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A conditional charged Penrose bound in spatial dimension at least four

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Assuming the stated neutral spacetime Penrose input in every spatial dimension $n\geq 4$, do the smooth one-ended purely electric charged DEC exteriors of the manuscript satisfy $m\geq |Q|$ and $X_A\leq m+\operatorname{sqrt}(m^2-Q^2)$, with $X_A=(A_min/\omega _{n-1})^((n-2)/(n-1))$ and the manuscript-normalized total electric charge Q? Include strict positivity and timelikeness of ADM mass and the large-area branch $m\geq (X_A+Q^2/X_A)/2$ for $X_A>|Q|$.

Short English statement

Assuming the stated neutral spacetime Penrose input in every spatial dimension $n\geq 4$, do the smooth one-ended purely electric charged DEC exteriors of the manuscript satisfy $m\geq |Q|$ and $X_A\leq m+\operatorname{sqrt}(m^2-Q^2)$, with $X_A=(A_min/\omega _{n-1})^((n-2)/(n-1))$ and the manuscript-normalized total electric charge Q? Include strict positivity and timelikeness of ADM mass and the large-area branch $m\geq (X_A+Q^2/X_A)/2$ for $X_A>|Q|$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-50-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-50

For purely electric data in dimensions $n\geq4$, a charged upper-area inequality and Reissner--Nordstrom--Tangherlini equality classification are claimed under the specified decay and trapping conditions. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

A conditional charged Penrose bound in spatial dimension at least four

OpenAI's claim

OpenAI presents a proof in the selected manuscript. For purely electric data in dimensions $n\geq4$, a charged upper-area inequality and Reissner--Nordstrom--Tangherlini equality classification are claimed under the specified decay and trapping conditions.

Mathematical statement of the claimed result

Result 1.
Fix $n\ge4$, $k=n-1$, $\omega=|\mathbb S^k|$. Take a smooth connected oriented $n$-dimensional exterior, complete with its nonempty compact boundary included and one Euclidean end outside a compact set. Let $g,K,E$ be smooth and purely electric, with $\tau=\operatorname{tr}_gK$ and \[ 2\mu_m=R_g+\tau^2-|K|_g^2-k(k-1)|E|_g^2,\quad J_m=\operatorname{div}_g(K-\tau g),\quad \mu_m\ge|J_m|_g,\quad\operatorname{div}_gE=0. \] Assume $\mu_m,|J_m|\in L^1$, $g-\delta=O_2(r^{-q})$, $K=O_1(r^{-1-q})$ for $q>(n-2)/2$, and $E=O_1(r^{-k})$. Let ADM energy $\mathcal E$ and momentum $P$ be the finite fluxes with factors $(2k\omega)^{-1}$ and $(k\omega)^{-1}$, and $Q=\omega^{-1}\lim_R\int_{r=R}g(E,\nu_R)dA_g$. Each boundary component satisfies $H+\epsilon_a\operatorname{tr}_{S_a}K\le0$ for a chosen sign $\epsilon_a\in\{-1,1\}$. 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 $a_g(S)=\inf_\Gamma\operatorname{Area}_g(\Gamma)$. Put $r_A=(a_g(S)/\omega)^{1/k}$ and $X_A=r_A^{k-1}$. The stated premise is the neutral numerical theorem for all $d\ge3$ with $O_6(r^{-q_0})$ metric decay, $O_5(r^{-1-q_0})$ tensor decay, $(d-2)/2<q_0<d-2$, positive full-cut area, future-timelike ADM vector and uniform future trapping. It supplies the neutral bound $\sqrt{\mathcal E^2-|P|^2}\ge\frac12(A_*/\omega_{d-1})^{(d-2)/(d-1)}$. Under that premise the charged extension proves \[ 0<a_g(S)\le|S|_g,\quad\mathcal E>|P|,\quad m=\sqrt{\mathcal E^2-|P|^2}\ge|Q|,\quad X_A\le m+\sqrt{m^2-Q^2}. \] For $X_A>|Q|$, this equals $m\ge\frac12(X_A+Q^2/X_A)$; for $0<X_A\le|Q|$, it imposes only $m\ge|Q|$. At $Q=0$, $m\ge X_A/2$. The electric field normalization in spacetime is $E^\flat=[k(k-1)/2]^{-1/2}\iota^*(i_N\mathcal F)$, with vanishing pullback magnetic two-form.

Outline of the argument

Area positivity and end preparation first put the data in the neutral companion scope. A coupled scalar system converts electric flux into a controlled energy decrease while retaining enclosure bounds; equality of that conversion reconstructs the original spacelike exterior with vanishing induced magnetic form.

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.

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