also vanish. At the equator \(a_0=\Delta_{\mathbb S^2}a_0=0\), while \[\nabla_{\mathbb S^2}a_0=a_0'(0)e_z,\qquad \partial_{\omega_z}W_2=Y_z.\] The product rule in \(T_2=-a_0B_Q^{-1}QW_2\phi\) proves [eq:equatorial-coefficient], with its stated sign and factor. The slope is nonzero by [eq:simple-angular-zero]. Moreover \(Y_z\phi\) is ionic-odd, so \(QY_z\phi=Y_z\phi\). This vector cannot vanish: the zero set of the multiplier \(Y_z\) is a measure-zero hyperplane, which cannot support a nonzero \(L^2\) vector. Injectivity of \(B_Q^{-2}\) proves nonvanishing. Continuity of the localized resolvents and vectors gives the uniform lower bound for small \(\lambda\). All operators in the recursion preserve the distinguished spin, so each \(T_j\) has spin up.

We next prove the remainder with its asserted uniformity. The retained vector \(bG\) solves the homogeneous \(Q\)-equation at \(\lambda=0\). Since \(QG=G\), its perturbed residual is \[ \begin{split} \mathcal L_{\lambda,Q}(bG) &=bQ\left\{\lambda\left(|y_1-y_2|^{-1}+W\right) -E_3(\lambda)+E_3(0)\right\}G\\ &=O(\lambda b), \end{split} \tag{50}\] also through two ordinary \(x\)-derivatives. Hardy’s inequality controls the ionic Coulomb factor on \(G\); the other \(x\)-dependent factors are smooth in the exterior. By [eq:channel-rate-separation,eq:fast-orbital-expansion], \[ b/A\leq C_D e^{-(\gamma-k_0)r/2}r^{C_D}. \tag{51}\] Thus the residual is bounded by \(C_D\lambda Ar^{-5}\), uniformly in \(\lambda\). No equality of perturbed and unperturbed excited thresholds is being assumed.

The matched slow terms have residual \(O(\lambda Ar^{-5})\). The same bound holds through two Cartesian derivatives: finite multipole remainders have symbol bounds on the localized coefficients, and the scalar remainder is multiplied by \(\lambda QW\phi=O(\lambda r^{-2})\). The three bounds in [eq:scalar-remainder-derivatives] suffice; one does not need the stronger assertion \(\partial_r^2\mathcal E=O(r^{-5})\). The source \(QR\) is exponentially smaller. Hence \(\mathcal E_Q\) satisfies a \(Q\)-equation with a source of this size, including two derivatives. On a fixed large starting sphere its value is \(O(\lambda)\), since \(QF=bG\) at zero coupling and all fixed-radius data converge at that rate. The positive \(Q\)-gap makes a sufficiently large multiple of \(\lambda Ar^{-5}\) a scalar norm supersolution which dominates both the source and boundary value. Exterior comparison gives the undifferentiated remainder bound.

For the derivative upgrade it is important to include the unbounded ionic operator. For compactly supported \(v\) with values in \(\operatorname{Dom}B_Q\), Fourier transform in \(x\) and the spectral resolution of \(B_Q\geq0\) give \[ \|D_x^2v\|_2+\|B_Qv\|_2 \leq C\|(-\Delta_x/2+B_Q)v\|_2. \tag{52}\] Indeed \(|\xi|^2\) and a spectral value \(b\geq0\) are each bounded by a fixed multiple of \(|\xi|^2/2+b\). For the local estimate, first control \(\nabla_xv\) by the cutoff energy identity and then apply [eq:positive-ionic-elliptic-estimate] with a second cutoff. The remaining coefficient \(k^2/2+u+\lambda QWQ\) and its \(x\)-derivatives are bounded operators on exterior unit balls, uniformly in their centers. Since \(B_Q\) is independent of \(x\), difference quotients and two further local estimates give \[\|\mathcal E_Q\|_{H_x^4(B(x,1/2);\mathcal K)} \leq C_D\lambda A(r)r^{-5}.\] The source derivatives and the already proved pointwise bound control the right side on a larger ball; \(A\) varies by a bounded factor there. Hilbert-valued Sobolev embedding in dimension three, \(H^4% \BeginAccSupp{pdfliteral=direct,method=hex,unicode,ActualText=21AA}% \KSOriginalHookrightarrow\EndAccSupp{pdfliteral=direct}C^{2,\alpha}\) for \(\alpha<1/2\), gives the two pointwise derivative bounds in [eq:complementary-expansion]. ◻

Proof of 6. Put \(t=\omega_z\) and \[T_{\mathrm{eq}}=T_4|_{t=0},\qquad M(x)=\phi f(x)+b(x)G+\lambda A(r)r^{-4}T_{\mathrm{eq}}.\] Axial symmetry and [eq:equatorial-coefficient] make \(T_{\mathrm{eq}}\) independent of equatorial azimuth. Smoothness and the zeros of \(T_2,T_3\) give \[\|T_2(\omega)\|+\|T_3(\omega)\| +\|T_4(\omega)-T_{\mathrm{eq}}\|\leq C_D|t|.\] The estimate holds on the whole sphere by compactness away from the equator. For any distinguished-spin-up \(Q\)-vector \(S\), [eq:core-spin-components] implies \[|\langle bG,S\rangle| \leq |b|\,\|S\|=2^{-1/2}\|bG\|\,\|S\|.\] Therefore \[ \begin{split} (1-2^{-1/2})(2b^2+\|S\|^2) &\leq\|bG+S\|^2\\ &\leq(1+2^{-1/2})(2b^2+\|S\|^2). \end{split} \tag{53}\] The \(P\)-component is orthogonal to both these terms. The simple angular zero and positive lower bound on \(\|T_{\mathrm{eq}}\|\) give \[\|M\|\geq c_D(A|t|+b+\lambda Ar^{-4}).\] The complementary expansion gives \[\|F-M\|\leq C_D\lambda A(r^{-2}|t|+r^{-5})=o_D(\|M\|),\] uniformly in small \(\lambda\) as the radius increases. Consequently \[ \|F\|\geq c_D(A|\omega_z|+b+\lambda Ar^{-4}). \tag{54}\] In addition, there is the absolute bound \[ b/\|F\|\leq2. \tag{55}\] The distinguished-down-spin component of \(M\) has norm \(b\). Increase the radius so that \(\|F-M\|\leq\|M\|/2\); then \[\|F\|\geq\|M\|-\|F-M\|\geq\|M\|/2\geq b/2.\] Only the radius, not the constant \(2\), depends on \(D\). The cutoff errors are relatively negligible because \[e^{-2\sigma r}/b^2\longrightarrow0\qquad(\sigma>\gamma).\] This proves exterior positivity of \(n\); positivity on the remaining compact set follows from 5 after decreasing \(\lambda_D\). Symmetry, compact convergence, and exterior smoothness have already been proved.

We verify differentiated comparisons through the moving crossover. The projected remainders and their first two radial derivatives are odd, so their angular derivative bounds supply the factor \(|t|\). Their errors can thus be divided by the first lower scale in [eq:three-scale-lower-bound]. For the complementary error, \[\frac{\|\partial_x^\nu\mathcal E_Q\|}{\|F\|} \leq C_D/r\quad(|\nu|\leq2,\ \lambda>0),\] using the third lower scale. At \(\lambda=0\) this error is zero. Differentiating the explicit slow terms with \(\partial_r+k\) leaves only inverse powers of \(r\). For the fast orbital, \[(\partial_r+k)b=(k-\gamma)b+O_D(b/r),\qquad (\partial_r+k)^2b=(k-\gamma)^2b+O_D(b/r).\] It follows, with \(\delta_\lambda=k-\gamma\), that \[ \begin{split} (\partial_r+k)F&=\delta_\lambda bG+o_D(\|F\|),\\ (\partial_r+k)^2F&=\delta_\lambda^2bG+o_D(\|F\|), \end{split} \tag{56}\] where the ratios denoted by \(o_D(1)\) tend to zero uniformly in the chosen coupling interval.

At fixed \(z\), \(\partial_p\) has radial part \((p/r)\partial_r\) and angular part of size \(O(|t|/r)\) on \(0\leq z\leq p/2\). Applying it once or twice to smooth odd angular coefficients retains a factor \(|t|\), together with the corresponding inverse powers of \(r\). Thus \[\left(\partial_p+k\frac p r\right)F =\delta_\lambda\frac p r\,bG+o_D(\|F\|).\] On \(0\leq z\leq L\), \(1-p/r=O_L(p^{-2})\) is negligible. With \(D_p=\partial_p+k\), \[ D_pF=\delta_\lambda bG+o_D(\|F\|),\qquad D_p^2F=\delta_\lambda^2bG+o_D(\|F\|). \tag{57}\] These estimates include angular derivatives of the displayed \(T_j\): their zeros are retained for \(T_2,T_3\), whereas the extra angular factor for \(T_4\) is bounded using the first lower scale. The ordinary derivative bounds handle \(\mathcal E_Q\) directly.

By [eq:absolute-core-ratio], \(\|bG\|/\|F\|\leq2\sqrt2\). The identity \[(\partial_r+2k)\|F\|^2 =2\operatorname{Re}\langle F,(\partial_r+k)F\rangle\] and its horizontal version bound the normalized first derivatives by \(4\sqrt2|\delta_\lambda|+o_D(1)\). Also \[(\partial_p+2k)^2\|F\|^2 =2\|D_pF\|^2+ 2\operatorname{Re}\langle F,D_p^2F\rangle\] gives the normalized second-derivative bound \((16+4\sqrt2)\delta_\lambda^2+o_D(1)\) in the strip. The differentiated cutoff errors remain relatively negligible. In particular, there is \(\rho_D(R)\downarrow0\), independent of small \(\lambda\), such that \[ \begin{split} \max\Bigg\{& \sup_{r\geq R}|\partial_r\log n+2k|,\ \sup_{\substack{r\geq R\\0\leq z\leq p/2}} \left|\partial_p\log n+2k\frac p r\right|,\\ &\sup_{\substack{p\geq R\\0\leq z\leq L}} \frac{|m_p|+|m_{pp}|}{m}\Bigg\} \leq100(|\gamma-k|+|\gamma-k|^2)+\rho_D(R). \end{split} \tag{58}\] All \(D\)-dependent coefficients other than the fast exponential rate have entered the vanishing function \(\rho_D\). Since \(\gamma-k_0\to0\) as \(D\to\infty\) and \(k\to k_0\) as \(\lambda\downarrow0\), this proves the derivative assertions in [eq:density-drift,eq:strip-data].

For the vertical ratio the expansions give, uniformly on \(0\leq z\leq L\), \[ \begin{split} F(p,z)={}&\phi a_0'(0)A(p)z/p+b(p,0)G +\lambda A(p)p^{-4}T_{\mathrm{eq}}\\ &+o_D\bigl(A(p)z/p+b(p,0)+\lambda A(p)p^{-4}\bigr). \end{split} \tag{59}\] The last coefficient tends to zero uniformly in \(\lambda,z\), and the error is relatively small compared with the norm of the three-term comparison vector by [eq:spin-noncancellation]. Indeed \(r-p=O_L(p^{-1})\); after including the lower radial terms, the projected factor is \(a_0'(0)z/p+O_D(z/p^2)\); and \(b(p,z)/b(p,0)=1+O_D(p^{-1})\). The \(T_2,T_3\) terms are \(O_D(\lambda A z/p^3)\); all other errors have the displayed bound by [eq:three-scale-lower-bound]. The squared comparison norm lies between \(1-1/\sqrt2\) and \(1+1/\sqrt2\) times \[|a_0'(0)|^2A(p)^2z^2/p^2+ 2b(p,0)^2+\lambda^2A(p)^2p^{-8}\|T_{\mathrm{eq}}\|^2.\] This expression is nondecreasing in \(z\). The ratio of the constants is \(3+2\sqrt2<6\); after absorbing uniformly small errors, the ratio in [eq:strip-data] is at most \(20\).

Now fix \(\lambda>0\). On the plane \(f=0\) and \(T_2=T_3=0\). Since \(b=o(Ap^{-4})\), the complementary expansion yields \[F(p,0)=\lambda A(p)p^{-4}T_{\mathrm{eq}}+o(A(p)p^{-4}).\] The coefficient is nonzero and the density cutoff error is exponentially smaller, proving [eq:plane-tail].

For plane-centered \(x_j=p_j\omega_j\), put \(r_j(x)=|x_j+x|\). Uniformly on compact displacement sets, \[r_j(x)-p_j\to\omega_*\cdot x,\qquad p_jz/r_j(x)\to z,\qquad A(r_j(x))/A(p_j)\to e^{-k\omega_*\cdot x}.\] The projected expansion and its simple angular zero imply \[\frac{PF(x_j+x)}{A(p_j)/p_j} \longrightarrow\phi a_0'(0)z e^{-k\omega_*\cdot x}.\] The complementary part divided by this normalization tends uniformly to zero. Its fast term is exponentially smaller; the \(T_2,T_3\) terms retain the angular zero and the plane term is \(O(Ap_j^{-4})\). The cutoff error is negligible. Squaring gives [eq:plane-translate] with \(a=|a_0'(0)|>0\).

Finally suppose \(r_j=|x_j|\to\infty\) and \(|z_j|\to\infty\). Reflection reduces the proof to \(z_j>0\). If \(\omega_{j,z}\) stays bounded below, the conclusion follows directly from the smooth nonzero projected angular coefficient. Otherwise \(t_j=z_j/r_j\to0\), and \[\frac{b(r_j)}{A(r_j)t_j}\to0,\qquad \frac{\lambda A(r_j)r_j^{-4}}{A(r_j)t_j} =\frac{\lambda}{r_j^3z_j}\to0.\] The \(T_2,T_3\) terms are relatively negligible as well. For bounded \(x\), \(t_j(x)/t_j=1+O(1/z_j+1/r_j)\). Since \(a_0(t)/t\) is continuous and bounded away from zero through the equator, the angular amplitude has relative variation tending to one. The radial ratio tends to \(e^{-k\omega_*\cdot x}\); all lower terms are uniformly negligible. Squaring proves [eq:bulk-translate]. ◻

Proof. Local regularity uses only the Coulomb eigenvalue equation, and exterior smoothness follows from the cluster argument in 7. The isolated ionic ground supplies a positive \(Q\)-gap and [eq:weighted-ionic-inverse], with constants now allowed to depend on \(t\).

The configuration-space lower bound [eq:configuration-exterior-bound] remains valid because \(t\geq0\). The energy hypothesis permits weights with exponent arbitrarily close to \(\sqrt{0.98}\). Since \[\sqrt{0.98}\sqrt{10/9}=\sqrt{49/45}>\gamma\] and \(k_0>\sqrt{0.98}\), the true and unperturbed states admit the same cutoff construction with residual exponent \(\sigma>\gamma\). There is no need to estimate the difference uniformly as \(t\downarrow0\): at this fixed positive \(t\), its finite weighted norms and the fixed energy difference divided by \(t\) are part of the constants. Thus [eq:cutoff-channel-equation,eq:channel-residual,eq:density-cutoff-error] hold at the chosen parameter.

The norm comparisons, angular induction, and integrating-factor argument use only the positive ionic gap, bounded exterior coefficients, and exponential residual. They give [eq:projected-channel-expansion,eq:scalar-remainder-derivatives] without a perturbative assertion about \(a_0\). Also \(\|F_{\mathrm{true}}-F\|=O(e^{-\sigma r})\), by the same weighted local estimates on the tail support. For each fixed \(\omega_z>0\), the liminf in [eq:fixed-channel-hypotheses] is therefore \(|a_0(\omega)|/\omega_z\). Its positive infimum and the smooth odd extension imply \[\inf_{\omega_z>0}\frac{|a_0(\omega)|}{\omega_z}>0,\qquad |a_0'(0)|>0.\] The sign of \(a_0\) is constant on the upper hemisphere and does not affect the density.

Repeat the complementary recursion [eq:complementary-recursion]. The strict inequality \(k<\gamma\) absorbs the residual of \(bG\) into \(tAr^{-5}\); the stronger uniform separation in [eq:channel-rate-separation] is unnecessary. On a fixed starting sphere the remainder is finite; dividing its bound by the positive profile \(tA_t(r)r^{-5}\) is absorbed in the allowed \(t\)-dependent comparison constant. The norm and differentiated comparisons give [eq:complementary-expansion] with \(t\)-dependent constants. Finally \(QY_z\phi=Y_z\phi% \BeginAccSupp{pdfliteral=direct,method=hex,unicode,ActualText=2260}% \KSOriginalNe\EndAccSupp{pdfliteral=direct}0\), so the coefficient in [eq:equatorial-coefficient] is nonzero without continuity from zero coupling. The fixed-coupling parts of the preceding density proof give the plane lower bound and both translation limits. ◻

Proof. On the upper half-space the Lipschitz map \(t% \BeginAccSupp{pdfliteral=direct,method=hex,unicode,ActualText=21A6}% \KSOriginalMapsto\EndAccSupp{pdfliteral=direct}\arcsin|\sin t|\) preserves \(\sin^2t\), preserves zero trace, and does not increase gradient energy. Extend the folded function oddly. Thus every competitor can be replaced by one taking values in \([0,\pi/2]\) above the plane.

The admissible class is nonempty. Smooth odd angles that equal \(\pi/2\) on increasingly large compact subsets of \(\Omega_+\) have constraint values approaching \(3\). For one such angle the constraint is greater than \(1\); multiplying it by a parameter in \([0,1]\) and using continuity produces the required value \(1\). A folded minimizing sequence is bounded in \(H^1\) on every compact set, since \(n\) has a positive local lower bound. Local compactness gives an odd limit with zero trace on the plane. The bound \(\sin^2\theta\le1\) and integrability of \(n\) preserve the constraint under exhaustion of space. Lower semicontinuity proves minimality. The bound on the angle also gives its membership in \(L^2(n)\).

Folding any perturbed competitor shows that a folded minimum is a minimum for unrestricted odd variations. The constraint derivative is nonzero on some compact smooth odd variation. Otherwise \(\sin(2\theta)=0\) almost everywhere on \(\Omega_+\). A locally Sobolev function with values in the discrete set \(\{0,\pi/2\}\) has zero gradient and is constant on that connected set; the trace and constraint exclude both possibilities. The Lagrange-multiplier rule, normalized by a factor of two, therefore gives \[\int n\nabla\theta\cdot\nabla\varphi =d\int n\sin(2\theta)\varphi\] for compact odd tests. The same identity for even tests is zero on both sides by parity, proving the full-space equation.

Testing with cutoffs times the bounded function \(\theta\) gives \[ \int n|\nabla\theta|^2 =d\int n\theta\sin(2\theta). \tag{64}\] The cutoff error tends to zero by finite gradient energy and \(\int n<\infty\). The left side is positive. The integral on the right without \(d\) is positive by folding, oddness, and the preceding nonconstancy argument. Hence \(d>0\).

Here are the local estimates needed below. Positivity of \(n\) and \(n\in H^2_{\mathrm{loc}}\) imply \(\nabla\log n\in L^6_{\mathrm{loc}}\). Rewrite [eq:angle-euler] by testing with \(\varphi/n\), where \(\varphi\) is compactly supported and smooth. The positive local lower bound and \(H^2_{\mathrm{loc}}\) regularity of \(n\) make this a legitimate local \(H^1\) test. Thus, in distributions, \[\Delta\theta =-(\nabla\log n)\cdot\nabla\theta-d\sin(2\theta).\] Its right side initially lies in \(L^{3/2}_{\mathrm{loc}}\). Interior estimates for the ordinary Laplacian on nested compact sets [22] give \[W^{2,3/2}_{\mathrm{loc}} \ \Longrightarrow\ \nabla\theta\in L^3_{\mathrm{loc}} \ \Longrightarrow\ H^2_{\mathrm{loc}} \ \Longrightarrow\ W^{2,3}_{\mathrm{loc}}.\] The last space gives \(\nabla\theta\in L^q_{\mathrm{loc}}\) for every finite \(q\), and a final application of the equation gives \(W^{2,p}_{\mathrm{loc}}\) for every \(p<6\). These estimates are uniform when the local \(H^2\) bounds and positive lower bounds for \(n\), the local angle energy, and \(d\) are uniform. They apply across the median plane because the equation has already been extended to full space.

Finally, \(\theta\ge0\) solves an equation with bounded coefficient \(d\sin(2\theta)/\theta\), and \(\delta=\pi/2-\theta\ge0\) on \(\Omega_+\) satisfies \[ L_n\delta=q\delta,\qquad q=d\,\frac{\sin(2\delta)}{\delta}\in[0,2d], \tag{65}\] with the continuous value \(2d\) at \(\delta=0\). On each compact ball the divergence-form coefficient \(nI\) is bounded and uniformly elliptic, and the zeroth-order coefficients are bounded. The local Harnack inequality therefore excludes an interior zero of either nonnegative function [17]: the resulting constant would violate the trace or the constraint. ◻

Proof. At \(\lambda=0\), let \(g=\sqrt{n_0}\cos\vartheta\) and \(h=\sqrt{n_0}\sin\vartheta\) for an admissible folded angle. Their squared norms are \(2\) and \(1\); they are orthogonal by parity. The determinant with occupied spin orbitals \((g/\sqrt2)\uparrow,(g/\sqrt2)\downarrow,h\uparrow\) has density \(n_0\) and kinetic energy \[\frac12\int\bigl(|\nabla g|^2+|\nabla h|^2\bigr) =\frac12\int|\nabla\sqrt{n_0}|^2 +\frac12\int n_0|\nabla\vartheta|^2.\] Adding the fixed integral \(\int u n_0\) and using the simple three-electron ground state in the indicated spin sector shows that the minimum is attained uniquely by \(g=\sqrt2 b,\ h=e\), with the signs fixed by folding. Indeed, equality fixes the occupied spin-orbital subspace; its one-dimensional down-spin part fixes \(g\), and the remaining up-spin orbital then fixes \(h\). This gives precisely \(\theta_*\).

To obtain a recovery sequence for small \(\lambda\), first multiply the bounded \(\theta_*\) by radial cutoffs. Its finite angle energy and the tail of \(n_0\) make the resulting energy and constraint errors tend to zero. Approximate on the remaining compact set by smooth odd functions. A fixed compact odd variation with nonzero constraint derivative corrects the constraint exactly, both at zero and at nearby \(\lambda\). Local convergence of \(n_\lambda\) then gives the energy upper bound for minima. Folding the corrected competitor, if necessary, only decreases its energy.

The minimizing angles consequently have uniformly bounded local \(H^1\) norms. Common tightness of \(n_\lambda\), supplied by 5, preserves the constraint for any local limit. Lower semicontinuity and the recovery bound make that limit a minimum at zero coupling, hence \(\theta_*\). For a fixed compact odd \(\varphi\) with nonzero limiting constraint derivative, the weak equation gives \[d_\lambda =\frac{\int n_\lambda\nabla\theta_\lambda\cdot\nabla\varphi} {\int n_\lambda\sin(2\theta_\lambda)\varphi}.\] Its numerator and denominator converge. The limiting orbital equations for \(b,e\) identify the quotient as \(d_*\). The uniform local estimates in 11, together with compact Sobolev embedding, now give \(C^1\) convergence on compact sets. Applying this argument to every sequence of minimizers proves the asserted uniformity over all folded choices. ◻

Proof. The norm, parity, and positivity statements follow from the constraint and 11. The Sobolev chain rule gives \[ |\nabla g|^2+|\nabla h|^2 =|\nabla\sqrt n|^2+n|\nabla\theta|^2, \tag{70}\] so both orbitals belong to \(H^1\). Positivity and local \(H^2\) regularity of \(n\), together with the angle estimates, give local \(H^2\) regularity of the orbitals. Differentiating [eq:angle-orbitals] and using [eq:angle-euler] proves [eq:formal-potential,eq:orbital-equations]. Since the continuous function \(g\) has a positive lower bound on each compact set, \(\Delta g/(2g)\) is locally square integrable. ◻

Proof. Solve backward with final flux equal to one. If \(J\) is the positive flux magnitude, the unnormalized solution and its flux obey \[ \begin{aligned} \widetilde U(p,z)&=\int_z^L\frac{J(p,s)}{m(p,s)}\,ds,\\ J(p,z)&=1+\mu^2\int_z^L\int_t^L \frac{m(p,t)}{m(p,s)}J(p,s)\,ds\,dt. \end{aligned} \tag{74}\] The successive-integral series is positive. Its term of order \(l\) is bounded by \[\frac{(\mu^2 C)^l(L-z)^{2l}}{(2l)!}.\] It therefore converges uniformly, defines \(J\ge1\), and gives the unique solution after normalization \(U(p,z)=\widetilde U(p,z)/\widetilde U(p,0)\). The formulas give \(U>0\) and \(U_z<0\).

For a kernel \(K=m(p,t)/m(p,s)\), \[|K_p|\le2\eta K,\qquad |K_{pp}|\le6\eta K.\] The product of \(l\) such kernels has derivatives of either positive order at most two bounded by \(20(1+l)^2\eta\) times that product. Termwise differentiation of the series is consequently justified. For \(C\le100\), the undifferentiated terms are bounded by \(10^{2l}/(2l)!\), and \[\sum_{l\ge0}20(1+l)^2\frac{10^{2l}}{(2l)!}<10^8.\] Since \(J\ge1\), these estimates bound its first two \(p\)-derivatives relative to \(J\). Also \(|(m^{-1})_p|\le\eta m^{-1}\) and \(|(m^{-1})_{pp}|\le3\eta m^{-1}\). Integrating the positive expression for \(\widetilde U\), and then differentiating its quotient by \(\widetilde U(p,0)\), proves [eq:strip-horizontal-derivatives]; the displayed numerical bounds give \(C_2<10^{30}\) with ample slack. For general fixed \(C\) the same convergent series gives a finite constant. Only the one-sided ratios in [eq:strip-comparison-data] were used. ◻

Proof. Set \(\delta=\pi/2-\theta\) on \(\Omega_+\). By [eq:delta-equation], \[L_n\delta=q\delta,\quad 0\le q\le2d,\qquad \frac4{\pi^2}n\delta^2\le g^2=n\sin^2\delta\le n\delta^2.\] Choose \(D\) large enough that \(d_*\) and \(\gamma-k_0\) are small. The multiplier convergence and the density estimates [eq:density-drift,eq:strip-data] allow us to choose the small constants below uniformly for sufficiently small \(\lambda\).

First, \(U\) from 14 is a subsolution in the strip. In cylindrical coordinates, \[L_nU=\mu^2U+U_{pp} +(p^{-1}+\partial_p\log n)U_p.\] The horizontal coefficient is bounded sufficiently far out by [eq:density-drift]. Thus [eq:strip-horizontal-derivatives], with \(\eta\) small, and \(q\le2d\), with \(d\) small, imply \[ (L_n-q)U\ge0. \tag{76}\]

We next supply an inner boundary for comparison. At zero coupling put \(\delta_*=\pi/2-\theta_*\). On \(z>0\), \[\delta_*=\arctan(\sqrt2 b/e).\] The scalar exterior estimates in [eq:unperturbed-strip-ratio] show that, uniformly for \(0<z\le L\), the first two horizontal logarithmic derivatives of \(b/e\) are small when \(D\) and then the starting radius are large. This remains true down to \(z=0\): the simple odd factor of \(e\) is canceled before taking a horizontal derivative. If \(t>0\), the ratios of \(t(\arctan t)'\) and \(t^2(\arctan t)''\) to \(\arctan t\) are bounded. Consequently \[\frac{|(\delta_*)_p|+|(\delta_*)_{pp}|}{\delta_*}\] is uniformly small in the strip. Combining this with its angle equation gives, for a fixed sufficiently large \(P_1\), \[ n_0^{-1}\partial_z(n_0\partial_z\delta_*) <\mu^2\delta_* \quad(p=P_1,\ 0\le z\le L). \tag{77}\] The strict inequality follows by making the horizontal error and \(2d_*\) smaller than \(\mu^2\). One-dimensional comparison with \(U_0(P_1,\cdot)\) now gives \(\delta_*/(\pi/2)>U_0\) for \(0<z\le L\), and the derivative of their difference at \(z=0\) is strictly positive. Its quotient by \(z\) therefore has a positive minimum on \(0<z\le L\). By 12 and continuity of the one-dimensional problem in its positive coefficient, this boundary ordering persists for small \(\lambda\), uniformly over all folded minimizers and all azimuths. For this use of continuity, [eq:strip-volterra] gives uniform convergence of \(U_\lambda\) and \((U_\lambda)_z\) on the fixed segment from uniform convergence and a positive lower bound for its density.

It follows that \[ \frac{\delta(x)}{\pi/2}\ge U(p,z) \quad(p\ge P_1,\ 0\le z\le L). \tag{78}\] Here is a comparison proof that also addresses infinity. The difference \(v=\delta/(\pi/2)-U\) satisfies \((L_n-q)v\le0\) and has nonnegative finite-boundary values. Test this inequality with a horizontal cutoff squared times its negative part, with the appropriate negative sign. The resulting Caccioppoli inequality is \[\int n\chi_R^2|\nabla v_-|^2 +\int nq\chi_R^2v_-^2 \le4\int n v_-^2|\nabla\chi_R|^2.\] The right side tends to zero because \(v_-\) is bounded and \(n\) is integrable. Thus \(v_-\) is constant and its zero boundary trace makes it zero. No value at infinity has been imposed. This proof also makes clear that \(\theta\) need not be axisymmetric: the coefficient \(q\) may depend on azimuth. Differentiating the boundary inequality at \(z=0\) gives \[ -\frac{\delta_z}{\delta}\le-U_z(p,0). \tag{79}\]

For a real smooth test \(\xi\) compactly supported sufficiently far out, put \(F_1=\delta\xi\). Integration by parts, with outward normal \(-e_z\) on the plane, yields \[ \int_{\Omega_+}n\delta^2|\nabla\xi|^2 =\int_{\Omega_+}n\bigl(|\nabla F_1|^2+q|F_1|^2\bigr) -\int_{z=0}n\left(-\frac{\delta_z}{\delta}\right)|F_1|^2. \tag{80}\] The energy of the solution of [eq:strip-ode] is \[\int_0^L n(U_z^2+\mu^2U^2)\,dz=-n(p,0)U_z(p,0).\] It minimizes this energy among functions with the same endpoint values. Use the trial function \(\cos(az)F_1\), \(a=\pi/(2L)\), for each fixed horizontal point. The exact identity \[|F_1'|^2+a^2|F_1|^2 -|(\cos(az)F_1)'|^2 =|\sin(az)F_1'+a\cos(az)F_1|^2\] gives \[ n(p,0)(-U_z(p,0))|F_1(0)|^2 \le\int_0^L n\bigl( |\partial_zF_1|^2+(\mu^2+a^2)|F_1|^2\bigr)\,dz. \tag{81}\] Thus the boundary term spends at most the vertical gradient energy in the strip, together with the indicated \(L^2\) cost.

The remaining gradient energy controls \(\int_{\Omega_+}n|Y\cdot\nabla F_1|^2\), where \(Y\) is the horizontal radial unit vector for \(z\le p/4\), the spatial radial unit vector for \(z\ge p/2\), and a smooth convex interpolation between them. More explicitly, choose \(\zeta(t)\in[0,1]\), equal to zero for \(t\le1/4\) and one for \(t\ge1/2\), with \(|\zeta'|\le8\), and set \[Y=(1-\zeta(z/p))\frac{(x_1,x_2,0)}p +\zeta(z/p)\frac{x}{r}.\] On the axis use \(Y=x/r\). The field has norm at most one and is tangential to the median plane. On the interpolation region, \(|\nabla(z/p)|\le5/(4r)\) and \(1/p\le\sqrt5/(2r)\). The product rule therefore gives \(\lvert\operatorname{div}Y\rvert\le100/r\). For sufficiently distant support it is horizontal throughout \(0\le z\le L\). In the interpolation region \(p/r\ge2/\sqrt5\); the two drift bounds in [eq:density-drift], with \(k\) close to one, therefore imply \[ n^{-1}\operatorname{div}(nY)\le-1 \tag{82}\] after increasing the fixed starting radius. There is no boundary contribution when integrating \(\operatorname{div}(nY|F_1|^2)\). Cauchy–Schwarz gives \[\int_{\Omega_+}n|Y\cdot\nabla F_1|^2 \ge\frac14\int_{\Omega_+}n|F_1|^2.\] Dropping the nonnegative \(q\) term in [eq:delta-transform-boundary] and using [eq:strip-flux-bound,eq:strip-boundary-cost] proves \[\int_{\Omega_+}n\delta^2|\nabla\xi|^2 \ge\left(\frac14-\mu^2-\frac{\pi^2}{4L^2}\right) \int_{\Omega_+}n\delta^2|\xi|^2.\] The comparison of \(g^2\) with \(n\delta^2\) yields the coefficient \[ \frac2{\pi^2} \left(\frac14-\frac1{100}-\frac{\pi^2}{400}\right) >0.0436>0.03. \tag{83}\] This proves [eq:exterior-weighted-gap]. Real and imaginary parts give complex tests; the form-domain approximation described at the beginning of the section gives the general statement. ◻

Proof. Put \(w=g^2\). On the upper half-space, \(f=h/g>0\) satisfies \[-\frac1{2w}\operatorname{div}(w\nabla f)=df.\] For compact tests inside \(\Omega_+\), the ground-state transform [6] gives \[ \frac12\int_{\Omega_+}w|\nabla\xi|^2 -d\int_{\Omega_+}w|\xi|^2 =\frac12\int_{\Omega_+}wf^2 |\nabla(\xi/f)|^2\ge0. \tag{85}\] Odd tests have zero trace at the plane. The left side of [eq:odd-ground-transform] is continuous in the weighted form norm. Cutoff and local zero-trace approximation therefore extend its nonnegativity to all odd functions in \(H^1(w)\), proving the odd part of the desired gap.

We prove the even part by controlling possible loss of weighted norm at infinity. Let \(w_0=2b^2\). The positive-orbital transform for \(b\) identifies \(\mathcal E_{w_0}\) with the quadratic form of \(h_D-\epsilon_b\), under \(\xi% \BeginAccSupp{pdfliteral=direct,method=hex,unicode,ActualText=21A6}% \KSOriginalMapsto\EndAccSupp{pdfliteral=direct}\sqrt2b\,\xi\). The identity first holds for compact smooth tests. Their density in \(H^1(w_0)\), together with closedness of the Schrödinger form, extends it to the full weighted domain: the transformed approximants are Cauchy in the form norm, and their \(L^2\) limit is \(\sqrt2b\,\xi\). No global bound on \(\nabla\log b\) is needed. By 3, there is an even spectral gap \(\Delta_{\mathrm{ev}}>0.2\): \[ \mathcal E_{w_0}(\xi)\ge \Delta_{\mathrm{ev}}\int w_0|\xi|^2 \quad\hbox{if \(\xi\) is even and }\int w_0\xi=0. \tag{86}\] In particular it is strictly greater than \(d_*\), and we choose \(D\) so that \(d_*<0.03\).

If the even assertion failed arbitrarily close to zero coupling, there would be \(\lambda_j\downarrow0\), folded minimizers, and real even tests with \[\int w_j\xi_j=0,\qquad \int w_j|\xi_j|^2=1,\qquad \limsup_j\mathcal E_{w_j}(\xi_j)\le d_*, \quad w_j=g_{\lambda_j}^2.\] Local uniform convergence to \(w_0\), and its positive local lower bounds, give a subsequence converging weakly in local \(H^1\) and strongly in local \(L^2\) to an even \(\xi\). Write \(s=\int w_0|\xi|^2\le1\). The limit has mean zero, since \[\left|\int_{|x|>R}w_j\xi_j\right| \le\left(\int_{|x|>R}w_j\right)^{1/2},\] and the tails on the right are uniformly small: \(w_j\le n_{\lambda_j}\), and the latter densities are tight. No tightness of \(w_j|\xi_j|^2\) is assumed.

Choose a smooth radial quadratic partition \(\chi_R^2+\rho_R^2=1\), with \(\chi_R=1\) on \(r\le R\), \(\chi_R=0\) on \(r\ge2R\), and derivatives of size \(O(R^{-1})\). Expanding gradients gives the weighted localization identity, as in the usual IMS formula [16]: \[\mathcal E_{w_j}(\xi_j) =\mathcal E_{w_j}(\chi_R\xi_j) +\mathcal E_{w_j}(\rho_R\xi_j) -\frac12\int w_j (|\nabla\chi_R|^2+|\nabla\rho_R|^2)|\xi_j|^2.\] The last term is \(O(R^{-2})\). For \(R\ge R_0\), 15, applied to both half-spaces, bounds the second term below by \(0.03\|\rho_R\xi_j\|_{L^2(w_j)}^2\). Local lower semicontinuity and local strong convergence therefore give \[\liminf_j\mathcal E_{w_j}(\xi_j) \ge \mathcal E_{w_0}(\chi_R\xi) +0.03\left(1-\int w_0|\chi_R\xi|^2\right)-O(R^{-2}).\] Local lower semicontinuity also gives finite global energy to the limit. Letting \(R\to\infty\) and using [eq:unperturbed-even-gap], we obtain \[d_*\ge\mathcal E_{w_0}(\xi)+0.03(1-s) \ge\Delta_{\mathrm{ev}}s+0.03(1-s)>d_*,\] a contradiction for every \(s\in[0,1]\). This proves the even assertion uniformly for all folded minimizers. Finally, reflection separates the even and odd parts orthogonally in the weighted norm and energy. The odd part has zero weighted mean. Combining the two inequalities, and then the real and imaginary parts, proves [eq:weighted-gap]. ◻

Proof. The strip derivative constant in 14 is less than \(10^{30}\). The drift coefficient in the strip has absolute value less than \(3\), so the horizontal error in \(L_nU\) is at most \(4\cdot10^{30}\eta_0U\). Since \(q\le2d<0.002\) and \(\mu^2=0.01\), this proves [eq:strip-subsolution] for every \(p\ge M\). At the inner cylinder the function \[\frac2\pi\delta-U -\left(\frac2\pi\delta_*-U_0\right)\] vanishes at \(z=0\) and has \(z\)-derivative of absolute value less than \(\alpha/2\). Its reference term is at least \(\alpha z\). Thus the required inner boundary ordering is strict. The same cutoff comparison proves [eq:strip-angle-comparison] for \(p\ge M\).

For \(r\ge S\), the vector field used above is horizontal throughout the strip. Its weighted divergence is bounded by \[-2k\,\frac2{\sqrt5}+\eta_0+\frac{100}{r}<-1.\] All boundary support lies at \(p>M\). The flux calculation and [eq:exterior-gap-constant] consequently give the exterior constant \(0.03\) from radius \(S\).

It remains to prove the even part of the full gap without a perturbative compactness argument. Choose a nonnegative smooth radial quadratic partition \(\chi^2+\rho^2=1\) between \(S\) and \(2S\), with \[\frac12\bigl(|\nabla\chi|^2+|\nabla\rho|^2\bigr) \le\frac{10}{S^2}.\] For an even test \(\xi\in H^1(g^2)\), put \(w=g^2\), \(w_0=2b^2\), and choose \[c=\frac{\int w_0\chi\xi}{\int w_0\chi}, \qquad f=\xi-c.\] The denominator is positive and \(\int w_0\chi f=0\). By the local weight comparison and [eq:unperturbed-even-gap], \[\begin{aligned} \mathcal E_w(\chi f) &\ge\frac12\mathcal E_{w_0}(\chi f) \ge\frac{\Delta_{\mathrm{ev}}}{2} \|\chi f\|_{L^2(w_0)}^2\\ &\ge\frac{\Delta_{\mathrm{ev}}}{4} \|\chi f\|_{L^2(w)}^2 \ge0.05\|\chi f\|_{L^2(w)}^2. \end{aligned}\] The exterior estimate gives \(\mathcal E_w(\rho f)\ge0.03\|\rho f\|_{L^2(w)}^2\). Although \(f\) need not have compact support, it belongs to \(H^1(w)\), because \(\int w=2\); thus the established cutoff extension applies. Localization yields \[\mathcal E_w(\xi)=\mathcal E_w(f) \ge\left(0.03-\frac{10}{S^2}\right)\|f\|_{L^2(w)}^2 \ge d\,\mathop{\mathrm{Var}}_w(\xi).\] The last step uses \(d<0.001\), the stated lower bound on \(S\), and the fact that subtracting any constant gives squared norm at least the weighted variance. The odd ground-state transform [eq:odd-ground-transform] does not require small coupling and completes the proof, including complex tests. ◻

Proof. We first verify the absolute integrability required to use \(V\). By 5, \(\Delta n\in L^1(\mathbf R^3)\) and \(\sqrt n\in H^1(\mathbf R^3)\). Local positivity and regularity give the almost-everywhere identity \[\sqrt n\,\Delta\sqrt n =\frac12\Delta n-|\nabla\sqrt n|^2.\] Using the formula for \(V\) in 13, the finite angle energy and \(\int n\sin^2\theta=1\), we obtain \[ \int n|V| \le \frac14\|\Delta n\|_1 +\frac12\|\nabla\sqrt n\|_2^2 +\frac12\int n|\nabla\theta|^2+d <\infty. \tag{95}\] The local assertion \(V\in L^2_{\mathrm{loc}}\) was proved in 13.

For a scalar orbital \(f\), consider the quadratic expression \[q_V[f]=\frac12\int|\nabla f|^2+\int V|f|^2, \qquad f\in\mathcal E_V :=\left\{f\in H^1(\mathbf R^3):\int|V||f|^2<\infty\right\}.\] Only this expression on its indicated test class is needed; we do not assume an ordinary Hamiltonian or a closed form with potential \(V\). For \(f\in C_c^\infty(\mathbf R^3;\mathbf C)\), the function \(f/g\) is admissible in [eq:weighted-gap], since \(g\) is positive and bounded below on each compact set. Testing \((-\Delta/2+V)g=0\) locally with \(|f|^2/g\) gives the positive-orbital identity \[ q_V[f] =\frac12\int g^2\left|\nabla\left(\frac f g\right)\right|^2 \ge d\left(\|f\|_2^2-|\langle\widehat g,f\rangle|^2\right). \tag{96}\] The identity uses real parts in the gradient expansion and therefore holds for complex \(f\) as well. This is the positive ground-state representation; see [6].

We record the extension of [eq:ensemble-orbital-bound] carefully. On a fixed compact set, \(H^1\) approximation by smooth functions passes through the potential integral: \(V\in L^2\) there and strong \(H^1\) convergence implies strong \(L^4\) convergence, hence convergence of the squared moduli in \(L^2\). Thus the inequality holds for compactly supported \(H^1\) orbitals. Choose real cutoffs \(0\le\chi_R\le1\) that equal one on \(|x|\le R\), vanish on \(|x|\ge2R\), and satisfy \(|\nabla\chi_R|\le C/R\). For \(f\in\mathcal E_V\), \(\chi_Rf\to f\) in \(H^1\), while dominated convergence gives \(\int V|\chi_Rf|^2\to\int V|f|^2\). The projection term also converges in \(L^2\). This proves [eq:ensemble-orbital-bound] on all of \(\mathcal E_V\).

Apply this scalar inequality to both components of a spinor \(\psi\). With \(\widetilde P_g=P_g\otimes\mathop{\mathrm{id}}_{\mathbf C^2}\), the resulting estimate is \[ \frac12\|\nabla\psi\|_2^2+\int V|\psi|_{\mathbf C^2}^2 \ge d\bigl(\|\psi\|_2^2 -\langle\psi,\widetilde P_g\psi\rangle\bigr). \tag{97}\] Now let \(\Gamma\) have density \(n\) and finite kinetic expectation. By [eq:ensemble-density,eq:ensemble-V-integrability] and Tonelli’s theorem, \[\sum_j\nu_j\int|V||\psi_j|_{\mathbf C^2}^2=\int|V|n<\infty.\] Hence every occupied natural spin orbital belongs componentwise to \(\mathcal E_V\), and the potential terms can be summed absolutely. Summing [eq:ensemble-spinor-bound] and using [eq:ensemble-pauli] gives \[\begin{align*} T[\Gamma]+\int Vn &\ge d\left(\mathop{\mathrm{Tr}}\gamma_\Gamma -\mathop{\mathrm{Tr}}(\gamma_\Gamma\widetilde P_g)\right)\\ &\ge d(3-2)=d. \end{align*}\] Here \(\widetilde P_g\) has rank two and \(0\le\gamma_\Gamma\le\mathop{\mathrm{id}}\). No rank bound on \(\Gamma\) is used.

Finally, the orbitals \(\widehat g,h\) belong to \(\mathcal E_V\) by [eq:ensemble-V-integrability]. If \(f\) denotes either of them, with respective energy \(\varepsilon=0,d\), testing its distributional equation by \(\chi_R^2 f\) gives \[q_V[\chi_R f] =\varepsilon\|\chi_R f\|_2^2 +\frac12\int|\nabla\chi_R|^2|f|^2.\] All terms converge as above, so \(q_V[\widehat g]=0\) and \(q_V[h]=d\). The determinant in [eq:ensemble-determinant] therefore satisfies \[T[|\Phi_\theta\rangle\langle\Phi_\theta|]+\int Vn =2q_V[\widehat g]+q_V[h]=d.\] This proves the minimum over all ensembles with finite kinetic energy; ensembles with infinite kinetic expectation cannot lower it. ◻

Proof. The potential class supplies, for every \(\varepsilon>0\), a constant \(C_\varepsilon\) such that \[ \int|v||f|^2 \le\varepsilon\,\frac12\|\nabla f\|_2^2 +C_\varepsilon\|f\|_2^2, \qquad f\in H^1(\mathbf R^3). \tag{99}\] Indeed, write \(v=v_1+v_\infty\) and truncate \(v_1\) at a sufficiently large amplitude. Its remaining \(L^{3/2}\) tail has arbitrarily small norm and is controlled by the Sobolev inequality; the truncated part and \(v_\infty\) are bounded. Summing [eq:ensemble-standard-form-bound] over spins and particles gives \[q_{H_v}[\Xi] \ge\frac{1-\varepsilon}{2}\sum_{i=1}^3\|\nabla_i\Xi\|_2^2 -3C_\varepsilon\|\Xi\|_2^2\] on the many-electron \(H^1\) form domain. If \(E=\inf\sigma(H_v)\), every normalized vector in its ground eigenspace consequently has kinetic expectation at most \((E+3C_\varepsilon)/(1-\varepsilon)\) for any \(0<\varepsilon<1\). Expanding \(\Gamma_s\) in its trace-class spectral decomposition proves that \(T[\Gamma_s]<\infty\), even for an infinite-dimensional ground eigenspace. The absolute potential expectation is finite by the same form bound, and \[E=T[\Gamma_s]+\int vn.\]

By 18, the determinant has no larger kinetic expectation and has the same density. Thus \[E\le q_{H_v}[\Phi_\theta] =T[|\Phi_\theta\rangle\langle\Phi_\theta|]+\int vn \le T[\Gamma_s]+\int vn=E.\] The square-root representation of the closed form [10] shows that equality at its bottom implies \((H_v-E)^{1/2}\Phi_\theta=0\). Hence \(\Phi_\theta\) is itself a ground-state eigenvector.

It remains to extract a spatial eigenfunction without any assumption on the other ground states or on the multiplicities of one-particle levels. Denote the sesquilinear form of \(h_v\) by \[q_v(f_1,f_2) =\frac12\int\nabla\overline{f_1}\cdot\nabla f_2 +\int v\overline{f_1}f_2,\] linear in its second argument. For \(f\in H^1(\mathbf R^3)\), replace the single down-spin orbital in [eq:ensemble-determinant] and set \[\Xi_f=(\widehat g\uparrow)\wedge(f\downarrow)\wedge(h\uparrow).\] This vector lies in the many-electron form domain. Expanding the one-body form between these determinants, spin orthogonality and \(\langle\widehat g,h\rangle=0\) give \[\begin{aligned} q_{H_v}(\Phi_\theta,\Xi_f) &=q_v(\widehat g,f)\\ &\quad+\langle\widehat g,f\rangle \bigl(q_v(\widehat g,\widehat g)+q_v(h,h)\bigr). \end{aligned}\] If \(\langle\widehat g,f\rangle=0\), then \(\Xi_f\) is orthogonal to \(\Phi_\theta\), and the ground-state form equation makes the left side zero. Therefore \[q_v(\widehat g,f)=0 \qquad\text{whenever }\langle\widehat g,f\rangle=0.\] Put \(e_g=q_v(\widehat g,\widehat g)\in\mathbf R\). Decomposing an arbitrary \(f\in H^1\) into its component parallel to \(\widehat g\) and its orthogonal component yields \[q_v(\widehat g,f)=e_g\langle\widehat g,f\rangle \qquad(f\in H^1).\] By Hermitian symmetry this is equivalent to \(q_v(f,\widehat g)=e_g\langle f,\widehat g\rangle\). The closed-form representation theorem [10] therefore gives \(\widehat g\in\mathop{\mathrm{Dom}}(h_v)\) and \(h_v\widehat g=e_g\widehat g\).

Subtracting \((-\Delta/2+V)g=0\) now gives \((v-V-e_g)g=0\) in distributions. The product is locally integrable: \(v\in L^{3/2}_{\mathrm{loc}}\), \(V\in L^2_{\mathrm{loc}}\), and \(g\in H^2_{\mathrm{loc}}\) is locally bounded. Since \(g\) is continuous and strictly positive, division on each compact set proves \(v=V+e_g\) almost everywhere. Adding a constant preserves the stated potential class, completing the proof. ◻

Proof. Every fixed translated ball eventually lies in an arbitrarily distant tail of \(V_{\mathrm c}\), proving its norm convergence. Weak-star compactness and a diagonal subsequence on expanding balls give \(W\).

To prove the assertion about solutions, let \(\chi\) be a smooth compact cutoff. Testing the equation with \(\chi^2u_j\) and taking the real part gives the usual cutoff identity. Its possibly negative critical-potential term is bounded by \[\int |V_{\mathrm c}(y_j+x)|\,|\chi u_j|^2\,dx \le C_{\mathrm S}\|V_{\mathrm c}(y_j+\cdot)\|_{L^{3/2}(\mathop{\mathrm{supp}}\chi)} \|\nabla(\chi u_j)\|_2^2.\] For large \(j\) this is absorbed into the kinetic term. The bounded potential, the eigenvalue, and the cutoff derivative are controlled by the assumed local \(L^2\) bounds. Hence the sequence is bounded in local \(H^1\). Rellich compactness gives the stated convergences.

For a compact smooth test \(\varphi\), the critical-potential term tends to zero by Hölder’s and Sobolev’s inequalities. The bounded-potential term passes to the limit because \(u_j\varphi\to u\varphi\) in \(L^1\) and the multipliers are uniformly bounded, while their pairing with the fixed function \(u\varphi\) converges weak star. The gradient term passes by weak convergence. This proves the limiting equation. ◻

Proof. Centers on the median plane. First let \(y_j=p_j\omega_j\), where \(p_j\to\infty\), \(\omega_j\cdot e_z=0\), \(|\omega_j|=1\), and, after subselection, \(\omega_j\to\omega_*\). Put \[c_j=\frac{A(p_j)}{p_j},\qquad g_j(x)=\frac{g(y_j+x)}{c_j},\qquad h_j(x)=\frac{h(y_j+x)}{c_j}.\] By [eq:plane-translate], for a constant \(a_*>0\), \[ g_j^2+h_j^2=\frac{n(y_j+\cdot)}{c_j^2} \longrightarrow a_*^2z^2e^{-2k\omega_*\cdot x} \quad\text{locally uniformly}. \tag{106}\] In particular these orbitals are locally bounded. Take any subsequential bounded-potential limit \(W\). By 20, a further subsequence has limits \(g_\infty,h_\infty\) satisfying the two orbital equations with potential \(W\) and energies \(0,d\). Their strong local \(L^2\) convergence and [eq:plane-normalized-orbitals] give \[0\le g_\infty(x)\le a_*|z|e^{-k\omega_*\cdot x},\qquad g_\infty^2+h_\infty^2=a_*^2z^2e^{-2k\omega_*\cdot x} \quad\text{a.e.}\] Since \(W\) is bounded, local elliptic regularity makes \(g_\infty\) continuous. It vanishes on the interior plane \(z=0\). The local Harnack inequality for a nonnegative solution with bounded zeroth-order coefficient [17] forces \(g_\infty\equiv0\); no sign condition on \(W\) is needed. The sign of \(h\) on each half-space therefore determines \[h_\infty(x)=a_*z e^{-k\omega_*\cdot x}.\] Because \(\omega_*\cdot e_z=0\), its Laplacian is \(k^2h_\infty\). The limiting equation gives \((W-d-k^2/2)h_\infty=0\). This determines \(W=c_\infty\) almost everywhere; the plane where \(h_\infty=0\) has zero volume.

A positive angle bound above the plane. We next show \[ \inf_{x_1,x_2\in\mathbf R}\theta(x_1,x_2,1)>0. \tag{107}\] On each compact subset of the open upper half-space, the normalized density in [eq:plane-normalized-orbitals] is uniformly bounded above and below by positive constants. The translated angle solves \[\operatorname{div}\!\left(\frac{n(y_j+x)}{c_j^2} \nabla\theta(y_j+x)\right) =-d\frac{n(y_j+x)}{c_j^2}\sin\bigl(2\theta(y_j+x)\bigr).\] On each such compact set the scalar coefficients have uniform positive lower and finite upper bounds, and the right-hand sides are uniformly bounded. Local energy estimates and the interior Hölder estimate for uniformly elliptic divergence-form equations [17] therefore give local uniform compactness of the bounded angles. These estimates require no bounds on derivatives of the normalized densities. The identity \[g_j=\sqrt{n(y_j+\cdot)/c_j^2}\cos\theta(y_j+\cdot),\] together with \(g_\infty=0\), implies that every such angle limit equals \(\pi/2\). Consequently \(\theta(x_1,x_2,1)\to\pi/2\) as \(x_1^2+x_2^2\to\infty\). On the remaining compact part it has a strictly positive minimum, proving [eq:angle-plane-parallel-lower-bound].

Choose \(m\in(0,\pi/2)\) below this infimum. In \(z>1\), the angle is a supersolution of \(-\operatorname{div}(n\nabla\theta)\ge0\). For \(w=(m-\theta)_+\), whose trace on \(z=1\) is zero, test with \(\chi_R^2w\), where \(\chi_R\) is a spatial cutoff with \(|\nabla\chi_R|\le C/R\). The resulting estimate is \[\int_{z>1} n\chi_R^2|\nabla w|^2 \le4\int_{z>1}n w^2|\nabla\chi_R|^2\longrightarrow0.\] Here \(w\) is bounded and \(n\) is integrable. Thus \(w\) is constant on this connected half-space, and its zero trace makes it zero. We have proved \(\theta\ge m\) throughout \(z\ge1\).

Centers whose heights diverge. Set \(\delta=\pi/2-\theta\) on the upper half-space. On \(z\ge1\), \[0\le\delta\le\frac\pi2-m,\qquad \operatorname{div}(n\nabla\delta)=d n\sin(2\delta),\qquad d\sin(2\delta)\ge c_0\delta\] for some \(c_0>0\). The last assertion follows by minimizing the continuous positive function \(d\sin(2s)/s\) on \([0,\pi/2-m]\), with its value at zero defined as \(2d\).

Now let \((y_j)_3\to+\infty\), and pass to a subsequence with \(y_j/|y_j|\to\omega_*\). By [eq:bulk-translate], \[N_j(x)=\frac{n(y_j+x)}{n(y_j)} \longrightarrow N_\infty(x)=e^{-2k\omega_*\cdot x} \quad\text{locally uniformly}.\] The same local energy and Hölder estimates give, along a subsequence, local uniform and weak local \(H^1\) convergence of \(\delta(y_j+\cdot)\) to a bounded nonnegative function \(w\). Passing the divergence equation to the limit requires only the displayed uniform convergence of the coefficients, not convergence of their derivatives. We obtain \[\operatorname{div}(N_\infty\nabla w)=dN_\infty\sin(2w),\qquad (\Delta-2k\omega_*\cdot\nabla)w=d\sin(2w)\ge c_0w.\] Local elliptic regularity makes \(w\) smooth. A short maximum-principle argument shows that \(w=0\). Indeed, for \(\varepsilon>0\) subtract \(\varepsilon\rho(x)\), where \(\rho(x)=\sqrt{1+|x|^2}\). The boundedness of \(w\) implies that \(w-\varepsilon\rho\) attains a global maximum at some \(x_\varepsilon\). At that point, \[c_0w(x_\varepsilon) \le (\Delta-2k\omega_*\cdot\nabla)w(x_\varepsilon) \le\varepsilon(3+2k),\] since \(|\nabla\rho|\le1\) and \(\Delta\rho\le3\). Comparing the maximum with any fixed point \(x_0\) gives \(w(x_0)\le\varepsilon\rho(x_0)+\varepsilon(3+2k)/c_0\). Letting \(\varepsilon\downarrow0\) proves the assertion.

It follows that \(\delta(y_j+\cdot)\to0\) locally uniformly along every such sequence. Hence \[\frac{h(y_j+x)}{\sqrt{n(y_j)}} =\sqrt{N_j(x)}\cos\delta(y_j+x) \longrightarrow e^{-k\omega_*\cdot x} \quad\text{locally uniformly}.\] Apply 20 to this normalized orbital along any subsequence on which the bounded potentials converge. Its limiting eigenfunction equation again identifies \(W=d+k^2/2\), since the limiting exponential is everywhere positive. Reflection treats centers with heights tending to \(-\infty\). Let \(\mathcal R\) denote reflection in the median plane. Since \(V\) is even, its arbitrary decomposition satisfies \[V_{\mathrm b}(\mathcal R x)-V_{\mathrm b}(x) =V_{\mathrm c}(x)-V_{\mathrm c}(\mathcal R x).\] Escaping translates of the right side tend to zero in local \(L^{3/2}\). The left side is uniformly bounded, so its local weak-star limits are also zero. Reflecting the center and local coordinates therefore gives the same constant limit for negative heights.

All remaining sequences. Given any escaping sequence, its heights have a subsequence that tends to \(+\infty\), tends to \(-\infty\), or converges to a finite number. The first two cases were just proved. In the third, project the centers onto the median plane. Their horizontal radii tend to infinity, and the original centers differ from these plane centers by convergent bounded displacements. The plane-centered conclusion therefore applies. To justify the moving displacement for weak-star convergence, translate each fixed compactly supported \(L^1\) test by the negative displacement; these tests converge in \(L^1\), while the multipliers stay uniformly bounded. Thus their limit remains the same constant. These cases exhaust every subsequential bounded-potential limit. ◻

Proof. Fix a ball \(B_L\). We first claim \[ \sup_{\substack{u\in H^1_0(B_L)\\\|u\|_{H^1(B_L)}\le1}} \left|\int_{B_L}\bigl(V(y+x)-c_\infty\bigr)|u(x)|^2\,dx\right| \longrightarrow0\qquad (|y|\to\infty). \tag{109}\] Sobolev’s inequality and the vanishing local \(L^{3/2}\) norm prove the claim for the critical part. If the bounded part failed, choose escaping centers and tests violating it. Extract a weak-star subsequence of the bounded multipliers and a subsequence of the tests converging strongly in \(L^2(B_L)\). Their squares converge in \(L^1\), because \[\bigl\||u_j|^2-|u|^2\bigr\|_1 \le(\|u_j\|_2+\|u\|_2)\|u_j-u\|_2.\] Uniform boundedness of the multipliers handles this changing test, and their weak-star limit is \(c_\infty\). This contradicts the selected violation and proves [eq:local-form-uniformity].

Choose once and for all a smooth, locally finite quadratic partition \(\sum_\ell\chi_\ell^2=1\) with uniformly bounded overlap, each \(\chi_\ell\) supported in a ball of one fixed radius, and \(\sum_\ell|\nabla\chi_\ell|^2\le K\). For any prescribed \(\eta>0\), [eq:local-form-uniformity] applies with error \(\eta\) to all these balls whose centers are sufficiently distant. If \(u\) is supported sufficiently far out, every nonzero \(\chi_\ell u\) has such a center. Summing the local potential estimates gives \[\begin{align*} \left|\int(V-c_\infty)|u|^2\right| &\le\eta\sum_\ell\|\chi_\ell u\|_{H^1}^2\\ &\le\eta\|\nabla u\|_2^2+\eta(1+K)\|u\|_2^2. \end{align*}\] Here the cross terms cancel because \(\sum_\ell\chi_\ell\nabla\chi_\ell=0\). Choosing \(\eta\le\varepsilon/(1+K)\) proves [eq:exterior-form-coercivity]. This argument uses a uniform local form estimate, not a pointwise limit of \(V\). ◻

Proof. Assume first that [eq:admissible-formal-potential] holds. [prop:potential-limits,lem:exterior-form-limit] give [eq:exterior-form-coercivity] with \(c_\infty=d+k^2/2\). Since \(d>0\), choose \[ k<a<\sqrt{k^2+2d},\qquad 0<\varepsilon<\min\left\{\frac12, c_\infty-\frac{a^2}{2}\right\}. \tag{110}\] We use the exponential-weight method [1], spelling out the estimate and its \(H^1\) conclusion for the form bound obtained here.

Let \(\eta\) be a smooth cutoff which vanishes on the ball required by [eq:exterior-form-coercivity] and equals one outside a slightly larger ball. With \(\rho(x)=\sqrt{1+|x|^2}\), put \[f_T(x)=a\min\{\rho(x),T\},\qquad u_T=\eta e^{f_T}g.\] The function \(f_T\) is bounded and Lipschitz with \(|\nabla f_T|\le a\). Thus \(u_T\in H^1\), and the weak eigenfunction equation for \(g\) may be tested with \(\eta^2e^{2f_T}g\). Expanding the gradient gives the exact identity \[ q_V[u_T]=\frac12\int g^2 \bigl|\nabla(\eta e^{f_T})\bigr|^2. \tag{111}\] The part with \(\eta^2|\nabla f_T|^2\) is at most \(a^2\|u_T\|_2^2\). All terms involving \(\nabla\eta\), including the cross term, are supported in a fixed compact annulus, where \(f_T\le a\rho\) independently of \(T\). Consequently \[q_V[u_T]\le\frac{a^2}{2}\|u_T\|_2^2+C\] with \(C\) independent of \(T\). Combining this with the exterior form bound yields \[\left(\frac12-\varepsilon\right)\|\nabla u_T\|_2^2+ \left(c_\infty-\varepsilon-\frac{a^2}{2}\right)\|u_T\|_2^2\le C.\] Both coefficients are positive by [eq:agmon-exponent-choice]. Monotone convergence of the squared weighted functions gives the \(L^2\) bound as \(T\to\infty\), and weak \(H^1\) compactness identifies the limit as \(\eta e^{a\rho}g\). Adding back the compact interior part proves \[ e^{a\rho}g\in H^1(\mathbf R^3),\qquad e^{a|x|}g\in H^1(\mathbf R^3). \tag{112}\] The second assertion also follows by multiplying the first function by the bounded Lipschitz factor \(e^{a(|x|-\rho(x))}\).

The trace map from \(H^1(\mathbf R^3)\) to \(L^2(\mathbf R^2)\) is bounded. The local continuity of \(g\), \(n\), and \(\theta\), together with the zero angle trace, identifies the trace on \(z=0\) as \[g(x_1,x_2,0)=\sqrt{n(x_1,x_2,0)}.\] Thus [eq:weighted-ground-orbital] implies \(\int_{z=0}e^{2ap}n(p,0)<\infty\). On the other hand, [eq:plane-tail] gives a fixed \(c_*>0\) and \(p_*<\infty\) such that \[\begin{align*} \int_{z=0}e^{2ap}n(p,0)\,dx_1dx_2 &\ge2\pi c_*^2\int_{p_*}^\infty e^{2(a-k)p}p^{2\beta-7}\,dp\\ &=\infty, \end{align*}\] because \(a>k\). This contradicts the trace bound and proves that [eq:admissible-formal-potential] is impossible.

Finally, the weighted gap and the analytic properties above give the ensemble kinetic certificate of 18. If an admissible potential represented \(n\) by a ground-state ensemble, 19 would identify it with \(V\) up to a constant. The sum space is invariant under adding constants, so this would imply [eq:admissible-formal-potential], already contradicted. ◻

Proof. First choose \(D\) large enough for [prop:two-centre-spectrum,prop:density-properties,prop:exterior-gap]. For this fixed \(D\), take \(\lambda>0\) sufficiently small that [prop:small-coupling,prop:angle-convergence,prop:weighted-gap] apply as well. The resulting three-electron state is an absolute bound ground state, and its density and angle satisfy every hypothesis of 23. The planar coefficient is nonzero because the coupling is positive. All these properties hold throughout some interval \(0<\lambda<\lambda_0(D)\). Choose the integer \(Z_0(D)\) so that \(1/Z_0(D)<\lambda_0(D)\), set \(\lambda=1/Z\), and apply the scaling equivalence of 2. ◻

Proof. First, all sufficiently small \(t\ge0\) are regular. The ground-level and symmetry conditions follow from 4. At zero coupling, \[2\epsilon_b-E_3(0)=-\epsilon_e=\frac{k_0^2}{2}>0.49, \qquad 0.99<k_0<\gamma,\] where 3 and the fixed value of \(D\) give the strict numerical inequalities. They persist for small \(t\). Positivity of the density and the direct-amplitude bound are supplied by the uniform channel construction in [prop:density-properties,lem:channel-expansion].

We record why the fixed numbers in [eq:fixed-parameter-numbers] are sufficient for the uniform exterior tests. Let \(\delta_0=\gamma-k_0>0\). The estimate [eq:quantitative-density-drift] supplies one remainder \(\rho_D(R)\downarrow0\), independent of all sufficiently small \(t\), for the first three unnormalized expressions in [eq:parameter-H]. By [eq:two-centre-numerics], \[ \delta_0<\frac{200}{D},\qquad 100(\delta_0+\delta_0^2)<10^{-195}\ll\eta_0 . \tag{120}\] For the last line of [eq:parameter-H], the uniform eventual vertical ratio is at most \(20\), by [eq:strip-data]. Consequently, for all sufficiently large fixed integers \(j\), \[\begin{split} &\max\left\{H_0(j),\,\limsup_{t\downarrow0}H_t(j)\right\}\\ &\qquad\le \max\left\{ \frac{100(\delta_0+\delta_0^2)+\rho_D(j)}{\eta_0}, \frac15\right\}<1. \end{split}\] The common remainder is obtained with \(D\) fixed, and the finite radius \(j\) is then chosen before taking the small-coupling limsup. This uses uniform exterior estimates throughout.

We next verify the last condition in [eq:parameter-radius]. The scalar exterior expansions for \(b,e\), including the simple odd factor of \(e\), imply that, uniformly for \(0<z\le L\), \[\partial_p\log(b/e)=-\delta_0+o(1),\qquad \partial_p^2\log(b/e)=o(1)\qquad(p\to\infty).\] The apparent singularity \(-\log z\) has no \(p\)-derivative. For \(q>0\), both \[\frac{q}{(1+q^2)\arctan q}, \qquad \frac{q(1-q^2)}{(1+q^2)^2\arctan q}\] have absolute value at most \(1\); this follows from \(q/(1+q^2)\le\arctan q\). Applying these bounds to \(\delta_*=\arctan(\sqrt2 b/e)\) therefore gives \[\frac{|(\delta_*)_p|}{\delta_*}\le\delta_0+o(1),\qquad \frac{|(\delta_*)_{pp}|}{\delta_*}\le\delta_0^2+o(1).\] The bounds extend to \(z=0\) by continuity. The cylindrical drift coefficient satisfies \(\lvert p^{-1}+\partial_p\log n_0\rvert\le3\) for sufficiently large \(p\). Since \(L_{n_0}\delta_*=q_*\delta_*\) with \(0\le q_*\le2d_*\), it follows that \[\frac{\partial_z(n_0\partial_z\delta_*)}{n_0\delta_*} \le 2d_*+3\delta_0+\delta_0^2+o(1)<\mu^2\] uniformly in the strip, once \(p\) is sufficiently large. The last strict inequality uses \(d_*<100/D\) and [eq:parameter-drift-slack]. Thus the defining set for \(M\) is nonempty.

On the segment \(p=M\), let \[f(z)=\frac2\pi\delta_*(M,z),\qquad v(z)=f(z)-U_0(M,z).\] The defining strict inequality for \(M\) gives \[\frac1{n_0(M,z)} \partial_z\bigl(n_0(M,z)\partial_zv\bigr)-\mu^2v<0, \qquad v(0)=0,\quad v(L)>0.\] One-dimensional comparison and its boundary point conclusion give \(v(z)>0\) for \(0<z\le L\) and \(v'(0)>0\). The quotient \(v(z)/z\) consequently extends to a continuous strictly positive function on the compact segment \([0,L]\). Its minimum is positive, proving \(\alpha>0\). ◻

Proof. By 26, small couplings are regular and the fixed-radius condition \(H_t(M)<1\) holds throughout some interval \(0<t<\tau_1\). We must check the tests for all minimizers, not merely for a selected branch.

The convergence in 12 holds for every sequence of folded minimizers as \(t\downarrow0\): their angles converge locally uniformly to \(\theta_*\), their multipliers to \(d_*\), and their derivatives converge uniformly on fixed exterior compact sets. The same statements are therefore uniform over the collection of folded minimizers for sufficiently small \(t\); otherwise a sequence violating uniform convergence would contradict that proposition. Since \(0<d_*<100/D<0.001\), the multiplier test passes for all minimizers.

The compact set \(\{|x|\le2S\}\) is now fixed. Its reference weight \(2b^2\) is bounded below by a positive constant. Density convergence and angle convergence give \[\frac{n_t\cos^2\theta}{2b^2}\longrightarrow1\] uniformly there, for all folded minimizers. This proves the second test. On the fixed cylinder \(p=M\), density convergence also gives \(U_t(M,\cdot)\to U_0(M,\cdot)\) in \(C^1([0,L])\). For example, this follows directly from the positive integral series in 14: \(n_t\) converges uniformly to a positive function on this compact segment, and the series and its final-flux formula converge uniformly. Together with the \(C^1\) angle convergence and \(\alpha>0\), this proves the last test.

Consequently \(\mathcal B\cap(0,\tau)=\varnothing\) for some \(\tau>0\). If \[C_{\mathcal B} =\sup_{0<t\le1}\frac1t\,\mathbf1_{\mathcal B}(t),\] then \(0\le C_{\mathcal B}\le1/\tau\); in the empty-bad-set case the supremum is zero. Thus \(Z=2+\lceil C_{\mathcal B}\rceil\) is finite and \(Z>C_{\mathcal B}\). If \(t_*=1/Z\) belonged to \(\mathcal B\), its contribution to the supremum would be \(Z\), a contradiction. Hence \(t_*% \BeginAccSupp{pdfliteral=direct,method=hex,unicode,ActualText=2209}% \KSOriginalNotin\EndAccSupp{pdfliteral=direct}\mathcal B\), as claimed. ◻

Proof of 1. Work first with the scaled system at \(t_*=1/Z>0\). By 28, its regularity gives the specified normalizable ground state in \(S_z=1/2\), and its energy is an absolute ground energy.

It is useful to check explicitly that the fixed-parameter channel theorem applies here. We know that the selected coupling passes the tests; we do not need to know that it belongs to the particular perturbation interval used earlier. Its ionic isolation supplies a positive \(Q\)-gap. The inequality \(2\epsilon_b-E_3(t_*)>0.49\) permits a full-configuration exponential weight with exponent \(a=0.98\), since \(a^2/2=0.4802<0.49\). On the cutoff region \(|y|\ge r/3\), the full radius is at least \(\sqrt{10/9}\,r\); hence, with room for the derivative estimates, an exterior error exponent \(\sigma=1.02>\gamma\) is available. Here \(\gamma<1.01\) follows from [eq:two-centre-numerics]. Moreover \(k_{t_*}<\gamma\) makes the retained unperturbed fast term exponentially smaller than \(A_{t_*}\) times any fixed inverse power. Finally, [eq:regular-amplitude-test] supplies the nonzero equatorial slope in the scalar channel coefficient. The complementary plane coefficient is nonzero because \[Q(y_{1z}+y_{2z})\phi=(y_{1z}+y_{2z})\phi% \BeginAccSupp{pdfliteral=direct,method=hex,unicode,ActualText=2260}% \KSOriginalNe\EndAccSupp{pdfliteral=direct}0,\] by even ionic parity and the fact that a nonzero \(L^2\) vector cannot be supported on the hyperplane \(y_{1z}+y_{2z}=0\). The injective reduced inverse preserves nonzeroness. These are exactly the fixed-parameter hypotheses of 10. That proposition gives the plane lower tail and both density translation limits at \(t_*\).

Choose any folded minimizing angle \(\theta\) for \(n_{t_*}\). All tests in [eq:parameter-angle-tests] hold for it, and \(H_{t_*}(M)<1\) supplies the drift and strip bounds with \(\eta_0=10^{-60}\) and vertical ratio at most \(100\). Also \(0.99<k_{t_*}<\gamma<1.01\) and \(M>10D\). The hypotheses of 17 are therefore satisfied. To indicate the numerical margins, its strip derivative constant is below \(10^{30}\), so its horizontal error is at most \(4\cdot10^{-30}U\), whereas \(\mu^2-2d>0.008\). At the inner cylinder, the reference gap is at least \(\alpha z\); the derivative test reduces it by less than \(\alpha z/2\). The exterior gap is \(0.03\), while the localization error \(10/S^2\) leaves a gap far larger than \(d<0.001\). Thus the full weighted gap holds for the constructed orbitals \(g,h\).

The fixed-density obstruction in 23, using the ensemble certificate and potential identification from 5, rules out every representing potential in the stated class for this scaled density. Finally, 2 transfers the obstruction to the physical system with charges \(Z,Z\). The number \(Z\) is finite, so the nuclear positions \((0,0,\pm D/Z)\) are finite and distinct. This establishes 1. ◻