Prove the Penrose inequality or present a counterexample

Prove the Penrose inequality or present a counterexample

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: In dimensions three and four, trapped exteriors are claimed deformable to nonnegative-scalar-curvature metrics with protected enclosing area and arbitrarily small ADM-energy loss; The numerical Riemannian mass-area inequality is claimed in every dimension for full outer-minimizing, locally perimeter-minimizing frontiers, including singular and disconnected ones; Equality in the neutral spacetime bound is claimed to reconstruct the original exterior as a Schwarzschild--Tangherlini slice under the precise horizon and decay assumptions; 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 the designated axisymmetric electrovacuum class, the claimed bound is $m^2\geq A/(16\pi)+Q^2/2+\pi(Q^4+4J^2)/A$ with conserved total angular momentum; The proposed neutral spacetime Penrose inequality compares invariant ADM mass with minimum enclosing area in all dimensions $n\geq3$, under the listed energy, decay, and weak future-trapping hypotheses; 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 provide the following support within the scopes described below.

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. If $E>|P|$, the exterior theorem asserts \[ \sqrt{E^2-|P|^2}\ge\sqrt{a_g(S)/(16\pi)}. \] The preparatory output treats the stronger subclass with compactly supported $K$, $g-\delta=O_j(r^{-1})$ for every fixed $j$, $P=0$, $E>0$, $\theta_+<0$, $A_*=a_g(S)>0$, and, for a smooth radius extension $\varrho\ge1$, \[ R_g=O(r^{-3-\delta}),\qquad \mu-|J|_g\ge c\varrho^{-3-\delta}\quad(c>0,\ 0<\delta<1). \] For sufficiently small fixed $\epsilon>0$ and large $N$, a smooth one-ended subexterior has a complete boundary-included metric $\widehat g_{\epsilon,N}$ satisfying \[ R_{\widehat g}\ge0,\quad H_{\partial\Omega}(\widehat g)<0,\quad \widehat g\ge e^{-4\epsilon}g,\quad E_{\widehat g}\le E+\eta_{\epsilon,N},\quad |T|_{\widehat g}\ge e^{-4\epsilon}A_* \] for every full cut $T$. At fixed $\epsilon$, $0\le\eta_{\epsilon,N}\le C(1+N)^a(e^{-\epsilon N}+e^{-\epsilon N/2})\to0$. The output metric has $O_2(r^{-1})$ decay and scalar curvature $O(r^{-3-\delta'})$ for some $\delta'>0$.

Result 2.

For $n\ge3$, let $(E,h)$ be a connected complete exterior with one Euclidean end and compact complement, including its complete possibly singular frontier $\Sigma$. Completeness is measured within each finite-distance class of its intrinsic length metric. The metric extends smoothly through the frontier and satisfies \[ R_h\ge0,\quad R_h\in L^1(E,dV_h),\quad R_h=O(r^{-n-\beta})\ (\beta>0),\qquad h-\delta=O_2(r^{-\tau})\ (\tau>(n-2)/2). \] The full frontier is the perimeter support of a filled side, outer minimizing and locally perimeter minimizing on both sides. Its regular part is smooth, embedded and minimal; its singular part satisfies $\dim_{\mathrm H}\Sigma_{\mathrm{sing}}\le n-8$. With $\omega_{n-1}=|\mathbb S^{n-1}|$ and the usual ADM energy normalization $[2(n-1)\omega_{n-1}]^{-1}$, \[ m_{\mathrm{ADM}}(h)\ge\frac12\left(\frac{\mathcal H_h^{n-1}(\Sigma)}{\omega_{n-1}}\right)^{(n-2)/(n-1)}. \] Every frontier component contributes to the area; no spin or frontier-connectedness hypothesis is imposed.

Result 3.

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 $a_g(S)=\inf_\Gamma\operatorname{Area}_g(\Gamma)$. Suppose $g-\delta=O_6(r^{-q})$, $K=O_5(r^{-1-q})$, $(n-2)/2<q<n-2$, and $E>|P|$. Require connected $S$, $\theta_+(S)=0$, and $\operatorname{Area}_g(\Gamma)\ge A=|S|_g>0$ for every full cut. Exclude every compact smooth embedded two-sided full enclosing hypersurface wholly in $\operatorname{int}\Omega$ with $\theta_+\le0$, allowing disconnected excluded surfaces. If \[ m=\sqrt{E^2-|P|^2}=\frac12(A/\omega_{n-1})^{(n-2)/(n-1)}, \] then the entire original pair $(g,K)$, including the boundary and future normal, is induced by a smooth spacelike embedding into Schwarzschild--Tangherlini of mass $m$, using its regular extension across the future horizon. The interior lies in its domain of outer communication, the end reaches spatial infinity, and $S$ maps to a full future-horizon section or the bifurcation sphere. Conversely, admissible spacelike model slices with these same intrinsic hypotheses attain equality and have $A=\omega_{n-1}(2m)^{(n-1)/(n-2)}$. Time-symmetric and nonzero-$K$ examples exist in each $n\ge3$. 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 4.

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 5.

Let the smooth oriented connected exterior be diffeomorphic to $\mathbb R^3$ minus an open ball, with spherical smooth compact boundary $S$, complete metric including $S$, a single AF end, and smooth $g,K,\mathcal E,\mathcal B$. A smooth effective $U(1)$ action preserves all data and the boundary, with rotation generator of period $2\pi$ agreeing with ordinary axial rotation at infinity. Impose source-free electrovacuum constraints \[ 16\pi\mu=R_g+(\operatorname{tr}_gK)^2-|K|^2, \quad 8\pi J_{\rm con}=\operatorname{div}_g(K-(\operatorname{tr}_gK)g), \] \[ 8\pi\mu=|\mathcal E|^2+|\mathcal B|^2,\quad 4\pi J_{\rm con}=(\mathcal E\times\mathcal B)^\flat,\quad \operatorname{div}_g\mathcal E=\operatorname{div}_g\mathcal B=0. \] Use $g-\delta=O_4(r^{-1})$, $K=O_3(r^{-3})$, Coulomb leading fields $\mathcal E^i=Q_e x^i/r^3+O_3(r^{-3})$, $\mathcal B^i=Q_b x^i/r^3+O_3(r^{-3})$, integrable constraints and finite ADM energy, momentum, angular momentum and charge fluxes. Require $P_{\rm ADM}=0$, $m=E_{\rm ADM}>0$, $Q^2=Q_e^2+Q_b^2$, and let $J$ denote conserved angular momentum, with its electromagnetic boundary contribution included; it agrees with $J_{\rm ADM}$. The boundary is a future MOTS; no wholly exterior enclosing smooth surface is weakly outer trapped; $\operatorname{Area}_g(\Gamma)\ge A=|S|_g>0$ for every full cut $\Gamma$; and \[A\ge4\pi\sqrt{Q^4+4J^2}.\] \[ m^2\ge\frac{A}{16\pi}+\frac{Q^2}{2}+\frac{\pi(Q^4+4J^2)}{A}. \] Equality on the strict area branch $A>4\pi\sqrt{Q^4+4J^2}$ identifies the entire original data with a smooth spacelike slice of subextremal dyonic Kerr--Newman with matching mass, angular momentum and charges, including smooth attachment and future normal at the horizon and an end at spatial infinity. Conversely, admissible such slices attain equality. The numerical endpoint is included, but extremal equality is not classified. Here every full cut is the entire boundary of an end-side exterior region, including all boundary-coincident components, and $O_j$ includes coordinate derivatives through order $j$.

Result 6.

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 $a_g(S)=\inf_\Gamma\operatorname{Area}_g(\Gamma)$. For $g-\delta=O_6(r^{-q})$, $K=O_5(r^{-1-q})$ with $(n-2)/2<q<n-2$, $\theta_+\le0$, $a_g(S)>0$ and $E>|P|$, the numerical bound is \[ \sqrt{E^2-|P|^2}\ge\frac12\left(\frac{a_g(S)}{\omega_{n-1}}\right)^{(n-2)/(n-1)}. \] Its endpoint corollary derives $E>|P|$ from the other hypotheses. In $n\in\{3,4\}$, the lower-regularity theorem only requires $g-\delta=O_2(r^{-q})$, $K=O_1(r^{-1-q})$ with $q>(n-2)/2$; it proves $a_g(S)>0$ and $E>|P|$ as well as the same inequality. The coefficient is sharp. Numerical assumptions do not require connected boundary, maximality, vacuum, outermostness, spin, or outer area minimization. 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 $\sqrt{E^2-|P|^2}\ge\sqrt{A_{\min}(S)/(16\pi)}$.

Result 7.

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

Collars and weak boundary supports set the graph barrier. Elliptic estimates exclude contact with the floor and solve the graph-conformal deformation; its negative inner mean curvature supplies the boundary route to the neutral mass-area bound. A conformal flow is built from smooth obstacles while preserving frontier area. Regular-patch estimates control forcing, and distance-median separation handles the singular region; mass monotonicity and the limiting comparison give the stated numerical inequality. Minimizing enclosures are treated through their area envelope. Saturation yields a constrained minimum in the original data, whose static rigidity equations produce the spacetime embedding and smooth horizon attachment; in three dimensions the analysis also rules out equality for disconnected horizons.

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. Axial variables are compared with a Kerr--Newman family at prescribed conformal capacity. Riemannian scale comparison and electromagnetic angular-momentum bookkeeping give the mass bound; saturation on the strict physical-area branch reconstructs an admissible dyonic spacetime slice. End preparation removes ADM momentum before a coupled graph and conformal scalar construction. Filled-equation estimates and height separation protect enclosing cuts, while the scalar identity bounds the change in mass; the Riemannian comparison then returns the inequality to the original data.

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, Boundary graph deformations for the spacetime Penrose inequality, 2026-09-27; citation key OAI:Boundary-graph-deformations-for-the-spacetime-Penrose-inequality-September-27-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: Theorem 2.4 (n3:intro:exterior-inequality); Statement source: Theorem 2.3 (n3:prepared:output).
• OpenAI, Conformal flow and the Riemannian Penrose inequality with minimizing frontiers, 2026-09-27; citation key OAI:Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: Theorem 1.1 (thm:rpi).
• OpenAI, Equality and rigidity in the spacetime Penrose inequality, 2026-09-27; citation key OAI:Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: Theorem 2.5 (thm:rigidity); Statement source: Theorem 2.6 (thm:converse); Statement source: Theorem 8.2 (q3:intro:rigidity); Statement source: Theorem 8.3 (q3:intro:connected-rigidity).
• 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 Kerr–Newman Penrose Inequality for Axisymmetric Electrovacuum Exteriors, 2026-10-05; citation key OAI:The-Kerr-Newman-Penrose-Inequality-for-Axisymmetric-Electrovacuum-Exteriors-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: Theorem 2.2 (m:main).
• OpenAI, The spacetime Penrose inequality and enclosing area, 2026-09-27; citation key OAI:The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: Theorem 1.3 (thm:main); Statement source: Corollary 1.4 (cor:automatic-timelikeness); Statement source: Theorem 1.5 (thm:low); Statement source: Theorem 7.14 (bridge:bridge:timelike / thm:weak3); Statement source: Theorem 7.16 (bridge:intro:global-penrose).
• 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.

Prove the Penrose inequality or present a counterexample

Definitions and mathematical statement

An initial data set consists of a Riemannian $3$-manifold $(M,g)$ and symmetric tensor $k$, its second fundamental form. In geometric units the constraint densities obey \[ 16\pi\mu=R_g+(\operatorname{tr}_gk)^2-|k|_g^2,\qquad 8\pi J=\operatorname{div}_g(k-(\operatorname{tr}_gk)g). \] The dominant energy condition is $\mu\ge|J|_g$. At an asymptotically Euclidean end, ADM energy is defined by the limiting metric-flux integral; the mass convention and decay hypotheses are those of the cited formulation. If $A$ is the infimum of areas enclosing the specified outermost apparent horizon, the desired bound is \[ m_{\rm ADM}\ge\sqrt{A/(16\pi)}, \] with the source's Schwarzschild rigidity condition. An apparent horizon is characterized by vanishing appropriate outgoing null expansion. The full spacetime statement does not assume $k=0$.

Formulation and concept references: [S1].

Short English statement

Does the mass of an asymptotically flat spacetime initial state always dominate the area-based lower bound from its outermost horizon?

Research attempt

Current frontier.

The selected problem is the sharp spacetime inequality, with the least enclosing area and Schwarzschild rigidity. The Riemannian $k=0$ case and spherical cases must be kept separate. Bray--Khuri develop a generalized Jang approach and prove its spherical existence theory. [E1]. Allen--Bryden--Kazaras--Khuri's Theorem 1.1 supplies a universal but suboptimal constant; it cannot be changed to $16\pi$ by rescaling the data. [S3]. Ellithy's May 2026 manuscript retains an explicit quasi-final-state hypothesis. [S2]. The abstract, formulation passages, and relevant analytic passages of Xu's 510-page v5 were inspected, not the entire proof. Its extension from an area-maximizing MOTS to the outermost MOTS is explicitly conditional. The recorded target is preserved, but neither that conditional extension nor its different area functional is silently adopted as a theorem about the target's least enclosing area. [S1].

Attack route.

A spacetime monotonic mass should connect the horizon area to the ADM end. I first derive the sharp mechanism in a spherical exterior without setting $k=0$, then ask which term loses its sign outside symmetry. The alternative Jang/conformal strategy must control both divergence terms and singular limits. A source-audit test below isolates an endpoint estimate that cannot justify such a limit on its own.

Attempt.

Work in the explicitly restricted exterior $[s_0,\infty)\times S^2$ with \[ g=ds^2+r(s)^2d\Omega^2,\qquad k=\kappa(s)ds^2+\tau(s)r(s)^2d\Omega^2. \] Assume the source's asymptotically flat end conditions, $r\to\infty$, $r'\to1$, $r^3\tau^2\to0$, and existence of the ADM flux limit. Put $J_s=J(\partial_s)$. Direct evaluation of the constraints gives \begin{align*} 16\pi\mu&=-4r''/r+2(1-r'^2)/r^2+4\kappa\tau+2\tau^2,\\ 8\pi J_s&=2(r'/r)(\kappa-\tau)-2\tau'. \end{align*} For example $\operatorname{tr}k=\kappa+2\tau$ and $|k|^2=\kappa^2+2\tau^2$, producing $4\kappa\tau+2\tau^2$. Define the spherical spacetime Hawking mass \[ m(s)=\frac r2(1-r'^2+r^2\tau^2). \] Differentiating, substituting the two constraint identities, and canceling $\kappa\tau$ yields the exact identity \begin{equation}\label{wsref-0-72-33-37-3a-69-64-65-6e-74-69-74-79} m'=4\pi r^2(\mu r'-r\tau J_s). \end{equation} This is checked symbolically in the companion code as well as by the displayed substitution. The formula is valid for either sign of $\tau$.

Assume a future apparent-horizon sphere at $s_0$, so in this sign convention $r'(s_0)+r(s_0)\tau(s_0)=0$, and impose the additional untrapped-exterior condition \begin{equation}\label{wsref-0-72-33-37-3a-65-78-74-72-61} r'>|r\tau|\quad(s>s_0). \end{equation} This is an explicit restriction, not a consequence claimed for general outermost-MOTS data. The dominant energy condition gives \[ m'\ge4\pi r^2\mu(r'-|r\tau|)\ge0, \qquad m(s_0)=r(s_0)/2. \] In areal coordinates the metric is $dr^2/r'^2+r^2d\Omega^2$; its ADM energy is $\lim r(1-r'^2)/2$ under the stated decay, and spherical symmetry gives zero ADM momentum. The $\tau$ term tends to zero, hence $m_{\rm ADM}=\lim m(s)\ge r(s_0)/2$.

This really controls the least enclosing area in this restricted exterior, not merely the area of a preferred coordinate surface. Write $r_0=r(s_0)$. The closed form $\omega=r_0^2\sin\theta\,d\theta\wedge d\phi$ has comass $r_0^2/r(s)^2\le1$, since (r37:extra) makes $r$ increasing. Every smooth oriented enclosing surface homologous to the horizon has $\int_\Sigma\omega=4\pi r_0^2$ by Stokes, so $|\Sigma|\ge4\pi r_0^2$. The horizon, or spheres approaching it from outside, attains that infimum. Thus $A=4\pi r_0^2$ and \[ m_{\rm ADM}\ge\sqrt{A/(16\pi)} \] in this class. This recovers a known kind of spherical result by a direct calculation; it is not claimed as a new theorem resolving general data. If equality holds, (r37:extra) and continuity force $\mu=J_s=0$ outside the horizon. The calculation itself establishes that necessary vacuum condition, not a complete nonspherical Schwarzschild rigidity argument.

The sharp constant cannot be recovered from a suboptimal one by changing units. Under the rescaling of all lengths by $c>0$, mass is multiplied by $c$ and area by $c^2$, leaving $m/\sqrt A$ unchanged. Likewise (r37:identity) explains the real obstruction to extending this proof: an arbitrary exterior has no globally defined areal-radius foliation for which the corresponding momentum and geometric terms reduce to $\mu r'-r\tau J_s$ with $r'>|r\tau|$. Existence of such a controlled foliation is additional analysis.

Endpoint stress test for a different proposed route. Xu v5, pp. 342--343, states an $L^{3/2}\to L^\infty$ bound for the Newtonian potential in its discussion of smoothing, before mentioning an upgraded $L^2$ estimate. The endpoint assertion alone is false already in Euclidean space. [S1]. To check this independently, let \[ f_N(x)=\frac{1_{\{e^{-N}<|x|<e^{-1}\}}} {|x|^2\log(1/|x|)}\qquad(N>2). \] Then spherical integration gives \[ \|f_N\|_{3/2}^{3/2}=4\pi\int_1^N t^{-3/2}\,dt\le8\pi, \qquad \frac1{4\pi}\int\frac{f_N(x)}{|x|}\,dx=\log N. \] Thus there is no uniform $L^{3/2}\to L^\infty$ operator bound. The same conclusion holds with smooth nonnegative cutoffs equal to one on a slightly smaller annulus: the norm stays bounded and the value still grows like $\log N$. Scaling the functions can even make their $L^{3/2}$ norms tend to zero while the potential at zero diverges. This tests one analytic sentence, not all subsequent estimates: an independently established stronger $L^2$ or Lorentz-space control could avoid it. No claim to have refuted the whole 510-page manuscript is made.

Stress test.

The spherical proof used (r37:extra); dropping it removes the sign of (r37:identity), even if the dominant energy condition remains true. Choosing a larger non-outer-minimizing horizon area would also change the problem. The source's equality convention refers to a slice of Schwarzschild spacetime, not necessarily the time-symmetric spatial metric; the attempt has not silently replaced one by the other.

For Jang-type routes, nonnegative scalar curvature after a deformation cannot be assumed merely because a divergence integrates formally to zero. Jaracz constructs spherical data obstructing the specific generalized-Jang/zero- divergence system with the desired asymptotics. That result constrains that coupling, not every Jang formulation or the Penrose inequality itself. [E2]. Boundary flux, horizon area comparison, asymptotic mass, and rigidity must all survive the same limit. None is supplied here in general.

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Outcome.

Outcome: Exploratory approach with unresolved gap.

The attempt supplies an exact mass derivative and a sharp least-enclosing-area proof in a stated spherical untrapped subclass, plus a rigorous counterexample to an endpoint analytic estimate used in a source discussion. Neither is a counterexample to the Penrose inequality. A general construction of a suitable monotone deformation or foliation, with all flux, area, and rigidity controls, remains unproved; the requested unrestricted theorem has not been obtained.

Sources

• S1 D. Xu, 'The Spacetime Penrose Inequality: Conditional Results for Stable MOTS and General Trapped Surfaces,' arXiv:2512.04137v5 $2025$. https://arxiv.org/abs/2512.04137. Catalogue formulation source.
• S2 The spacetime Penrose inequality under a quasi final state hypothesis. https://arxiv.org/abs/2605.18730. Catalogue research/status reference; not a proof endorsement.
• S3 Proof of the Spacetime Penrose Inequality With Suboptimal Constant in the Asymptotically Flat and Asymptotically Hyperboloidal Regimes. https://arxiv.org/abs/2504.10641. Catalogue research/status reference; not a proof endorsement.
• E1 Hubert L. Bray and Marcus A. Khuri, A Jang equation approach to the Penrose inequality, Discrete and Continuous Dynamical Systems 27 $2010$, 741--766, spherical existence theorem and proposed nonspherical coupling. https://arxiv.org/abs/0910.4785.
• E2 Jaroslaw S. Jaracz, Nonexistence of Solutions to the Coupled Generalized Jang Equation/Zero Divergence System, Classical and Quantum Gravity 40 $2023$, 195013. https://arxiv.org/abs/2304.09332. Consulted 27 September 2026; only the specified coupled system is obstructed.