Equatorial symmetry and a Jacobian-jet test for local Kerr caustic cusps

Equatorial symmetry and a Jacobian-jet test for local Kerr caustic cusps

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 notation.

Kerr space-time models a rotating vacuum black hole; Schwarzschild is its nonrotating counterpart. Lensing maps initial null-ray directions to source locations. Critical curves are loci where the lens-map derivative is singular; caustics are their images. The strong-deflection limit concerns rays passing near the trapped photon region and requires a specified expansion parameter. Higher-order images correspond to rays making additional turns.

Catalogue wording retained for traceability.

Retain source questions $1$--$13$: obtain analytic descriptions of relativistic caustics beyond their equatorial intersections; describe lensing off the equatorial plane; understand higher-order caustics analytically and numerically; decide observational accessibility of strong-field lensing; find caustic shapes in the strong-deflection limit; describe off-equatorial strong-deflection lensing and additional images; identify effective coordinates for Kerr lensing; formulate projects and non-disk lensing events for MAXIM; determine whether accretion-disk gas trajectories need fully general-relativistic or magnetohydrodynamic/Kerr codes; compare time delays in Kerr and shifted Schwarzschild and determine the order of agreement; and compare caustics and critical curves at first order in the strong-field limit with shifted Schwarzschild.

Scope and source qualification.

Several items are observational or computational research objectives rather than formal conjectures. MAXIM is retained as the historical project named by the source. No unspecified coordinate shift, expansion order or observation criterion is silently fixed. The catalogue title/status was paired with a machine-generated research attempt; that attempt is not peer-reviewed evidence of settlement. The statement covers the original source questions rather than only the title’s subcase.

Short English statement

Develop analytic strong-field Kerr lensing, caustics, image and timing descriptions, with the observational and modelling questions in the source.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Derived a reflection-symmetric local cusp and an explicit symmetry-compatible degeneracy, giving a concrete nondegeneracy test. No actual Kerr caustic or any bundled observational/strong-field comparison is computed.

Sources and literature check.

Read the supplied complete definition, catalogue wording, scope qualifications and source list for this canonical ID. The original-source citations are inherited as supplied references; no fresh primary-source or novelty search was performed for this attempt. No current open-status claim is inferred from this calculation.

A local cusp test compatible with reflection symmetry.

Let a smooth two-dimensional lens map $F(s,t)=(X,Y)$ obey $X(s,-t)=X(s,t)$ and $Y(s,-t)=-Y(s,t)$, as a chosen equatorial-symmetry chart would. The elementary normal-form example \[F(s,t)=(s,t^3+st) \] has Jacobian determinant $J=3t^2+s$. Its critical set is $s=-3t^2$, and the caustic image is $(-3t^2,-2t^3)$, which satisfies $27Y^2+4X^3=0$. At nonzero $t$, the critical image has nonzero derivative and is a fold arc; at zero both first derivatives vanish and the cubic cusp is visible explicitly.

A symmetry-only failure.

The map $F_0(s,t)=(s,st)$ has the same reflection parity but critical set $s=0$, whose entire image is one point. Thus reflection symmetry and a vanishing Jacobian do not force a stable cusp; a nonzero cubic transverse derivative and a suitable mixed derivative are genuine extra conditions. Nor does this local cusp calculation produce a global astroid or count higher-order sheets.

Remaining Kerr scope.

One must compute these derivatives from an actual Kerr null-geodesic endpoint map in specified observer/source coordinates and establish nondegeneracy in a declared strong-deflection regime. That is the first gap. Off-equatorial and higher-order images, global caustic shapes, effective coordinates, observational accessibility, the historical MAXIM project, gas/MHD modelling, and comparison of time delays and critical curves with shifted Schwarzschild remain outside this local test. No numerical Kerr integration or observational prediction was performed.

Verification and adversarial audit.

Checked critical set, cusp elimination equation, fold derivative, degenerate symmetric example and all broad lensing/model/observational limitations. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked critical set, cusp elimination equation, fold derivative, degenerate symmetric example and all broad lensing/model/observational limitations. Scope remains limited: Evaluate and control the relevant Jacobian jets for the actual Kerr endpoint map in specified coordinates and strong-deflection scaling.

Coordinator checked the cusp determinant, critical image equation and symmetry-preserving degenerate comparison map. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.

Python 3 and local shell used to read supplied files and serialize this manuscript; mathematics derived symbolically in the text, with no numerical experiment or formal proof assistant. 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. Evaluate and control the relevant Jacobian jets for the actual Kerr endpoint map in specified coordinates and strong-deflection scaling.

Completion Estimate

COMPLETION: 0% The calculation is an elementary benchmark or conditional reduction; it establishes no claimed new portion of the complete source target.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 20002518 (AIM-PHYSICS-0026). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Gravitational lensing in the Kerr spacetime geometry, Questions 1–13. https://aimath.org/WWN/lensing/lensing.pdf.