RN horizon and escape deductions for specified model

If $Q^2>0$ and $M^2\ge Q^2$, the quadratic numerator factors as $r^{2}f=(r-r_{+})(r-r_{-})$; both roots are positive because $M>0$ and $\sqrt{M^2-Q^2}<M$. If $Q^2=0$, then $f=1-2M\mathrm{/}r$ and its only positive root is $2M$. If $Q^2>M^2$, then $r^{2}f=(r-M{)}^2+(Q^2-M^2)>0$ for every $r>0$, so there is no positive root.

The inverse of the $v,r$ block in [F1] is $\left(\begin{array}{cc}0& 1\\ 1& f\end{array}\right)$, so $g^{-1}(dr,dr)=f$. The vector $\xi ={\mathrm{\partial }}_v$ is Killing because the metric coefficients do not depend on $v$, and ${\xi }^{\mathrm{♭}}=-fdv+dr$ equals $dr$ at a root. Therefore each root is null and $\xi$ is its null normal. Since ${\xi }^2=-f$, $d({\xi }^2)=-f^{\mathrm{\prime }}(r_H)dr=-2{\kappa }_H{\xi }^{\mathrm{♭}}$ there. For two distinct roots, differentiating $f=(r-r_{+})(r-r_{-})\mathrm{/}{r}^2$ gives $f^{\mathrm{\prime }}(r_{\pm })=(r_{\pm }-r_{\mp })\mathrm{/}{r}_{\pm }^2$; at $Q^2=M^2$ this derivative is zero. For $Q^2=0$, $f^{\mathrm{\prime }}(2M)=1\mathrm{/}(2M)$ and $\kappa =1\mathrm{/}(4M)$. The product definition [F1] keeps the metric nondegenerate at each root.

For $Q^2>0$ and $M^2\ge Q^2$, [L1] gives the claimed root crossing and, when $M^2>Q^2$, the local bifurcation chart. If $Q^2=M^2$, [L1] gives the regular ingoing crossing but explicitly does not assert that simple-root chart. For $Q^2=0$, put $r_H=2M$, $\kappa =1\mathrm{/}(4M)$, and write $f=(r-r_H)b(r)$ with $b(r)=1\mathrm{/}r$ and $b(r_H)=2\kappa >0$. Then $\frac{1}{f}-\frac{1}{2\kappa (r-r_H)}=\frac{2\kappa -b(r)}{2\kappa b(r)(r-r_H)}=-\frac{1}{2\kappa b(r)}{\int }_0^1{b}^{\mathrm{\prime }}(r_H+s(r-r_H))ds,$ so this difference extends smoothly across $r_H$. Integrate it to $h$ with $h(r_H)=0$, and define $K(r)=(r-r_H){\mathrm{\ell }}_{\ast }^{-1}{e}^{2\kappa h(r)}$. Then $K^{\mathrm{\prime }}(r_H)={\mathrm{\ell }}_{\ast }^{-1}>0$ and differentiation gives $f{K}^{\mathrm{\prime }}=2\kappa K$ on both sides and at the root. By [F3], shrink the radial interval so $K^{\mathrm{\prime }}>0$ and $K$ has a smooth local inverse $\rho$ about zero. On the selected ingoing model define $V=e^{\kappa v},U=-K(r)e^{-\kappa v}\mathrm{.}$ Thus $UV=-K(r)$, with $U<0$ for $r>r_H$ and $U>0$ for $r<r_H$, and the Jacobian is $\kappa K^{\mathrm{\prime }}(r)>0$ across the root. The inverse is $r=\rho (-UV)$ and $v={\kappa }^{-1}\log V$ on $V>0$. Differentiating gives $dv=dV\mathrm{/}(\kappa V)$ and $dr=-(VdU+UdV)\mathrm{/}{K}^{\mathrm{\prime }}(r)$. Substitution in the original ingoing block $-fd{v}^2+2dvdr$ cancels its $d{V}^2$ coefficient by $f{K}^{\mathrm{\prime }}=2\kappa K$, leaving $-\frac{1}{\kappa K^{\mathrm{\prime }}(\rho (-UV))}(dU\otimes dV+dV\otimes dU)\mathrm{.}$ This coefficient is smooth and nonzero, and the sphere term is $\rho (-UV{)}^2\gamma$. Choose $\epsilon >0$ so these formulas are defined with $\rho (-UV)>0$ on $(-\epsilon ,\epsilon {)}^2$. They give a smooth Lorentzian metric on the full rectangle times $S^2$, agreeing with the selected ingoing metric on its entire $V>0$ half, including both $r<r_H$ and $r>r_H$. Glue that half to its old coordinate preimage. Any overlap sequence tending to a new point on $V=0$ has $v={\kappa }^{-1}\log V\to -\mathrm{\infty }$, so cannot approach a finite old point; the other rectangle edges are not new points. This gives a Hausdorff local extension. The field is zero in this uncharged case. The old Killing field becomes $\kappa (V{\mathrm{\partial }}_V-U{\mathrm{\partial }}_U)$; it extends smoothly, preserves the displayed metric, is null and normal on either axis away from their intersection, and vanishes on the spacelike sphere $U=V=0$. Thus this is the required two-sided local bifurcation chart.

For $Q^2>0$ and $M^2\ge Q^2$, apply [L2] using the exact observable [L3]. Its current Definition and manifest placement now exist and were checked against this use; the escape conclusion remains conditional on the current draft mathematical proof, with no maximal-extension or conformal-completion inference.

If $Q^2>M^2$, then $f>0$ by step 1.1 and the future null field $n$ of [F1] satisfies $dr\mathrm{/}dv=f\mathrm{/}2>0$. For any starting $r_0>0$, the function $v(r)=v_0+{\int }_{r_0}^{r}2\mathrm{/}f(s)ds$ is smooth and strictly increasing on $[r_0,\mathrm{\infty })$; since $f(r)\to 1$, $v(r)\to +\mathrm{\infty }$ and the corresponding integral curve reaches arbitrarily large $r$. Hence every point belongs to $\mathcal{E}$.

If $Q^2=0$, then $f<0$ below $2M$, $f=0$ at $2M$, and $f>0$ above. The product splitting in [F1] gives a unique decomposition of every tangent as $Z=\alpha \mathrm{\ell }+\beta n+W$ with $W$ tangent to the sphere. Since the sphere term is orthogonal to $\mathrm{\ell },n$ and positive, causality gives $g(Z,Z)=-2\alpha \beta +g(W,W)\le 0$ and hence $\alpha \beta \ge 0$. The timelike condition is $-2\alpha \beta +g(W,W)<0$, so a timelike vector has $\alpha \beta >0$ and its coefficients have the same nonzero sign. Each of the two sign branches is connected: scale $W$ to zero within the timelike cone and then vary the positive (respectively negative) coefficients while keeping them equal. Since $N=\mathrm{\ell }+n$ has coefficients $(1,1)$, the selected future timelike component is the positive branch. Every future causal vector lies in its closure, so $\alpha ,\beta \ge 0$. Therefore $dr(Z)=-\alpha +(f\mathrm{/}2)\beta \le 0$ for $r\le 2M$. On a collar of $2M$, put $\tau =v-r$; for nonzero future causal $Z$, $d\tau (Z)=\alpha +(1-f\mathrm{/}2)\beta \ge \alpha +\beta \mathrm{/}2>0$ after shrinking the collar so $\mathrm{\mid }f\mathrm{\mid }<1$. Smoothness and $f(2M)=0$ give $C<\mathrm{\infty }$ with $f(r)\le C\mathrm{\mid }r-2M\mathrm{\mid }$ there, hence $dr(Z)\le C\mathrm{\mid }r-2M\mathrm{\mid }d\tau (Z)$. On each $C^1$ path piece in the collar we may use $\tau$ as parameter. If $r<2M$, then $w=2M-r$ obeys $dw\mathrm{/}d\tau \ge -Cw$, so $d(e^{C\tau }w)\mathrm{/}d\tau \ge 0$ and $w(\tau )\ge w({\tau }_0)e^{-C(\tau -{\tau }_0)}>0$ for every finite $\tau$; a path starting below the root cannot reach it. If a path starts at the root, $u=(r-2M{)}_{+}$ obeys $du\mathrm{/}d\tau \le Cu$ almost everywhere on the collar: for $r>2M$ this is the displayed cone bound, for $r<2M$ its derivative is zero, and on the root the cone bound gives $dr\mathrm{/}d\tau \le 0$. Thus $d(e^{-C\tau }u)\mathrm{/}d\tau \le 0$ and $u({\tau }_0)=0$ forces $u=0$, so no outward excursion starts at the root. The same estimates apply across finitely many piece joins by continuity. Every point with $r>2M$ escapes along the future null field $n$, because $dr\mathrm{/}dv=f\mathrm{/}2>0$ and $f(r)\to 1$. Thus the uncharged escape boundary is $r=2M$.

Steps 1.1-2.5 prove the stated geometric roots, signed surface-gravity convention, regular-crossing cases, and selected escape conclusions on the chosen product, conditional on the exact current draft mathematical interfaces [L1]-[L3] and on the equation-based physical-model input [F2]. Their authoring-ready records do not constitute independent proof acceptance. The inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ premise [A2] is stated without invoking full choice; no event-horizon, maximal-extension, or empirical claim follows.