Proof. For smooth constrained data, the electric and magnetic charges are exactly the fluxes in 14 . Proposition 3 shows that these fluxes are continuous in the non-degenerate energy norm, and hence the maps \(U\mapsto q_E(U)\) and \(U\mapsto q_B(U)\) extend to the finite-energy completion. Lemma 3 constructs smooth stationary Kerr-Newman Maxwell fields \(F_e\) and \(F_m\) with normalized charges \((1,0)\) and \((0,1)\); their Cauchy data are \(U_e,U_m\).

Now, take \(U\in\mathcal{H}_{\mathrm{Max}}^{(k)}(\Sigma_0)\) and set \(U_0=U-q_E(U)U_e-q_B(U)U_m\). The normalization of \(U_e,U_m\), together with the linearity of the charges, gives \(q_E(U_0)=q_B(U_0)=0\). Thus \(U_0\in\mathcal{H}_{\mathrm{Max},0}^{(k)}\), which proves the decomposition. If \(c_eU_e+c_mU_m\) also belongs to the charge-free space, the two charge identities force \(c_e=c_m=0\); the two parts therefore meet only at the origin. Proposition 6 gives boundedness of both projections, so the direct sum is topological. Evolving the three pieces by the source-free Maxwell equation gives ?? . Finally, Corollary 2 shows that every nonzero element of the stationary charge sector keeps positive local energy on some compact radial set for all time. It cannot converge locally to zero. ◻

Proof. Let \[G=F_{\mathrm{rad}},\qquad u=\mathfrak M G.\] By Theorem 1, \(G\) is already charge-free and is uniquely determined by \(F\). Definition 9(A1) gives the initial comparison \[\label{eq:intro95transfer95chain951} \mathcal{E}_M^{(k)}[u](0)\le C_R\mathcal{E}_{\mathrm{Max}}^{(k)}[G](0).\tag{6}\] Proposition 31, applied to the same compatible solution \(u\), gives the master estimate of Definition 10(M1): \[\label{eq:intro95transfer95chain952} \lVert u\rVert_{\mathcal{X}^{(k)}_M(0,\tau)}^2 \le C_M\mathcal{E}_M^{(k)}[u](0).\tag{7}\] Since \(G=\mathfrak R\mathfrak M G=\mathfrak R u\) in the charge-free class, Definition 9(A4) and Definition 10(M2) yield \[\label{eq:intro95transfer95chain953} \lVert G\rVert_{\mathcal{X}^{(k)}_{\mathrm{Max}}(0,\tau)}^2 =\lVert\mathfrak R u\rVert_{\mathcal{X}^{(k)}_{\mathrm{Max}}(0,\tau)}^2 \le C_R\lVert u\rVert_{\mathcal{X}^{(k)}_M(0,\tau)}^2.\tag{8}\] Combining 6 8 proves 4 . The fixed equivalence constants in the norm definitions are absorbed into \(C\); with this normalization one may take \(C=C_R^2C_M\). Reversing the foliation gives the past estimate. For data in the finite-energy completion, choose smooth charge-free approximants as in Lemma 4. The preceding estimate is uniform along that approximating sequence, and continuity of the solution map (Proposition 7), together with lower semicontinuity of the spacetime norm, passes the bound to the limit as in Proposition 48.

For radiation fields, define on smooth charge-free data \[\label{eq:intro95maxwell95trace95factorization} \mathscr S_{\mathrm{Max}}^{\pm}G =\mathcal{R}_\infty^{\pm}\,\mathscr S_M^{\pm}(\mathfrak M G).\tag{9}\] The three factors in 9 are bounded by Definition 9(A1), (A5), and the same-order reconstruction bounds. The trace therefore extends to \(\mathcal{H}_{\mathrm{Max},0}^{(k)}\). If the right-hand side vanishes, the master radiation field vanishes; the kernel statement in (A5) then gives \(u=0\), and hence \(G=\mathfrak R u=0\). Thus the trace is injective. On the dense class of smooth radiation data, the inverse has the explicit form \[\label{eq:intro95maxwell95wave95operator} \mathscr W_{\mathrm{Max}}^{\pm}\rho =\mathfrak R\,\mathscr W_{M,0}^{\pm}\big((\mathcal{R}_\infty^{\pm})^{-1}\rho\big),\tag{10}\] which is bounded by (A4)-(A5) and satisfies \(\mathscr S_{\mathrm{Max}}^{\pm}\mathscr W_{\mathrm{Max}}^{\pm}\rho=\rho\). By density, 10 extends to the radiation Hilbert space. Therefore \(\mathscr S_{\mathrm{Max}}^{\pm}\) are bounded isomorphisms, and \(\mathscr S_{\mathrm{Max}}=\mathscr S_{\mathrm{Max}}^+(\mathscr S_{\mathrm{Max}}^-)^{-1}\) is bounded. The two stationary charge coordinates do not belong to this radiative scattering map; Theorem 1 has already split them off. If (A6) is available, Proposition 53 applies the finite commuted hierarchy and Sobolev embedding on the regular slices, giving ?? . ◻

Proof. Once the fixed-background spin-one reduction and the estimates listed in conditions (A1)-(A5) of Definition 9 have been proved for the chosen Kerr-Newman parameter range, the conditions of Theorem 2 are exactly met. The theorem then gives boundedness, integrated decay, radiation fields, wave operators, and scattering for the stationary-subtracted Maxwell field. If the finite-order hierarchy (A6) is also available, the pointwise conclusion follows from the final part of the theorem. For \(Q=0\), the required analytic estimates are supplied by the Kerr Maxwell and Teukolsky theory cited in Corollaries 6 and 13, and the same argument recovers the slowly rotating Kerr subcase. ◻

Proof. Work in an orthonormal frame \((f_0,f_1,f_2,f_3)\) with \(f_0=n_{\Sigma_\tau}\). Writing \(E_i=G(f_0,f_i)\) and \(B_i=\tfrac12\epsilon_{ijk}G(f_j,f_k)\), a direct computation from ?? gives \(\mathbf{T}_{\mu\nu}[G]f_0^\mu f_0^\nu=\tfrac12(\lvert E\rvert^2+\lvert B\rvert^2)\) and \(\mathbf{T}_{\mu\nu}[G]f_0^\mu f_i^\nu=(E\times B)_i\), so \(\mathbf{T}[G](f_0,\cdot)\) is a causal future-directed covector and the dominant energy condition holds. Since \(N\) is uniformly future timelike on compact radial sets, \(N=A f_0+\sum_i A^i f_i\) with \(A\ge c>0\) and \(A^2-\sum_i(A^i)^2\ge c^2\). Using \(|E\times B|\le\tfrac12(\lvert E\rvert^2+\lvert B\rvert^2)\), \[\mathbf{T}_{\mu\nu}[G]N^\mu f_0^\nu =A\,\tfrac12(\lvert E\rvert^2+\lvert B\rvert^2)+\textstyle\sum_i A^i(E\times B)_i \ge \tfrac{c^2}{A+|A'|}\cdot\tfrac12(\lvert E\rvert^2+\lvert B\rvert^2)\ge c'(\lvert E\rvert^2+\lvert B\rvert^2),\] where \(|A'|=(\sum_i (A^i)^2)^{1/2}\). The upper bound is immediate from boundedness of the frame coefficients. Near \(\mathcal{H}^+\) the red-shift construction of [4] keeps \(N\) uniformly timelike in regular coordinates, so the bound persists there. All constants are uniform on compact parameter sets because \(g_\mathrm{KN}\), the foliation and \(N\) depend smoothly on \((M,a,Q)\). ◻

Proof. The source-free equations 2 give \(\nabla^\mu\mathbf{T}_{\mu\nu}=G_{\nu}{}^{\alpha}\nabla^\mu G_{\mu\alpha} +G^{\mu\alpha}\nabla_{[\mu}G_{\nu\alpha]}=0\), since \(\nabla^\mu G_{\mu\alpha}=0\) is \(\mathrm d\star_{g}G=0\) and \(\nabla_{[\mu}G_{\nu\alpha]}=0\) is \(\mathrm dG=0\). Hence \(\nabla^\mu J_\mu^X=\mathbf{T}_{\mu\nu}\nabla^\mu X^\nu =\tfrac12\mathbf{T}^{\mu\nu}\pi^X_{\mu\nu}\) by symmetry of \(\mathbf{T}\). ◻

Proof. Lemma 2 gives \(\nabla^\mu J^X_\mu[G]=\tfrac12\mathbf{T}^{\mu\nu}[G]\pi^X_{\mu\nu}\). Integrating this identity over the oriented slab \(\mathcal{D}_R(\tau_1,\tau_2)\) and applying the divergence theorem gives the sum of the outward boundary fluxes. The flux on \(\Sigma_{\tau_2}\) appears with the future normal, while the flux on \(\Sigma_{\tau_1}\) has the opposite orientation and is moved to the right-hand side. What remains are the radial cutoff boundary, possible horizon pieces, and possible null-infinity approximants.

For a null boundary, approximate it first by spacelike or timelike hypersurfaces that converge to the null hypersurface in regular coordinates. For smooth \(G\), the Maxwell stress tensor is smooth, and \(X\) is smooth up to the horizon and in the asymptotic chart used for the truncation. The induced flux densities converge in \(L^1\) on compact portions of the null boundary. The limit is the usual null flux, which gives ?? . ◻

Proof. Let \(\Omega\) be an oriented smooth three-chain with boundary \(\partial\Omega=S_2-S_1\). Stokes’ theorem and the Maxwell equations give \[\int_{S_2}F-\int_{S_1}F=\int_\Omega \mathrm dF=0, \qquad \int_{S_2}\star_gF-\int_{S_1}\star_gF=\int_\Omega \mathrm d\star_gF=0.\] The two fluxes therefore depend only on the homology class of the sphere. Large coordinate spheres on two slices are homologous through the domain of outer communications after adding the timelike cylinder at large radius, and the finite-energy decay in the asymptotic end makes the limit in 14 independent of the chosen large radius sequence. The charges are therefore independent of \(\tau\). ◻

Proof. Treat \(q_E\); \(q_B\) is identical. The constraint \(\operatorname{div}E=0\) and Stokes’ theorem give that \(4\pi q_E(U)=\int_{S_{\tau,r}}E(\nu_{\tau,r})\, \mathrm d\mu_{S_{\tau,r}}\) is independent of \(r\ge R_0\), where \(\nu_{\tau,r}\) is the outward unit normal of \(S_{\tau,r}\). By Cauchy-Schwarz and \(|S_{\tau,r}|\le Cr^2\), \[16\pi^2|q_E(U)|^2\le |S_{\tau,r}|\int_{S_{\tau,r}}|E(\nu_{\tau,r})|^2\,\mathrm d\mu_{S_{\tau,r}} \le C r^2\int_{S_{\tau,r}}|E|^2\,\mathrm d\mu_{S_{\tau,r}},\] so \(\int_{S_{\tau,r}}|E|^2\ge C^{-1}r^{-2}|q_E(U)|^2\). Integrating in \(r\) over \([R_0,2R_0]\), using \(r^{-2}\ge(2R_0)^{-2}\) and \(\mathrm d\mu_{\Sigma_\tau}\simeq\mathrm d r\,\mathrm d\mu_{S_{\tau,r}}\), gives \(\int_{\Sigma_\tau\cap\{R_0\le r\le2R_0\}}|E|^2\ge C^{-1}R_0^{-1}|q_E(U)|^2\). The same bound for \(B\) gives ?? . The right-hand side is dominated by the order-zero energy, so \(q_E,q_B\) are bounded linear functionals and extend uniquely. ◻

Proof. Put \(f=r/\Sigma\). Then \[\label{eq:explicit95coulomb95field} \begin{align} \widehat F_e &=\mathrm d\bigl(-f\,\mathrm dt+a f\sin^2\theta\,\mathrm d\phi\bigr) \\ &=-\partial_r f\,\mathrm dr\wedge\mathrm dt-\partial_\theta f\,\mathrm d\theta\wedge\mathrm dt +a\sin^2\theta\,\partial_r f\,\mathrm dr\wedge\mathrm d\phi \\ &\quad +a\bigl(\sin^2\theta\,\partial_\theta f +2f\sin\theta\cos\theta\bigr)\mathrm d\theta\wedge\mathrm d\phi. \end{align}\tag{16}\] Thus \(\mathrm dF_e=0\) identically. If \(Q\ne0\), the Kerr-Newman background Maxwell field is \(F^{\mathrm{bg}}=Q\widehat F_e\) with the same orientation convention; it satisfies the source-free Maxwell equation on the fixed background, hence \(\mathrm d\star_g\widehat F_e=0\). Both the coefficients in 16 and the operator \(F\mapsto\mathrm d\star_gF\) depend smoothly on \(Q\), so the identity extends to \(Q=0\) by taking the limit. In four space-time dimensions, \(\star_g\) maps closed and co-closed two-forms to closed and co-closed two-forms; therefore \(F_m=\star_gF_e\) also solves 2 .

The same formula also fixes the normalization and the asymptotics. Since \[\label{eq:fe95asymptotics} f=r^{-1}+O(r^{-3}),\qquad \partial_r f=-r^{-2}+O(r^{-4}),\qquad \partial_\theta f=O(r^{-3}),\tag{17}\] we have, on large coordinate spheres, \[\label{eq:coulomb95flux95asymptotics} \widehat F_e=r^{-2}\,\mathrm dr\wedge\mathrm dt +2a r^{-1}\sin\theta\cos\theta\,\mathrm d\theta\wedge\mathrm d\phi +O(r^{-3})_{\mathrm{orth}},\tag{18}\] up to the global sign fixed before ?? . The magnetic flux of the angular term vanishes because \(\int_0^\pi\sin\theta\cos\theta\,\mathrm d\theta=0\), while the electric flux is \[\label{eq:electric95flux95normalization} \frac{1}{4\pi}\int_{S_{\tau,r}}\star_g\widehat F_e =\pm\frac{1}{4\pi}\int_0^{2\pi}\!\int_0^\pi \sin\theta\,\mathrm d\theta\mathrm d\phi+O(r^{-1})=\pm1+O(r^{-1}).\tag{19}\] We choose the sign for which the limit is \(+1\). Thus \(q_E[F_e]=1\) and \(q_B[F_e]=0\). Since \(F_m=\star_gF_e\), and \(\star_g^2=-1\) on two-forms in the Lorentzian exterior with our convention, \(q_B[F_m]=q_E[F_e]=1\) and \(q_E[F_m]=-q_B[F_e]=0\) after the same orientation choice.

The leading term in 18 is the Coulomb part in an asymptotically orthonormal frame. The purely angular coefficient in 16 is \(O(r^{-1})\) as a coordinate coefficient, but it becomes \(O(r^{-3})\) in the orthonormal frame because \(\mathrm d\theta\wedge\mathrm d\phi\) carries the factor \(r^{-2}\sin^{-1}\theta\) after orthonormalization. The other mixed angular terms are \(O(r^{-3})\) or better. Hence \(|F_e|=O(r^{-2})\). Regular angular derivatives preserve this order, \(r\partial_r\) derivatives preserve the symbolic order, and covariant radial derivatives gain a factor of \(r^{-1}\). Hodge duality is a uniformly bounded zeroth-order operation in the asymptotically flat frame, so the corresponding bounds hold for \(F_m\). This establishes ?? . In the final step, \[\int_R^\infty r^{-4-2j}r^2\,\mathrm dr<\infty\qquad (j\ge0),\] so the Cauchy data of \(F_e\) and \(F_m\) have finite non-degenerate energy after any finite number of commutations in \(\mathbb{D}_k\). ◻

Proof. The normalized representative \(F_{\mathrm{stat}}^{\mathrm{KN}}(q_E,q_B)=q_EF_e+q_BF_m\) is source-free by Lemma 3. Since 2 is linear, \(F_{\mathrm{rad}}=F-F_{\mathrm{stat}}^{\mathrm{KN}}(q_E[F],q_B[F])\) is source-free. The charge functionals are linear and the representatives have the normalization ?? ; hence \[q_E(F_{\mathrm{rad}})=q_E(F)-q_E(F)q_E(F_e)-q_B(F)q_E(F_m)=q_E(F)-q_E(F)=0,\] and the same computation with \(q_B\) gives \(q_B(F_{\mathrm{rad}})=0\). ◻

Proof. We show that \(F_{\mathrm{stat}}^{\mathrm{KN}}(q_E,q_B)=q_EF_e+q_BF_m\) is pointwise nonzero on the exterior; the corollary then follows from stationarity. First, 16 gives \[F_e(\partial_r,\partial_t)=-\partial_rf =\frac{r^2-a^2\cos^2\theta}{\Sigma^2},\qquad f=\frac{r}{\Sigma}.\] In the exterior \(r>r_+>M\), and \(M^2>a^2+Q^2\) forces \(M>|a|\); hence \(r>|a|\ge|a\cos\theta|\) and \(F_e\ne0\) at every exterior point. Second, on real two-forms in Lorentzian signature \(\star_g^2=-\mathrm{Id}\), so \(\star_g\) has no real eigen-two-forms: if \(c_eF_e+c_m\star_gF_e=0\) at a point with \((c_e,c_m)\ne(0,0)\), applying \(\star_g\) gives \(c_e\star_gF_e-c_mF_e=0\), and eliminating \(\star_gF_e\) between the two relations gives \((c_e^2+c_m^2)F_e=0\), contradicting \(F_e\ne0\). Hence \(q_EF_e+q_BF_m=q_EF_e+q_B\star_gF_e\) is nonzero at every exterior point whenever \((q_E,q_B)\ne(0,0)\). Its non-degenerate energy density is therefore strictly positive on every compact radial set \(K\) by Lemma 1, and stationarity makes the integral of that density over \(\Sigma_\tau\cap K\) a positive constant independent of \(\tau\). That rules out local decay to zero in any non-degenerate local \(L^2\) norm. ◻

Proof. For smooth \(U\), Proposition 4 gives \(U_0=U-q_E(U)U_e-q_B(U)U_m\) with zero electric and magnetic charges. Hence \(\mathcal{J}\) maps \(\mathcal{C}_{\mathrm{sm}}^{(k)}\) into \(\mathbb{R}^2\oplus\mathcal{D}_0^{(k)}\) whenever \(U_0\) is smooth, and into the closure \(\mathbb{R}^2\oplus\mathcal{H}_{\mathrm{Max},0}^{(k)}\) in general. Since \(U_e\) and \(U_m\) have finite order-\(k\) energy, \[\label{eq:J95upper95bound} \|U_0\|_{\mathcal{H}_{\mathrm{Max},0}^{(k)}} \le \|U\|_{\mathrm{ch},k} +|q_E(U)|\,\|U_e\|_{\mathcal{H}_{\mathrm{Max}}^{(k)}} +|q_B(U)|\,\|U_m\|_{\mathcal{H}_{\mathrm{Max}}^{(k)}} \le C\|U\|_{\mathrm{ch},k}.\tag{24}\] Thus \(\mathcal{J}\) is bounded. Conversely, for smooth charge-free \(U_0\), \(U=q_EU_e+q_BU_m+U_0\) is smooth constrained data with charges \((q_E,q_B)\), and \[\label{eq:J95inverse95bound} \|U\|_{\mathrm{ch},k} \le C\big(|q_E|+|q_B|+\|U_0\|_{\mathcal{H}_{\mathrm{Max},0}^{(k)}}\big).\tag{25}\] We obtain boundedness of the inverse on the dense subspace \(\mathbb{R}^2\oplus\mathcal{D}_0^{(k)}\). The two estimates extend both maps to the completions. For smooth data, the normalization ?? gives \[\mathcal{J}(q_EU_e+q_BU_m+U_0)=(q_E,q_B,U_0),\qquad (\text{the map }\eqref{eq:charge95adapted95inverse})\circ\mathcal{J}(U)=U.\] By continuity the same two identities hold on the completed spaces. ◻

Proof. For \(U\in\mathcal{H}_{\mathrm{Max}}^{(k)}\) write, according to the definition of the full energy space, \(U=c_eU_e+c_mU_m+U_0\) with \(U_0\in\mathcal{H}_{\mathrm{Max},0}^{(k)}\). The normalization of the stationary representatives gives \(q_E(U)=c_e\) and \(q_B(U)=c_m\), because the charge-free component has both charges equal to zero. Hence \(\Pi_{\mathrm{stat}}U=c_eU_e+c_mU_m\) and \(\Pi_0U=U_0\). The norm 23 immediately gives \[\|\Pi_{\mathrm{stat}}U\|_{\mathcal{H}_{\mathrm{Max}}^{(k)}}+ \|\Pi_0U\|_{\mathcal{H}_{\mathrm{Max}}^{(k)}} \le C\|U\|_{\mathcal{H}_{\mathrm{Max}}^{(k)}}.\] If a vector belongs to both \(\operatorname{span}\{U_e,U_m\}\) and \(\mathcal{H}_{\mathrm{Max},0}^{(k)}\), applying \(q_E\) and \(q_B\) gives both coefficients equal to zero. The sum is therefore direct and, because the projections are bounded, it is a topological direct sum. ◻

Proof. Let \(U\in\mathcal{H}_{\mathrm{Max},0}^{(k)}\) and choose \(U_n\in D\) with \(U_n\to U\) in the full energy norm. Since the charge functionals are continuous and \(U\) is charge-free, \(q_E(U_n)\to0\) and \(q_B(U_n)\to0\). Define \[V_n=U_n-q_E(U_n)U_e-q_B(U_n)U_m.\] Because \(D\) contains \(U_e,U_m\) and is a linear space of smooth constrained data, \(V_n\in D\). The normalization of \(U_e,U_m\) gives \(q_E(V_n)=q_B(V_n)=0\), so \(V_n\in D\cap\mathcal{H}_{\mathrm{Max},0}^{(k)}\). Finally, \(\|V_n-U\|\le\|U_n-U\|+|q_E(U_n)|\|U_e\|+|q_B(U_n)|\|U_m\|\to0\). ◻

Proof. In horizon-regular coordinates write \(g=-N_L^2\mathrm d\tau^2+h_{ij}(\mathrm dx^i+\beta^i\mathrm d\tau)(\mathrm dx^j+\beta^j\mathrm d\tau)\) with lapse \(N_L\), shift \(\beta\) and induced metric \(h\) smooth up to \(\mathcal{H}^+\) on \([0,T]\) and stationary in \(\tau\) after pullback by the Killing flow. With \(U=(E,B)\), Maxwell’s equations are \[\label{eq:maxwell95symhyp} \partial_\tau E=\mathcal{L}_\beta E+\operatorname{curl}_h(N_LB),\qquad \partial_\tau B=\mathcal{L}_\beta B-\operatorname{curl}_h(N_LE),\tag{26}\] with constraints 20 . The principal symbol is skew in \((E,B)\) and symmetric after pairing with \(h(E,E')+h(B,B')\), so 26 is symmetric hyperbolic with smooth coefficients on each finite slab; smooth constrained data yield smooth solutions. The \(J^N\)-identity gives \[\label{eq:N95energy95gronwall} E_N[F](\tau)+\mathcal{F}_{\mathcal{H}^+}(0,\tau)+\mathcal{F}_{\infty}(0,\tau) =E_N[F](0)+\int_{\mathcal{D}(0,\tau)}\mathbf{T}_{\alpha\beta}[F]\nabla^\alpha N^\beta\,\mathrm d\mu_g,\tag{27}\] with nonnegative horizon and null-infinity fluxes by the dominant energy condition. Since \(\nabla N\) is bounded and the \(N\)-energy density is equivalent to \(|E|^2+|B|^2\) by Lemma 1, Gronwall gives the order-zero bound; commuting with \(\mathbb{D}_k\) gives a symmetric-hyperbolic system with bounded lower-order sources, and induction gives ?? . Taking \(\operatorname{div}_h\) of 26 and using \(\operatorname{div}_h\operatorname{curl}_h=0\) shows \((\operatorname{div}_hE,\operatorname{div}_hB)\) solves a homogeneous linear system, so zero constraints persist. For finite-energy data, decompose \(U_0=\Pi_{\mathrm{stat}}U_0+\Pi_0U_0\); the stationary part evolves as \(F_{\mathrm{stat}}^{\mathrm{KN}}\) and the charge-free part is approximated by Lemma 4, the estimate on differences making the approximants Cauchy. The limit is independent of the sequence, satisfies ?? , and inherits ?? by weak lower semicontinuity. Time reversal gives the past statement. ◻

Proof. Let \((U_n)\) and \((V_n)\) be two smooth approximating sequences for the same datum \(U\). The corresponding smooth solutions satisfy the energy estimate for their difference, whose initial data are \(U_n-V_n\to0\) in \(\mathcal{H}_{\mathrm{Max}}^{(k)}\). Thus the difference of the two solution sequences tends to zero in the finite-slab energy norm on every slab. The limit is therefore independent of the approximation. If two finite-energy solutions have the same datum, their difference is obtained as the limit of smooth solutions with vanishing initial data and hence has zero energy on each finite slab. Linearity then gives uniqueness. ◻

Proof. The change from an orthonormal frame adapted to \(\Sigma_\tau\) to \((\widehat e_3,\widehat e_4,e_A)\) is smooth and uniformly invertible on each regular patch, and the listed components are the entries of \(F\) in the second frame. By Lemma 1, \(\mathbf{T}[F](N,n_{\Sigma_\tau}) \simeq|E|^2+|B|^2\), and the frame change is bounded with bounded inverse; commuting by \(\mathbb{D}_k\) preserves this since the commutator fields are regular with bounded coefficients. ◻

Proof. Let \(T\) be the future unit normal to the coordinate sphere inside the timelike normal two-plane and let \(R\) be the outward unit spacelike normal, so that \(e_4=T+R\) and \(e_3=T-R\). Then \[\rho_F=\frac{1}{2}F(e_3,e_4)=F(T,R),\qquad \sigma_F=\frac{1}{2}F(e_1,e_2).\] Let us fix the orientation for which \(\mathrm d\mu_{S_{\tau,r}}=e^1\wedge e^2\) and the space-time volume form is \(T^\flat\wedge R^\flat\wedge\mathrm d\mu_{S_{\tau,r}}\). Evaluating on \((e_1,e_2)\) gives the exact restrictions \[\label{eq:flux95middle95relation} F|_{S_{\tau,r}}=2\sigma_F\,\mathrm d\mu_{S_{\tau,r}},\qquad (\star_gF)|_{S_{\tau,r}}=\rho_F\,\mathrm d\mu_{S_{\tau,r}}.\tag{28}\] The opposite global orientation changes both signs but not the zero-mean conclusion. Using the charge convention 14 , we therefore obtain the explicit charge-mean identities \[\label{eq:middle95charge95matrix} \int_{S_{\tau,r}}\rho_F\,\mathrm d\mu_{S_{\tau,r}}=4\pi q_E[F],\qquad \int_{S_{\tau,r}}\sigma_F\,\mathrm d\mu_{S_{\tau,r}}=2\pi q_B[F].\tag{29}\] For the stationary representatives normalized in ?? , Lemma 3 shows \[(q_E[F_e],q_B[F_e])=(1,0),\qquad (q_E[F_m],q_B[F_m])=(0,1).\] Thus \(F_{\mathrm{rad}}=F-q_E[F]F_e-q_B[F]F_m\) has both charges equal to zero, and 29 proves ?? . Charge conservation propagates the two identities to every homologous sphere on every slice of the solution. ◻

Proof. condition (A1) of Definition 9 supplies the fixed-background master variables and the compatible matrix equation. Lemma 10 and Proposition 14 give the scalar wave principal part for the closed spin-one operator specified in (A1). We therefore compare the principal wave operator and the lower-order coefficient classes with their \(a=0\) values. The inverse metric obeys, on compact radial sets, \[\label{eq:metric95symbol95difference} g_{\mathrm{KN}}^{\mu\nu}(M,a,Q)-g_{\mathrm{RN}}^{\mu\nu}(M,Q)=a\,G_1^{\mu\nu}(r,\theta)+a^2G_2^{\mu\nu}(r,\theta,a,Q),\tag{30}\] with smooth uniformly bounded coefficients; the linear term is carried by the \(\mathrm dt\,\mathrm d\phi\) cross entry, producing the stationary-axial principal perturbation \(a\,\mathcal{G}^{\mu\nu}\nabla_\mu\nabla_\nu\). The first-order Teukolsky terms at \(a=0\) are included in \(\mathcal{P}_{\mathrm{RN},Q}\). Their difference from the \(a=0\) coefficients is smooth in \(a\) and has linear part produced by the rotation of the principal frame, the \(t\phi\) coupling and the spin coefficients; after the regular horizon and infinity weights this difference contributes \(a r^{-2}\mathcal{C}^\mu\nabla_\mu\); the spheroidal-spherical harmonic discrepancy and the remaining curvature terms contribute \(a r^{-3}\mathcal{D}\). The even quadratic-in-\(a\) principal, first-order and zeroth-order remainders form the stationary second-order operator \(a^2\mathcal{Q}_{a,Q}^{(2)}\), which satisfies the same short-range perturbative bounds and is smaller than the displayed linear perturbation in the slow-rotation range. At \(a=0\) all coefficients are spherically symmetric and are collected in \(\mathcal{P}_{\mathrm{RN},Q}\); the Reissner-Nordström curvature is quadratic in \(Q\) and appears in \(\mathcal{V}_Q,\mathcal{W}_Q\). The horizon and infinity weights are smooth and nonzero in the regular exterior, so they conjugate \(\mathcal{P}_b\) by bounded factors, preserving the principal comparison and changing only the displayed lower-order coefficients. After time freezing and multiplication by \(h^2\), the linear-in-\(a\) first-order matrix part becomes an order-\(h\) subprincipal endomorphism of size \(O(|a|/M)\) on compact normalized conic patches, while the quadratic remainder contributes \(O(a^2/M^2)\). The diagonal part is precisely the spin-weighted radial skew symbol \(q_s\) displayed in ?? ; every non-diagonal or frame-mixing contribution is therefore included in \(B_{\mathrm{mat},a,Q}\) and satisfies ?? . Smoothness in \((a,Q)\) gives the uniform bounds and the short-range decay in \(r\), and 30 gives the stated orders. ◻

Proof. The coefficients of \(\mathcal{P}_b\) are stationary and axisymmetric, so the Killing fields \(T\) and \(\Phi\) commute with the principal part and only meet lower-order coefficient matrices. The regular angular fields and \(r\partial_r\) do not commute with the wave operator, but their commutators are again differential operators of order at most two with coefficients obtained by differentiating the metric, frame and potential coefficients. At \(a=0\) these terms belong to the commuted Reissner-Nordström model hierarchy. The difference between the Kerr-Newman and Reissner-Nordström coefficients is \(O(|a|/M)\) in the symbol classes of 30 ; applying finitely many fields in \(\mathbb{D}_k\) preserves the same decay and the same small factor. Induction on \(|Z|\) gives ?? , with the displayed first- and zeroth-order coefficient families absorbing the lower-order commutators. ◻

Proof. The charge trace bound of Proposition 3 extends \(q_E,q_B\) continuously to the energy completion. The stationary representatives are the smooth fields of Lemma 3; their decay places their Cauchy data in every finite-order energy space. The boundedness and direct-sum properties of \(\Pi_{\mathrm{stat}}\) and \(\Pi_0\) are exactly Proposition 6. Lemma 4 provides smooth charge-free approximants, while Propositions 7 and 8 give finite-energy existence, uniqueness, continuity of the solution map and constraint propagation. These statements are the charge and Cauchy properties recorded here. ◻

Proof. Starting with a smooth charge-free Maxwell solution \(G\), (A1) supplies the master field \(u=\mathfrak M G\), the energy comparison 33 , and the scalar-principal-symbol system. The red-shift estimate near the horizon, the \(r^p\) identity in the far region, and the trapped-set commutator produce the a priori inequality \[\|u\|_{\mathcal{X}^{(k)}_M(\tau_1,\tau_2)}^2 \le C\mathcal{E}_M^{(k)}[u](\tau_1) +C\|\mathcal{E}_{a,Q}u\|_{LE^{*,k}}^2 +C\|u\|_{LE^0_{\mathrm{comp},k}}^2.\] At commuted order \(j\le k\), the perturbation bound in (A2) and 36 imply \[X_j\le C E_j(\tau_1)+C\frac{|a|}{M}X_j+ \eta X_j+C_{j,\eta}X_{j-1}+C\|u\|_{LE^0_{\mathrm{comp},j}}^2,\] where \(X_j\) denotes the order-\(j\) master spacetime norm and \(E_j\) the corresponding initial energy. Choose \(\eta\) and then \(|a|/M\) small enough, according to Proposition 30, to absorb the two \(X_j\) terms. The induction begins at \(j=0\), where 36 has no lower-order term. The remaining compact term is removed by the Fredholm alternative encoded in (A3): a violating sequence has either bounded temporal frequency, where limiting absorption and absence of outgoing real resonances force the limit to vanish, or high frequency, where the semiclassical no-defect alternative rules out residual local mass. For that reason, (M1) follows from (A1)-(A3). Applying (A4) then gives \[\|G\|_{\mathcal{X}^{(k)}_{\mathrm{Max}}}^2 \le C_R\|u\|_{\mathcal{X}^{(k)}_M}^2 \le C\mathcal{E}_M^{(k)}[u](\tau_1) \le C\mathcal{E}_{\mathrm{Max}}^{(k)}[G](\tau_1),\] which is the charge-free Maxwell estimate. The trace estimates from the red-shift and \(r^p\) fluxes yield bounded radiation maps. With (A5), the abstract Hilbert trace criterion, Lemma 31, turns these maps into bounded isomorphisms; Proposition 51 constructs the required dense right inverse for the fixed-background test field from the resolvent. To finish, no preceding estimate gives pointwise decay without an additional finite commuted hierarchy; that hierarchy is exactly (A6), and Corollary 7 is the only place where it is used. ◻

Proof. After \(k\) is fixed, choose \(\varepsilon_Q(k)\) so that the Reissner-Nordström horizon, photon-sphere collar, red-shift collar and far-field constants remain in compact non-degenerate ranges. This fixes the spherical local-energy constant \(C_{\mathrm{RN},k}\), the red-shift and \(r^p\) constants, the angular Hodge constants, and the trapped-set constants appearing in the semiclassical normalization. At commuted order \(j\le k\) the perturbative estimate has the form \[\label{eq:finite95order95closure95ineq} X_j\le A_jE_j+B_j\frac{|a|}{M}X_j+\eta_jX_j +D_j\sum_{\ell<j}X_\ell+K_j,\tag{46}\] where \(X_j\) denotes the order-\(j\) master spacetime norm, \(E_j\) is the corresponding energy, and \(K_j\) is the compact remainder. First choose \(\eta_j>0\) so that \(\eta_jX_j\) is absorbable. Then choose \(\varepsilon_a(k)\) small enough that \(B_j|a|M^{-1}\le 1/4\) for every \(j\le k\). The induction starts at \(j=0\), where the lower-order sum is absent, and then proceeds upward. At each stage the compact term \(K_j\) is removed by the Fredholm/no-defect argument in (A3): bounded total frequencies are excluded by the limiting-absorption real-axis argument, while unbounded conic frequencies are excluded by the localized high-frequency estimate ?? after the geometric verification of Proposition 43 and the finite-rank-bundle threshold verification of Proposition 37. That gives the order-\(j\) estimate without invoking any order greater than \(j\). The charge decomposition is Theorem 1, applied before forming the master variables. The hierarchy (A6) appears only in Proposition 53; it is not present in the estimates used for boundedness, local energy decay, radiation traces or the Hilbert-space scattering construction. The constants obtained after the last induction step are collected into \(C_k\). ◻

Proof. The estimate 47 in (M1) is stated on arbitrary slabs \([\tau_1,\tau_2]\). Taking \(\tau_1=0\) and \(\tau_2=\tau\) gives \(\|u\|_{\mathcal{X}^{(k)}_M(0,\tau)}^2\le C_M\mathcal{E}_M^{(k)}[u](0)\). Applying the same estimate to the time-reversed foliation gives the corresponding bound on past slabs. ◻

Proof. Suppose instead that reconstruction were known only in the form \(\|G\|_{\mathcal{X}^{(k)}_{\mathrm{Max}}}\le C\|u\|_{\mathcal{X}^{(k+1)}_M}\). The order-\(k\) master estimate controls \(\|u\|_{\mathcal{X}^{(k)}_M}\) but gives no uniform control of the top order derivatives appearing in \(\|u\|_{\mathcal{X}^{(k+1)}_M}\). A compactness or interpolation argument cannot recover this missing derivative with constants uniform on the energy space, since high angular or radial frequencies can keep the order-\(k\) norm fixed while the order-\((k+1)\) norm diverges. Accordingly, an order-\(k\) Maxwell estimate follows from an order-\(k\) master estimate only when the reconstruction is bounded at the same order, as in 48 . ◻

Proof. If \(G\ne0\) and \(\mathfrak M G=0\), then every radiation field obtained by first passing to master variables assigns to \(G\) the same radiation data as to the zero solution. Such a radiation map cannot be injective on the charge-free energy space. The inverse identities in 42 rule out this situation for smooth solutions, and by density for the finite-energy completion, because \(G=\mathfrak R\mathfrak M G\) in the completed energy norm. ◻

Proof. Use the Bianchi identity in the form \(\nabla_\lambda F_{\mu\nu}+\nabla_\mu F_{\nu\lambda}+\nabla_\nu F_{\lambda\mu}=0\) and apply \(\nabla^\lambda\). The two differentiated divergence terms vanish after commuting derivatives because \(\nabla^\lambda F_{\lambda\mu}=0\). The commutators acting on a covariant two-form are \[[\nabla_\alpha,\nabla_\beta]F_{\mu\nu} =-R_{\alpha\beta\mu}{}^\gamma F_{\gamma\nu} -R_{\alpha\beta\nu}{}^\gamma F_{\mu\gamma}.\] Contracting the commutator terms gives exactly the Ricci and Riemann curvature endomorphism on the right of ?? . Since these terms contain no derivatives of \(F\), the top-order part is \(\nabla^\lambda\nabla_\lambda\) acting diagonally on the six components of a two-form, and its principal symbol is ?? . This computation is independent of separation and is used only to fix the top-order hyperbolic character; the closed extreme-variable system used below is the structural condition (A1). ◻

Proof. The Newman-Penrose Maxwell equations 54 show that the two extreme scalars and the middle scalar form a first-order system in which the principal directions enter through the repeated null frame and the coefficients are smooth spin coefficients. Under the stated closed-reduction assumption, the chosen regular weights conjugate the two extreme equations by nonvanishing multipliers; such a conjugation changes only first- and zeroth-order terms. The second-order part is therefore exactly the second-order part of ?? . Lemma 10 independently gives the scalar principal symbol of the full Maxwell wave equation on two-forms, so every Ricci, Weyl, spin-coefficient and weight contribution in the closed extreme system is lower order. The compatibility class is defined as the closure of the smooth range of this map in Definition 11; therefore no reconstruction theorem is used to define the master domain. The remaining components of \(F\) are recovered from the Maxwell equations by the Hodge and null-transport argument of Section 12, not by the closed extreme equations alone. ◻

Proof. Expand the radial principal term, \[\label{eq:radial95expand} \Delta^{-s}\partial_r\bigl(\Delta^{s+1}\partial_r\,\psi\bigr) =\partial_r(\Delta\,\partial_r\psi)+s\,\Delta'\,\partial_r\psi =\partial_r(\Delta\,\partial_r\psi)+2s(r-M)\,\partial_r\psi,\tag{55}\] using \(\Delta'=2(r-M)\). The term \(2s(r-M)\partial_r\) is first order and proportional to \(s\); collecting it into \(\mathcal{B}_s^{(1)}\), the second-order part of \(-\mathcal{T}_s\) is exactly \[\begin{align} &\partial_r(\Delta\,\partial_r\,\cdot\,) +\frac{1}{\sin\theta}\partial_\theta(\sin\theta\,\partial_\theta\,\cdot\,) \\ &\quad -\Bigl[\frac{(r^2+a^2)^2}{\Delta}-a^2\sin^2\theta\Bigr]\partial_t^2 -\frac{2a(2Mr-Q^2)}{\Delta}\partial_t\partial_\phi \\ &\quad -\Bigl[\frac{a^2}{\Delta}-\frac{1}{\sin^2\theta}\Bigr]\partial_\phi^2, \end{align}\] which is precisely \(\Sigma\,\Box_{g_{\mathrm{KN}}}\) by 80 . Hence the second-order part of \(\mathcal{T}_s\) is \(-\Sigma\,\Box_{g_{\mathrm{KN}}}\) and ?? follows, since the principal symbol of \(\Box_{g_{\mathrm{KN}}}\) is \(g_{\mathrm{KN}}^{\mu\nu}\xi_\mu\xi_\nu\). The remaining spin-dependent first-order terms (the usual weighted \(\partial_t,\partial_\phi\) contributions) and all spin-dependent zeroth-order angular and curvature multipliers are collected in \(\mathcal{B}_s^{(1)},\mathcal{B}_s^{(0)}\); for \(s=\pm1\) they are lower order and their curvature coefficients decay like \(r^{-3}\) in the asymptotic frame. On the exterior \(\{r>r_+\}\) one has \(\Sigma=r^2+a^2\cos^2\theta\ge r_+^2>0\), smooth, so the normalized operator \((-\Sigma)^{-1}\mathcal{T}_s\) has principal symbol \(g_{\mathrm{KN}}^{\mu\nu}\xi_\mu\xi_\nu\); the sign and the smooth nonvanishing factor are fixed once and for all in Definition 11. ◻

Proof. Part (i) is Lemma 10. Part (ii) is the definition of the regular variables and of the closed compatible class. Part (iii) follows because multiplication by the nonvanishing weights \(w_\pm\) and by \(-\Sigma\) changes no second-order symbol, while Proposition 14 computes the second-order part of the spin-weighted model as \(-\Sigma\Box_{g_{\mathrm{KN}}}\). For (iv), the \((a,Q)\)-dependence of ?? sits in the coefficients \(\Delta^{-1}(r^2+a^2)^2\), \(a^2\sin^2\theta\), \(2a(2Mr-Q^2)\Delta^{-1}\), \(a^2\Delta^{-1}\), \(\Delta^{s+1}\), and in \(\mathcal{B}_s^{(1)},\mathcal{B}_s^{(0)}\). By Lemma 32 the second-order part differs from its \(a=0\) value by \(aG_1^{t\phi}2\partial_t\partial_\phi+O(a^2)\) with \(G_1^{t\phi}=O(r^{-3})\). The assumed coefficient bounds place the rotational first-order terms in \(ar^{-2}\mathcal{C}^\mu\nabla_\mu\) and the rotational zeroth-order terms in \(ar^{-3}\mathcal{D}\), while the \(O(a^2)\) second-order terms are absorbed in \(a^2\mathcal{Q}^{(2)}_{a,Q}\). This is the class in ?? ; after applying the local-energy duality in Definition 8 it gives 35 . Part (v) is the frame equivalence Lemma 5 composed with the boundedness of the regular weights. ◻

Proof. For a scalar wave \(\psi\) the red-shift deformation satisfies, in a horizon collar, \[\label{eq:redshift95deformation} \mathbf{T}^{\mu\nu}[\psi]\,\pi^N_{\mu\nu}\ge c_0\,|\nabla\psi|^2, \qquad c_0\propto\kappa_+=\frac{r_+-r_-}{2(r_+^2+a^2)}>0,\tag{57}\] positive in the sub-extremal range; this is the computation of [4] for the scalar stress tensor. Summing over \(\psi_+,\psi_-\) gives the leading term. The rotational first-order part of ?? contributes, by Cauchy-Schwarz, \(|\sum_j\mathbf{T}[\psi_j]\cdot(\text{first-order error})|\le C|a|M^{-1}|\nabla\Psi|^2 +C_QM^{-2}|\Psi|^2\), the small factor coming from the rotational coefficients evaluated at \(r\simeq r_+(Q)\simeq M\). Choosing \(\mathcal{U}_H\) and the rotation size so that \(C|a|/M<c_0/4\) absorbs the top-order error. The remaining \(M^{-2}|\Psi|^2\) is controlled by the one-dimensional Hardy inequality along the transversal red-shift direction, \[\label{eq:horizon95hardy} \int_{\mathcal{U}_H}M^{-2}|\Psi|^2\,\mathrm d\mu \le C\int_{\mathcal{U}_H}|\nabla_{\widehat e_3}\Psi|^2\,\mathrm d\mu+C\!\!\int_{\partial\mathcal{U}_H}\!\!|\Psi|^2,\tag{58}\] the boundary term being a lower commuted energy. Summing over \(\mathbb{D}_k\) gives ?? . ◻

Proof. Apply the divergence theorem to the red-shift current on a collar bounded by \(\Sigma_{\tau_1}\), \(\Sigma_{\tau_2}\), a timelike hypersurface approaching \(\mathcal{H}^+\), and the outer boundary of the collar. Proposition 15 controls the non-degenerate bulk in the collar. Passing the timelike boundary to the horizon gives the horizon flux as a monotone nonnegative limit, by the positivity of the red-shift density in Lemma 1. The Hardy inequality in the transversal direction controls the lower-order terms, and the remaining boundary pieces are bounded by the commuted energies on the initial and final slices. The resulting horizon flux controls all tangential derivatives to the stated order and one red-shift transversal derivative. ◻

Proof. Conjugating by \(r\), the scalar far-field part of ?? reads \(\mathcal{P}_b(r\psi)=-L\underline L(r\psi)+r^{-2}\mathop{}\!\mathchoice{\Delta\mkern-12mu/}{\Delta\mkern-12mu/}{\Delta\mkern-9mu/}{\Delta\mkern-8mu/}\,(r\psi)+O(r^{-2})\partial(r\psi) +O(r^{-3})(r\psi)\), with \(O(\cdot)\) including the Reissner-Nordström spherical terms and the rotational \(a/r^2\) matrix terms. Multiply by \(r^pL(r\psi)\) and integrate by parts in \((t,r_*,\omega)\). The identity \(-2\Re(r^pL(r\psi)\overline{L\underline L(r\psi)})=\partial_{r_*}(r^p|L(r\psi)|^2) -pr^{p-1}|L(r\psi)|^2+\partial_t(\cdots)\) gives the radial term \(p\int r^{p-1}|L(r\psi)|^2\); the angular term integrates on \(\mathbb{S}^2\) to \((2-p)\int r^{p-1}|\mathop{}\!\nabla\mkern-13mu/\,(r\psi)|^2\) plus a total derivative. Short-range terms are bounded by Cauchy-Schwarz, \[\label{eq:rp95error95absorb} \Big|\!\int r^pL(r\Psi)\cdot O(r^{-1})\nabla(r\Psi)\Big| \le \delta\!\int r^{p-1}|L(r\Psi)|^2+C_\delta\!\int r^{-1-p_0}|\nabla(r\Psi)|^2,\tag{59}\] the last integral being part of the local-energy norm for \(R\) large; the rotational errors carry the extra factor \(|a|/M\) and are absorbed after fixing the parameters, while the spherical charge terms are controlled by the Reissner-Nordström model. Summing over \(\psi_\pm\) and \(\mathbb{D}_k\) gives ?? . ◻

Proof. The \(p=2\) estimate controls \(\int r|L(r\Psi)|^2\) with the dyadic weight of the Friedlander argument. For fixed \(u=t-r_*\), \(r_2\Psi(u+r_{*,2},r_2,\omega)-r_1\Psi(u+r_{*,1},r_1,\omega)=\int_{r_1}^{r_2}L(r\Psi)\,\mathrm ds_*\), and Cauchy-Schwarz with the \(p=2\) weight bounds the \(L^2_{u,\omega}\) norm by \((\sum_{n\ge n_0}2^{-n})^{1/2}\) times the hierarchy norm, \(\to0\) as \(r_1,r_2\to\infty\). Thus \(r\Psi\) is Cauchy in \(L^2_{u,\omega}\) and the limit exists. ◻

Proof. The Maxwell estimate of Sterbenz-Tataru [2] holds on every stationary spherically symmetric black hole with a non-degenerate horizon and a single normally hyperbolic photon sphere; non-extremal Reissner-Nordström is in this class for \(|Q|<M\). After the electric and magnetic spherical means are removed, the Maxwell field decomposes into vector spherical harmonics with \(\ell\ge1\); the spin-one variables follow by a fixed angular Hodge transform and a regular radial weight, an isomorphism of the charge-free tensor-field and spin-one energies at the same order by Lemma 5 and the Hodge estimate on \(\mathbb{S}^2\). The horizon term is the red-shift current of [4] and the far-field term is the \(r^p\) current of [5]. The bulk degenerates only at \(r_{\mathrm{ph}}(Q)\), where the Regge-Wheeler potential has a single non-degenerate maximum for each \(\ell\ge1\); the charge-free projection removes the \(\ell=0\) Coulomb mode. Choosing \(\varepsilon_Q<1/4\) keeps \(r_+(Q),\kappa_+(Q),r_{\mathrm{ph}}(Q)\) and the relevant potential barriers separated from all degeneracies, so the constant is uniform. The reduction of the charge-free Maxwell field to the spin-one radial system and the proof that every non-extremal Reissner-Nordström exterior meets the conditions of [2] are detailed in Section 9. ◻

Proof. By 30 , \(p_{a,Q}\) is a \(C^2\)-small stationary perturbation of \(p_{0,Q}\) after dividing by \(|\xi|^2\). The Reissner-Nordström photon sphere is normally hyperbolic: the linearized flow has one expanding and one contracting normal direction and neutral tangential directions; indeed it is \(r\)-normally hyperbolic for every \(r\) in the sense of [13], and in the Schwarzschild case this is the explicit computation at \(r=3M\), \(\xi_{r_*}=0\) of [13]. By [11], [13], [28] \(r\)-normal hyperbolicity is open under \(C^1\) perturbations of the Hamiltonian vector field, so \(K_{a,Q}\) and its stable/unstable bundles persist and depend \(C^1\) on \((a,Q)\); ?? are the uniform expansion/contraction inequalities after shrinking \(\varepsilon_a,\varepsilon_Q\). The quantitative version, with the escape function used below, is in Section 10. ◻

Proof. Since \(K'=2r\omega\) and \(\Delta'=2(r-M)\), \[\mathcal{R}'(r)=2KK'-\Delta'\lambda =4r\omega K-2(r-M)\lambda.\] The equation \(\mathcal{R}(r_t)=0\) gives \(K_t^2=\Delta_t\lambda\). Since \(\Delta_t>0\) and \(\lambda>0\), \(K_t\ne0\). The equation \(\mathcal{R}'(r_t)=0\) gives \(2r_t\omega K_t=(r_t-M)\lambda\). Combining the two identities and dividing by \(K_t\) gives ?? . Differentiating once more, \[\mathcal{R}''=2(K')^2+2KK''-\Delta''\lambda =8r^2\omega^2+4\omega K-2\lambda.\] Substituting ?? and \(\lambda=4r_t^2\omega^2\Delta_t/(r_t-M)^2\) gives \[\begin{align} \mathcal{R}''(r_t) &=8r_t^2\omega^2+\frac{8r_t\omega^2\Delta_t}{r_t-M} -\frac{8r_t^2\omega^2\Delta_t}{(r_t-M)^2} \\ &=\frac{8r_t\omega^2}{(r_t-M)^2} \bigl(r_t(r_t-M)^2-M\Delta_t\bigr). \end{align}\] The bracket equals \[r_t(r_t-M)^2-M(r_t^2-2Mr_t+a^2+Q^2) =(r_t-M)^3+M(M^2-a^2-Q^2),\] which proves ?? ; it is positive because \(r_t>r_+>M\) and \(M^2-a^2-Q^2>0\).

For the superradiant factor, we solve ?? for \(am\): \[am=(r_t^2+a^2)\omega-K_t =-\omega\frac{P(r_t)}{r_t-M}, \qquad P(r)=r^3-3Mr^2+(a^2+2Q^2)r+a^2M.\] Since \(K(r_+)=(r_+^2+a^2)\varpi\), \[\omega\varpi=\frac{\omega K(r_+)}{r_+^2+a^2} =\frac{\omega^2\bigl((r_+^2+a^2)(r_t-M)+P(r_t)\bigr)}{(r_t-M)(r_+^2+a^2)}.\] Using the horizon relation \(a^2+Q^2=2Mr_+-r_+^2\), the numerator polynomial becomes \[(r_+^2+a^2)(r-M)+P(r) =(r-r_+)\bigl(r^2+(r_+-3M)r+Mr_+\bigr).\] The discriminant of the quadratic factor is \((r_+-M)(r_+-9M)<0\), while its leading coefficient is positive. Hence the quadratic factor is positive for all real \(r\); because \(r_t>r_+\), this proves ?? .

Finally substitute the exact trapped value of \(K_t\) into the imaginary radial spin-one coefficient: \[\operatorname{Im}V^{(s)}_{\mathrm{rad}}(r_t) =-\frac{2s(r_t-M)}{\Delta_t}\frac{2r_t\omega\Delta_t}{r_t-M} +4s\omega r_t =0.\] This cancellation is algebraic, not an estimate, and it is uniform up to the subextremal constants appearing in the preceding positivity statement. Its more detailed semiclassical translation into the vector-bundle subprincipal symbol is Lemma 39. ◻

Proof. The finite-frequency separated radial operator has imaginary potential ?? . Multiplying the stationary equation by \(h^2\) and writing \(\hat{\omega}=h\omega\), \(\hat{m}=hm\) gives the order-\(h\) skew-adjoint part \(ihB_h\); the real diagonal symbol is ?? . The real second-order part gives the scalar principal symbol \(p_{a,Q,\hat{\omega}}I_2\), and all remaining coefficients are subprincipal or lower-order terms bounded in Lemma 38. The compatible two-component reduction may contain a matrix skew-subprincipal part. Its size is the coefficient comparison ?? , transferred to the frozen semiclassical normal form, and gives ?? on a fixed trapped collar.

For a scalar commutant \(A_h\) the symbolic calculus gives \[\label{eq:main95body95diag95commutator95identity} \begin{align} & \frac{i}{h}\langle [P_{h,\mathrm{sa}},A_h^*A_h]v,v\rangle -2\langle A_h\operatorname{diag}(b_{+1},b_{-1})A_hv,v\rangle \\ &\qquad =\langle \operatorname{Op}_h\bigl( H_{p_{a,Q,\hat{\omega}}}(\beta^2)I_2 -2\beta^2\operatorname{diag}(b_{+1},b_{-1}) \bigr)v,v\rangle +O(h)\lVert v\rVert_{H_h^1}^2 . \end{align}\tag{60}\] At a trapped point the principal trapping relations give \[\label{eq:main95body95Khat95trapped95for95threshold} \hat{K}_t=\frac{2r_t\hat{\omega}\Delta_t}{r_t-M}.\tag{61}\] Substitution in ?? gives \[\label{eq:main95body95diag95skew95zero95computation} b_s(r_t)= -\frac{2s(r_t-M)}{\Delta_t}\frac{2r_t\hat{\omega}\Delta_t}{r_t-M} +4s\hat{\omega} r_t=0, \qquad s=\pm1,\tag{62}\] which proves ?? . Since \(\operatorname{diag}(b_{+1},b_{-1})\) is smooth, its difference from its value on \(K_{a,Q}\) is \(O(|y|+|\nu|)\) in normal stable/unstable coordinates. In the second-microlocal trapped calculus, the region \(|y|+|\nu|\ge c h^{1/2}\) is controlled by the positive normal escape form, while the core \(|y|+|\nu|\le c h^{1/2}\) is a lower trapped remainder propagated along the stable and unstable directions. The symbolic Cauchy inequality therefore gives ?? .

It remains to add the matrix part \(B_{h,\mathrm{mat}}\). On \(K_{a,Q}\) the diagonal part is zero, and the matrix part obeys ?? . Since \(\lambda_0\) has a positive uniform lower bound on the normalized trapped shell, decreasing \(\varepsilon_a(k)\) gives ?? . The off-trapped variation of \(B_{h,\mathrm{mat}}\) is smooth in the normal variables and is estimated by the same second-microlocal Cauchy inequality as the diagonal variation. Combining this with the scalar escape lower bound of Proposition 19 gives ?? . The constants depend only on the fixed differentiability order and on the compact normalized frequency shell. ◻

Proof. The reduced radial principal equation has the form \[\label{eq:main95body95reduced95radial95symbol} p_{a,Q,\hat{\omega}}=c_1(r,\theta)\eta^2 -c_2(r,\theta)\mathcal{R}(r)+O(|(y,\eta)|^3)\tag{63}\] near a trapped point, where \(c_1,c_2\) are positive smooth factors on the regular exterior. The trapped equations are \(\mathcal{R}(r_t)=\mathcal{R}'(r_t)=0\), and Proposition 17 gives \(\mathcal{R}''(r_t)>0\). Taylor expansion therefore gives ?? with \(A_t=c_1(r_t,\theta_t)>0\) and \(B_t=\frac{1}{2}c_2(r_t,\theta_t)\mathcal{R}''(r_t)>0\), after absorbing harmless positive factors into the normalized covariable \(\eta\). The Hamilton equations in the radial normal variables are \[\label{eq:main95body95radial95hamilton95matrix} \dot{y}=\partial_\eta p=2A_t\eta+O(|(y,\eta)|^2), \qquad \dot{\eta}=-\partial_y p=2B_t y+O(|(y,\eta)|^2).\tag{64}\] The linear matrix has eigenvalues \(\pm2\sqrt{A_tB_t}\), which is ?? . The relation with ?? follows from the explicit radial Hamiltonian normalization: the coefficient of \(\eta^2\) contributes a positive multiple of \(\Delta(r_t)(r_t^2+a^2)^{-2}\), while the radial potential contributes a positive multiple of \(\mathcal{R}''(r_t)(r_t^2+a^2)^{-2}\). Positivity and compactness of the normalized shell give the uniform lower bound \(\lambda_0\).

The variables \(x_u,x_s\) diagonalize the linear part of 64 , giving ?? . Differentiating ?? along the flow gives \[\begin{align} -H_pG_h &=-\frac{x_sH_px_s}{x_s^2+h} +\frac{x_uH_px_u}{x_u^2+h} \\ &=\lambda_t\left( \frac{x_s^2}{x_s^2+h}+\frac{x_u^2}{x_u^2+h}\right) +O\left(\frac{|(x_s,x_u)|^3}{x_s^2+x_u^2+h}\right). \end{align}\] If the collar is chosen so that \(|(x_s,x_u)|\le\delta\), the error is bounded by \(C\delta\bigl(x_s^2/(x_s^2+h)+x_u^2/(x_u^2+h)\bigr)+C_\delta h\), which proves ?? after decreasing \(\delta\). In the final step, Proposition 18 applies this normal form to the full spin-one subprincipal endomorphism. The diagonal part vanishes on \(K_{a,Q}\) by the explicit computation 62 ; its off-trapped variation is absorbed by the second-microlocal normal quadratic form, and the matrix part is kept below the finite-rank-bundle threshold by the choice of \(\varepsilon_a(k)\). This is the threshold inequality used in the logarithmic trapped estimate. ◻

Proof. We give the calculation at top order and then sum over \(\Gamma^I\in\mathbb{D}_k\); Lemma 6 supplies the lower commutator terms produced by this summation.

(i) Reissner-Nordström commutant. On compact radial sets away from the red-shift and far-field collars, the Reissner-Nordström estimate ?? is equivalent to the existence of a self-adjoint first-order commutant \(A_Q\) such that \[\label{eq:rn95commutator95symbol} \big\langle \tfrac{1}{2i}(P_{\mathrm{RN},Q}^*A_Q-A_Q^*P_{\mathrm{RN},Q})u,u\big\rangle \ge c\lVert u\rVert_{LE^1_{\mathrm{deg}}}^2-C\lVert u\rVert_{LE^0(K_{0,Q})}^2 -C\lVert P_{\mathrm{RN},Q}u\rVert_{LE^*}^2 .\tag{65}\] The principal symbol of \(A_Q\) is the monotone radial multiplier constructed in Lemma 18; it is positive away from the photon sphere and has the quadratic trapping degeneracy recorded in ?? .

(ii) Transport to Kerr-Newman. Lemma 14 allows one to transport the radial commutant to a symbol \(a_{a,Q}=a_Q+a a_1\) centered at \(K_{a,Q}\) and satisfying \[\label{eq:perturbed95commutator95symbol} H_{p_{a,Q}}a_{a,Q}\ge (c-C|a|/M)m_{a,Q}-C\rho_{K_{a,Q}}^2 -C|a|M^{-1}r^{-3}|\xi|^2 .\tag{66}\] Here \(m_{a,Q}\) is the principal Morawetz density, and the last two terms are supported in the compact trapped collar or in the far region where the \(r^p\) identity applies. A symmetric quantization with a finite phase-space partition therefore gives the same bulk positivity as in 65 , up to a compact \(LE^0\) term, far-field errors controlled by Proposition 16, and horizon-collar terms controlled by the red-shift estimate of Proposition 15. The trapped algebra used in transporting the commutant is the calculation of Proposition 17; in particular trapping is separated from superradiance and the radial Hessian is strictly hyperbolic. Proposition 18 shows that the diagonal spin-one imaginary subprincipal contribution vanishes on \(K_{a,Q}\); Proposition 37 then verifies the finite-rank-bundle threshold for the full two-component skew part. For that reason, the trapped commutator has the same sign as the scalar normally hyperbolic commutator, up to the compact local remainder left explicit below.

(iii) Rotational and matrix errors. Write \(\mathcal{P}_b=P_{\mathrm{RN},Q}I_2+\mathcal{E}_{a,Q}\) as in Proposition 10. The principal part of \(\mathcal{E}_{a,Q}\) is \(a\mathcal{G}_{a,Q}^{\mu\nu}\nabla_\mu\nabla_\nu\), with short-range coefficient size \(O(|a|M^{-1}r^{-1})\) in the asymptotic frame, while the first- and zeroth-order terms have the displayed \(ar^{-2}\) and \(ar^{-3}\) decay. Pairing with the commutant and integrating by parts gives \[\label{eq:matrix95error95bound95morawetz} |\langle \mathcal{E}_{a,Q}u,A_{a,Q}u\rangle| \le \delta\lVert u\rVert_{LE^1_{\mathrm{deg}}}^2 +C_\delta (|a|/M)^2\lVert u\rVert_{LE^0_{\mathrm{comp}}}^2 +C_\delta\lVert u\rVert_{LE^0_{\mathrm{low}}}^2 .\tag{67}\] The Hardy and elliptic patching estimates of Lemma 16 place the lower-order exterior pieces in the displayed norms. The commutators \([\mathcal{P}_b,\Gamma^I]\) have the same short-range structure and are controlled by 36 ; at top order this contributes \(\eta X_j\), and the strict lower-order terms are included in the induction.

Adding the compact commutator estimate to the red-shift and far-field currents, integrating over \(\mathcal{D}(\tau_1,\tau_2)\), and taking \(\varepsilon_a(k)\) small enough to absorb the \(O(|a|/M)\) top-order terms gives ?? . The remaining boundary terms are endpoint energies and nonnegative fluxes through the horizon and the outgoing radial boundary. This establishes the raw estimate with the compact local remainder left explicit. ◻

Proof. Apply the raw estimate ?? . The only term not already present in the desired estimate is the compact local norm \(\|\Psi\|_{LE^0_{\mathrm{comp},k}}^2\). Proposition 22 gives, for any \(\delta>0\), \[\label{eq:compact95term95removed95use} \|\Psi\|_{LE^0_{\mathrm{comp},k}(\tau_1,\tau_2)}^2 \le \delta\|\Psi\|_{LE^1_{\mathrm{deg},k}(\tau_1,\tau_2)}^2 +C_\delta\sum_{|I|\le k}\|\mathcal{P}_b\Gamma^I\Psi\|_{LE^*}^2 +C_\delta\bigl(\mathcal{E}_M^{(k)}[\Psi](\tau_1)+\mathcal{E}_M^{(k)}[\Psi](\tau_2)\bigr).\tag{68}\] The degenerate local-energy norm in the first term is one of the coercive components of the Morawetz density \(\mathcal{M}_k\), supplemented by the red-shift collar and far-field pieces already included in the full spacetime norm. Choose \(\delta\) so that the first term on the right of 68 is absorbed into the left side of ?? ; the remaining terms have exactly the source and endpoint-energy form appearing in ?? . This yields the estimate with constants uniform after the finite-order slow-weak thresholds have been fixed. ◻

Proof. At infinity the one-dimensional Hardy inequality gives \(\int_{R_1}^\infty r^{-2}|v|^2\,\mathrm dr_*\le 4\int_{R_1}^\infty|\partial_{r_*}v|^2\,\mathrm dr_*+CR_1^{-1} \int_{R_1}^{2R_1}|v|^2\,\mathrm dr_*\), the annulus term included in the compact region. In the horizon collar \(s=r-r_+(Q)\) is smooth in ingoing coordinates and the red-shift field is uniformly transversal; a Poincaré inequality along its integral curves bounds the \(L^2\) norm in \(r_+\le r\le R_0\) by the transversal derivative and the trace on \(r=R_0\), the trace controlled by the compact annulus energy. Smoothness and non-degeneracy of \(r_+(Q),\kappa_+(Q)\) for \(|Q|\le\varepsilon_QM\) give uniform constants; the slow-weak case follows from uniform frame equivalence for \(|a|/M\) small. ◻

Proof. For \(a=0\) the charge-free spin-one reduction is diagonal in vector spherical harmonics and each coefficient satisfies \[\Big(-\partial_{r_*}^2+f_Q(r)\frac{\ell(\ell+1)}{r^2}-\omega^2\Big)u_\ell=f_\ell, \qquad \ell\ge1.\] On the compact set \(K'\), the coefficient \(f_Qr^{-2}\) is bounded below by a positive constant depending only on \(K'\) and the weak-charge range. For \(\ell\ge L_0(I,K')\), the angular term dominates the bounded multiplier \(\omega^2\). Pairing the equation with \(\overline{u_\ell}\), integrating by parts after a cutoff equal to one on \(K\), and absorbing the cutoff commutators by the larger compact set \(K'\) gives \[\|u_\ell\|_{H^1_{-\sigma}(K)} \le C\|f_\ell\|_{H^{-1}_{\sigma}(K')} +C\ell^{-1}\|u_\ell\|_{H^1_{-\sigma}(K')}.\] Summation over \(\ell\ge L\) proves ?? for \(a=0\). For \(|a|\ll M\), Lemma 32 and Proposition 14 show that the slow-rotation perturbation of the principal angular operator is \(O(|a|/M)\) and that the remaining new terms are first order or lower order with bounded coefficients on \(K'\). Choosing \(\varepsilon_a\) small and then \(L_0\) large absorbs the principal perturbation into the left side, while commutators of \(\Pi_{\ge L}\) with the smooth coefficient perturbations have size \(O(L^{-1})\) in the displayed norms by the angular pseudodifferential calculus on \(\mathbb{S}^2\). Hardy at the two ends supplies the same estimate for the weighted collars. The last assertion follows by applying ?? to a normalized defect sequence whose residual tends to zero and then sending \(L\to\infty\). ◻

Proof. The characteristic set of \(g_{\mathrm{RN}}^{\mu\nu}\xi_\mu\xi_\nu\) agrees with that of \(p^{\sharp}_{0,Q}\) away from the horizon because the two symbols differ by the positive factor \(f_Q\). The rescaled flow has \(H_{p^{\sharp}_{0,Q}}r_*=2\xi_{r_*}\) and \(H_{p^{\sharp}_{0,Q}}\xi_{r_*}=-\partial_{r_*}(f_Qr^{-2})|\xi_\omega|^2\). The radial potential \(f_Qr^{-2}|\xi_\omega|^2\) has a unique critical point \(r_{\mathrm{ph}}(Q)\), and its second \(r_*\)-derivative there is negative of size \(\simeq M^{-4}|\xi_\omega|^2\), uniformly for \(|Q|\le\varepsilon_QM\). Choose \(b_Q(r)\) increasing through zero at \(r_{\mathrm{ph}}(Q)\) with \(b'_Q>0\) in the collar, and set \(a_Q=b_Q(r)\xi_{r_*}\) times a compact radial cutoff. Then \[H_{p^{\sharp}_{0,Q}}(b_Q\xi_{r_*}) =2b'_Q\xi_{r_*}^2-b_Q\partial_{r_*}(f_Qr^{-2})|\xi_\omega|^2\] plus cutoff terms. The first term controls \(\xi_{r_*}^2\), and since \(b_Q\) and \(\partial_{r_*}(f_Qr^{-2})\) have opposite signs across the photon sphere, the second is non-negative and degenerates quadratically at trapping. Cutoff errors sit in compact regions and are bounded by the displayed controlled trapped-collar term. Smoothness in \(Q\) gives uniform constants. ◻

Proof. By Proposition 10 and Lemma 6, \(\mathcal{P}_b-P_{\mathrm{RN},Q}I\) is \(a\mathcal{G}^{\mu\nu}\nabla_\mu\nabla_\nu+ar^{-2}C^\mu \nabla_\mu+ar^{-3}D+\text{lower commutators}\). The scalar principal perturbation is paired in divergence form and is the symbol perturbation already in the transported commutator; \(A_{a,Q}\) is first order with symbol supported where the compact and far-field controls apply, so \(|\langle ar^{-2}C^\mu\nabla_\mu w,A_{a,Q}w\rangle|\le C|a|M^{-1}\lVert w\rVert_{LE^1_{\mathrm{deg}}}(\lVert w\rVert_{LE^1_{\mathrm{deg}}} +\lVert w\rVert_{LE^0_{\mathrm{comp}}})\). Cauchy’s inequality gives the first two terms; the zeroth-order part is handled by Hardy at infinity and elliptic patching, and the \(\Gamma^I\)-commutators either keep the small top-order coefficient or have fewer derivatives, giving \(\lVert w\rVert_{LE^0_{\mathrm{low}}}\). ◻

Proof. Let \(u_n\) be a contradiction sequence with \(\lVert u_n\rVert_{LE^0(K_0)}=1\) and with the right-hand side of the compact-remainder estimate tending to zero faster than \(n^{-2}\). Multiplying by a scalar time cutoff which is one on the middle half of the slab produces \(\chi_nu_n\) with \[\label{eq:localized95compact95cutoff} \lVert\mathcal{P}_b(\chi_nu_n)\rVert_{LE^*([T_1,T_2]\times K_1)} +\lVert(1-\chi_n)u_n\rVert_{LE^0(K_0)}\to0, \qquad \lVert\chi_nu_n\rVert_{LE^0(K_0)}\ge \frac{1}{2}.\tag{69}\] Plancherel in the stationary time variable gives a Fourier profile whose frozen residual is small relative to its local \(H^1\) mass. Decompose the Fourier side further into dyadic total-frequency shells \(\Lambda\simeq h^{-1}\) and into a fixed finite conic partition adapted to the scalar principal symbol. If a bounded set of shells carries a positive fraction of the mass, a measurable selection in those shells, followed by normalization, gives ?? .

If the mass escapes to \(\Lambda\to\infty\), the selection is made using the quotient whose numerator is precisely the square of the residual appearing in the normally hyperbolic resolvent normalization: \[\label{eq:localized95hf95selection95quotient} \mathcal{Q}_h(v)= \frac{h^{-2}\log(1/h)^{2}\lVert P_h v\rVert^{2}_{L^2(K_1)} +\lVert P_h v\rVert^{2}_{H^{-1}_h(K_1)}}{\lVert v\rVert^{2}_{H^1_h(K_0)}}.\tag{70}\] The factor \(h^{-2}\log(1/h)^2\) is essential: after taking square roots it is precisely the term \(h^{-1}\log(1/h)\|P_hv\|_{L^2}\) in ?? . Were 70 bounded below by a positive number on every sufficiently large dyadic shell and every conic cell, Plancherel, the finite overlap of the conic partition, and the uniform equivalence between the dyadic \(H^1_h\) norm and the local-energy density on compact sets would yield \[\lVert\chi_nu_n\rVert_{LE^0(K_0)}^2 \le C\Big(\lVert\mathcal{P}_b(\chi_nu_n)\rVert_{LE^*([T_1,T_2]\times K_1)}^2 +o(1)\lVert\chi_nu_n\rVert_{LE^0(K_0)}^2\Big),\] contradicting 69 after the last term is absorbed. Therefore a subsequence of conic cells has \(\mathcal{Q}_h(v)\to0\). Normalizing by \(\lVert v\rVert_{H^1_h(K_0)}=1\) gives exactly ?? ?? . The \(L^2\) residual is obtained by smoothing the selected packet at scale \(h\); the commutator of the smoothing with \(P_h\) is one order lower in the semiclassical calculus and is absorbed by the \(H^{-1}_h\) term in 70 .

The boundary and charge conditions pass to the selected profiles at the level at which they are used. The time cutoffs are taken with vanishing extra flux through their collars, and the radiation trace maps are continuous on the local-energy class, so the outgoing or incoming convention is inherited by the limit. The dyadic total-frequency selection is made inside the original closed compatible graph space; the phase-space cutoffs used to locate a conic cell are only test operators for the defect measure and are not asserted to preserve the Maxwell constraints. Electric and magnetic charges are zero for the original compatible sequence, and high-angular pieces have no Coulomb component because the charge is the \(\ell=0\) spherical mode already removed by the charge-free projection.

When this lemma is used to choose the slow-rotation threshold, the usual contradiction argument lets one replace a failure for all thresholds by a sequence with \(|a_n|/M\to0\). After the threshold has been fixed, the conclusion is the stronger fixed-range statement written above: the parameters only have to remain in the chosen slow-weak compact set. ◻

Proof. The compatible class \(\mathcal{C}_h\) is defined by the frozen Maxwell constraints, the closed range of the extreme-variable map, and the chosen outgoing or incoming radial convention; it is closed in the local resolvent graph norm by Lemma 48. Scalar temporal Fourier selection commutes with the stationary equation. Scalar angular/frequency cutoffs are not used to create new exact Cauchy data. Therefore the finite conic partition is used only in the weak form: it identifies a phase-space region on which the defect measure of \(v_n\) is nonzero. Choosing \(\chi=1\) on the base projection of that region gives \[\|A_{j,h_n}v_n\|_{L^2}\le C\|\chi v_n\|_{L^2}+O(h_n^\infty)\|v_n\|_{H^1_{h_n}},\] and the localized resolvent estimate controls \(\chi v_n\) directly in terms of \(P_{h_n}v_n\), which remains compatible. Accordingly, the conic localization is a support argument for the associated defect measure, not a projection of the Maxwell constraint equations. The charge-free condition is also preserved at the level used here: the Coulomb charge is the \(\ell=0\) flux coordinate removed before the master map is formed, and high-frequency angular pieces cannot carry it. ◻

Proof. Let \(\mathcal{C}\) denote the compatible graph space on a fixed finite slab: its norm consists of the local-energy norm of the Maxwell field, the local graph norm of the two master variables, and the distributional Maxwell constraint norm. The Maxwell constraints and the charge functionals are continuous in this topology by Proposition 3 and the construction of Definition 11; therefore the charge-free compatible subspace is closed.

Multiplication by a scalar time cutoff preserves the algebraic relations defining the extreme Newman-Penrose components and changes the master equation only by commutators supported in the cutoff collars. These collar terms are precisely the source terms which tend to zero in 69 . The stationary Fourier transform is unitary from the time-localized graph space into the direct integral of its frozen graph spaces, so a measurable frequency selection may be made inside the compatible fibre. The dyadic total-frequency projection is taken in that direct integral and therefore also leaves the selected profile in the compatible fibre.

The finite conic pseudodifferential partition is used after this selection to show where a normalized semiclassical measure can concentrate. It is not used to replace the selected profile by an arbitrary microlocally cut-off profile in the Maxwell constraint equations. Thus the limiting bounded-frequency profile and the normalized high-frequency sequence remain eligible for the two exclusion results cited in the compactness argument. ◻

Proof. Suppose that ?? fails for some fixed \(\delta>0\). After normalizing the compact local norm to one and letting the right side tend to zero, Lemma 20, together with Proposition 21, produces either a bounded-total-frequency defect or an unbounded compatible high-frequency defect.

In the bounded branch, the selected sequence satisfies ?? . Proposition 45 excludes such a sequence after the slow-rotation threshold has been chosen. Its proof is a Fredholm compactness argument using Lemma 25, Lemma 17, closedness of the compatible class, and the Reissner-Nordström real-axis exclusion; the limiting profile would be an outgoing or incoming real resonance, which is zero by Proposition 24.

In the unbounded branch, angularly elliptic packets are controlled by Lemma 17. The remaining characteristic packets satisfy the exact normalized high-frequency residual ?? , by ?? . Proposition 44, which uses the localized normally hyperbolic estimate after the geometric verification of Proposition 43, forces the local semiclassical \(H^1\) mass on \(K_0\) to vanish. This contradicts the normalization ?? . Both frequency branches are impossible, and ?? holds. Substitution into the raw estimate and absorption of the first term yield the compact-free estimate. ◻

Proof. Decompose into vector spherical harmonics. Charge subtraction removes \(\ell=0\), so each radial coefficient satisfies, with source \(\mathcal{S}_\ell= P_{\mathrm{RN},Q}^{\mathrm{stat}}\Psi\) projected to the harmonic, \[\label{eq:rw95stationary95model} -\partial_{r_*}^2u_\ell+V_{\ell,Q}(r)u_\ell=\mathcal{S}_\ell, \qquad V_{\ell,Q}(r)=f_Q(r)\,\frac{\ell(\ell+1)}{r^2},\tag{71}\] \(f_Q=1-2M/r+Q^2/r^2\). For \(\varepsilon_Q<1/4\), \(V_{\ell,Q}\ge0\) and is strictly positive for \(r>r_+(Q)\) when \(\ell\ge1\), uniformly. Pairing with \(\overline{u_\ell}\) and integrating in \(r_*\), using horizon regularity and finite energy at infinity, \[\int\big(|\partial_{r_*}u_\ell|^2+V_{\ell,Q}|u_\ell|^2\big)\,\mathrm dr_* =\Re\int \mathcal{S}_\ell\overline{u_\ell}\,\mathrm dr_*.\] The left side is bounded below by \(c_0\!\int(|\partial_{r_*}u_\ell|^2+r^{-2}|u_\ell|^2)\) after a one-dimensional Hardy inequality at infinity (using \(\ell\ge1\)) and compact elliptic control across the horizon collar; summing the harmonics gives the weighted energy on the left of ?? , while the right side is bounded by Cauchy-Schwarz by \(\|P_{\mathrm{RN},Q}^{\mathrm{stat}}\Psi\|_{\dot{\mathcal{X}}_0^*}\) times the square root of the same energy. The homogeneous case is immediate. ◻

Proof. The first assertion is the zero-frequency case \(\omega=0\) of the real-axis exclusion in Definition 9. For the perturbative proof criterion, let \(\Psi\) be stationary (\(T\Psi=0\)), charge-free and finite-energy with \(\mathcal{P}_b\Psi=0\). Since \(T\Psi=0\), the stationary-axial principal term in ?? drops; the remaining principal difference is \(O(a^2)\) and the first- and zeroth-order rotational terms have the form \(ar^{-2}\mathcal{C}^\mu_{a,Q}\nabla_\mu+ar^{-3}\mathcal{D}_{a,Q}\). Hence \[P_{\mathrm{RN},Q}^{\mathrm{stat}}\Psi=-(\mathcal{P}_b-P_{\mathrm{RN},Q}^{\mathrm{stat}})\Psi=:\mathcal{S}, \qquad \|\mathcal{S}\|_{\dot{\mathcal{X}}_0^*}\le C\frac{|a|}{M}\|\Psi\|_{\dot{\mathcal{X}}_0},\] with \(\|\Psi\|_{\dot{\mathcal{X}}_0}^2=\int(|\nabla_{r,\omega}\Psi|^2+r^{-2}|\Psi|^2)w_0\), by Cauchy-Schwarz and Lemma 16. Lemma 22 gives \[\|\Psi\|_{\dot{\mathcal{X}}_0}^2\le c_0^{-1}C\frac{|a|}{M}\|\Psi\|_{\dot{\mathcal{X}}_0}^2.\] Choosing \(\varepsilon_a\) so that \(c_0^{-1}C\varepsilon_a<1\) forces \(\Psi=0\). The constants remain uniform for \(|Q|\le\varepsilon_QM\) by the weak-charge choice in Proposition 30. ◻

Proof. The case \(\omega=0\) is Lemma 22. For \(\omega\ne0\), each harmonic satisfies \(-u_\ell''+V_{\ell,Q}u_\ell=\omega^2u_\ell\) (\(\ell\ge1\)) with \(V_{\ell,Q}\) smooth, real and short-range at \(r_*=\pm\infty\). Short-range asymptotics give \(u_\ell=A_\pm e^{i\omega r_*}+B_\pm e^{-i\omega r_*}+o(1)\) as \(r_*\to\pm\infty\). Finite non-degenerate energy makes \(u_\ell,u_\ell',\omega u_\ell\) square-integrable in both ends, so the oscillatory coefficients vanish, \(A_\pm=B_\pm=0\). The Wronskian \(\partial_{r_*}\operatorname{Im}(\overline{u_\ell} u_\ell')=0\) then has zero constant, and unique continuation from either end gives \(u_\ell\equiv0\). Summing the harmonics gives \(u_\omega=0\). ◻

Proof. The case \(\omega=0\) is the stationary coercivity of Lemma 22. Assume \(\omega\ne0\) and decompose into vector spherical harmonics. For each \(\ell\ge1\) the radial coefficient solves \[-u_\ell''+V_{\ell,Q}(r)u_\ell=\omega^2u_\ell,\] where the potential is real and short-range at both ends in \(r_*\). The outgoing and ingoing conditions mean, for the future convention, \[u_\ell=A_+e^{i\omega r_*}+o(1),\quad r_*\to+\infty, \qquad u_\ell=A_-e^{-i\omega r_*}+o(1),\quad r_*\to-\infty,\] with the same limits for \(u_\ell'\) after differentiating the leading terms; the past convention reverses both signs and gives the identical conclusion below. Since \(V_{\ell,Q}\) is real, the Wronskian current \(W=\operatorname{Im}(\overline{u_\ell}u_\ell')\) is constant. For the future convention, \[W(+\infty)=\omega |A_+|^2, \qquad W(-\infty)=-\omega |A_-|^2.\] Equality of the two constants gives \(\omega(|A_+|^2+|A_-|^2)=0\), hence \(A_+=A_-=0\). The Volterra construction of Jost solutions, equivalently unique continuation for the radial ordinary differential equation from an end, then forces \(u_\ell\equiv0\). Summing in \(\ell\) gives \(v=0\). ◻

Proof. At \(\omega=0\) this is Lemma 22. For \(\omega\ne0\), each harmonic satisfies the one-dimensional Schrödinger equation 71 with \(\omega^2\), short-range real potential; the no-real-resonance Lemma 24 gives a zero homogeneous outgoing/ingoing kernel on the weighted spaces. The Fredholm alternative (the free resolvent maps \(H^{-1}_\sigma\to H^1_{-\sigma}\) for \(\sigma>1/2\), the potential is compact \(H^1_{-\sigma}\to H^{-1}_\sigma\) on bounded frequency intervals, and the kernel is zero) gives ?? , uniform in \(Q\) by smooth non-degenerate dependence of the horizon radius, surface gravity, photon-sphere radius and potential on \(Q\). Summing harmonics and the finite commutator family closes the estimate. The Wronskian and Fredholm details are in Section 11. ◻

Proof. In the bounded-total-frequency branch, pass to a subsequence with \(\omega_n\to\omega_\infty\) and \(Q_n\to Q_\infty\). The coefficient expansion of Proposition 10 gives convergence of the frozen operators from \(H^1_{-\sigma}(K_1)\) to \(H^{-1}_{\sigma}(K_1)\). Lemma 25, combined with the high-angular coercivity estimate Lemma 17, gives a uniform local \(H^1\) bound. Local elliptic regularity upgrades the Rellich limit to strong \(H^1(K_0)\) convergence, so the limiting profile is nonzero. Lemma 48 passes the compatibility, charges and Sommerfeld convention to the limit. The limit is a Reissner-Nordström outgoing or incoming real resonance, impossible by Lemma 24 for nonzero frequency and by Lemma 22 at zero frequency.

In the unbounded-total-frequency branch, the angularly elliptic part is removed by Lemma 17. The remaining conic characteristic packets satisfy the residual normalization 41 , which is the condition of Proposition 44. That proposition forces the local \(H^1_{h_n}\) mass to vanish, contradicting the normalization in 38 . No defect sequence can therefore exist. ◻

Proof. Because \(\mathcal{P}_b(e^{-i\omega t}\psi)=0\), the product \(\mathcal{P}_b(\chi_Te^{-i\omega t}\psi)\) consists only of terms in which at least one \(t\)-derivative falls on \(\chi_T\). These terms are supported where \(T\le |t|\le2T\). A first derivative of \(\chi_T\) contributes \(T^{-1}\) and a second derivative contributes \(T^{-2}\). Since the mode has time-independent local energy density, the \(LE^*\) norm over a time interval of length \(O(T)\) is bounded by \(CT^{-1}\) times the corresponding local energy, plus lower-order \(CT^{-2}\) terms. After multiplying by \(T^{-1}\) this tends to zero. The endpoint energies of the cutoff solution are the mode energy plus errors supported where \(\chi_T\) is not constant; divided by \(T\) these errors vanish, and the normalized endpoint averages converge to the mode energy. ◻

Proof. The vanishing is precisely the real-axis exclusion in Definition 9(A3). Conversely, suppose the resolvent closure of Lemma 26 holds and a nonzero real-frequency mode exists. For \(\omega=0\) this contradicts the perturbative stationary argument in Proposition 23, which uses Lemma 22 and the comparison bound. For \(\omega\ne0\), a nonzero finite-energy mode has nonzero local \(H^1\) norm on some compact radial set \(K_0\); rescaling gives a profile \(v\) with \(\|v\|_{H^1(K_0)}=1\) and \(\mathcal{L}_{a,Q}(\omega)v=0\). Since both branches of Lemma 26 are stated for arbitrary compact sets, that lemma applies with this \(K_0\) and gives a contradiction. In turn, the closure lemma is a sufficient proof route for the assumed real-axis exclusion. ◻

Proof. For the scalar estimate expand \(v=\sum_{\ell\ge1,m}v_{\ell m}Y_{\ell m}\). Since \(\lambda_1=2\) is the first nonzero eigenvalue of \(-\mathop{}\!\mathchoice{\Delta\mkern-12mu/}{\Delta\mkern-12mu/}{\Delta\mkern-9mu/}{\Delta\mkern-8mu/}\,\) on \(\mathbb{S}^2\), \[\lVert v\rVert_{H^{s+1}}^2\simeq \sum_{\ell\ge1,m}(1+\lambda_\ell)^{s+1}|v_{\ell m}|^2 \le C_s\sum_{\ell\ge1,m}(1+\lambda_\ell)^s\lambda_\ell |v_{\ell m}|^2 \simeq C_s\lVert\mathop{}\!\nabla\mkern-13mu/\,v\rVert_{H^s}^2.\] This establishes ?? . For a one-form, write \(Y=\mathop{}\!\nabla\mkern-13mu/\,\phi+{}^\star\!\mathop{}\!\nabla\mkern-13mu/\,\chi\) with mean-free potentials. Then \(\mathop{}\!\mathchoice{\Delta\mkern-12mu/}{\Delta\mkern-12mu/}{\Delta\mkern-9mu/}{\Delta\mkern-8mu/}\,\phi=\mathop{\mathrm{div}\mkern-16mu/\,}\nolimits Y\) and \(\mathop{}\!\mathchoice{\Delta\mkern-12mu/}{\Delta\mkern-12mu/}{\Delta\mkern-9mu/}{\Delta\mkern-8mu/}\,\chi=\mathop{\mathrm{curl}\mkern-19mu/\,}\nolimits Y\). The same spectral gap gives \(\lVert\phi\rVert_{H^{s+2}}+\lVert\chi\rVert_{H^{s+2}} \le C_s(\lVert f\rVert_{H^s}+\lVert h\rVert_{H^s})\), and differentiating once gives ?? . For that reason, the angular inversion uses exactly one angular derivative, the derivative already present in the \(LE^1\) part of the order-\(k\) master norm after the \(k\) commutations. ◻

Proof. Multiply the first equation by \(v\) and integrate along outgoing null hypersurfaces, the second along incoming ones. The red-shift frame makes the horizon boundary density positive; in the far field the Reissner-Nordström transport signs give the \(r^{-1}|v|^2\) contribution after passing to \(rv\). The \(O(|a|/M)\) pieces are absorbed for \(\varepsilon_a\) small. Zero spherical mean removes the non-decaying angular mode, so Poincaré on \(\mathbb{S}^2\) and Hardy in \(r\) control \(r^{-2}|v|^2\). Commuting with \(\mathbb{D}_k\) and inducting on the number of commutations closes ?? . ◻

Proof. In the regular null frame the Maxwell equations split into transport equations for the middle component and angular equations for its gradient. In complex notation, with \(\mathcal{D}\varphi=\mathop{}\!\nabla\mkern-13mu/\,\rho_F+{}^\star\!\mathop{}\!\nabla\mkern-13mu/\,\sigma_F\), the component identities, after collecting the lower-order connection terms, are \[\begin{align} \label{eq:middle95transport95hodge} e_4\varphi+(\operatorname{tr}\chi)\varphi &= \mathop{\mathrm{div}\mkern-16mu/\,}\nolimits\alpha-i\mathop{\mathrm{curl}\mkern-19mu/\,}\nolimits\alpha+C_4^0\varphi+C_4^1\cdot\alpha, \nonumber\\ e_3\varphi+(\operatorname{tr}\underline\chi)\varphi &=-\mathop{\mathrm{div}\mkern-16mu/\,}\nolimits\underline\alpha-i\mathop{\mathrm{curl}\mkern-19mu/\,}\nolimits\underline\alpha +C_3^0\varphi+C_3^1\cdot\underline\alpha,\\ \mathcal{D}\varphi &=-e_3\alpha+C_3^2\cdot\alpha+C_3^3\cdot\underline\alpha+C_3^4\varphi = e_4\underline\alpha+C_4^2\cdot\alpha+C_4^3\cdot\underline\alpha+C_4^4\varphi. \nonumber \end{align}\tag{72}\] Here the \(C_i^j\) are connection-coefficient tensors smooth in the regular exterior; at \(a=0\) they have the Reissner-Nordström signs and \(r^{-1}\) decay, and for \(|a|\ll M\) their difference from the spherical coefficients is \(O(|a|M^{-1}r^{-2})\) in the symbol classes used in the energy norms. These identities are the frame components of \(\mathrm dF=0\) and \(\mathrm d\star_gF=0\): the first two lines are obtained by inserting one \(e_4\) or \(e_3\) and two angular vectors, while the third is obtained by inserting one angular vector and the two null vectors.

The charge-free condition supplies the zero spherical means of \(\rho_F\) and \(\sigma_F\) by Proposition 9. Applying Lemma 28 to \(\rho_F\) and \(\sigma_F\) on each sphere gives \[\label{eq:middle95angular95from95extreme} \lVert\varphi\rVert_{H^{s+1}(S_{\tau,r})} \le C_s\lVert\mathcal{D}\varphi\rVert_{H^s(S_{\tau,r})}.\tag{73}\] The last line of 72 then bounds the angular part of \(\varphi\) by first derivatives of the extreme components plus lower-order terms. Since \(\alpha\) and \(\underline\alpha\) are smooth nonzero weights times the entries of \(\Psi\), and since the \(LE^1\) part of \(\mathcal{X}^{(k)}_M\) controls one space-time derivative after each commutation in \(\mathbb{D}_k\), the right side of 73 is bounded by \(C\lVert\Psi\rVert_{\mathcal{X}^{(k)}_M}+C\lVert\varphi\rVert_{LE^0_k}\).

The first two lines of 72 , after the rescaling \(\widetilde{\varphi}=r^2\varphi\), are exactly of the form covered by Lemma 29; the source terms are angular derivatives of \(\alpha,\underline\alpha\) and the same lower-order coefficients. The \(r^{-1}\) spherical terms are controlled by Hardy and the zero-mean Poincaré inequality, while the \(O(|a|/M)\) rotational terms are absorbed after decreasing \(\varepsilon_a(k)\). Thus \[\lVert\varphi\rVert_{\mathcal{X}^{(k)}_{\mathrm{Max}}} \le C\lVert\Psi\rVert_{\mathcal{X}^{(k)}_M}+C\eta\lVert\varphi\rVert_{\mathcal{X}^{(k)}_{\mathrm{Max}}}, \qquad \eta=|a|/M,\] and the last term is absorbed in the slow-weak range. This establishes ?? . If two middle components correspond to the same \(\Psi\), their difference has zero mean, homogeneous angular equation and homogeneous transport sources; the same estimate forces the difference to vanish, proving uniqueness. ◻

Proof. The frame decomposition \(F=\rho_Fe^3\wedge e^4+\sigma_Fe^1\wedge e^2+\sum_A \alpha_Ae^A\wedge e^4+\sum_A\underline\alpha_Ae^A\wedge e^3\) is algebraic; the extreme components are part of \(\Psi\) up to smooth nonzero weights and the middle component is controlled by Proposition 25, with the rescaled frame smooth and uniformly invertible, giving ?? . The operator \(\mathfrak R\) solves 72 with the zero means fixed by charge subtraction and assembles the two-form; applying \(\mathfrak M\) recovers the extreme master variables since the master equations are derivatives of the Maxwell system. Conversely, reconstructing from \(\mathfrak M F_{\mathrm{rad}}\) gives a charge-free solution with the same Cauchy data for all frame components, equal to \(F_{\mathrm{rad}}\) by Proposition 7. ◻

Proof. Apply the model estimate with \(P_0u=P_bu-\mathcal{Q}_bu\): \(\lVert u\rVert_X^2\le C_0\lVert u(0)\rVert_E^2+2C_0\lVert P_bu\rVert_Y^2+2C_0\lVert\mathcal{Q}_bu\rVert_Y^2 \le C(\lVert u(0)\rVert_E^2+\lVert P_bu\rVert_Y^2+\lVert u\rVert_{X_{\mathrm{low}}}^2)+2C_0\delta^2\lVert u\rVert_X^2\), using ?? and \((\delta a+b)^2\le2\delta^2a^2+2b^2\). With \(2C_0\delta^2<1/2\) the last term is absorbed; the lower norm is controlled by induction or by the no-kernel argument. ◻

Proof. Move the two absorbable terms in ?? to the left. By ?? , \[\label{eq:finite95order95one95step} X_j\le \frac{4}{3} A_j+\frac{4}{3} d_jX_{j-1}.\tag{74}\] For \(j=0\) this gives \(X_0\le (4/3)A_0\), because the strict lower-order term is absent at order zero. Assume the conclusion has been proved through order \(j-1\). Substituting the inductive claim into 74 gives \[X_j\le \frac{4}{3} C_j(E_j+F_j)+\frac{4}{3}d_jC_{j-1,k}\sum_{i=0}^{j-1}(E_i+F_i) \le C_{j,k}\sum_{i=0}^{j}(E_i+F_i).\] The assertion follows by induction. The constants are finite because only finitely many commutators \(\Gamma^I\), \(|I|\le k\), and finitely many symbol seminorms enter. The choice of \(\varepsilon_a(k)\) in Proposition 30 makes \(b_j\le1/8\), while \(\eta_j\le1/8\) is fixed in 36 ; therefore the smallness condition is precisely the finite-order slow-rotation smallness used in the transfer theorem. ◻

Proof. Apply the Reissner-Nordström model estimate ?? to each commuted field \(\Gamma^Iu\), \(|I|\le j\), after writing \[\label{eq:complete95commuted95source95split} P_{\mathrm{RN},Q}\Gamma^Iu =\Gamma^If+[\mathcal{P}_b,\Gamma^I]u-(\mathcal{P}_b-P_{\mathrm{RN},Q})\Gamma^Iu .\tag{75}\] The first term contributes \(F_j\). The perturbation term \((\mathcal{P}_b-P_{\mathrm{RN},Q})\Gamma^Iu\) is bounded by the structural estimate 35 ; after Cauchy’s inequality and summation over \(|I|\le j\) it contributes \(\mu_j(a)X_j\), with \(\mu_j(a)\le A_j|a|/M\), plus terms of strict lower order. The commutator term is estimated by Lemma 6 and 36 : for the chosen Cauchy parameter \(\eta_j\), \[\label{eq:complete95commutator95bound} \sum_{|I|\le j}\|[\mathcal{P}_b,\Gamma^I]u\|_{LE^*}^2 \le \eta_jX_j+D_jX_{j-1},\tag{76}\] with the convention \(X_{-1}=0\). The horizon and far-field pieces of the model norm are replaced by the Kerr-Newman red-shift and \(r^p\) currents, ?? and ?? ; the trapped part is first estimated by the raw commutator inequality ?? . The only unabsorbed term at this stage is the compact local remainder, denoted by \(K_j\). This establishes ?? .

The compact term is removed by Proposition 22. In its quantitative form, a failure of ?? would produce the bounded- or unbounded-frequency defect sequence of Lemma 20; the bounded branch is ruled out by the real-axis limiting-absorption alternative, and the unbounded branch is ruled out by the normally hyperbolic estimate after the geometric verification of Proposition 39. Hence ?? holds for every \(\theta>0\).

Substitute ?? into ?? and move the terms in ?? to the left. We obtain \[\label{eq:complete95one95step} X_j\le \frac{4}{3}\Big((A_j+C_{j,\theta})(E_j+F_j) +(D_j+C_{j,\theta})X_{j-1}\Big).\tag{77}\] The case \(j=0\) gives \(X_0\le C(E_0+F_0)\). Inductively substituting the already obtained bounds for \(X_0,\ldots,X_{j-1}\) proves ?? . If \(\mathcal{P}_bu=0\), then \(f=0\) and all external source terms \(F_i\) vanish. The commutator terms do not vanish; they are precisely the terms estimated in 76 and have already been absorbed or passed to lower order in the induction. Since \(E_i\le C E_k\) for \(i\le k\) by the definition of the commuted energy, ?? follows. The induction is upward in \(j\), so no derivative above the current order is used. ◻

Proof. Proposition 10 and Lemma 6 place the Kerr-Newman master operator in the perturbative form used in Proposition 27, with \(\delta\simeq |a|/M\) and constants uniform for \(|Q|\le\varepsilon_QM\). The model Reissner-Nordström estimate gives the red-shift, Morawetz and far-field control. Proposition 22 reduces any failure of the compact part of the estimate to a normalized real-frequency defect profile. Definition 9(A3), equivalently the no-kernel conclusion of Proposition 24, excludes such profiles. Proposition 28 then gives the finite-order algebra: after choosing the compact-removal parameter, the strict lower-order Cauchy parameter and finally \(\varepsilon_a(k)\), all top-order terms are absorbed and the induction closes without using derivatives above order \(k\). For a homogeneous compatible solution the external source norms \(F_i\) in ?? vanish, while the commutator terms have already been handled by the order-by-order induction; therefore the result is exactly 47 . ◻

Proof. Lemma 50 proves the Hilbert-space statement in full. Here is the argument. The bound on \(W_0\) allows it to extend uniquely by continuity from the dense subspace \(E\subset R_+\) to a bounded operator \(W_+:R_+\to H\). Since \(S_+W_0\rho=\rho\) on \(E\) and \(S_+\) is bounded, passing to the closure gives \(S_+W_+=\mathrm{Id}_{R_+}\). Hence \(S_+\) is surjective. If \(h\in\ker S_+\), the zero-kernel condition gives \(h=0\), so \(S_+\) is also injective. Finally, for any \(h\in H\), \(S_+(W_+S_+h-h)=0\), hence \(W_+S_+h=h\) by injectivity. Therefore \(W_+=S_+^{-1}\) and \(\|S_+^{-1}\|\le C\). ◻

Proof. Boundedness of the future trace follows from Lemma 12 at null infinity and Corollary 4 at the horizon, both controlled by 47 ; the past trace is its time reversal. The dense radiation class, bounded right inverses and zero-kernel condition are precisely the content of (A5). Applying Lemma 31 gives the master wave operators. The Maxwell maps compose the master maps with the same-order reconstruction (Proposition 26) and the bounded radiation identifications in (A5); charge subtraction fixes the two Coulomb means, so the charge-free radiation map has no finite-dimensional kernel or cokernel. Time reversal gives the past isomorphism. ◻

Proof. When \(Q=0\) the comparison background is Schwarzschild and the perturbation in the master operator is measured only by \(|a|/M\). The red-shift, far-field, photon-sphere, zero-mode and Hodge reconstruction arguments above specialize without the charged spherical terms. The fixed-background Maxwell estimates on slowly rotating Kerr, together with the corresponding Teukolsky estimates and middle-component reconstruction, are supplied by [6], [7], [21], [22]. These estimates verify Definition 9 in the Kerr subcase, and Theorem 2 then gives the stated stationary-subtracted boundedness, decay, radiation and scattering conclusions. ◻

Proof. Choose \(\varepsilon_Q(k)\) so the Reissner-Nordström horizon is uniformly non-degenerate, \(r_{\mathrm{ph}}(Q)\) stays in a fixed compact interval, and the spin-one potentials keep the charge-free angular barrier for all \(\ell\ge1\); smooth dependence of \(r_+(Q),\kappa_+(Q),r_{\mathrm{ph}}(Q)\) and the potentials on \(Q\) gives uniform constants in Lemmas 13, 16, 22, and 25. With \(\varepsilon_Q(k)\) fixed, collect the finitely many constants multiplying \(|a|/M\) in the red-shift error, 59 , ?? , the transport errors and ?? , and choose \(\varepsilon_a(k)\) so each product is below one tenth of the available positive constant. Increasing a fixed \(C_{\mathrm{sw}}\) to dominate the endpoint, trace, density and wave-operator constants gives a uniform range at order \(k\). ◻

Proof. Commuting by \(\Gamma^I\in\mathbb{D}_k\) produces either terms in the order-\(k\) master norm or strict lower-order coefficient terms controlled by Lemma 6. The red-shift controls the transversal horizon derivatives, the \(r^p\) hierarchy the far-field weighted derivatives, and the compact Morawetz estimate the remaining local derivatives up to the trapping degeneration. Proposition 26 reconstructs the Maxwell tensor at the same order. The Sobolev step uses \(k_0\) derivatives; \(k\ge k_0+2\) leaves room for the commutators and dyadic weights. ◻

Proof. The charge and finite-energy Cauchy parts are Proposition 11. The perturbative estimates collected in this section show how the Reissner-Nordström model estimate is used once the spin-one operator has the scalar principal symbol and short-range perturbation structure of Proposition 10. The horizon estimate is Proposition 15; the \(r^p\) identity and radiation trace are Proposition 16 and Lemma 12; the trapped-set estimate is Proposition 20, with the model estimate and compact-error closure of Subsections 7.3 and 7.4. The stationary and real-frequency kernel exclusions are supplied by the (A3) limiting-absorption/no-defect statement and are used in Propositions 23 and 24. Under these exclusions the absorption Proposition 27 gives 47 , hence (M1). Same-order reconstruction is Lemma 28, Proposition 25, and Proposition 26, and proves (M2). The radiation maps are Proposition 29, which uses (A5) in the trace criterion and therefore proves (M3). The constants are fixed in Proposition 30. If (A6) is supplied, Corollary 7 transfers it at the same order and proves (M4). Accordingly, each analytic conclusion is attached to the precise estimate listed in Proposition 12, and no pointwise decay is claimed without the hierarchy condition. ◻

Proof. Each contravariant component in 78 is a rational function of \(r,\cos^2\theta\) and of \(a^2\), except for \(g_{\mathrm{KN}}^{t\phi}\) which is \(a\) times such a function. Indeed \(\Sigma=r^2+a^2\cos^2\theta\) and \(\Delta=\Delta_0+a^2\) depend on \(a\) only through \(a^2\), and the numerators \((r^2+a^2)^2-a^2\Delta\sin^2\theta\) and \(\Delta-a^2\sin^2\theta\) are polynomials in \(a^2\). Hence \(g_{\mathrm{KN}}^{tt},g_{\mathrm{KN}}^{\phi\phi},g_{\mathrm{KN}}^{rr},g_{\mathrm{KN}}^{\theta\theta}\) are even in \(a\) and differ from their \(a=0\) values, which are \(g_{\mathrm{RN}}^{tt}=-1/f_Q\), \(g_{\mathrm{RN}}^{\phi\phi}=1/(r^2\sin^2\theta)\), \(g_{\mathrm{RN}}^{rr}=f_Q\), \(g_{\mathrm{RN}}^{\theta\theta}=1/r^2\), by \(O(a^2)\). The only odd part is \(g_{\mathrm{KN}}^{t\phi}\) of 79 ; expanding \(\Sigma^{-1}\Delta^{-1}=(r^2\Delta_0)^{-1}(1+O(a^2))\) gives \[g_{\mathrm{KN}}^{t\phi}=-\frac{a(2Mr-Q^2)}{r^2\Delta_0}\bigl(1+O(a^2)\bigr) =a\,G_1^{t\phi}+O(a^3),\] which is ?? ; the \(O(a^3)\) is absorbed in \(a^2 G_2\). As \(r\to\infty\), \(\Delta_0=r^2 f_Q\sim r^2\) and \(2Mr-Q^2\sim2Mr\), so \(G_1^{t\phi}\sim-2M/r^3=O(r^{-3})\). On every set \(r\ge r_+(Q)+\eta\), smoothness and uniform bounds follow because \(f_Q\) and \(\Delta\) are bounded away from zero uniformly for \(|Q|\le\varepsilon_QM\). In a horizon collar the Boyer-Lindquist formula is not used as a coordinate estimate; after passing to the fixed horizon-regular coordinates of Section 2, the metric coefficients are smooth functions of \((a,Q)\) across \(\mathcal{H}^+\), so the same first-order coefficient comparison holds there by Taylor expansion in \(a\). The far-field statement for \(G_2\) follows from the analogous expansions of the even components, each contributing a difference \(O(a^2 r^{-2})\) relative to the flat metric. ◻

Proof. The fixed-background master equation used here is the spin-one operator of Section 6. The purpose of this appendix is to record the order in \(a\) and \(r\) of its coefficients and to separate the explicit metric-symbol calculation from the lower-order spin-one estimates. The spin-one Teukolsky calculus beginning with Teukolsky [27] and the Kerr-Newman Carter structures of Giorgi [17], [18] provide the model for these coefficient classes.

The scalar principal part is 80 , giving the scalar symbol and, by Lemma 32, the principal perturbation \(a\,\mathcal{G}^{\mu\nu}\nabla_\mu\nabla_\nu\) plus an \(O(a^2)\) second-order remainder. The supplied spin-one first-order terms are linear combinations of \(\partial_t\), \(\partial_\phi\) and the spin coefficients of the principal null frame. The \(a\)-dependence enters through (i) the cross term \(-2a(2Mr-Q^2)\Delta^{-1}\partial_t\partial_\phi\) already counted in the principal part, and (ii) the rotation of the principal null directions, which is \(O(a)\) and multiplies a first-order operator whose coefficients carry the spin-coefficient decay \(O(r^{-1})\); after the regular horizon and infinity weights, which are smooth and nowhere vanishing on the exterior, these contribute the term \(a\,r^{-2}\mathcal{C}^\mu\nabla_\mu\). The zeroth-order spin-one potential is a curvature expression; its part that is odd and linear in \(a\) has radial weight \(O(r^{-3})\) by the asymptotic flatness of the curvature, giving \(a\,r^{-3}\mathcal{D}\). The remaining even \(a\)-dependence, including the \(O(a^2)\) principal part, is collected in the second-order operator \(a^2\mathcal{Q}^{(2)}_{a,Q}\). At \(a=0\) all coefficients are spherically symmetric and assembled into \(\mathcal{P}_{\mathrm{RN},Q}\) of ?? ; the Reissner-Nordström curvature is quadratic in \(Q\) and enters \(\mathcal{V}_Q,\mathcal{W}_Q\). Smoothness of \(g_{\mathrm{KN}}\), of the frame, and of the weights in \((a,Q)\) gives the uniform bounds. The explicit radial weights give the short-range decay, and the \(a^2\mathcal{Q}^{(2)}_{a,Q}\) contribution obeys the same perturbative local-energy bound as 35 after possibly decreasing \(\varepsilon_a(k)\). ◻

Proof. We carry out the spherical reduction with the radial weight displayed. Write \[g_{\mathrm{RN}}=h_{ab}\,\mathrm dx^a\mathrm dx^b+r^2\gamma_{AB}\,\mathrm d\omega^A\mathrm d\omega^B, \qquad x^a=(t,r_*), \qquad h=f_Q(-\mathrm dt^2+\mathrm dr_*^2),\] so that \(|g_{\mathrm{RN}}|^{1/2}=f_Qr^2|\gamma|^{1/2}\), \(g^{tt}=-f_Q^{-1}\), \(g^{r_*r_*}=f_Q^{-1}\), and \(g^{AB}=r^{-2}\gamma^{AB}\). For \(\ell\ge1\) let \(Y=Y_{\ell m}\) and \(\mathcal{X}_A=\epsilon_A{}^{B}\nabla_B Y\). The odd sector may be represented locally by the gauge potential \[A^o=a_{\ell m}(t,r_*)\mathcal{X}_A\,\mathrm d\omega^A.\] Then \[\label{eq:odd95field95from95potential} F^o=\partial_ta_{\ell m}\,\mathrm dt\wedge \mathcal{X} +\partial_{r_*}a_{\ell m}\,\mathrm dr_*\wedge\mathcal{X} -\ell(\ell+1)a_{\ell m}Y\,\mathrm d\mu_{\mathbb{S}^2},\tag{81}\] where \(\mathrm d_\omega\mathcal{X}=-\ell(\ell+1)Y\mathrm d\mu_{\mathbb{S}^2}\) fixes the sign convention. The Bianchi equation is therefore automatic. If the tensor expansion is written with an angular coefficient \(q_{\ell m}Y r^2\mathrm d\mu_{\mathbb{S}^2}\), then \[\label{eq:rn95radial95weight95relation} r^2q_{\ell m}=-\ell(\ell+1)a_{\ell m}, \qquad \Phi^o_{\ell m}=\sqrt{\ell(\ell+1)}\,a_{\ell m} =-\frac{r^2q_{\ell m}}{\sqrt{\ell(\ell+1)}}.\tag{82}\] This yields the smooth nonzero radial weight that appears in the statement.

The remaining Maxwell equation is the \(A\)-component of \(\nabla^\mu F_{\mu\nu}=0\). Using the inverse metric above and the spherical identity \(\nabla^B_\gamma(\mathrm d_\omega\mathcal{X})_{BA}=\ell(\ell+1)\mathcal{X}_A\) with the sign convention of 81 , we compute \[\begin{align} \label{eq:odd95divergence95computation} 0&=|g|^{1/2}\nabla^\mu F^o_{\mu A} \notag\\ &=|\gamma|^{1/2}\Bigl(-\partial_t^2a_{\ell m}+ \partial_{r_*}^2a_{\ell m} -f_Q(r)\frac{\ell(\ell+1)}{r^2}a_{\ell m}\Bigr)\mathcal{X}_A. \end{align}\tag{83}\] Hence \(\Phi^o_{\ell m}=\sqrt{\ell(\ell+1)}a_{\ell m}\) satisfies \[\label{eq:odd95master95derivation} -\partial_t^2\Phi^o_{\ell m}+\partial_{r_*}^2\Phi^o_{\ell m} -f_Q\frac{\ell(\ell+1)}{r^2}\Phi^o_{\ell m}=0,\tag{84}\] which is equivalent to ?? . The even sector is obtained by applying the four-dimensional Hodge star, which commutes with the source-free Maxwell system and exchanges exact and coexact vector harmonics, or by repeating 83 with \(\mathop{}\!\nabla\mkern-13mu/\,Y\). Hence both polarizations have the same potential ?? .

The calculation shows that the charge parameter enters the fixed-background test problem only through the warping factor \(f_Q\) and the tortoise relation \(\mathrm dr=f_Q\mathrm dr_*\). The unit-sphere Laplacian contributes exactly \(\ell(\ell+1)\); no separate curvature potential is produced for the conformally invariant Maxwell two-form. Extra \(Q\)-dependent potentials arise in the coupled Einstein-Maxwell perturbation system, or after non-conformal master changes, not for the gauge-invariant test-field variables above.

To finish, the map from the tensor field to \((\Phi^e_{\ell m},\Phi^o_{\ell m})\) uses only \(-\mathop{}\!\mathchoice{\Delta\mkern-12mu/}{\Delta\mkern-12mu/}{\Delta\mkern-9mu/}{\Delta\mkern-8mu/}\,^{-1}\) on \(\ell\ge1\), the explicit radial weight in 82 , and the first-order constraints. The spectral gap \(\ell(\ell+1)\ge2\) and Lemma 5 therefore give two-sided equivalence between the charge-free Maxwell energy and the sum of the spin-one energies at every finite commuted order. ◻

Proof. Since \(f_Q>0\) on \(\{r>r_+(Q)\}\) and \(\ell(\ell+1)\ge2\), the potential \(V_{\ell,Q}\) in ?? is positive on \(\{r>r_+(Q)\}\), and \(V_{\ell,Q}\) vanishes where \(f_Q\) does and decays like \(r^{-2}\) at infinity. Compute \[\frac{\mathrm d}{\mathrm dr}\Bigl(\frac{f_Q}{r^2}\Bigr) =\frac{\mathrm d}{\mathrm dr}\Bigl(\frac{1}{r^2}-\frac{2M}{r^3}+\frac{Q^2}{r^4}\Bigr) =-\frac{2}{r^3}+\frac{6M}{r^4}-\frac{4Q^2}{r^5} =\frac{-2r^2+6Mr-4Q^2}{r^5}.\] The numerator vanishes at \(r^2-3Mr+2Q^2=0\), i.e.at \(r=\frac{1}{2}(3M\pm\sqrt{9M^2-8Q^2})\); the larger root ?? lies in \((r_+(Q),\infty)\) and the smaller one is inside the horizon for \(|Q|<M\). Since \(f_Q/r^2\to0\) at both ends and is positive in between, the unique exterior critical point is a maximum, and \(\frac{\mathrm d^2}{\mathrm dr_*^2}(f_Q r^{-2})=f_Q\frac{\mathrm d}{\mathrm dr}(f_Q\frac{\mathrm d}{\mathrm dr}(f_Q r^{-2}))<0\) there. Smooth non-degenerate dependence of \(r_{\mathrm{ph}}(Q)\), \(r_+(Q)\) on \(Q\) gives uniformity for \(|Q|\le\varepsilon_QM\). ◻

Proof. The horizon \(r_+(Q)\) is non-degenerate because \(f_Q'(r_+)=2\kappa_+(Q)>0\) for \(|Q|<M\). Lemma 34 gives the single photon sphere; its normal hyperbolicity is the content of Section 10. These are exactly the structural conditions under which [2] proves a Maxwell integrated local-energy estimate with a single photon-sphere degeneracy. In the charge-free \(\ell\ge1\) sector the spin-one scalars ?? are equivalent to the Maxwell field by Lemma 33, so the estimate transfers to the spin-one energy. The non-degenerate horizon term is supplied by the red-shift multiplier of [4] (Proposition 15 at \(a=0\)), and the weighted far-field fluxes for \(0\le p\le2\) by the \(r^p\) identity of [5] (Proposition 16 at \(a=0\)); the photon-sphere-degenerate bulk is the Morawetz density ?? . Summing over the finite commutator family \(\mathbb{D}_k\) and over \(\ell\ge1\) gives ?? . Choosing \(\varepsilon_Q<1/4\) keeps \(r_+(Q),\kappa_+(Q),r_{\mathrm{ph}}(Q)\) and the potentials in fixed compact non-degenerate ranges, so the constant \(C_{\mathrm{RN},k}\) is uniform. ◻

Proof. At \(K_{0,Q}\) the radial linearization of the flow is governed by \(-\tfrac12\partial_{r_*}^2(f_Q r^{-2})|\xi_\omega|^2>0\) by Lemma 34, so the radial pair \((r_*,\xi_{r_*})\) has one positive and one negative Lyapunov exponent, while the symplectically orthogonal angular directions are tangent to \(K_{0,Q}\) and neutral. This is the \(r\)-normal hyperbolicity of the photon sphere; for Schwarzschild it is the explicit computation at \(r=3M\), \(\xi_{r_*}=0\) of [13], and the smooth incoming/outgoing tails of codimension one are identified there. By the structural-stability theorem for \(r\)-normally hyperbolic invariant manifolds [28], used in this microlocal setting by [13] and in the full subextremal Kerr range by [11], any \(C^1\)-small perturbation of the Hamiltonian vector field preserves \(K\), \(\Gamma_\pm\) and the exponent bounds. The perturbation of the rescaled principal symbol from \(p^{\sharp}_{0,Q}\) to the corresponding Kerr-Newman rescaling \(p^{\sharp}_{a,Q}\) is \(C^2\)-small by Lemma 32 after dividing by \(|\xi|^2\), so \(K_{a,Q}\) is a \(C^1\) graph over \(K_{0,Q}\) and the expansion/contraction estimate ?? holds uniformly after shrinking \(\varepsilon_a,\varepsilon_Q\). ◻

Proof. Using \(H_{p^{\sharp}_{0,Q}}r_*=2\xi_{r_*}\) and \(H_{p^{\sharp}_{0,Q}}\xi_{r_*}=-\partial_{r_*}(f_Q r^{-2})|\xi_\omega|^2\) on \(\{p^{\sharp}_{0,Q}=0\}\), \[H_{p^{\sharp}_{0,Q}}(b_Q\xi_{r_*}) =2b_Q'\,\xi_{r_*}^2-b_Q\,\partial_{r_*}(f_Q r^{-2})\,|\xi_\omega|^2.\] The first term controls \(\xi_{r_*}^2\) since \(b_Q'>0\) in the collar. For the second, \(\partial_{r_*}(f_Q r^{-2})\) is positive for \(r<r_{\mathrm{ph}}(Q)\) and negative for \(r>r_{\mathrm{ph}}(Q)\) (Lemma 34), while \(b_Q\) is negative for \(r<r_{\mathrm{ph}}(Q)\) and positive for \(r>r_{\mathrm{ph}}(Q)\); therefore the product \(-b_Q\partial_{r_*}(f_Q r^{-2})\ge0\), and a Taylor expansion at \(r_{\mathrm{ph}}(Q)\) gives the quadratic degeneracy \(\operatorname{dist}(r,r_{\mathrm{ph}}(Q))^2\). Cutoff terms are supported in the fixed photon-sphere collar and are absorbed into \(-C\chi_{K_{0,Q}}\tau^2\) on the characteristic set. Smoothness in \(Q\) gives the uniform \(c\). ◻

Proof. If \(\omega=0\), then \(K=-am\) is independent of \(r\). Hence \[\mathcal{R}'(r)=-\Delta'(r)\lambda=-2(r-M)\lambda.\] In the exterior \(r_t>r_+>M\), and \(\lambda>0\), so \(\mathcal{R}'(r_t)<0\), contradicting the trapped condition \(\mathcal{R}'(r_t)=0\). ◻

Proof. Lemma 37 first gives \(\omega\ne0\). The horizon identity \(K(r_+)=(r_+^2+a^2)\varpi\) gives \(\omega\varpi=\omega K(r_+)/(r_+^2+a^2)\). Eliminating \(am\) through 87 gives \(K(r_+)=\omega\bigl[(r_+^2+a^2)(r_t-M)+P(r_t)\bigr]/(r_t-M)\), which is ?? with \(\Phi(r)=(r-M)(r_+^2+a^2)+P(r)\). Expanding and using the horizon relation \(a^2+Q^2=2Mr_+-r_+^2\), the linear coefficient of \(\Phi\) becomes \(4Mr_+-r_+^2\) and the constant \(-Mr_+^2\), so all explicit \(a,Q\) cancel and \(\Phi(r)=r^3-3Mr^2+(4Mr_+-r_+^2)r-Mr_+^2\); long division by \((r-r_+)\) has zero remainder and gives ?? . The discriminant and sign of \(g\) are immediate, and \(r_t>r_+>M\) for subextremal parameters. ◻

Proof. By Lemma 37, \(\omega\ne0\) at the trapped point. Also \(\mathcal{R}''=2(K')^2+2KK''-\Delta''\lambda=8r^2\omega^2+4\omega K-2\lambda\). Substituting \(\lambda=2r_t\omega K_t/(r_t-M)\) and \(K_t=2r_t\omega\Delta_t/(r_t-M)\) gives \(\mathcal{R}''(r_t)=\frac{8r_t\omega^2}{(r_t-M)^2}\,[\,r_t(r_t-M)^2-M\Delta_t\,]\). Since \(r_t(r_t-M)^2-M\Delta_t=(r_t-M)^3+M(M^2-a^2-Q^2)\) and \(M^2-a^2-Q^2=(r_+-M)^2\), ?? follows; positivity is immediate for \(r_t>M\). ◻

Proof. The normalized Teukolsky calculation of Proposition 14 identifies the second-order part of each extreme equation with the scalar wave symbol \(g_{\mathrm{KN}}^{\mu\nu}\xi_\mu\xi_\nu\). Freezing time by \(\partial_t\mapsto -i\omega\), replacing each spatial derivative on the conic patch by \(h^{-1}(hD)\), and multiplying the frozen operator by \(h^2\), the second-order part becomes exactly \(\operatorname{Op}_h(p_{a,Q,\hat{\omega}})I_2\) with \(p_{a,Q,\hat{\omega}}\) as in 89 . The principal cross term \(2g^{t\phi}_{\mathrm{KN}}\partial_t\partial_\phi\) contributes \(2g^{t\phi}_{\mathrm{KN}}\hat{\omega}\eta_\phi\), and the radial and angular second-order terms contribute \(g^{rr}_{\mathrm{KN}}\eta_r^2\), \(g^{\theta\theta}_{\mathrm{KN}}\eta_\theta^2\), and \(g^{\phi\phi}_{\mathrm{KN}}\eta_\phi^2\). This establishes the asserted principal symbol.

Every first-order coefficient in the spin-weighted equations, including the terms produced by the regular conjugating weights, has the form \(b^t\partial_t+\sum b^j\partial_j\) with smooth bounded coefficients and the short-range decay recorded in Proposition 10. After time freezing and multiplication by \(h^2\), the temporal part is \(h\,b^t\hat{\omega}\), while each spatial part is \(h\,b^j(hD_j)\). These contributions are therefore collected in \(hP_{1,h}^{(a,Q)}\), with the first bound in 90 . Zeroth-order curvature, spin and matrix terms acquire the factor \(h^2\) and form \(h^2P_{0,h}^{(a,Q)}\), with the second bound. The same order count gives, for every fixed compactly supported semiclassical cutoff \(A_h\in\Psi_h^0\), \[[P_h^{(a,Q)}(\hat{\omega}),A_h] =\frac{h}{i}\operatorname{Op}_h(\{p_{a,Q,\hat{\omega}},a_A\})I_2 +h^2R_{A,h},\] with \(R_{A,h}:H_h^1(K')\to H_h^{-1}(K)\) uniformly bounded. The smooth dependence on \(a,Q\), the compactness of the normalized frequency patch, and the finite commutation order yield uniform constants. Since \(P_{1,h}^{(a,Q)}\) and \(P_{0,h}^{(a,Q)}\) are strictly below the scalar second-order principal part, the Hamilton flow, trapped set, propagation support and normal rates are determined by \(p_{a,Q,\hat{\omega}}\) alone. ◻

Proof. The radial principal trapping equations are the equations for \[\mathcal{R}_h(r)=\widehat K(r)^2-\Delta(r)\lambda.\] Since \(\widehat K'=2r\hat{\omega}\) and \(\Delta'=2(r-M)\), \[\mathcal{R}_h'(r)=4r\hat{\omega}\widehat K-2(r-M)\lambda.\] At \(r=r_t\), ?? gives \[\lambda=\frac{\widehat K_t^2}{\Delta_t} =\frac{2r_t\hat{\omega}\widehat K_t}{r_t-M}.\] Because \(\Delta_t>0\) and \(\lambda>0\), \(\widehat K_t\ne0\), and division by \(\widehat K_t\) gives the exact trapped value \[\label{eq:semiclassical95Kt95for95skew} \widehat K_t=\frac{2r_t\hat{\omega}\Delta_t}{r_t-M}.\tag{91}\] Substitution into ?? gives \[q_s(r_t)= -\frac{2s(r_t-M)}{\Delta_t}\frac{2r_t\hat{\omega}\Delta_t}{r_t-M} +4s\hat{\omega} r_t=0,\] which proves ?? .

The remaining step is to translate this scalar radial calculation into the system symbol. Multiplication by the real weights used in Definition 11, and by the real factor used to put the stationary equation in semiclassical form, changes the self-adjoint real part and lower real potentials but does not create an imaginary principal contribution. The angular eigenvalue correction \(h^2\lambda_s-\lambda=O(h)\) is real. This implies that the imaginary part of the order-\(h\) subprincipal endomorphism is the diagonal matrix \(\operatorname{diag}(q_{+1},q_{-1})\) restricted to the trapped set, and this matrix is zero by ?? . This establishes ?? . ◻

Proof. The principal commutator identity is \[\begin{align} & \frac{i}{h}\langle [P_{h,\mathrm{sa}},A_h^*A_h]v,v\rangle -2\langle A_hB_{h,\mathrm{diag}}A_hv,v\rangle \\ &\quad =\left\langle \operatorname{Op}_h\!\left(H_p(\beta^2)I_2-2\beta^2\operatorname{diag}(q_{+1},q_{-1})\right)v,v \right\rangle +O(h)\|v\|_{H_h^1}^2 . \end{align}\] Lemma 39 gives \(\operatorname{diag}(q_{+1},q_{-1})|_{K_{a,Q}}=0\). Since the symbols are smooth, in normal coordinates \((z,y,\nu)\) near \(K_{a,Q}\) one has \[\operatorname{diag}(q_{+1},q_{-1})(z,y,\nu) =O(|y|+|\nu|).\] In the logarithmic escape proof the region \(|y|+|\nu|\ge c h^{1/2}\) is controlled by the positive normal quadratic form after choosing the collar small, while the core \(|y|+|\nu|\le c h^{1/2}\) contributes only to the lower second-microlocal remainder and is propagated out by the stable/unstable estimates. Equivalently, for every \(\varepsilon>0\) the symbolic Cauchy inequality in the second-microlocal calculus gives ?? . Substituting this bound into the preceding commutator identity shows that the diagonal term has zero threshold endomorphism on \(K_{a,Q}\) and is absorbed by the scalar normal expansion. This provides the threshold input used by Proposition 38; no pointwise positivity away from the second-microlocal decomposition is being assumed. ◻

Proof. By the spin-one decomposition recorded in the compatible reduction, \[\label{eq:matrix95threshold95decomposition95used} B_{a,Q}=\operatorname{diag}(q_{+1},q_{-1})+B_{\mathrm{mat},a,Q}, \qquad \|B_{\mathrm{mat},a,Q}\|\le C_k\bigl(|a|/M+a^2/M^2\bigr)\tag{94}\] on the normalized trapped shell. Lemma 39 gives \(\operatorname{diag}(q_{+1},q_{-1})|_{K_{a,Q}}=0\). Choose \(\varepsilon_a(k)\) so that \[C_k(\varepsilon_a(k)+\varepsilon_a(k)^2)\le \lambda_0/8 .\] Then 92 holds. This provides the threshold condition used in Hintz’s finite-rank-bundle normally hyperbolic estimate: the part of the skew-subprincipal symbol evaluated on \(K_{a,Q}\) is dominated by the normal expansion rate.

What is left is to account for points in the collar but off \(K_{a,Q}\). Let \(\pi(z,y,\nu)=z\) be the normal projection to \(K_{a,Q}\). Smoothness gives \[B_{a,Q}(z,y,\nu)-B_K(z)=O(|y|+|\nu|)\] in the finitely many symbol seminorms used at order \(k\), with constants uniform in the slow-weak range; the matrix part obeys the stronger small bound ?? throughout the collar. The same second-microlocal Cauchy estimate used in Proposition 36 bounds this variation by an arbitrarily small multiple of the normal escape quadratic form plus the propagation remainder. Combining that estimate with 92 and the scalar escape inequality furnished by Proposition 43 gives 93 . All constants involve only the finite order \(k\), the normalized frequency shell and the slow-weak compact parameter set, so the choice is uniform. ◻

Proof. This is the normally hyperbolic trapped resolvent estimate in the form needed here. The main steps are recorded to locate the point at which the bundle-valued skew-subprincipal term enters. For scalar operators (\(N=1\)) the localized \(h^{-1}\log(1/h)\) resolvent bound at an \(r\)-normally hyperbolic trapped set is the estimate of Wunsch-Zworski [13]; the refined analysis of the trapped model, including the role of the threshold constant \(\lambda_0/2\) and the notion of \(r\)-normally hyperbolic sets used in (N2), is due to Dyatlov [11], [12]. The extension to operators on a finite-rank Hermitian bundle whose principal symbol is a real scalar multiple of the identity, with the skew-adjoint subprincipal part controlled by exactly the threshold bound of (N3), is due to Hintz [14]. In the present application the diagonal spin-one part has zero threshold value on \(K\), while the possible matrix part is kept below the threshold by Proposition 37. The passage from the localized trapped estimate to the cutoff resolvent statement, by gluing with the elliptic parametrix, real-principal-type propagation, and the radial-point estimates at the two ends in (N1), is the propagation-of-singularities gluing of Datchev-Vasy [15]; the semiclassical radial-point and propagation estimates themselves are stated, in the form used here, in [30] and [31]. We now recall the finite-order mechanism.

Near \(K\) choose homogeneous symplectic coordinates \((z,y,\nu)\), where \(z\) are coordinates on \(K\) and \((y,\nu)\) are stable and unstable normal coordinates. The normal hyperbolicity condition gives, after shrinking the collar, \[H_py=-\lambda(z)y+O((|y|+|\nu|)^2),\qquad H_p\nu=\lambda(z)\nu+O((|y|+|\nu|)^2), \qquad \lambda(z)\ge\lambda_0>0.\] With the sign convention above, the outgoing logarithmic escape function is \[g_h=\frac{1}{2}\log\frac{\nu^2+h}{y^2+h}\] with smooth cutoffs; for the incoming convention one uses \(-g_h\). On the outgoing collar, \[H_pg_h =\lambda(z)\frac{\nu^2}{\nu^2+h} +\lambda(z)\frac{y^2}{y^2+h} +O(|(y,\nu)|)\frac{y^2+\nu^2}{y^2+\nu^2+h}-O(h) \ge c\frac{y^2+\nu^2}{y^2+\nu^2+h}-Ch,\] after the collar is shrunk. The error terms supported outside this collar are handled by real-principal-type propagation. Quantizing a real scalar order-zero commutant \(A_h=\operatorname{Op}_h(\alpha)\) constructed from \(g_h\) gives the trapped positive commutant. For the bundle operator one decomposes \(P_h=P_h^0+ihS_h+h^2R_h\), with \(P_h^0\) self-adjoint modulo \(h^2\Psi_h^0\) and \(S_h\) self-adjoint modulo \(h\Psi_h^0\). The order-\(h\) commutator identity is \[\begin{align} & \frac{i}{h}\langle [P_h^0,A_h^*A_h]v,v\rangle -2\langle A_hS_hA_hv,v\rangle \\ &\quad =2h^{-1}\operatorname{Im}\langle A_hP_hv,A_hv\rangle +O(1)\|v\|^2_{\mathrm{controlled}}+O(h)\|v\|_{H_h^1}^2. \end{align}\] The principal symbol of the left side is the scalar escape contribution \(H_p(\alpha^2)I_N\) minus the Hermitian skew-part endomorphism. The threshold bound in (N3) is precisely the inequality ensuring that this skew term is dominated by the normal expansion in the preceding escape estimate; if this Hermitian skew-part symbol vanishes on \(K\), it is absorbed after shrinking the collar. Combining the trapped estimate with elliptic estimates, real-principal-type propagation and the radial-point estimates at the two ends gives ?? ; the logarithm comes only from the regularized hyperbolic escape function.

The asymptotically flat end is used in the usual limiting-absorption form. With \(x=r^{-1}\) near null infinity, the rescaled characteristic operator has radial sets at fiber infinity corresponding to outgoing and incoming null directions. A semiclassical radial commutant with the limiting-absorption sign gives control in a collar of \(x=0\) by the residual and by an interior cutoff. The interior cutoff is propagated either to the trapped neighbourhood, where the logarithmic estimate above applies, or to an elliptic region. The horizon collar is treated in the same way, with the red-shift sign replacing the asymptotic radial sign. For that reason, the gluing involves a finite partition into the trapped collar, the two radial collars, nontrapped characteristic annuli and elliptic regions; no estimate at infinity is deduced from the trapped model itself.

For ?? , insert an elliptic graph multiplier of order \(s\) on the trapped patch before and after the same escape commutant. The commutator of this multiplier with \(P_h\) has graph order \(s-1\), producing the term \(\|\chi_1v\|_{H_h^{s-1}}\). The estimate is closed by induction in \(s\), with the \(s=0\) case furnished by ?? . Since the argument uses only finitely many symbol seminorms and finitely many radial/elliptic charts, the constants are uniform on compact families satisfying (N1)-(N4). The compatible-class version is read as an a priori estimate for vectors \(v\) already lying in the closed graph domain. The scalar cutoffs appear only in \(\chi v\) and \(\chi_1P_hv\); the proof does not require \(\chi v\) or a general phase-space cutoff of \(v\) to satisfy the Maxwell compatibility equations. If the resolvent is written as \(\chi(P_h\pm i0)^{-1}\chi\), this is shorthand for the same a priori inequality on compatible solutions with the chosen limiting-absorption convention. Because the graph class is closed and invariant under the stationary operator and the boundary convention, the ambient finite-rank-bundle estimate restricts to it with the same constant. ◻

Proof. The stationary constraints are first-order linear differential relations with smooth coefficients, and the extreme Teukolsky variables are obtained by closed first-order maps from the charge-free Maxwell graph norm to the master graph norm. Hence, if \(v_j\) is compatible and \(v_j\to v\) in the local graph norm while \(P_hv_j\to f\), the constraints pass to the limit distributionally and the two extreme variables of \(v\) are the corresponding closed limits. The outgoing or incoming radial convention is defined by the limiting-absorption graph topology; it is therefore closed under this convergence. Accordingly, the compatible space is a closed graph subspace invariant under the stationary operator and under the resolvent selected by the corresponding boundary convention. Scalar cutoffs in the localized resolvent are used on the original compatible profile. General pseudodifferential elements of the phase-space partition appear only in the compactness argument to identify the support of semiclassical defect measures, where exact preservation of the Maxwell constraints by each cutoff is neither claimed nor needed. ◻

Proof. The positive-commutator proof of Proposition 38 is applied to the compatible profile \(v\) itself. After the trapped commutant is localized by scalar cutoffs, all commutators with the cutoffs are supported either in the elliptic region or in a nontrapped characteristic collar. Those terms are controlled by the elliptic parametrix, real-principal-type propagation, and the radial-point estimates stated in Proposition 40. The compatibility equations are therefore used only to specify the graph domain, the limiting-absorption convention, and the closed class in which defect profiles are selected. Since the graph class is closed by Lemma 40, passing to weak or semiclassical limits preserves compatibility for the original sequence. That gives ?? , which is the form used in Propositions 41 and 44. ◻

Proof. Item (i) is Lemma 38, whose proof shows that all matrix and spin terms occur at semiclassical order \(h\) or lower. Items (ii) and (iii) are Proposition 43 combined with the explicit non-superradiance calculation of Proposition 34. Item (iv) is Lemma 39 for the diagonal Teukolsky skew part together with Proposition 37 for the full finite-rank-bundle threshold. For (v), Lemmas 40 and 41 give the closed graph property and the cutoff a priori form. Concretely, the compatible class is specified by the Maxwell constraints, by the two Teukolsky extreme variables, and by the chosen outgoing or incoming radial convention. If a sequence is compatible and converges in the local resolvent graph norm, the Maxwell constraints pass to the limit distributionally, the closed range definition of Definition 11 keeps both extreme variables in the compatible class, and the radial convention is closed by the limiting-absorption graph topology. In Lemma 43 the phase-space partition is used only to identify where a semiclassical measure may live. The actual trapped resolvent estimate is applied to the original compatible function with scalar cutoffs \(\chi\prec\chi_1\), as in ?? ; therefore the proof does not require arbitrary pseudodifferential cutoffs to preserve the Maxwell constraints exactly.

Proposition 38 gives the \(h^{-1}\log(1/h)\) bound once the listed geometric, radial, and subprincipal conditions have been verified. The preceding paragraphs verify those conditions for the compatible scalar-principal Kerr-Newman spin-one operator on the compatible graph class: scalar real principal symbol, clean normally hyperbolic trapped set, non-superradiant radial-point sign, diagonal trapped skew cancellation, finite-rank-bundle matrix threshold, and closed compatibility. Away from the trapped neighbourhood, the cutoffs are controlled by the elliptic parametrix and by real-principal-type propagation to the horizon or null infinity. These are exactly the extra pieces used in Lemma 43, so the displayed estimate reduces there to the localized resolvent bound ?? . The finite graph-norm estimate is ?? , which is the same statement as ?? with \(P_h=P_h^{(a,Q)}(\hat{\omega})\). ◻

Proof. Here the estimate is the radial-point propagation inequality for a real scalar principal symbol with the outgoing or incoming sign fixed at the radial set. In the present operator the principal symbol is \(p_{a,Q,\hat{\omega}}I_2\) by Lemma 38; the spin and matrix terms are subprincipal or lower order and therefore enter the positive-commutator proof as bounded lower-order terms. The non-degenerate horizon supplies the red-shift sign, while the asymptotically flat end supplies the outgoing estimate at null infinity. The physical-space counterparts are Propositions 15 and 16; ?? is their semiclassical radial-point form with the same sign convention.

For a sequence satisfying ?? , the stronger condition \(h^{-1}\log(1/h)\|P_hv_h\|_{L^2}\to0\) implies \(h^{-1}\|P_hv_h\|_{L^2}\to0\). Thus, the source term in ?? is negligible. The interior term is then propagated along nontrapped bicharacteristics and is controlled by finite-time real-principal-type propagation away from the radial sets. The commuted statement follows from Lemma 6; the commutator terms are strictly lower order at the current level or small perturbative terms, and 36 closes them. This is the radial-point step used in Lemma 43. ◻

Proof. We establish ?? by induction on \(s\). For \(s=0\), apply ?? to the pair \(\chi_{k+1}\prec\chi_{k+2}\). The lower Sobolev term is absent, the residual term tends to zero by ?? , and the \(O(h^\infty)\) term is negligible because \(v_h\) is bounded in \(H_h^{k+1}\). This yields \(\lVert\chi_{k+1}v_h\rVert_{L^2}\to0\).

Assume the claim has been proved at order \(s-1\), where \(1\le s\le k+1\). Apply ?? to the adjacent pair \(\chi_{k+1-s}\prec\chi_{k+2-s}\). The resolvent residual tends to zero by ?? . The lower graph term is \(\lVert\chi_{k+2-s}v_h\rVert_{H_h^{s-1}}\), which is precisely the induction conclusion at order \(s-1\). The \(O(h^\infty)\) remainder is again negligible by the uniform \(H_h^{k+1}\) bound. This establishes the order \(s\) statement.

The formulation matters here because the lossy estimate is applied to the finite graph norm of the compatible stationary operator itself. We do not place an identity of the form \(P_h\Gamma^Iv=\Gamma^IP_hv+[P_h,\Gamma^I]v\) inside the factor \(h^{-1}\log(1/h)\). Hence no uncontrolled term of size \(\log(1/h)\lVert\Gamma^Iv\rVert_{H_h^1}\) is generated. Physical commutators enter only in the preceding local-energy hierarchy, where top-order perturbative coefficients are made small by the slow-rotation choice and strict lower-order terms are removed by induction. ◻

Proof. The compact-remainder argument is Proposition 22. After the bounded-frequency branch is excluded by Proposition 45, the remaining branch has total stationary frequency \(h_n^{-1}\to\infty\) and is put in the semiclassical normalization of Lemma 38. Lemma 44 gives precisely the residual condition ?? . In Lemma 43, the elliptic piece is controlled by the parametrix, and the characteristic nontrapped piece is transported to the red-shift or far-field radial sets. The only remaining term is the trapped piece, for which the displayed resolvent estimate gives ?? . Since \(h_n^{-1}\log(1/h_n)\|P_{h_n}^{(n)}v_n\|_{L^2}\to0\), the trapped mass vanishes. This identifies the role of Definition 13: it is precisely the one stated above. ◻

Proof. For (i): the characteristic set \(\Sigma_p=\{p=0\}\) is smooth away from the zero section. On each component with fixed time orientation, the radial Hamilton equations give \(\dot{r}=\partial_{\xi_r}p=2g^{rr}\xi_r\), with \(g^{rr}\) a positive multiple of \(\Delta(r^2+a^2)^{-2}\) on the characteristic set after the conserved reduction, so \(\xi_r=0\) on \(K\); the second condition \(\partial_r p=0\) is the stationarity of the radial potential, which is 86 . The differentials \(d\xi_r\) and \(d(\partial_r p)\) are independent there because \(\partial_{\xi_r}(\partial_r p)=0\) while \(\partial_r(\partial_r p)=\partial_r^2 p\ne0\) by Proposition 35; therefore \(K\) is a smooth codimension-two submanifold of \(\Sigma_p\). Since \(\iota_{H_p}\omega=dp\), the restriction of the symplectic form to \(\Sigma_p\) is presymplectic with kernel spanned by \(H_p\); this is the expected flow direction, not a radial degeneracy. The normal block determined by \((\xi_r,\partial_rp)\) has Poisson pairing \(\{\xi_r,\partial_r p\}=\pm\partial_r^2p\ne0\) (the sign depends on the convention for the canonical bracket), so the radial normal bundle is symplectic and has a genuine hyperbolic splitting. After fixing the homogeneous scale and quotienting by \(H_p\), the remaining tangent variables are the angular action-angle variables. For (ii), set \(y=r-r_t\) and \(\eta=\xi_r\). The radial normal linearisation is \[\label{eq:radial95normal95linearization} \frac{d}{dt} \begin{pmatrix} y\\ \eta\end{pmatrix} = \begin{pmatrix}0&\partial_{\xi_r}^2p\\-\partial_r^2p&0\end{pmatrix} \begin{pmatrix} y\\ \eta\end{pmatrix},\tag{95}\] whose eigenvalues are \(\pm\lambda_0\) with \(\lambda_0^2=-(\partial_{\xi_r}^2p)(\partial_r^2p)\big|_{r_t}\). Here \(\partial_{\xi_r}^2p=2g^{rr}>0\) is a positive multiple of \(\Delta(r_t)(r_t^2+a^2)^{-2}\). The reduced radial equation has the form \(\xi_r^2-\mathcal{R}(r)/(r^2+a^2)^2=0\) up to a positive smooth factor; hence Proposition 35 gives \(\partial_r^2p=-c\,\mathcal{R}''(r_t)(r_t^2+a^2)^{-2}<0\) for a positive smooth factor \(c\). Thus \(\lambda_0^2\) is the positive multiple of \(\Delta(r_t)\mathcal{R}''(r_t)(r_t^2+a^2)^{-4}\) recorded in the statement, and \(\lambda_0>0\). The flow on \(K\) preserves \((\omega,m,\lambda)\) and is integrable in the remaining angular variables. On every compact normalized energy shell, the tangent dynamics is conjugate, after the usual action-angle parametrization away from coordinate caustics and by a finite chart argument across them, to translation with smoothly varying frequencies. Its derivative cocycle therefore has at most polynomial growth in the flow parameter, so all tangential Lyapunov exponents vanish and \(\mu=0\); thus \(\lambda_0>r\mu\) for every fixed finite \(r\). Part (iii) is the structural stability of \(r\)-normally hyperbolic trapped sets in the form used in [13], applied to the \(C^\infty\) family of symbols whose dependence on \((a,Q)\) is the smooth perturbation of Lemma 32; the rates are continuous and bounded below away from extremality. Part (iv) is Proposition 34. Items (i)-(iv) are the geometric conditions required for a normally hyperbolic trapped-set estimate. They do not by themselves prove the resolvent bound ?? . That bound is the high-frequency estimate fixed in Definition 13. What remains to check before using that estimate for the spin-weighted operator is the passage from the geodesic symbol to the frozen master operator. Lemma 38 gives the scalar principal normal form \(pI_2\). Lemma 39 gives zero threshold value for the diagonal Teukolsky skew part, and Proposition 37 gives the small finite-rank matrix threshold for the remaining skew part. Hence the geometry and the subprincipal threshold conditions required to apply the high-frequency estimate stated in Definition 13 are verified without imposing an additional hidden condition. ◻

Proof. For (a), if \(f=(f_+,f_-)\) and \(v=(P_h\pm i0)^{-1}f\), diagonality gives \[v_\pm=(P_{h,\pm}\pm i0)^{-1}f_\pm.\] Applying ?? to the two components and adding the two squared inequalities gives \[\lVert\chi v\rVert_{L^2\oplus L^2}^2 \le C^2h^{-2}\log(1/h)^2 \lVert\chi f\rVert_{L^2\oplus L^2}^2,\] which is ?? . This elementary step is used only for diagonal component estimates.

For (b), the stationary compatibility equations and the radial outgoing or incoming convention are closed under local \(L^2\) convergence and under the graph norms used for the frozen Maxwell system; this is the same closedness proved for the radiation class in Lemma 48. The restriction of a bounded inverse to a closed invariant subspace cannot increase its operator norm. In turn, the vector-bundle estimate for the full subprincipal matrix operator restricts to \(\mathcal{C}_h\) without changing the constant. No scalar diagonalization of the lower-order spin-one couplings is being assumed in this step.

For (c), choose a finite family \(\{\chi_j\}\) with \(\sum_j\chi_j^2=1\) on the trapped neighbourhood and with uniformly bounded overlaps. The estimate is applied to the original compatible solution with the scalar cutoffs \(\chi_j\) and the corresponding enlarged cutoffs. The commutators \([P_h,\chi_j]\) are first-order semiclassical operators supported in the same compact region and are bounded by the local \(H^1_h\) graph norm already present in the trapped estimate. Summing in \(j\) changes only the constant. The microlocal cutoffs used to decompose a defect measure are not themselves required to map the compatible class into itself; they enter only in the propagation and support argument of Lemma 43. ◻

Proof. Choose zeroth-order semiclassical cutoffs \[A_{\mathrm{ell}}+A_{\mathrm{nt}}+A_{\mathrm{tr}}=I+O(h^\infty) \quadmicrolocally on K_0,\] where \(A_{\mathrm{ell}}\) is supported in the elliptic region of the scalar principal symbol, \(A_{\mathrm{tr}}\) is supported in a small conic neighbourhood of the trapped set, and \(A_{\mathrm{nt}}\) is supported on the characteristic set away from that neighbourhood. The principal symbol is real and scalar by Proposition 14, and Lemma 38 places all spin and matrix terms in semiclassical subprincipal or lower orders; therefore the propagation of semiclassical measures is governed by the Hamilton flow of the scalar symbol. Angularly elliptic packets with bounded temporal frequency have already been removed by Lemma 17, so the present partition is a genuine total-frequency conic partition.

On the elliptic piece, \[\label{eq:hf95elliptic95piece} \|A_{\mathrm{ell}}v_h\|_{H^1_h} \le C\|P_h^{(a,Q)}(\hat{\omega})v_h\|_{H^{-1}_h(K_1)} +O(h^\infty)\|v_h\|_{H^1_h(K_1)},\tag{96}\] which tends to zero. On the nontrapped characteristic piece, propagation of singularities along \(H_p\) carries any defect mass to one of the two ends in finite bicharacteristic time. The radial-point/red-shift estimate at the horizon, with the sign selected by the boundary condition, and the outgoing far-field estimate at null infinity imply that no defect measure enters from the ends. For that reason, \(A_{\mathrm{nt}}v_h\) has no defect mass.

It remains to exclude mass in the trapped neighbourhood. Let \(\chi\prec\chi_1\) be compact cutoffs with \(\chi=1\) on the projection of \(\operatorname{WF}_h(A_{\mathrm{tr}})\). The normally hyperbolic resolvent estimate ?? gives \[\label{eq:hf95trapped95piece} \|\chi v_h\|_{L^2} \le C h^{-1}\log(1/h)\|\chi_1P_h^{(a,Q)}(\hat{\omega})v_h\|_{L^2} +O(h^\infty)\|v_h\|_{H^1_h(K_1)}.\tag{97}\] The right-hand side tends to zero by ?? . Thus, the trapped microlocal \(L^2\) mass tends to zero. Next we upgrade the three microlocal pieces to the local semiclassical \(H^1_h\) norm. The elliptic piece already satisfies 96 . For the nontrapped and trapped pieces the cutoffs have compact semiclassical wave-front sets in the finite-frequency region determined by the total-frequency normalization. Thus, for every first-order semiclassical differential operator \(B=hD_x\) supported in \(K_0\), the compositions \(BA_{\mathrm{nt}}\) and \(BA_{\mathrm{tr}}\) are uniformly bounded zeroth-order semiclassical pseudodifferential operators, modulo \(O(h^\infty)\) errors. Thus \[\|A_{\mathrm{nt}}v_h\|_{H^1_h(K_0)} +\|A_{\mathrm{tr}}v_h\|_{H^1_h(K_0)} \le C\bigl(\|A_{\mathrm{nt}}v_h\|_{L^2} +\|A_{\mathrm{tr}}v_h\|_{L^2}\bigr) +O(h^\infty)\|v_h\|_{H^1_h(K_1)}.\] The nontrapped \(L^2\) term is zero in the defect measure by propagation to the red-shift and far-field radial sets; the trapped \(L^2\) term is zero by 97 . Combining these two facts with the elliptic piece and the finite microlocal partition proves ?? . ◻

Proof. Suppose such a sequence exists. After passing to a subsequence, \(\hat{\omega}_n\to\hat{\omega}_\infty\) and \(Q_n\to Q_\infty\). The semiclassical symbols of \(P_{h_n}^{(n)}\) converge in \(C^2\) on compact phase-space sets to the normalized scalar symbol \(p^{\sharp}_{a_\infty,Q_\infty,\hat{\omega}_\infty}\), by Lemma 32, Proposition 14, and the semiclassical normal form of Lemma 38. The coefficients, boundary convention and compatible constraints are exactly those covered by Lemma 43. The residual condition ?? is ?? , and the trapped piece is controlled by the high-frequency estimate ?? ; Proposition 43 verifies the geometric and skew-subprincipal identities needed to apply that estimate to the present operator. Therefore \[\|v_n\|_{H^1_{h_n}(K_0)}\longrightarrow0,\] contradicting ?? . Accordingly, no high-frequency defect sequence exists. ◻

Proof. By definition, \(P_{h_n}^{(n)}v_n=h_n^2\mathcal{L}_{a_n,Q_n}(\omega_n)v_n\). Hence \[\label{eq:unscaled95to95scaled95l2} h_n^{-1}\log(1/h_n) \|P_{h_n}^{(n)}v_n\|_{L^2(K_1)} =h_n\log(1/h_n) \|\mathcal{L}_{a_n,Q_n}(\omega_n)v_n\|_{L^2(K_1)}\longrightarrow0.\tag{98}\] Also, the dual semiclassical norm is weaker after the factor \(h_n^2\): for \(\phi\in H^1_{h_n,0}(K_1)\), \[\label{eq:unscaled95to95scaled95dual} |\langle P_{h_n}^{(n)}v_n,\phi\rangle| \le h_n^2 \|\mathcal{L}_{a_n,Q_n}(\omega_n)v_n\|_{L^2(K_1)}\|\phi\|_{L^2(K_1)} \le h_n^2\|\mathcal{L}_{a_n,Q_n}(\omega_n)v_n\|_{L^2(K_1)} \|\phi\|_{H^1_{h_n}}.\tag{99}\] This establishes the \(H^{-1}_{h_n}\) convergence and hence ?? . ◻

Proof. The case \(\omega=0\) is Lemma 22: each harmonic obeys \(-u_\ell''+V_{\ell,Q}u_\ell=0\) with \(V_{\ell,Q}>0\) for \(\ell\ge1\) (Lemma 34), and pairing with \(\overline{u_\ell}\) forces \(u_\ell\equiv0\). For \(\omega\ne0\) each harmonic obeys \[\label{eq:app95radial95mode} -u_\ell''+V_{\ell,Q}u_\ell=\omega^2 u_\ell,\qquad \ell\ge1,\tag{100}\] with \(V_{\ell,Q}\) real, smooth, and short-range at \(r_*=\pm\infty\) (\(V_{\ell,Q}\to0\) exponentially as \(r_*\to-\infty\), like \(r_*^{-2}\) as \(r_*\to+\infty\)). The Jost solutions give \(u_\ell=A_\pm e^{i\omega r_*}+B_\pm e^{-i\omega r_*}+o(1)\) as \(r_*\to\pm\infty\). Finite non-degenerate energy requires \(u_\ell,u_\ell'\in L^2\) near both ends, which is incompatible with a nonzero oscillatory coefficient; hence \(A_\pm=B_\pm=0\). The Wronskian-type quantity \(W(r_*)=\operatorname{Im}(\overline{u_\ell}\,u_\ell')\) has \(W'=\operatorname{Im}(\overline{u_\ell}\,u_\ell'')=\operatorname{Im}((V_{\ell,Q}-\omega^2)|u_\ell|^2)=0\) since \(V_{\ell,Q}-\omega^2\) is real; therefore \(W\) is constant, and \(W\to0\) at both ends, so \(W\equiv0\). With both Jost coefficients vanishing at one end, unique continuation for the ordinary differential equation 100 gives \(u_\ell\equiv0\). Summation over \(\ell\ge1\) gives \(u_\omega=0\). ◻

Proof. For \(\omega=0\) use Lemma 22. For \(\omega\ne0\), each harmonic satisfies 100 . Under the future convention the Sommerfeld conditions give \[u_\ell=A_+e^{i\omega r_*}+o(1)\quad (r_*\to+\infty), \qquad u_\ell=A_-e^{-i\omega r_*}+o(1)\quad (r_*\to-\infty),\] and the differentiated asymptotics hold for \(u_\ell'\). Since \(V_{\ell,Q}\) is real, \(W=\operatorname{Im}(\overline{u_\ell}u_\ell')\) is constant. The two ends give \(W(+\infty)=\omega|A_+|^2\) and \(W(-\infty)=-\omega|A_-|^2\), hence \(A_+=A_-=0\). A Jost/Volterra uniqueness argument at either end, equivalently unique continuation for 100 , gives \(u_\ell=0\). The past convention reverses the signs at both ends and gives the same identity. Summing harmonics gives \(v=0\). ◻

Proof. At \(\omega=0\) this is Lemma 22. For \(\omega\ne0\) decompose into spherical harmonics. The \(\ell\)th coefficient satisfies \[\label{eq:app95lap95mode} P_{\ell,\omega,Q}u_{\ell}:= \Big(-\frac{d^2}{dr_*^2}+V_{\ell,Q}(r)-\omega^2\Big)u_{\ell}=f_{\ell}, \qquad V_{\ell,Q}=f_Q\frac{\ell(\ell+1)}{r^2}.\tag{101}\] For any fixed finite set of angular momenta, the one-dimensional outgoing resolvent estimate, compactness of the short-range potential as a map \(H^1_{-\sigma}\to H^{-1}_{\sigma}\), and Lemma 46 give the desired bound by the Fredholm alternative. The constant is uniform on compact \(\omega\)-intervals: otherwise a normalized sequence would converge to a non-zero outgoing/ingoing homogeneous solution, contradicting Lemma 46.

The remaining step is to justify summation in \(\ell\). Choose \(R_-\ll0\ll R_+\) in the tortoise coordinate \(r_*\), and choose cutoffs \(\chi_-\), \(\chi_0\), \(\chi_+\) with \(\chi_-\) supported near the horizon end, \(\chi_+\) supported near infinity, and \(\chi_0\) supported on the remaining compact annulus. On \(\operatorname{supp}\chi_0\) the quantity \(f_Qr^{-2}\) has a positive lower bound, uniformly for \(|Q|\le\varepsilon_QM\). Hence, for \(\ell\ge L_0(I)\), \[\label{eq:app95lap95high95l95barrier} f_Q(r)\frac{\ell(\ell+1)}{r^2}-\omega^2\ge c\ell^2, \qquad \omega\in I,\qquad r_*\in\operatorname{supp}\chi_0.\tag{102}\] Pairing 101 with \(\chi_0^2\overline{u_\ell}\) and integrating by parts gives \[\label{eq:app95lap95high95l95identity} \|\chi_0\partial_{r_*}u_\ell\|_{L^2}^2 +c\ell^2\|\chi_0u_\ell\|_{L^2}^2 \le C\|f_\ell\|_{H^{-1}_\sigma}\|u_\ell\|_{H^1_{-\sigma}} +C\|u_\ell\|_{H^1(\operatorname{supp}d\chi_0)}^2.\tag{103}\] On the two ends the potential is nonnegative and the missing \(L^2\) control is provided by the one-dimensional Hardy inequality \[\label{eq:app95lap95end95hardy} \|\langle r_*\rangle^{-\sigma}w\|_{L^2}^2 \le C_\sigma\|\langle r_*\rangle^{1-\sigma}\partial_{r_*}w\|_{L^2}^2 +C_\sigma\|w\|_{L^2(R_-<r_*<R_+)}^2, \qquad \sigma>\frac{1}{2}.\tag{104}\] Applying this to \(\chi_\pm u_\ell\), absorbing the transition terms into 103 , and using \(\ell(\ell+1)\ge2\) on the charge-free sector gives \[\label{eq:app95lap95high95l} \lVert u_{\ell}\rVert_{H^1_{-\sigma}}+\lVert\omega u_{\ell}\rVert_{L^2_{-\sigma}} \le C\lVert P_{\ell,\omega,Q}u_{\ell}\rVert_{H^{-1}_{\sigma}}, \qquad \ell\ge L_0, \quad \omega\in I,\tag{105}\] with \(C\) independent of \(\ell\). The finitely many modes \(1\le\ell<L_0\) are covered by the Fredholm step. Near \(\omega=0\) the stationary estimate applies uniformly because charge subtraction removes the \(\ell=0\) mode and the first allowed angular eigenvalue is \(2\). Smooth non-degenerate dependence of \(r_+(Q)\), \(\kappa_+(Q)\), \(r_{\rm ph}(Q)\) and \(V_{\ell,Q}\) on \(Q\) provides uniformity for \(|Q|\le\varepsilon_QM\). Summing the harmonic estimates and then the finite commutator family proves ?? . ◻

Proof. The compatible class is the closed range of the differential master map together with the Maxwell constraints and the zero charge conditions. In coordinates it is given by a finite system of linear equations with smooth coefficients in \((a,Q,\omega)\), consisting of the frozen Teukolsky equations, the angular constraint equations used in the Hodge reconstruction, the two zero-mean sphere conditions for the middle scalars, and the first-order transport equations along \(\mathcal{H}^+\) and \(\mathscr I^+\) (or their past analogues). Each equation is continuous from local \(H^1\) to distributions; therefore the equations and the two charge-free mean conditions pass to the limit. Equivalently, applying the bounded reconstruction operator of Proposition 26 on compact subdomains and using its uniqueness identities, the locally convergent limit reconstructs a source-free charge-free Maxwell field whose master variables are \(v_\infty\).

What is left is to record the boundary condition. Near infinity the outgoing condition has the form \[(\partial_{r_*}-i\omega_n)v_n=R^+_n, \qquad R^+_n\to0 \quad in the weighted trace space,\] after cutting off to the asymptotic end; near the future horizon it has the analogous ingoing equation \((\partial_{r_*}+i\omega_n)v_n=R^-_n\) in regular red-shift coordinates. The coefficients and the cutoffs converge smoothly, and the local elliptic bounds give convergence of the traces on finite collars; the remainders therefore converge distributionally and in the weighted trace topology to the corresponding remainders for \(\omega_\infty\), which vanish. This implies that the Sommerfeld convention is closed under the limit. ◻

Proof. Suppose no such thresholds existed. Then for a sequence \(\varepsilon_j\downarrow0\) one could find counterexamples with \(|a_j|\le\varepsilon_jM\). Relabelling this counterexample sequence gives \(a_n\to0\). Pass to \(\omega_n\to\omega_\infty\) and \(Q_n\to Q_\infty\). On every compact radial set \(K_2\), Proposition 32 and the smooth coefficient dependence give \[\mathcal{L}_{a_n,Q_n}(\omega_n)-\mathcal{L}_{0,Q_\infty}(\omega_\infty) \longrightarrow0 \quad asH^1(K_2)\to H^{-1}(K_2).\] The assumed local \(H^1\) boundedness and the residual convergence therefore imply \(\mathcal{L}_{0,Q_\infty}(\omega_\infty)v_n\to0\) in \(H^{-1}(K_2)\) for each \(K_2\). Rellich gives, after passing to a subsequence, strong \(L^2\) convergence on compact subdomains. To keep the normalization in \(H^1\), use the local elliptic estimate for the frozen elliptic stationary operator: if \(K_0\Subset K_1\Subset K_2\), then \[\label{eq:local95elliptic95upgrade95bounded95freq} \|w\|_{H^1(K_1)} \le C\Big(\|\mathcal{L}_{0,Q_\infty}(\omega_\infty)w\|_{H^{-1}(K_2)} +\|w\|_{L^2(K_2)}\Big).\tag{106}\] Applying 106 to \(w=v_n-v_m\) shows that \(v_n\) is Cauchy in \(H^1(K_1)\), hence converges strongly in \(H^1(K_0)\) to a limit \(v_\infty\) with \(\|v_\infty\|_{H^1(K_0)}=1\). Passing to the limit in the operator equation gives \(\mathcal{L}_{0,Q_\infty}(\omega_\infty)v_\infty=0\). Lemma 48 shows that \(v_\infty\) is charge-free, compatible, and satisfies the same outgoing/ingoing Sommerfeld convention. Thus \(v_\infty\) is a real outgoing resonance. This contradicts Lemma 46 if \(\omega_\infty\ne0\) and Lemma 22 if \(\omega_\infty=0\). The contradiction proves the stated thresholds. ◻

Proof. For the scalar estimate write \(v=\sum_{\ell\ge1,m}v_{\ell m}Y_{\ell m}\). Since \(-\mathop{}\!\mathchoice{\Delta\mkern-12mu/}{\Delta\mkern-12mu/}{\Delta\mkern-9mu/}{\Delta\mkern-8mu/}\,Y_{\ell m}=\ell(\ell+1)Y_{\ell m}\) and the first nonzero eigenvalue is \(2\), \[\lVert v\rVert_{H^{s+1}}^2\simeq \sum_{\ell\ge1,m}(1+\ell(\ell+1))^{s+1}|v_{\ell m}|^2 \le C_s\sum_{\ell\ge1,m}(1+\ell(\ell+1))^{s}\ell(\ell+1)|v_{\ell m}|^2 \simeq C_s\lVert\mathop{}\!\nabla\mkern-13mu/\,v\rVert_{H^s}^2.\] For one-forms use the Hodge decomposition \(Y=\mathop{}\!\nabla\mkern-13mu/\,\phi+{}^\star\!\mathop{}\!\nabla\mkern-13mu/\,\chi\) with mean-free potentials. Then \(\mathop{}\!\mathchoice{\Delta\mkern-12mu/}{\Delta\mkern-12mu/}{\Delta\mkern-9mu/}{\Delta\mkern-8mu/}\,\phi=\mathop{\mathrm{div}\mkern-16mu/\,}\nolimits Y\) and \(\mathop{}\!\mathchoice{\Delta\mkern-12mu/}{\Delta\mkern-12mu/}{\Delta\mkern-9mu/}{\Delta\mkern-8mu/}\,\chi=\mathop{\mathrm{curl}\mkern-19mu/\,}\nolimits Y\), and the scalar elliptic estimate on the orthogonal complement of the constants gives \(\lVert\phi\rVert_{H^{s+2}}+\lVert\chi\rVert_{H^{s+2}} \le C_s(\lVert f\rVert_{H^s}+\lVert h\rVert_{H^s})\). Taking one angular derivative proves ?? . The metrics of the rescaled spheres form a compact \(C^{s+2}\) family for \(|a|\le\varepsilon_aM\), \(|Q|\le\varepsilon_QM\), so the elliptic constants remain uniform after absorbing the smooth conformal factor into the radial weights. ◻

Proof. Pair 113 with \(\widetilde{\varphi}\) and integrate over the spacetime slab \(\{0\le\tau\le T\}\) against the volume form. The outgoing derivative contributes, after integration by parts in \(e_4\), \[\label{eq:app95transport95boundary} \tfrac12\int_{\Sigma_T}\lvert\widetilde{\varphi}\rvert^2\,\nu_4 -\tfrac12\int_{\Sigma_0}\lvert\widetilde{\varphi}\rvert^2\,\nu_4 +\tfrac12\int_{\mathcal{H}^+}\lvert\widetilde{\varphi}\rvert^2 +\tfrac12\int_{\mathscr I^+}\lvert\widetilde{\varphi}\rvert^2,\tag{117}\] all four boundary integrands nonnegative because the regular frame makes the horizon flux \(\int_{\mathcal{H}^+}\lvert\widetilde{\varphi}\rvert^2\) a positive multiple of the red-shift density and the far-field flux \(\int_{\mathscr I^+}\lvert\widetilde{\varphi}\rvert^2\) is the limit of the \(r^p\)-weighted density at \(p=0\). The right-hand side produces \(\int r^2(\mathop{\mathrm{div}\mkern-16mu/\,}\nolimits\alpha-i\mathop{\mathrm{curl}\mkern-19mu/\,}\nolimits\alpha)\,\overline{\widetilde{\varphi}}\), bounded by Cauchy-Schwarz by the first norm on the right of ?? times \(\lVert\widetilde{\varphi}\rVert_{LE^{k}}\). The source \(\widetilde{F}_4\) is, by the computation after 113 , a smooth \(O(r^{-2})\) multiple of \(\widetilde{\varphi}\) plus an \(O(\lvert a\rvert/M\,r^{-1})\) term; the first is absorbed by the Hardy inequality 58 (zero mean supplies the Poincaré constant on \(\mathbb{S}^2\)), the second by smallness of \(\varepsilon_a\). The incoming equation 115 is treated identically with \(e_3\) and the incoming null hypersurfaces; the regularity of \(\widetilde{F}_3\) across \(\mathcal{H}^+\) is what prevents a \(f_Q^{-1}\) blow-up. Summing the outgoing and incoming identities and moving the absorbed terms to the left gives the order-zero case of ?? . Commuting the system with the regular derivative algebra \(\mathbb{D}_k\) produces only lower-order commutators of the same Reissner-Nordström sign plus \(O(\lvert a\rvert/M)\) rotational commutators, so induction on the number of commutations closes the estimate at order \(k\). ◻

Proof of Proposition 25. We combine the angular and null estimates with the exact derivative count. Since \(\widetilde{\varphi}\) has zero spherical mean, Lemma 49 gives on each sphere \[\label{eq:app95angular95tilde95phi} \lVert\widetilde{\varphi}\rVert_{H^{s+1}(S_{\tau,r})} \le C_s\lVert\mathop{}\!\nabla\mkern-13mu/\,\widetilde{\varphi}\rVert_{H^s(S_{\tau,r})} =C_s r^2\lVert\mathop{}\!\nabla\mkern-13mu/\,\varphi\rVert_{H^s(S_{\tau,r})}.\tag{118}\] The angular equations in 110 express \(\mathop{}\!\nabla\mkern-13mu/\,\varphi\) as a linear combination of one regular null derivative of \(\alpha\) or \(\underline\alpha\) and lower-order smooth coefficient terms: \[\label{eq:app95angular95source95bound} \lVert\mathop{}\!\nabla\mkern-13mu/\,\varphi\rVert_{H^s(S_{\tau,r})} \le C_s\bigl(\lVert e_3\alpha\rVert_{H^s(S_{\tau,r})} +\lVert e_4\underline\alpha\rVert_{H^s(S_{\tau,r})} +\lVert\alpha\rVert_{H^s}+ \lVert\underline\alpha\rVert_{H^s} +r^{-1}\lVert\varphi\rVert_{H^s}\bigr).\tag{119}\] The first four terms on the right are controlled by the \(LE^1\) part of the master norm because \(\alpha\) and \(\underline\alpha\) differ from the master entries only by the fixed regular weights of Section 5.1. The last term is lower order; after integration in \((\tau,r)\) it is controlled by Hardy and by the Poincaré inequality coming from the zero mean, and the \(O(|a|/M)\) rotational perturbative part is absorbed by taking \(|a|/M\le\varepsilon_a(k)\).

Proposition 46 controls the null derivatives and the radial part of \(\widetilde{\varphi}\) from 113 115 ; its right-hand sides are \(r^2\) times angular derivatives of the extreme components plus the same lower-order terms. Combining 118 , 119 , and ?? , and then absorbing the lower-order term, gives \[\label{eq:app95middle95done} \lVert\widetilde{\varphi}\rVert_{\mathcal{X}^{(k)}_{\mathrm{Max}}} \le C\lVert\Psi\rVert_{\mathcal{X}^{(k)}_M}.\tag{120}\] Since \(\varphi=r^{-2}\widetilde{\varphi}\) and \(r^{-2}\) is a smooth bounded weight with bounded derivatives on \(r\ge r_+(a,Q)\) in the regular compactification, 120 is equivalent to ?? . For uniqueness, the difference of two middle components with the same \(\Psi\) has zero mean, zero angular data in 110 , homogeneous transport equations in 113 115 , and zero initial trace; the same estimate forces the difference to vanish. ◻

Proof. The estimate in ?? makes \(W_0\) uniformly continuous on \(E\), so by density it extends uniquely to a bounded map \(W_+:R_+\to H\) with \(\lVert W_+\rVert\le C_1\). Since \(S_+\) is bounded and \(S_+W_0\rho=\rho\) on \(E\), continuity gives \[S_+W_+=\mathrm{Id}_{R_+}.\] Hence \(S_+\) is surjective. By the assumed kernel condition it is also injective. Moreover, for \(h\in H\), \[S_+(W_+S_+h-h)=S_+W_+S_+h-S_+h=S_+h-S_+h=0,\] so \(W_+S_+h=h\). Hence \(W_+=S_+^{-1}\) and the inverse is bounded. ◻

Proof of Lemma 31. Take \(E\) to be the dense class of smooth compactly supported radiation fields. The trace estimate gives boundedness of \(S_+\). The backward construction supplies \(W_0\) on \(E\) with the energy bound ?? ; in geometric applications \(W_0\) is obtained as the limit of uniformly bounded finite-slab backward solutions. The condition that zero future radiation forces zero Cauchy data is exactly \(\ker S_+=\{0\}\). Lemma 50 therefore gives the bounded inverse wave operator. ◻

Proof. Let \(u=\mathfrak M G\). By Lemma 7, \[\|u\|_{\mathcal{X}^{(k)}_M(0,\tau)}^2\le C_M\mathcal{E}_M^{(k)}[u](0).\] The energy comparison in 48 gives \(\mathcal{E}_M^{(k)}[u](0)\le C_R\mathcal{E}_{\mathrm{Max}}^{(k)}[G](0)\). Since \(G\) is charge-free and smooth, the inverse identity gives \(G=\mathfrak R u\), and the same-order reconstruction bound gives \[\|G\|_{\mathcal{X}^{(k)}_{\mathrm{Max}}(0,\tau)}^2 \le C_R\|u\|_{\mathcal{X}^{(k)}_M(0,\tau)}^2.\] Combining the three inequalities together gives ?? , with \(C\) bounded by the displayed product after enlarging \(C_R\) once to cover both energy comparison and reconstruction. ◻

Proof. By Proposition 4, the field \(F_{\mathrm{rad}}=F-F_{\mathrm{stat}}^{\mathrm{KN}}(q_E[F],q_B[F])\) is source-free and has both charges zero. It therefore lies in the class to which Proposition 47 applies. Substituting \(G=F_{\mathrm{rad}}\) in ?? gives the stated estimate. ◻

Proof. The norm \(\mathcal{X}^{(k)}_{\mathrm{Max}}\) is a finite sum of non-negative terms of three kinds: slice energies, spacetime local-energy integrals with the weights of Definition 8, and far-field or null-boundary fluxes obtained as monotone limits of non-negative truncated fluxes. Strong convergence of the Cauchy data and Proposition 7 give weak convergence of each commuted component \(\Gamma^IG_n\) to \(\Gamma^IG\) in \(L^2_{\mathrm{loc}}\) on compact subslabs, after extracting subsequences if necessary; uniqueness identifies every subsequential limit with the same finite-energy solution. The spacetime terms are therefore lower semicontinuous by weak lower semicontinuity of weighted \(L^2\) norms on each compact radial truncation, followed by monotone convergence over the dyadic exterior annuli.

For the supremum of the slice energies, fix a rational time \(s\in[0,\tau]\). The trace map from the local-energy solution space to the energy space on \(\Sigma_s\) is continuous for smooth solutions and extends by the well-posedness estimate; hence \[\mathcal{E}_{\mathrm{Max}}^{(k)}[G](s) \le \liminf_{n\to\infty}\mathcal{E}_{\mathrm{Max}}^{(k)}[G_n](s).\] Taking the supremum over rational \(s\), and using the energy continuity in time for the finite-energy solution, gives the lower semicontinuity of the energy supremum. The far-field and horizon fluxes are defined by non-negative truncated boundary integrals; Fatou’s lemma and then the truncation limit give the same inequality for them. Summing the finitely many commuted components proves ?? . The argument for the master norm is identical, since its terms have the same Hilbert-space form. ◻

Proof. Let \(G[0]\in\mathcal{H}_{\mathrm{Max},0}^{(k)}(\Sigma_0)\). By Lemma 4 choose smooth charge-free data \(G_n[0]\) with \(G_n[0]\to G[0]\). Applying ?? to \(G_n-G_m\) shows that the corresponding smooth solutions are Cauchy in \(\mathcal{X}^{(k)}_{\mathrm{Max}}(0,\tau)\) for each finite \(\tau\). Their limit is independent of the approximating sequence by the same difference estimate and agrees with the finite-energy solution constructed in Proposition 7. Lemma 51 gives lower semicontinuity of the spacetime norm and therefore gives the bound for the limit. This defines the unique extension of the estimate to the completed charge-free energy space. ◻

Proof. In Proposition 47 the only estimates used are the order-\(k\) master bound, the order-\(k\) initial energy comparison, and the order-\(k\) same-order reconstruction estimate. The commuted equations are closed inside the finite family \(\mathbb{D}_k\), and no term is estimated by invoking an order-\((k+1)\) master norm. Thus, the Maxwell conclusion is at the same order as the assumed master estimate. ◻

Proof. The constant in ?? is obtained from finitely many constants: the master estimate constant, the reconstruction and energy-comparison constants, the trace constants, and the foliation constants used to compare regular frames. If these constants are locally bounded on a parameter set, then any finite product or sum of them is locally bounded. The estimate is therefore uniform on such a set. ◻

Proof. Let \(u=\mathfrak M G\). The radiation identification in Definition 10 gives \(\mathscr S_{\mathrm{Max}}^{\pm}G[0]=\mathcal{R}_\infty^{\pm}\mathscr S_M^{\pm}u[0]\). Hence \[\|\mathscr S_{\mathrm{Max}}^{\pm}G[0]\|_{\mathcal{R}_{\mathrm{Max},\pm}^{(k)}} \le C_\infty \|\mathscr S_M^{\pm}u[0]\|_{\mathcal{R}_{M,\pm}^{(k)}} \le C_\infty C_T \mathcal{E}_M^{(k)}[u](0)^{1/2}.\] The initial energy comparison in 48 then bounds this by \(C\mathcal{E}_{\mathrm{Max}}^{(k)}[G](0)^{1/2}\). Applying the same inequality to differences gives a continuous extension from smooth charge-free data to the completed charge-free energy space. ◻

Proof. Fix \(\sigma\in E\). Let \(\sigma_T\) be smooth cutoffs of the radiation data on the finite portions \(\mathscr I^+_T\cup\mathcal{H}^+_T\) of future null infinity and the future event horizon, chosen so that \(\sigma_T\to\sigma\) in \(\mathcal{R}_{M,+}^{(k)}\). On the truncated exterior region bounded by \(\Sigma_0\), \(\mathscr I^+_T\), \(\mathcal{H}^+_T\) and a final spacelike cap, solve the backward characteristic problem with boundary radiation data \(\sigma_T\) and zero data on the final cap away from the two null ends. Symmetric-hyperbolic energy estimates on the truncated region yield \[\label{eq:finite95slab95backward95bound} \mathcal{E}_M^{(k)}[u_T](0)+\|u_T\|_{LE^1_{\mathrm{deg},k}([0,T])}^2 \le C\|\sigma_T\|_{\mathcal{R}_{M,+}^{(k)}}^2+C\|\chi u_T\|_{L^2([0,T]H^1)}^2,\tag{121}\] where \(\chi\) is supported in a fixed compact radial set. We remove the compact term. Suppose absorption failed. Then there would be truncated backward solutions \(u_{T_n}\) with radiation norm tending to zero and \[\label{eq:backward95compact95normalization} \|\chi u_{T_n}\|_{L^2([0,T_n]H^1)}=1, \qquad \mathcal{E}_M^{(k)}[u_{T_n}](0)+\|u_{T_n}\|_{LE^1_{\mathrm{deg},k}}^2=O(1).\tag{122}\] Choose a time cutoff \(\eta_n\) equal to one on the middle half of \([0,T_n]\) and with derivatives supported in the two end quarters, and set \(w_n=\eta_nu_{T_n}\). The equation for \(w_n\) has source \([\mathcal{P}_{a,Q},\eta_n]u_{T_n}\) plus the vanishing boundary radiation error. After discarding the end intervals and applying Plancherel in the stationary time variable, there are real frequencies \(\omega_n\) and compatible frozen profiles \(v_n\) with a nonzero compact normalization. Normalize the selected packet according to its total stationary frequency \[\label{eq:backward95total95frequency95scale} \Lambda_n=1+|\omega_n|+\lambda_n.\tag{123}\] If \(\Lambda_n\) is bounded, set \[\label{eq:backward95frequency95defect95bf} \|v_n\|_{H^1(K_0)}=1, \qquad \mathcal{L}_{a,Q}(\omega_n)v_n\to0 \quadin H^{-1}_{\sigma,\mathrm{loc}}.\tag{124}\] If \(\Lambda_n\to\infty\), set \(h_n=\Lambda_n^{-1}\), \(\hat{\omega}_n=h_n\omega_n\), and make the same dyadic conic selection as in Lemma 20. The selected packet is normalized by \[\label{eq:backward95frequency95defect95hf} \|v_n\|_{H^1_{h_n}(K_0)}=1,\tag{125}\] and its residual satisfies \[\label{eq:backward95frequency95defect95hf95residual} h_n^{-1}\log(1/h_n) \|h_n^2\mathcal{L}_{a,Q}(\omega_n)v_n\|_{L^2_{\mathrm{comp}}} +\|h_n^2\mathcal{L}_{a,Q}(\omega_n)v_n\|_{H^{-1}_{h_n,\mathrm{loc}}} \to0.\tag{126}\] The outgoing/ingoing Sommerfeld convention is inherited from the prescribed null data and the zero final cap. In the bounded-total-frequency branch, the closedness of the compatible radiation class, Proposition 45, Lemma 17, and Lemma 47 rule out 124 . In the unbounded branch, angularly elliptic packets are controlled by Lemma 17. On the remaining conic packets, 126 is precisely the residual condition of Lemma 43; using ?? , that lemma forces \(\|v_n\|_{H^1_{h_n}(K_0)}\to0\), contradicting 125 . In turn, the compact term in 121 is absorbed, and \[\label{eq:backward95uniform95slab95bound} \mathcal{E}_M^{(k)}[u_T](0)+\|u_T\|_{LE^1_{\mathrm{deg},k}([0,T])}^2 \le C\|\sigma_T\|_{\mathcal{R}_{M,+}^{(k)}}^2\tag{127}\] with \(C\) independent of \(T\) and of the cutoff.

The uniform bound allows \(T\to\infty\) along a weakly convergent subsequence in the energy space and strongly on compact subregions by local compactness. The limit \(u\) is a global compatible solution, its future radiation trace is \(\sigma\) because the fluxes through \(\mathscr I^+_T\) and \(\mathcal{H}^+_T\) converge to the prescribed data and the cap flux tends to zero, and 127 gives ?? . This defines \(\mathscr W_0\sigma=u\) on the dense class \(E\). Linearity follows from uniqueness of the truncated problem and passage to the limit. The kernel condition needed to pass from a dense right inverse to a two-sided inverse is Lemma 46 in case (a) and Proposition 24 in case (b). Therefore the abstract Hilbert-space criterion, Lemma 50, extends \(\mathscr W_0\) continuously to \((\mathscr S_M^+)^{-1}\) with norm at most \(C_W\). The past construction is the same with the incoming characteristic problem and the incoming limiting-absorption resolvent. Thus (A5) is a consequence of the real-axis limiting-absorption resolvent and the real-frequency exclusion, rather than an independent condition. ◻

Proof. Boundedness is Proposition 50. Injectivity: if \(\mathscr S_{\mathrm{Max}}^+G[0]=0\) then, with \(u=\mathfrak M G\) and \(\mathcal{R}_\infty^+\) an isomorphism, \(\mathscr S_M^+u[0]=0\); the master radiation map is injective, so \(u[0]=0\) and the inverse identity gives \(G=\mathfrak R u=0\). Surjectivity: for \(\rho_+\in\mathcal{R}_{\mathrm{Max},+}^{(k)}\) set \(\sigma_+=(\mathcal{R}_\infty^+)^{-1}\rho_+\), \(u[0]=\mathscr W_M^+\sigma_+\); then \(G=\mathfrak R u\) has \(\mathscr S_{\mathrm{Max}}^+G[0]=\mathcal{R}_\infty^+\sigma_+=\rho_+\). The inverse \((\mathscr S_{\mathrm{Max}}^+)^{-1}=\mathfrak R\,\mathscr W_M^+(\mathcal{R}_\infty^+)^{-1}\) is bounded; the past case is identical and the scattering operator is their composition. ◻

Proof. The charge map \(F\mapsto(q_E[F],q_B[F])\) is conserved and continuous, and Theorem 1 gives the bounded splitting of the Cauchy datum into its stationary charge part and charge-free radiative part. On the radiative part, Theorem 3 gives a bounded isomorphism between Cauchy data and future, respectively past, radiation data. Taking the direct sum of the identity map on the two-dimensional charge sector with this radiation isomorphism gives the asserted bounded isomorphism for general finite-energy fields. ◻

Proof. The local-energy part of the estimate gives \[\int_T^{2T}\!\int_{\Sigma_s\cap\{r\le R\}} |G|^2\,\mathrm d\mu_{\Sigma_s}\,\mathrm ds \le C_R\mathcal{E}_{\mathrm{Max}}^{(k)}[G](0).\] If the asserted bound failed for every \(s\in[T,2T]\), the left-hand side would be strictly larger than \(T\cdot C_RT^{-1}\mathcal{E}_{\mathrm{Max}}^{(k)}[G](0)\) after increasing the constant slightly, contradicting the displayed integral estimate. Therefore at least one time \(\tau_T\in[T,2T]\) satisfies the stated compact decay bound. ◻

Proof. Cover \(\Sigma_\tau\) by finitely many compact coordinate patches, horizon red-shift patches, and dyadic annuli \(r\simeq R\) in the asymptotic region. On compact and horizon patches, the derivatives controlled in 49 include enough tangential and transversal derivatives for the ordinary Sobolev embedding on a three-dimensional slice, so the \(L^\infty\) norm is bounded by the square root of the commuted \(L^2\) energy, giving the factor \((1+\tau)^{-\gamma/2}\).

On a dyadic annulus write \(r=R\rho\) with \(\rho\in[1,2]\). The scale-invariant Sobolev inequality on the rescaled annulus gives \(\|G\|_{L^\infty(r\simeq R)}\lesssim R^{-1}\) times the appropriate rescaled \(H^2\) norm. The weighted far-field part of the hierarchy controls this rescaled norm uniformly in \(R\), producing the radial weight \(w(r)\); for the usual one radial weight and two angular derivatives, \(w(r)\simeq(1+r)^{-1}\). Combining the patch estimates gives ?? . ◻

Proof. Write the initial data as \(U=(E,B)\) on the regular slice \((\Sigma_0,h)\). The constraints are \[\label{eq:component95constraint95density95start} \operatorname{div}_hE=0,\qquad \operatorname{div}_hB=0\tag{128}\] in the distributional sense. Let \(\chi_j\) be radial cutoffs equal to one on \(\{r\le j\}\) and supported in \(\{r\le2j\}\), and mollify in a finite collection of regular coordinate charts after parallel transport to the tangent bundle. This produces smooth compactly supported pairs \((E'_j,B'_j)\) with \((E'_j,B'_j)\to(E,B)\) in the unconstrained \(H^k\) energy norm. The divergence errors \(f^E_j=\operatorname{div}_hE'_j\) and \(f^B_j=\operatorname{div}_hB'_j\) tend to zero in \(H^{k-1}_{\mathrm{loc}}\) and in the corresponding weighted dual norm. Moreover their total integrals over each large compact exhaustion vanish after subtracting a smooth compactly supported bump in the outer annulus, because the limiting fields satisfy 128 .

On a smooth exhaustion \(\Omega_j\Subset\Sigma_0\) containing the support of \((E'_j,B'_j)\) solve the Neumann problems \[\label{eq:component95neumann95projection} \Delta_h\phi^E_j=f^E_j,\qquad \Delta_h\phi^B_j=f^B_j, \qquad \partial_\nu\phi^E_j=\partial_\nu\phi^B_j=0\quadon \partial\Omega_j,\tag{129}\] with zero mean. Local elliptic estimates on the uniformly regular exhaustion and the weighted Hardy inequality on the asymptotic end imply \[\label{eq:component95projection95small} \|\nabla_h\phi^E_j\|_{H^k}+\|\nabla_h\phi^B_j\|_{H^k} \le C\bigl(\|f^E_j\|_{H^{k-1}_{\mathrm{w}}} +\|f^B_j\|_{H^{k-1}_{\mathrm{w}}}\bigr)\longrightarrow0.\tag{130}\] Set \[\label{eq:component95projected95data} \widehat E_j=E'_j-\nabla_h\phi^E_j, \qquad \widehat B_j=B'_j-\nabla_h\phi^B_j.\tag{131}\] Then \(\operatorname{div}_h\widehat E_j=\operatorname{div}_h\widehat B_j=0\) on \(\Omega_j\), the normal boundary condition lets the fields be extended by zero without introducing a boundary divergence, and 130 gives \(\widehat U_j=(\widehat E_j,\widehat B_j)\to U\) in \(\mathcal{H}_{\mathrm{Max}}^{(k)}\). This is the slice Hodge projection onto the closed constraint subspace, written explicitly.

The charge functionals are continuous by Proposition 3; hence \(q_E(\widehat U_j)\to q_E(U)=0\) and \(q_B(\widehat U_j)\to q_B(U)=0\). Define \[\label{eq:component95charge95projection} U_j=\widehat U_j-q_E(\widehat U_j)U_e-q_B(\widehat U_j)U_m.\tag{132}\] The stationary data \(U_e,U_m\) are smooth finite-energy constrained data and are normalized by ?? . Hence \(U_j\) is smooth, constrained and charge-free. The finite-rank stationary projection is bounded, so \[\label{eq:component95density95final95bound} \|U_j-U\|_{\mathcal{H}_{\mathrm{Max}}^{(k)}} \le \|\widehat U_j-U\|_{\mathcal{H}_{\mathrm{Max}}^{(k)}} +C\bigl(|q_E(\widehat U_j)|+|q_B(\widehat U_j)|\bigr)\longrightarrow0.\tag{133}\] The convergence of the corresponding solutions on finite slabs follows by applying the well-posedness estimate ?? to the difference of two solutions. ◻

Proof. After writing \(\gamma_{AB}=r^2(\gamma_{\mathbb{S}^2,AB}+O(r^{-2}))\) on the asymptotic end and using a finite atlas on compact \(r\)-regions, the estimate is the usual elliptic estimate for the Hodge Laplacian on \(\mathbb{S}^2\). The zero-mean condition removes the constants in the scalar case, and the absence of harmonic one-forms on \(S^2\) removes the Hodge kernel in the one-form case. The first positive eigenvalue is bounded away from zero uniformly on the parameter range; for the round metric it is \(2\) for one-forms and \(2\) for scalar functions with \(\ell\ge1\). Commuting the Hodge system with angular derivatives and summing gives ?? ; the perturbation of the sphere metric is absorbed into the left side by the same uniform elliptic estimate on the compactified family of metrics. ◻

Proof. The null Maxwell equations in the same regular frame used in Section 7.6 have the form \[\begin{align} \label{eq:null95middle95equations95component} e_4\varphi+\operatorname{tr}\chi\,\varphi &=\mathop{\mathrm{div}\mkern-16mu/\,}\nolimits\alpha-i\mathop{\mathrm{curl}\mkern-19mu/\,}\nolimits\alpha+A_4^0\varphi+A_4^1\cdot\alpha,\nonumber\\ e_3\varphi+\operatorname{tr}\underline\chi\,\varphi &=-\mathop{\mathrm{div}\mkern-16mu/\,}\nolimits\underline\alpha-i\mathop{\mathrm{curl}\mkern-19mu/\,}\nolimits\underline\alpha +A_3^0\varphi+A_3^1\cdot\underline\alpha,\nonumber\\ \mathop{}\!\nabla\mkern-13mu/\,\rho_G+{}^\star\!\mathop{}\!\nabla\mkern-13mu/\,\sigma_G &=-e_3\alpha+B_3^1\cdot\alpha+B_3^2\cdot\underline\alpha+B_3^3\varphi \nonumber\\ &=e_4\underline\alpha+B_4^1\cdot\alpha+B_4^2\cdot\underline\alpha+B_4^3\varphi. \end{align}\tag{134}\] All coefficients are smooth in the regular exterior, bounded on compact radial sets, and short-range in the asymptotic end; in the slow-weak range the difference from the Reissner-Nordström coefficients is perturbative. Charge subtraction gives zero spherical mean for both \(\rho_G\) and \(\sigma_G\) by Proposition 9. Applying Lemma 53 to the scalar functions \(\rho_G\) and \(\sigma_G\) on each \(S_{t,r}\) and using the third line of 134 controls the angular part of \(\varphi\) by one regular null derivative of the extreme components plus lower-order terms: \[\label{eq:component95angular95middle95bound} \|\nabla\mkern-13mu/\,\varphi\|_{LE^0_k} \le C\bigl(\|\alpha\|_{LE^1_k} +\|\underline\alpha\|_{LE^1_k} +\|\varphi\|_{LE^0_k}\bigr).\tag{135}\] The first two lines of 134 , integrated along the outgoing and incoming null directions after the fixed rescaling \(r^2\varphi\), yield the transport bound \[\label{eq:component95transport95middle95bound} \|e_4\varphi\|_{LE^0_k}+\|e_3\varphi\|_{LE^0_k}+\|\varphi\|_{LE^0_k} \le C\bigl(\|\alpha\|_{LE^1_k} +\|\underline\alpha\|_{LE^1_k} +\|G\|_{E_k(\tau_1)}\bigr) +C\eta\|\varphi\|_{LE^1_k},\tag{136}\] where \(\eta=|a|/M\). The zero-mean Poincaré inequality on the spheres and the Hardy inequality in the radial variable control the lower-order \(\|\varphi\|_{LE^0_k}\) terms by the left side away from the initial slice; the initial trace is included in \(\|G\|_{E_k(\tau_1)}\). Taking \(\varepsilon_a(k)\) so that \(C\eta<1/2\) absorbs the perturbative contribution in 136 . Combining 135 and 136 proves ?? . The weights relating \((\alpha,\underline\alpha)\) to \((\psi_+,\psi_-)\) are smooth, nonzero in the regular frame, and have bounded derivatives of the required order, so they do not change the derivative count. ◻

Proof. Let \(G_j\) be the smooth charge-free approximants of Lemma 52. The bounds 48 and ?? show that \(\mathfrak M G_j\) is Cauchy in the master energy space and that \(\mathfrak R\mathfrak M G_j\) is Cauchy in the Maxwell energy space. The limits are independent of the approximating sequence because the same estimates applied to differences give zero limit when the initial data tend to zero. This defines the continuous extension of \(\mathfrak M\) and of \(\mathfrak R\) on the closure of the compatible smooth class. The identities ?? hold on smooth solutions by Definition 9(A4) and pass to the limit by continuity. ◻

Proof. For smooth charge-free \(G\) the trace is \(\mathscr S_{\mathrm{Max}}^{\pm}G=\mathcal{R}_\infty^{\pm}\mathscr S_M^{\pm}\mathfrak M G\). The master trace bound, the radiation identification bound, and the energy comparison in 48 give \[\|\mathscr S_{\mathrm{Max}}^{\pm}G\|_{\mathcal{R}_{\mathrm{Max},\pm}^{(k)}} \le C\|G\|_{\mathcal{H}_{\mathrm{Max},0}^{(k)}(\Sigma_0)}.\] The map therefore extends by density. Conversely, for smooth radiation data \(\rho\) set \[\mathscr W_{\mathrm{Max}}^{\pm}\rho =\mathfrak R\mathscr W_M^{\pm}(\mathcal{R}_\infty^{\pm})^{-1}\rho.\] The bounds in (A5) and Proposition 55 show that \(\mathscr W_{\mathrm{Max}}^{\pm}\) is bounded on the dense class of smooth radiation data and hence extends to the radiation Hilbert space. The identity \(\mathscr S_{\mathrm{Max}}^{\pm}\mathscr W_{\mathrm{Max}}^{\pm}=I\) holds on the dense class and then by continuity. Injectivity follows from the zero-radiation statement in (A5) after applying the master map. For that reason, the inverse is the continuous extension of the smooth backward construction. ◻

Proof. The first two assertions are purely Maxwellian: the Coulomb representatives are normalized by ?? , and Proposition 5 identifies the finite-energy space with the closure of smooth constrained data with the two fluxes controlled. The closed master map itself is not derived from a Dudley-Finley decoupling assertion; it is precisely the structural condition (A1). Given (A1), Lemma 10, Proposition 14, and Corollary 3 show that the closed operator has the scalar wave principal symbol and the short-range perturbative form required in (A2). Bounded real frequencies are closed by compactness and the Reissner-Nordström limiting-absorption estimate; this is Proposition 45. Unbounded real frequencies are normalized semiclassically by Lemma 38, localized by Lemma 43, and excluded by Proposition 38 after the trapped-set verification and the application statement of Proposition 39. Thus (A3) follows from the normally hyperbolic high-frequency estimate together with the bounded-frequency argument proved in Section 11.

Proposition 26 proves (A4) by the Hodge and null-transport estimates of Section 12, under the same closed-reduction hypothesis that supplies the extreme master variables; the component approximation in Lemma 52 passes the resulting estimates to finite-energy charge-free data. Proposition 51 constructs the dense backward right inverse and then applies the Hilbert-space criterion, Lemma 50. Thus (A5) follows from the same limiting-absorption estimates. The additional pointwise condition (A6) is used only for the decay statement and nowhere in the energy, local-energy, or scattering proof. These are exactly the conditions used in Theorem 4. ◻

Proof. The boundedness constant is obtained by the three-step computation \[\|G\|_{\mathcal{X}^{(k)}_{\mathrm{Max}}}^2 \le C_R\|\mathfrak M G\|_{\mathcal{X}^{(k)}_M}^2 \le C_RC_M\mathcal{E}_M^{(k)}[\mathfrak M G](0) \le C_R^2C_M\mathcal{E}_{\mathrm{Max}}^{(k)}[G](0),\] which is Proposition 47. For the trace, write \[\mathscr S_{\mathrm{Max}}^\pm=\mathcal{R}_\infty^\pm\mathscr S_M^\pm\mathfrak M.\] The map \(\mathfrak M\) has energy norm at most \(C_R^{1/2}\) by 48 , giving the second line of ?? . For the inverse wave operator, \[(\mathscr S_{\mathrm{Max}}^\pm)^{-1}=\mathfrak R\mathscr W_M^\pm(\mathcal{R}_\infty^\pm)^{-1},\] and the reconstruction energy bound gives the third line. The scattering operator is the composition \(\mathscr S_{\mathrm{Max}}^+(\mathscr S_{\mathrm{Max}}^-)^{-1}\), giving the fourth line. The charged statement follows from the direct-sum decomposition of Theorem 1. ◻

Proof. The first inequality is the same-order reconstruction estimate in Definition 10(M2). The reason is that 48 is stated in the identical order-\(k\) spacetime norm, so the angular Hodge inversion and the null transport estimates of Proposition 26 do not require an order-\((k+1)\) master bound. The second inequality is the master local-energy estimate (M1) with \(\tau_1=0\), \(\tau_2=\tau\). Its compact remainder has already been removed in Proposition 22: bounded temporal frequencies are excluded by Proposition 45, and unbounded conic packets are excluded by Lemma 43 using exactly ?? . The third inequality is the initial energy comparison in 48 . Multiplying ?? ?? gives \[\|G\|_{\mathcal{X}^{(k)}_{\mathrm{Max}}(0,\tau)}^2 \le C_R^2C_M\mathcal{E}_{\mathrm{Max}}^{(k)}[G](0),\] which is 139 for smooth charge-free data.

Let now \(G_0\in\mathcal{H}_{\mathrm{Max},0}^{(k)}(\Sigma_0)\). Choose smooth charge-free data \(G_{0,n}\) converging to \(G_0\) by Lemma 4, and denote the solutions by \(G_n\). The estimate above applied to differences gives \[\|G_n-G_m\|_{\mathcal{X}^{(k)}_{\mathrm{Max}}(0,\tau)}^2 \le C_R^2C_M\mathcal{E}_{\mathrm{Max}}^{(k)}[G_{0,n}-G_{0,m}](0),\] so \((G_n)\) is Cauchy in the local-energy topology on every finite slab. The finite-energy well-posedness result, Proposition 7, identifies the limit with the Maxwell solution from \(G_0\), and Lemma 51 gives the corresponding bound for the limit. In the final step, if \(F\) has charges, Theorem 1 writes \(F=F_{\mathrm{stat}}^{\mathrm{KN}}+F_{\mathrm{rad}}\); the stationary part is not included in the local-energy decay statement, and \(F_{\mathrm{rad}}\) is charge-free, so the preceding argument applies to it. ◻

Proof. Items (i) and (ii) are the numbered conclusions in Proposition 57 and in the computations of Sections 6-13. The spherical local-energy, red-shift and \(r^p\) estimates are used only through Lemma 13, Proposition 15, and Proposition 16; after these inequalities are fixed, the absorption is the algebraic estimate of Proposition 27. The bounded-frequency real-axis part is closed by Propositions 22 and Proposition 45; a putative compact defect would converge to an outgoing or incoming resonance, which is excluded by Lemmas 46 and 25. The unbounded-frequency part is precisely the semiclassical alternative of Lemma 43; the estimate used there beyond the geometric computations is Proposition 38, which gives ?? . Same-order reconstruction is Proposition 26, and the radiation isomorphism is obtained from Proposition 51 and the Hilbert criterion Lemma 50. These alternatives cover every implication in the proof of Theorem 4; the hierarchy (A6) is cited only in Proposition 53. ◻

Proof. The decomposition is defined by 3 . Proposition 4 gives that \(F_{\mathrm{rad}}\) is source-free and has zero electric and magnetic charges. If two decompositions existed, their difference would be a stationary representative \(q_EF_e+q_BF_m\) with both charges zero; the normalization ?? forces \(q_E=q_B=0\), proving uniqueness.

Proposition 31 converts conditions (A1)-(A5) of Definition 9 into the analytic conclusions of Definition 10. Hence the master variables of \(F_{\mathrm{rad}}\) obey the master estimate and reconstruct the Maxwell tensor at the same order. Proposition 47, with the constant calculation made explicit in Proposition 58, then gives the energy and integrated local-energy bound 139 ; Proposition 48 extends the estimate from smooth charge-free data to the finite-energy completion. Proposition 50 gives the radiation trace bounds. Theorem 3 gives the bounded inverse wave operators and the scattering map by composing the master radiation isomorphism with the reconstruction isomorphisms. If the additional hierarchy condition (A6) is available, Proposition 53 applied to \(G=F_{\mathrm{rad}}\) gives ?? whenever the hierarchy has enough derivatives for Sobolev embedding. ◻

Proof. The corollary keeps the only non-Maxwell structural condition explicit. The closed master map and its compatible class are assumed in (A1). Once that map is present, Lemma 10, Proposition 14, Corollary 3, and Proposition 10 verify the scalar principal symbol, energy comparison, and short-range Reissner-Nordström perturbation structure. The bounded-frequency part of the real-axis exclusion is Proposition 45 together with Lemma 47. The high-frequency part of (A3) is the localized normally hyperbolic estimate, Proposition 38, applied through Proposition 39: Proposition 43 verifies the trapped-set geometry, Lemma 39 and Proposition 37 verify the finite-rank-bundle skew-subprincipal threshold, and Proposition 21 keeps the defect profiles in the closed compatible class. Proposition 26 proves (A4), Proposition 51 derives (A5), and Theorem 4 applies. For \(Q=0\), the Teukolsky reduction and the high-frequency estimate are also supplied by [7], [10], [21], [22]. ◻

Proof. The constants in the final estimate are built from a finite list: the constants in the Reissner-Nordström model estimate, the red-shift and far-field constants, the Hardy and elliptic constants, the perturbative absorption thresholds, and the reconstruction and trace bounds. These constants depend continuously, or upper semicontinuously after taking finite suprema, on \((a,Q)\) as long as the slow-weak inequalities are strict. On compact subsets of that range the finite list is bounded, and the formula in Proposition 58 gives uniform bounds for the Maxwell estimates and operator norms. ◻

Proof. Let us set \(Q=0\). Then the Reissner-Nordström comparison model becomes Schwarzschild, and all charged spherical terms in the perturbation formulas vanish. The stationary charge sector reduces to the electric and magnetic Kerr Coulomb representatives. The fixed-background Kerr Maxwell theory cited in the statement supplies the spin-one estimates, real-axis exclusion, wave operators and same-order reconstruction needed in Definition 9. Substitution of these estimates into Theorem 4 gives the claimed slowly rotating Kerr result. ◻

Proof. The coupled system has unknowns \((\dot{g},\dot{F})\), gauge freedom, constraints, pure-gauge modes, and linearized stationary Kerr-Newman modes. The fixed-background equation has only \(F\), its two charges, and its reconstruction problem. Setting \(\dot{g}=0\) does not recover free Maxwell, since the linearized Einstein equation constrains the variation of the electromagnetic energy-momentum tensor. A coupled theorem is useful here only after one constructs the fixed-background master map, proves the master estimate, excludes the fixed-background kernel, and reconstructs \(F\) at the same order, which are the tasks of Sections 7-15. ◻