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Exact reversible shooting criterion for halo-orbit bifurcation

Status Partially solved · unsolvedmath.comLLM status Open unverified

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 notation.
In the circular restricted three-body problem, two primaries follow prescribed circular motion and a test particle has negligible effect on them. In rotating coordinates the Jacobi constant combines kinetic energy and the effective gravitational/centrifugal potential. The Hill region is the set where the zero-velocity inequality permits motion. Its necks occur near collinear equilibria. In the spatial restricted problem, halo orbits are the named three-dimensional periodic families near the collinear libration-point region, with characteristic symmetries and out-of-plane motion. Bifurcation from a planar periodic family and existence of a numerically observed family are different assertions.
Catalogue wording retained for traceability.
Problem 27. Existence proof for the "halo orbits" of the three-dimensional restricted three-body problem. (Rick Moeckel)
Scope and source qualification.
The primary source includes a tentative bifurcation remark and states that the author knew numerical studies. That historical remark is not independently checked or promoted to a precise theorem. Parameters and family definition must be specified for a proof. The catalogue title/status was paired with a machine-generated research attempt; that attempt is not peer-reviewed evidence of settlement.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

Give a rigorous existence proof for the spatial restricted three-body halo-orbit family.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-185-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-185

Exact reversible shooting criterion for halo-orbit bifurcation

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.
In the circular restricted three-body problem, two primaries follow prescribed circular motion and a test particle has negligible effect on them. In rotating coordinates the Jacobi constant combines kinetic energy and the effective gravitational/centrifugal potential. The Hill region is the set where the zero-velocity inequality permits motion. Its necks occur near collinear equilibria. In the spatial restricted problem, halo orbits are the named three-dimensional periodic families near the collinear libration-point region, with characteristic symmetries and out-of-plane motion. Bifurcation from a planar periodic family and existence of a numerically observed family are different assertions.
Catalogue wording retained for traceability.
Problem 27. Existence proof for the "halo orbits" of the three-dimensional restricted three-body problem. (Rick Moeckel)
Scope and source qualification.
The primary source includes a tentative bifurcation remark and states that the author knew numerical studies. That historical remark is not independently checked or promoted to a precise theorem. Parameters and family definition must be specified for a proof. 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

Give a rigorous existence proof for the spatial restricted three-body halo-orbit family.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Proved the odd-residual factorization and a conditional even halo-branch shooting criterion with its second derivative. No actual CR3BP seed or parameter certificate is verified.
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.
An exact local symmetry reduction, conditionally.
Suppose a smooth reversible shooting construction for the spatial restricted problem yields residuals $F_1(a,b,c)$, $F_2(a,b,c)$, where c is an initial vertical displacement and a,b are two planar shooting variables. Assume vertical reflection gives $F_1(a,b,-c)=F_1(a,b,c)$ and $F_2(a,b,-c)=-F_2(a,b,c)$. Then \[F_2(a,b,c)=cK(a,b,c),\qquad K(a,b,c)=\int_0^1\partial_cF_2(a,b,tc)dt, \] and K is even in c. At c=0 it equals the vertical linearized residual, so the planar trivial factor is removed without division at zero.
Conditional branch calculation.
Let $G=(F_1,K)$, and suppose $G(a_*,b_*,0)=0$ and $J=D_{(a,b)}G(a_*,b_*,0)$ is invertible. The local implicit-function theorem gives unique smooth a(c),b(c) solving G=0. Reflection and uniqueness make them even, so their first derivatives vanish at zero; differentiating twice yields \[\binom{a''(0)}{b''(0)}=-J^{-1}\partial_c^2G(a_*,b_*,0). \] For nonzero c the full residuals vanish and vertical displacement is nonzero. If the shooting setup's two endpoints are fixed by the reversing involution, time reversal joins the orbit segment to a periodic orbit. This last geometric correspondence is a required hypothesis, not inferred from arbitrary residual names.
Gap.
No actual restricted-three-body seed, mass ratio, return-time transversality or interval bound proving invertibility is supplied. Establishing the displayed finite certificate in the genuine flow is the first missing step. Consequently the calculation gives neither unconditional halo existence nor a family-wide parameter range. The tentative historical bifurcation remark is retained as source context and no freshness or novelty claim is made.
Verification and adversarial audit.
Checked factorization at c=0, evenness, implicit-function derivatives, nonplanarity for c nonzero and the required endpoint-reversibility geometry. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked factorization at c=0, evenness, implicit-function derivatives, nonplanarity for c nonzero and the required endpoint-reversibility geometry. Scope remains limited: Find a genuine reversible planar seed with transverse return, vanishing reduced residual and a rigorously invertible shooting Jacobian.

Coordinator checked the integral factorization of the odd residual, evenness and the second derivative of the conditional implicit branch. 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. Find a genuine reversible planar seed with transverse return, vanishing reduced residual and a rigorously invertible shooting Jacobian.

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

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