gpt 6 astra ultra attempt on Phys-86

Separatrix Splitting under Quasiperiodic Forcing

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Fix $J,g>0$ and even zero-average trigonometric polynomials $f$ on $(\mathbb R/2\pi\mathbb Z)^2$ and $m$ on $\mathbb R/2\pi\mathbb Z$. For actions $A\in\mathbb R^2,I\in\mathbb R$ and angles $\alpha\in(\mathbb R/2\pi\mathbb Z)^2,\varphi\in\mathbb R/2\pi\mathbb Z$, set \[H=\omega(\eta)\cdot A+\frac{I^2}{2J}+Jg^2(\cos\varphi-1)+\varepsilon f(\alpha)(\cos\varphi-1)+\varepsilon^{\prime}m(2\alpha_2)(\cos\varphi-1),\] where $\omega(\eta)=(\eta^{1/2}\overline\omega_1,\eta^{-1/2}\overline\omega_2)$. Let $\eta_n\downarrow0$ satisfy $|\omega(\eta_n)\cdot\nu|>\eta_n^a|\nu|^{-\tau}$ for every $0\ne\nu\in\mathbb Z^2$, with fixed $a,\tau>0$. The invariant torus has $A=0,I=0,\varphi=0$. On the section $H=0,\varphi=\pi$, write its unstable and stable manifolds as graphs $A^u(\alpha),A^s(\alpha)$ when these exist and put $\Delta(\varepsilon,\varepsilon^{\prime},\eta)=\det[\partial_\alpha(A^u-A^s)]_{\alpha=0}$. The evenness gives the indicated symmetric homoclinic point.

Statement recorded in the supplied source.

With $\varepsilon=\eta^c$ for a fixed sufficiently large $c>0$, determine the asymptotic splitting $\Delta(\eta^c,\varepsilon^{\prime},\eta)$ as $\eta=\eta_n\to0$ when $\varepsilon^{\prime}=O(1)$, or at least obtain a positive lower bound for its absolute value in the nondegenerate cases. This includes understanding the transition in the splitting asymptotics when $\varepsilon^{\prime}=\eta^T$ crosses a critical exponent $T$, despite convergence of the perturbation series throughout this larger parameter region. [S2]

Short English statement

Find the exponentially small separatrix splitting when one quasiperiodic forcing component is small and the fast component remains of bounded size, including the change in asymptotics inside the perturbation series convergence region.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Computed the two-coupling first-order Hessian determinant and verified an exact rank-one degeneracy. The bounded-fast-forcing asymptotic, positive lower bounds and exponent transition are not proved.

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 first-order calculation and its limitation.

The question concerns a bounded fast forcing, so expansion at zero in both couplings is only a diagnostic. For the unperturbed positive separatrix choose $\varphi_0(t)=4\arctan e^{gt}$; then $\cos\varphi_0-1=-2\operatorname{sech}^2(gt)$. Write $f(\alpha)=\sum_k f_k\cos(k\cdot\alpha)$, taking one representative of each opposite Fourier pair, and similarly $m(2\alpha_2)=\sum_l m_l\cos(2l\alpha_2)$. The integral of the perturbing potential along this orbit is \[ L(\alpha)=-2\varepsilon\sum_k f_k W(k\cdot\omega)\cos(k\cdot\alpha)-2\varepsilon'\sum_l m_l W(2l\omega_2)\cos(2l\alpha_2),\quad W(v)=\int_{\mathbb R}\operatorname{sech}^2(gt)\cos(vt)\,dt. \] All sums are finite, and exponential decay permits differentiation. Thus its Hessian at zero is the real symmetric matrix $M=2\varepsilon\sum_k f_kW(k\cdot\omega)kk^T+8\varepsilon'\sum_l l^2m_lW(2l\omega_2)e_2e_2^T$, where $e_2=(0,1)^T$. If the first term is $\varepsilon C$ and the second is $\varepsilon'De_2e_2^T$, elementary expansion gives \[\det M=\varepsilon^2\det C+\varepsilon\varepsilon'D C_{11}.\] This identifies precisely the two competing terms in this linear-response determinant; it does not identify the determinant of the true splitting for order-one $\varepsilon'$.

Countercheck and failed extension.

If every Fourier vector of $f$ is parallel to $e_2$, then the full Hamiltonian is independent of $\alpha_1$. Hamilton's equation gives $\dot A_1=0$; both invariant-manifold graphs have $A_1=0$, and their splitting determinant vanishes identically whenever those graphs exist. Thus the nondegeneracy proviso cannot be dropped. Even when $C_{11}\ne0$, cancellation between the two displayed determinant terms is possible. Bounding a higher-order error merely by $O(\varepsilon^2+\varepsilon'^2)$ is useless against an exponentially small determinant. A uniform error smaller than the actual candidate leading term along the stated Diophantine sequence, with bounded fast forcing, is the first missing step. The source's critical-exponent transition and a positive lower bound remain unresolved; this is a first-order diagnostic, not a new splitting theorem.

Verification and adversarial audit.

Checked factors from cos(phi)-1, the rank-one determinant expansion, zero couplings and forcing independent of alpha_1; did not identify the Melnikov approximation with an exact splitting. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked factors from cos(phi)-1, the rank-one determinant expansion, zero couplings and forcing independent of alpha_1; did not identify the Melnikov approximation with an exact splitting. Scope remains limited: Control the nonlinear invariant-manifold remainder relative to the exponentially small leading determinant uniformly for bounded fast forcing.

Coordinator checked the separatrix profile, Fourier Hessian coefficients, rank-one determinant expansion and exact alpha-one invariance obstruction. Uniform nonlinear splitting errors remain unproved. 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. Control the nonlinear invariant-manifold remainder relative to the exponentially small leading determinant uniformly for bounded fast forcing.

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: A Curated Collection of Open Mathematics Problems, record 7800009 (AMR-077-0009). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Giovanni Gallavotti, Separatrix Splitting for a Quasi Periodically Forced Pendulum (6 November 1998), Hamiltonian, definition of Delta, Diophantine sequence, and final Question. https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9805.separatrix.tex.