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Homology-only escape and a strict zero-winding choreography gap

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 Newtonian $n$-body problem, positions $q_i\in\mathbb R^d$ and masses $m_i>0$ obey $m_i\ddot q_i=\nabla_{q_i}U$, where $U(q)=\sum_{i<j}m_im_j/|q_i-q_j|$ and the gravitational constant is normalized to one. Here $T=\tfrac12\sum_i m_i|\dot q_i|^2$, $E=T-U$, and a collision has $q_i=q_j$. The center of mass can be fixed at zero; moment of inertia is $I=\sum_i m_i|q_i|^2$. For a fixed period $P$, the action is $\mathcal A(q)=\int_0^P(T+U)\,dt$. A collision-free minimizer in an admissible loop class satisfies the Newtonian equations when variations are permitted. The period, constraints and closure of the class must be specified: a minimizing sequence can escape or approach collisions. For a planar collision-free periodic loop, identify the plane with $\mathbb C$ and define $k_{ij}=\deg(t\mapsto(q_i(t)-q_j(t))/|q_i(t)-q_j(t)|)\in\mathbb Z$. These pairwise winding numbers describe homology information; they do not determine the full braid homotopy class. The figure-eight is the named equal-mass planar three-body choreography. Its pairwise winding triple is $(0,0,0)$. A claim of minimization requires comparison with every admissible loop in the stated class, rather than only symmetric perturbations.
Catalogue wording retained for traceability.
Problem 5. Is the figure-8 orbit a minimizer with homology class (0, 0, 0)? (Alain Chenciner)
Scope and source qualification.
The source asks for the homology class, which is weaker than a braid or symmetry class. Adding the figure-eight symmetries would change the target. The catalogue title/status was paired with a machine-generated research attempt; that attempt is not peer-reviewed evidence of settlement.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

Determine whether the three-body figure-eight minimizes action in its full zero-winding class.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-163-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-163

Homology-only escape and a strict zero-winding choreography gap

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 Newtonian $n$-body problem, positions $q_i\in\mathbb R^d$ and masses $m_i>0$ obey $m_i\ddot q_i=\nabla_{q_i}U$, where $U(q)=\sum_{i<j}m_im_j/|q_i-q_j|$ and the gravitational constant is normalized to one. Here $T=\tfrac12\sum_i m_i|\dot q_i|^2$, $E=T-U$, and a collision has $q_i=q_j$. The center of mass can be fixed at zero; moment of inertia is $I=\sum_i m_i|q_i|^2$. For a fixed period $P$, the action is $\mathcal A(q)=\int_0^P(T+U)\,dt$. A collision-free minimizer in an admissible loop class satisfies the Newtonian equations when variations are permitted. The period, constraints and closure of the class must be specified: a minimizing sequence can escape or approach collisions. For a planar collision-free periodic loop, identify the plane with $\mathbb C$ and define $k_{ij}=\deg(t\mapsto(q_i(t)-q_j(t))/|q_i(t)-q_j(t)|)\in\mathbb Z$. These pairwise winding numbers describe homology information; they do not determine the full braid homotopy class. The figure-eight is the named equal-mass planar three-body choreography. Its pairwise winding triple is $(0,0,0)$. A claim of minimization requires comparison with every admissible loop in the stated class, rather than only symmetric perturbations.
Catalogue wording retained for traceability.
Problem 5. Is the figure-8 orbit a minimizer with homology class $0, 0, 0$? (Alain Chenciner)
Scope and source qualification.
The source asks for the homology class, which is weaker than a braid or symmetry class. Adding the figure-eight symmetries would change the target. 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

Determine whether the three-body figure-eight minimizes action in its full zero-winding class.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Proved unattained zero action infimum in the literal unrestricted fixed-period zero-winding loop class, including nonconstant least-period loops. This is a formulation audit; any historically intended choreography or other extra class remains unanswered.
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.
Testing the literal full homology class.
Fix the positive period $P$ and equal positive masses. Choose any distinct planar points $a_i$ with center of mass zero. For $R>0$, the constant loops $q_i^{(R)}(t)=Ra_i$ are collision free and have each pairwise winding equal to zero. Their kinetic action is zero, and homogeneity of the Newtonian potential gives \[\mathcal A(q^{(R)})=P\,U(a)/R\longrightarrow0. \] Every finite collision-free loop has strictly positive action, since $U(q(t))>0$ and $T\ge0$. Consequently the infimum on the entire zero-winding class is zero and is not attained. Any fixed figure-eight orbit has positive action and is beaten by these loops for sufficiently large $R$.
Boundary and constraint audit.
The competitors are constant loops, not Newtonian solutions. They are nevertheless admissible in action minimization over loops; requiring competitors themselves to solve the equations would change the variational question. If nonconstant loops were demanded, take $q_i(t)=Ra_i+R^{-1}v_i\sin(2\pi t/P)$, with constants $v_i$ of zero mass-weighted sum and at least one nonzero. The full loop has least period $P$. For large $R$ every pair remains in a disk avoiding zero, so its degree stays zero; kinetic action is $O(R^{-2})$ and potential action is $O(R^{-1})$. Thus even a nonconstant least-period requirement does not prevent escape.
Scope-review outcome.
Under the literal unrestricted fixed-period homology class in the supplied definition, this elementary escape calculation gives a negative answer to minimization. It does not apply to a choreography, fixed-size, braid or symmetry constraint added to the admissible set; none is silently imposed here. No novelty is claimed. Coordinator scope review checked the supplied definition and cached primary excerpt; the metadata remains unresolved with zero completion because this is a formulation audit, not a claim to settle a historically intended extra class. Whether the historical question intended an additional implicit compactness constraint requires source-scope review, not an invented repair.
Verification and adversarial audit.
Checked center of mass, every pairwise winding, collision avoidance, action scaling, positive figure-eight action and nonconstant approximating competitors; withheld solved metadata for separate full-scope review. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked center of mass, every pairwise winding, collision avoidance, action scaling, positive figure-eight action and nonconstant approximating competitors; withheld solved metadata for separate full-scope review. Scope remains limited: No mathematical gap in the elementary escape argument for the literal class; separate scope review must determine whether any implicit historical constraint changes the admissible set. The principal drafting agent supplied the constant-loop escape argument. The GPT-6-based Codex coordinator supplied the explicit nonconstant R a_i + R^{-1} v_i sin(2 pi t/P) refinement; the drafting agent verified its center-of-mass, least-period, winding, collision-clearance and action-scaling properties.

Coordinator checked the canonical definition and cached primary source item 5, and verified the constant-loop zero-infimum calculation. The coordinator additionally supplied the explicit nonconstant sinusoidal escape family; the drafting agent verified its least period, zero winding and action bounds. These competitors are variational loops, not Newtonian periodic solutions or choreographies, and the historical source interpretation remains qualified. 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. The elementary escape argument has no gap for the literal class; a historical extra admissibility condition, if intended, requires a separately specified problem and proof.

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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