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Phys-175 · physics · mathematics · Physics

Closure-area limit sets and geometric phase on the shape sphere

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 the planar three-body problem, a shape is a configuration modulo translation, rotation and positive scaling. Its normalized shape space is a sphere, with collisions marked and collinear configurations on an equator. A syzygy is a collinear configuration; its symbol records which body is in the middle. For a trajectory, its $\omega$-limit set consists of accumulation points as forward time tends to its endpoint at infinity; its $\alpha$-limit set is defined backward in time. A limit cycle is a closed dynamical orbit. A shape curve is the image of a motion in the shape sphere; its closure has area measured with the source’s sphere normalization, whose total area is $\pi$.
Catalogue wording retained for traceability.
For the three-body shape space: determine whether a solution can have an $\alpha$- or $\omega$-limit set that is a limit cycle. Prove or refute that for every $0<A_0\le\pi$ there is a solution whose shape-curve closure has area $A_0$. Determine whether two closed shape curves in the same homotopy class can be joined through closed shape curves of solutions; the source conjectures no, and asks for the connected components.
Scope and source qualification.
The last question concerns realizable dynamical shape curves, not arbitrary loops for which homotopy already provides a deformation. The source leaves the topology and deformation class to be specified. The damaged area notation A0 is repaired. The catalogue title/status was paired with a machine-generated research attempt; that attempt is not peer-reviewed evidence of settlement.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

Study shape-space limit cycles, possible closure areas and connected components of closed three-body shape curves.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-175-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-175

Closure-area limit sets and geometric phase on the shape sphere

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 the planar three-body problem, a shape is a configuration modulo translation, rotation and positive scaling. Its normalized shape space is a sphere, with collisions marked and collinear configurations on an equator. A syzygy is a collinear configuration; its symbol records which body is in the middle. For a trajectory, its $\omega$-limit set consists of accumulation points as forward time tends to its endpoint at infinity; its $\alpha$-limit set is defined backward in time. A limit cycle is a closed dynamical orbit. A shape curve is the image of a motion in the shape sphere; its closure has area measured with the source’s sphere normalization, whose total area is $\pi$.
Catalogue wording retained for traceability.
For the three-body shape space: determine whether a solution can have an $\alpha$- or $\omega$-limit set that is a limit cycle. Prove or refute that for every $0<A_0\le\pi$ there is a solution whose shape-curve closure has area $A_0$. Determine whether two closed shape curves in the same homotopy class can be joined through closed shape curves of solutions; the source conjectures no, and asks for the connected components.
Scope and source qualification.
The last question concerns realizable dynamical shape curves, not arbitrary loops for which homotopy already provides a deformation. The source leaves the topology and deformation class to be specified. The damaged area notation A0 is repaired. 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

Study shape-space limit cycles, possible closure areas and connected components of closed three-body shape curves.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Distinguished zero-area smooth images from positive-area closures and constructed kinematic dense bands of area pi a. No Newtonian shape realization, limit-cycle theorem or component classification.
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.
Smooth image versus closure area.
A smooth curve on a two-dimensional sphere has area zero on every compact time interval: in a coordinate chart its Lipschitz image can be covered by $O(1/\delta)$ squares of side $O(\delta)$, giving total area $O(\delta)$. A countable union over bounded intervals still has zero area. This does not imply that its closure has zero area, which is the actual source target.
An explicit kinematic dense-band example.
On a sphere of radius $1/2$, whose total area is $\pi$, use coordinates $\theta\pmod{2\pi}$ and normalized height $z\in[-1,1]$. The curve $\theta(t)=t\pmod{2\pi}$, $z(t)=a\sin(\sqrt2t)$, $0<a<1$, is smooth and dense in the band $|z|\le a$. To verify density, times $t=\theta_0+2\pi n+o(1)$ reduce the second phase to an irrational circle rotation; its orbit is dense, as follows by the pigeonhole approximation of a nonzero arbitrarily small rotation step and its multiples. The area element is $\tfrac14d\theta\,dz$, so the closure has area $\pi a$. These curves are kinematic examples, not three-body shape trajectories.
Gap.
Thus no purely dimension-of-a-curve argument refutes the requested positive closure areas. Conversely the ability to draw a dense curve gives no Newtonian realization. One must construct a physical invariant set or orbit whose projected closure has the specified area, while controlling collisions and conserved quantities. Shape-space limit cycles and connected components of closed realizable shape curves are also unproved; homotopy through arbitrary curves does not supply a deformation through solutions. The first missing step is a dynamical realization with a specified topology and reconstruction condition.
Verification and adversarial audit.
Checked image-area covering, sphere area normalization, irrational-phase density, band area and explicit non-dynamical status of the example. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked image-area covering, sphere area normalization, irrational-phase density, band area and explicit non-dynamical status of the example. Scope remains limited: Construct actual three-body invariant sets/orbits realizing the desired closure geometry and reconstruction conditions.

Coordinator checked the irrational-flow density argument and area of the latitude band on the radius-one-half sphere; the example is kinematic. 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. Construct actual three-body invariant sets/orbits realizing the desired closure geometry and reconstruction conditions.

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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