gpt 6 astra ultra attempt on Phys-409

Teukolsky Energy Decay on Extremal Kerr Spacetimes

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.

Kerr parameters are the mass $M>0$ and rotation parameter $a$; extremality is $|a|=M$. Teukolsky equations govern particular spin-field components on this background. A fixed azimuthal mode has angular dependence $e^{im\varphi}$ for a fixed integer $m$. Outgoing and ingoing components are the source's null-frame components at the future horizon.

Catalogue wording retained for traceability.

On an extremal Kerr spacetime with $|a|=M$, prove energy boundedness and energy decay for fixed-azimuthal-mode solutions of the Teukolsky equations, with the predicted pointwise decay away from the future event horizon, decay of the outgoing component on the horizon, and generic growth of the ingoing component there.

Scope and source qualification.

Decay away from the horizon, decay of the outgoing horizon component and generic growth of the ingoing component are different required conclusions.

Short English statement

Prove the stated boundedness, decay and horizon-growth behavior for fixed angular modes of the Teukolsky fields on extremal Kerr.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Proved a conserved horizon derivative and forced higher-derivative growth in a degenerate auxiliary transport equation, with a precise conditional genericity statement.

Sources and literature check.

Read the full supplied definition, catalogue wording, source qualifications and listed references for 30006046. No fresh status or novelty search was performed. The argument below proves its auxiliary statements directly and does not import an unverified external theorem.

A degenerate-horizon transport test.

To isolate why bounded exterior fields need not control transverse horizon derivatives, consider the auxiliary smooth equation \[ 2\partial_v\partial_r\psi+\partial_r(r^2\partial_r\psi)=0, \qquad r\geq0. \] This is a model equation, not the spin-dependent Teukolsky equation. Restricting to $r=0$ gives $\partial_v(\partial_r\psi)=0$, hence $\partial_r\psi(v,0)=H$, fixed by initial data. Differentiating once in $r$ and then restricting gives \[ 2\partial_v\partial_r^2\psi(v,0)+2H=0, \quad \partial_r^2\psi(v,0)=\partial_r^2\psi(0,0)-Hv. \] A nonzero conserved horizon derivative therefore forces linear growth of the next derivative in this model.

Genericity and norm check.

On any initial-data space where evaluation $\psi\mapsto\partial_r\psi(0,0)$ is a nonzero continuous linear functional, the condition $H=0$ is a closed codimension-one kernel; its complement is open and dense. This is a precise model meaning of generic growth. It says nothing about decay of $\psi(v,0)$, which this horizon identity does not determine, or about energy norms with degenerate weights.

Remaining extremal Kerr target.

The actual Teukolsky components, fixed azimuthal number and spin-dependent couplings must be analyzed in the source null frame. The first gap is obtaining a coercive global estimate compatible with the degenerate horizon and identifying the actual horizon transport constants. Energy boundedness and decay, pointwise decay away from the horizon, outgoing horizon decay and ingoing generic growth are separate conclusions; none is established here for extremal Kerr by the scalar model calculation.

Verification and adversarial audit.

Checked differentiation at r=0, linear growth sign, the continuous nonzero functional hypothesis for genericity and the absence of any asserted equivalence to Teukolsky components. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked differentiation at r=0, linear growth sign, the continuous nonzero functional hypothesis for genericity and the absence of any asserted equivalence to Teukolsky components. First remaining gap: Derive coercive global estimates and the actual spin-dependent horizon transport identities for fixed-mode Teukolsky fields on extremal Kerr. The inherited source status is preserved; original-target completion remains zero.

Coordinator checked the two horizon transport identities and conditional codimension-one genericity statement. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.

Python 3 and local file tools used to read definitions and serialize manuscripts. Calculations are symbolic reasoning; 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. Derive coercive global estimates and the actual spin-dependent horizon transport identities for fixed-mode Teukolsky fields on extremal Kerr.

Completion Estimate

COMPLETION: 0% This is an elementary auxiliary calculation or an explicit test of a restricted strategy. No new portion of the full source target is established.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 30006046 (OWR-14298591-010). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mathematical Aspects of General Relativity $2024$, Oberwolfach Reports, DOI 10.4171/owr/2024/36, pp. 2033--2112. Original report and mathematical context. Incorporated report material remains under the source CC BY-SA 4.0 terms unless otherwise noted. https://doi.org/10.4171/owr/2024/36.