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Bartnik’s Stationary Vacuum Extension Conjecture

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and conventions.
Initial data $(g,k)$ consist of a spatial Riemannian metric and second fundamental form; their constraint quantities are energy density $\mu$ and momentum density $J$. The dominant energy condition is $\mu\geq|J|_g$. On a boundary surface the outgoing expansion is $\theta=H_g+\operatorname{tr}_{S^2}k$ with a chosen outward-normal convention. Stationary vacuum data embed in a vacuum spacetime admitting the required stationary Killing field; asymptotic flatness fixes the exterior decay. Bartnik boundary data encode the source's metric, mean-curvature and extrinsic-curvature boundary components. The primary report defines $B(g,k)=(g|_\Sigma,H_g,(k-(\operatorname{tr}_gk)g)(\nu,\cdot))$, where $\nu$ is the unit normal and $\Sigma$ is the matching boundary. Its exterior is $\mathbb R^3\setminus B^3$.
Target retained from the canonical source catalogue.
Catalogue wording retained for traceability.
Bartnik's stationary vacuum extension conjecture: let $(B^3,g_0,k_0)$ satisfy the dominant energy condition, with boundary $S^2$ whose expansion $\theta=H_{g_0}+\operatorname{tr}_{S^2}k_0$ is not everywhere nonpositive. Does there exist a unique asymptotically flat stationary vacuum initial-data set $(\mathbb{R}^3\setminus B^3,g,k)$ whose Bartnik boundary data $B(g,k)$ agree with $B(g_0,k_0)$ on $S^2$?
Source scope and qualification.
The full boundary-data formula and Conjecture 5 were recovered from PDF page 31. Uniqueness must use the appropriate coordinate/gauge equivalence; the expansion condition is not everywhere nonpositive, rather than pointwise positive. The inherited assessment wrongly says the current extraction omits the statement, so it is retained solely as dated provenance and flagged as inconsistent with the current formulation.

Short English statement

Do admissible ball data have a unique asymptotically flat stationary vacuum exterior with matching Bartnik boundary information?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Phys-360-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-360

Bartnik’s Stationary Vacuum Extension Conjecture

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and conventions.
Initial data $(g,k)$ consist of a spatial Riemannian metric and second fundamental form; their constraint quantities are energy density $\mu$ and momentum density $J$. The dominant energy condition is $\mu\geq|J|_g$. On a boundary surface the outgoing expansion is $\theta=H_g+\operatorname{tr}_{S^2}k$ with a chosen outward-normal convention. Stationary vacuum data embed in a vacuum spacetime admitting the required stationary Killing field; asymptotic flatness fixes the exterior decay. Bartnik boundary data encode the source's metric, mean-curvature and extrinsic-curvature boundary components. The primary report defines $B(g,k)=(g|_\Sigma,H_g,(k-(\operatorname{tr}_gk)g)(\nu,\cdot))$, where $\nu$ is the unit normal and $\Sigma$ is the matching boundary. Its exterior is $\mathbb R^3\setminus B^3$.
Target retained from the canonical source catalogue.
Catalogue wording retained for traceability.
Bartnik's stationary vacuum extension conjecture: let $(B^3,g_0,k_0)$ satisfy the dominant energy condition, with boundary $S^2$ whose expansion $\theta=H_{g_0}+\operatorname{tr}_{S^2}k_0$ is not everywhere nonpositive. Does there exist a unique asymptotically flat stationary vacuum initial-data set $(\mathbb{R}^3\setminus B^3,g,k)$ whose Bartnik boundary data $B(g,k)$ agree with $B(g_0,k_0)$ on $S^2$?
Source scope and qualification.
The full boundary-data formula and Conjecture 5 were recovered from PDF page 31. Uniqueness must use the appropriate coordinate/gauge equivalence; the expansion condition is not everywhere nonpositive, rather than pointwise positive. The inherited assessment wrongly says the current extraction omits the statement, so it is retained solely as dated provenance and flagged as inconsistent with the current formulation.

Short English statement

Do admissible ball data have a unique asymptotically flat stationary vacuum exterior with matching Bartnik boundary information?

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Determined the unique scalar-flat radial exterior mass from positive round mean curvature in the time-symmetric ansatz; not general stationary existence/uniqueness.
Sources and literature check.
Read the supplied definition, catalogue wording, source qualifications and reference entries for 30004802. The historical source/context and inherited status are retained; no fresh literature or novelty certification is claimed. No external theorem is used without proof in this attempt; explicit model assumptions are distinguished from facts about the source model.
A round time-symmetric exterior calculation.
Restrict to $k=0$ and a spherical exterior metric $g=dr^2/f(r)+r^2d\Omega^2$, $f>0$. Computing its scalar curvature yields $R_g=2(1-f-rf')/r^2$. Thus the time-symmetric vacuum constraint gives $(rf)'=1$, or \[ f(r)=1-2m/r. \] At a round boundary of radius $R$, the unit normal is $\sqrt f\,\partial_r$ and the area derivative gives $H=2\sqrt{f(R)}/R$. Accordingly the boundary pair $(R,H)$, with $H>0$, determines \[ m=\frac R2\left(1-\frac{H^2R^2}{4}\right). \] The radial family is asymptotically flat and has $f>0$ for $r\ge R$. The momentum boundary component is zero since $k=0$. This verifies uniqueness of the mass parameter within this restricted radial ansatz.
Limits of the reduction.
It proves neither uniqueness among nonspherical stationary exteriors nor an extension for arbitrary source boundary tensors. The source allows expansion that is merely not everywhere nonpositive, substantially weaker than the pointwise $H>0$ used here with $k=0$. Also regular dominant-energy interiors may constrain the allowed round data; none is constructed for every algebraic pair.
First gap.
The unrestricted stationary boundary-value system needs existence and uniqueness modulo boundary-compatible diffeomorphisms, with all components of the supplied Bartnik data. The elementary radial ODE does not provide those elliptic/gauge estimates. The recovered full boundary formula and its source qualification are retained, with no fresh general-conjecture status claim.
Verification and adversarial audit.
Checked scalar curvature, outward-normal mean curvature, f positivity and the stronger restrictions imposed by spherical time symmetry. This is a GPT-6 Astra Ultra self-audit, not independent human verification. Prompt A drafting followed by prompt B scope/dependency/boundary self-audit. Checked scalar curvature, outward-normal mean curvature, f positivity and the stronger restrictions imposed by spherical time symmetry. The remaining gap is: Solve the full stationary vacuum boundary problem modulo gauge for the source expansion condition and complete Bartnik data. Unresolved status and zero original-target completion are retained.

Coordinator checked the radial scalar-curvature equation, boundary mean curvature and restricted mass determination. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 and local file-reading/writing tools for manuscript serialization and structural checks; mathematical calculations were symbolic reasoning, with no numerical experiment or theorem prover. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.
Final status and first remaining gap.
UNRESOLVED. Solve the full stationary vacuum boundary problem modulo gauge for the source expansion condition and complete Bartnik data.

Completion Estimate

COMPLETION: 0% The proved calculation is an elementary auxiliary result or a test of a restricted strategy; no new portion of the full source target is established.

Sources

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