A charged enclosing-area bound for weakly asymptotically Euclidean exteriors

A charged enclosing-area bound for weakly asymptotically Euclidean exteriors

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: A combined manuscript claims neutral three- and four-dimensional bounds, three-dimensional charged upper-area bounds, original-data rigidity, and a local anti-de Sitter perturbation inequality; For one-ended three-dimensional source-free electric--magnetic data, the proposed sharp bound is $m\geq Q$ and $r_A\leq m+\sqrt{m^2-Q^2}$, with Reissner--Nordstrom rigidity in the stated horizon subclass. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.

Let $\Omega^n$, $n\ge3$, be smooth, connected and orientable, with nonempty compact smooth boundary $S$, complete with that boundary included, and exactly one end, with Euclidean coordinates and compact complement. The tensors $g,K$ extend smoothly to $S$. With $\tau=\operatorname{tr}_gK$, use \[ 2\mu=R_g+\tau^2-|K|_g^2,\qquad J=\operatorname{div}_g(K-\tau g),\qquad \mu\ge|J|_g,\quad\mu,|J|_g\in L^1(\Omega,dV_g). \] The finite ADM fluxes use $\omega_{n-1}=|\mathbb S^{n-1}|$ and \[ E=\frac1{2(n-1)\omega_{n-1}}\lim_{R\to\infty}\int_{|x|=R} (\partial_jg_{ij}-\partial_ig_{jj})n_\delta^i\,dA_\delta,\quad P_i=\frac1{(n-1)\omega_{n-1}}\lim_{R\to\infty}\int_{|x|=R}(K_{ij}-\tau g_{ij})n_\delta^j\,dA_\delta. \] Orient boundary normals towards the end, with $\theta_+=H+\operatorname{tr}_SK$. Here $O_j(r^{-b})$ controls coordinate derivatives through order $j$, of respective sizes $O(r^{-b-i})$. Choose a connected smooth end-side region $D\subset\Omega$, closed in $\Omega$, containing the distant end and having its manifold interior in $\operatorname{int}\Omega$. A full cut consists of all of its intrinsic boundary $\Gamma=\partial D$; require this boundary to be compact, smoothly embedded and two-sided. All components and portions coinciding with $S=\partial\Omega$ count. Put $a_g(S)=\inf_\Gamma\operatorname{Area}_g(\Gamma)$. In dimension three, suppose $g-\delta=O_2(r^{-q})$, $K=O_1(r^{-1-q})$ for $q>1/2$, and $\theta_+\le0$ on every boundary component. The neutral result gives $\sqrt{E^2-|P|^2}\ge\sqrt{a_g(S)/(16\pi)}$ when $E>|P|$, and a separate endpoint argument proves $E>|P|$ without imposing that sign initially. In dimension three on the same smooth one-ended complete exterior, let the electric and magnetic fields $\mathcal E,\mathcal B$ be smooth, divergence free, and $O_1(r^{-2})$. Suppose $g-\delta=O_2(r^{-q})$, $K=O_1(r^{-1-q})$ for $q>1/2$, finite ADM and charge fluxes, and integrable geometric densities $2\mu=R_g+\tau^2-|K|^2$, $J=\operatorname{div}_g(K-\tau g)$. The non-electromagnetic matter condition is \[ \mu_m=\mu-|\mathcal E|^2-|\mathcal B|^2\ge|J_m|,\qquad J_m=J-2(\mathcal E\times\mathcal B)^\flat. \] Set $Q_E=(4\pi)^{-1}\lim_R\int_{|x|=R}g(\mathcal E,\nu_R)dA_g$, define $Q_B$ similarly, and put $Q=\sqrt{Q_E^2+Q_B^2}$, $r_A=\sqrt{a_g(S)/(4\pi)}$. If $\theta_+\le0$ on the entire boundary, the charged conclusion is \[ E>|P|,\qquad m=\sqrt{E^2-|P|^2}\ge Q,\qquad r_A\le m+\sqrt{m^2-Q^2}. \] For $r_A>Q$ this is equivalent to $m\ge(r_A+Q^2/r_A)/2$; for $0<r_A\le Q$, the mass-charge bound is the active restriction. The combined paper's numerical charged theorem does not assert unrestricted equality rigidity. There is also a complete-data formulation on a smooth connected orientable boundaryless three-manifold $M$, with complete $g$, a compact core and finitely many AF ends. Each end has $g-\delta=O_2(r^{-q})$, $K=O_1(r^{-1-q})$, $q>1/2$, and finite ADM limits; the constraint densities are globally integrable and satisfy DEC. Choose an end-side open region $D^+$, possibly disconnected or containing other ends, whose full boundary $S$ is nonempty, compact, smooth, embedded and two-sided, with $\theta_+\le0$ towards $D^+$. Define \[ A_{\min}(S)=\inf_{D'^+\subseteq D^+}\operatorname{Area}_g(\partial D'^+), \] where smooth open competitors contain the distant designated end and have compact smooth full boundary, permitting disconnected competitors, other retained ends, and coincidence with $S$. The conclusion is $E>|P|$ and $\sqrt{E^2-|P|^2}\ge\sqrt{A_{\min}(S)/(16\pi)}$. In the three-dimensional complete-data equality theorem, use a smooth connected orientable boundaryless $M$ with complete metric, compact core and finitely many AF ends, each obeying $O_2/O_1$ decay with $q>1/2$, finite ADM limits, integrable densities and DEC. Take $S=\bigsqcup_{i=1}^{\ell}S_i$, $1\le\ell<\infty$, with disjoint connected compact smooth two-sided components bounding a connected closed exterior $\Omega$ with exactly the designated end. Require $\theta_+(S_i)=0$ and $\operatorname{Area}_g(\Gamma)\ge A=\sum_i|S_i|_g>0$ for every full enclosing cut. Exclude distinct enclosing cuts whose components each coincide with an entire $S_i$ or lie wholly outside, with at least one component of the latter kind, and with $\theta_+\le0$ on all components. If $E>|P|$ and $\sqrt{E^2-|P|^2}=\sqrt{A/(16\pi)}$, then $\ell=1$, $S\cong S^2$, and the entire original exterior admits a proper smooth spacelike Schwarzschild embedding of mass $\sqrt{E^2-|P|^2}$ inducing $g,K$ and the prescribed future normal. Its end reaches spatial infinity and its boundary is a full future-horizon section or bifurcation sphere, with smooth continuation through that boundary. For connected $S$, the connected-horizon corollary replaces the cut exclusion by exclusion of every wholly exterior weakly future-trapped enclosing surface, including disconnected ones. Thus $\ell\ge2$ forces strict inequality in the finite-component subclass.

Result 2.

Let $\Omega^n$, $n\ge3$, be smooth, connected and orientable, with nonempty compact smooth boundary $S$, complete with that boundary included, and exactly one end, with Euclidean coordinates and compact complement. The tensors $g,K$ extend smoothly to $S$. With $\tau=\operatorname{tr}_gK$, use \[ 2\mu=R_g+\tau^2-|K|_g^2,\qquad J=\operatorname{div}_g(K-\tau g),\qquad \mu\ge|J|_g,\quad\mu,|J|_g\in L^1(\Omega,dV_g). \] The finite ADM fluxes use $\omega_{n-1}=|\mathbb S^{n-1}|$ and \[ E=\frac1{2(n-1)\omega_{n-1}}\lim_{R\to\infty}\int_{|x|=R} (\partial_jg_{ij}-\partial_ig_{jj})n_\delta^i\,dA_\delta,\quad P_i=\frac1{(n-1)\omega_{n-1}}\lim_{R\to\infty}\int_{|x|=R}(K_{ij}-\tau g_{ij})n_\delta^j\,dA_\delta. \] Orient boundary normals towards the end, with $\theta_+=H+\operatorname{tr}_SK$. Here $O_j(r^{-b})$ controls coordinate derivatives through order $j$, of respective sizes $O(r^{-b-i})$. Choose a connected smooth end-side region $D\subset\Omega$, closed in $\Omega$, containing the distant end and having its manifold interior in $\operatorname{int}\Omega$. A full cut consists of all of its intrinsic boundary $\Gamma=\partial D$; require this boundary to be compact, smoothly embedded and two-sided. All components and portions coinciding with $S=\partial\Omega$ count. Put $a_g(S)=\inf_\Gamma\operatorname{Area}_g(\Gamma)$. In dimension three on the same smooth one-ended complete exterior, let the electric and magnetic fields $\mathcal E,\mathcal B$ be smooth, divergence free, and $O_1(r^{-2})$. Suppose $g-\delta=O_2(r^{-q})$, $K=O_1(r^{-1-q})$ for $q>1/2$, finite ADM and charge fluxes, and integrable geometric densities $2\mu=R_g+\tau^2-|K|^2$, $J=\operatorname{div}_g(K-\tau g)$. The non-electromagnetic matter condition is \[ \mu_m=\mu-|\mathcal E|^2-|\mathcal B|^2\ge|J_m|,\qquad J_m=J-2(\mathcal E\times\mathcal B)^\flat. \] Set $Q_E=(4\pi)^{-1}\lim_R\int_{|x|=R}g(\mathcal E,\nu_R)dA_g$, define $Q_B$ similarly, and put $Q=\sqrt{Q_E^2+Q_B^2}$, $r_A=\sqrt{a_g(S)/(4\pi)}$. Each boundary component may satisfy either $H+\operatorname{tr}_SK\le0$ or $H-\operatorname{tr}_SK\le0$, with the choice made separately. The numerical assertion is \[ a_g(S)>0,\quad E>|P|,\quad m=\sqrt{E^2-|P|^2}>0,\quad E\ge\sqrt{|P|^2+Q^2},\quad m\ge Q,\quad r_A\le m+\sqrt{m^2-Q^2}. \] The polynomial expression $m\ge(r_A+Q^2/r_A)/2$ applies on $r_A>Q$; at $0<r_A\le Q$ the active condition is $m\ge Q$. In the connected future-horizon subclass, impose $\theta_+=0$, outer area minimization and exclusion of every wholly exterior weakly trapped full cut. If $m>Q$ and $\sqrt{A/(4\pi)}=m+\sqrt{m^2-Q^2}$, the entire original data, including fields, arise from a smooth spacelike slice of dyonic Reissner--Nordstrom with parameters $(m,Q_E,Q_B)$ and smooth future-horizon attachment. Matter densities vanish on that exterior. Conversely admissible model slices with actual mass $M>\sqrt{q_E^2+q_B^2}$ and charges $(q_E,q_B)$ attain \[ a_g(S)=|S|_g=4\pi\left(M+\sqrt{M^2-q_E^2-q_B^2}\right)^2. \] Extremal equality is not classified.

Outline of the argument

Prepared-end graph deformations protect the minimum area and compare invariant masses. Conformal flow supplies the Riemannian step, electromagnetic flux converts charge into the scalar comparison, and equality conditions reconstruct the original static spacetime data; the anti-de Sitter result uses a separate local conformal expansion.

A coupled filled scalar deformation retains enclosing area while accounting for electromagnetic flux in the mass balance. The neutral comparison gives the upper-area branch, and saturation of the comparison identities reconstructs a dyonic spacetime slice carrying the given metric and extrinsic curvature.

Formal verification

For family 260, the only selected family entry is the CKS end-replacement construction and Schwarzschild equality examples. A connected-boundary inequality is conditional on the flat Penrose input; the general Bondi or spacetime inequalities are not formalized there. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

• OpenAI, Spacetime Penrose inequalities: enclosing area, charge, and rigidity, 2026-10-05; citation key OAI:Spacetime-Penrose-inequalities-enclosing-area-charge-and-rigidity-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: Theorem 2.2 (n3:intro:exterior-inequality); Statement source: ch:intro:main; Statement source: Corollary 9.4 (bridge:bridge:timelike); Statement source: Theorem 9.2 (bridge:intro:global-penrose); Statement source: Theorem 10.2 (n3:intro:rigidity).
• OpenAI, The spacetime Penrose inequality with charge and original-data rigidity, 2026-10-05; citation key OAI:The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: Theorem 2.3 (thm:numerical); Statement source: Theorem 2.5 (thm:rigidity); Statement source: Theorem 2.6 (thm:converse).
• Family 260: Lean scope.
• OpenAI catalogue entry.