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Phys-240-A1 · revision r1 · on Phys-240

gpt 6 astra ultra attempt on Phys-240

Exploration · LLM-generated · gpt 6 astra ultra ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Efficient Reduced Free-Energy Computation for Atomistic Systems
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Definitions and conventions.
  5. Source scope and qualifications.
  6. Catalogue mathematical target.
  7. Catalogue wording retained for traceability.
  8. Saved literature context.
  9. Short English statement
  10. Research attempt
  11. Model, date and investigated scope.
  12. Sources and literature check.
  13. Context and investigated scope.
  14. Exact Gaussian elimination with retained atoms.
  15. A precise computational subcase.
  16. Beyond the harmonic case.
  17. Verification and adversarial audit.
  18. Tools, process and contributions.
  19. Final status and first remaining gap.
  20. Completion Estimate
  21. Sources

Efficient Reduced Free-Energy Computation for Atomistic Systems

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 conventions.
$X_r\in\mathbb R^{dN_r}$ are retained representative-atom coordinates and $X_c\in\mathbb R^{dN_c}$ are the coordinates to be integrated out; $N=N_r+N_c$ and $\beta>0$ is inverse temperature. $V$ is the atomistic potential energy and the integral is conditional on the fixed retained coordinates. The normalized free energy $A(X_r)/N_c$ is measured per eliminated particle. Existence of a finite integral requires confinement or the boundary and translation-fixing conventions of the atomistic model.
Source scope and qualifications.
Efficiency and the allowed potentials are not quantitatively specified. The report describes a method for chain-like nearest- and next-nearest-neighbor systems even when atom coordinates are multidimensional. Its passage closely repeats ID 30001115; the latter clean formulation explicitly restricts to a one-dimensional system, so the potentially broader catalogue record is retained with this relation disclosed.
Catalogue mathematical target.
Catalogue wording retained for traceability.
For $A(X_r)=-\beta^{-1}\log\int_{\mathbb R^{dN_c}}e^{-\beta V(X_r,X_c)}\,dX_c$, can $A(X_r)/N_c$ be computed efficiently as $N_c\to+\infty$?
Saved literature context.
The supplied catalogue's assessment dated 2026-08-21 records: The extracted question asks for an efficient large-Nc computation and no general algorithm or asymptotic resolution is identified in the cited literature. The assessment is retained as dated provenance; it is not a new comprehensive literature or open-status certification.

Short English statement

Compute the reduced free energy per removed atom efficiently as the number of removed atoms grows.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Derives the Schur-complement Gaussian free energy and an exact-arithmetic linear-cost banded harmonic subcase.
Sources and literature check.
The complete supplied definition, including source qualifications and the dated literature assessment, was reviewed. The calculation below is an elementary restricted attempt motivated by [S2]; it does not claim novelty or a fresh literature-status determination.
Context and investigated scope.
The complete supplied definition, including source qualifications and the dated literature assessment, was reviewed. The calculation below is an elementary restricted attempt motivated by [S2]; it does not claim novelty or a fresh literature-status determination.
Exact Gaussian elimination with retained atoms.
Restrict to a quadratic potential in the eliminated vector $y\in\mathbb R^n$, where $n=dN_c$: \[ V(x,y)=\tfrac12y^TKy+b(x)^Ty+c(x),\qquad K=K^T>0. \] Completing the square and changing variables gives \[ A(x)=c(x)-\tfrac12b(x)^TK^{-1}b(x) +\frac1{2\beta}\log\det K-\frac n{2\beta}\log(2\pi/\beta). \] The positivity hypothesis is essential. If $K$ has a zero mode with no confining higher-order term, the integral along that direction diverges, irrespective of whether the linear coefficient there vanishes.
A precise computational subcase.
For a fixed-block-bandwidth positive-definite $K$ with block size $d$ fixed, band Cholesky elimination needs a number of arithmetic operations proportional to $N_c$: each row updates only a bounded number of blocks. It obtains $\log\det K$ from twice the sum of logarithms of its positive Cholesky diagonal and obtains the quadratic form by triangular solves. This is an exact-arithmetic operation count, not a uniform floating-point error theorem; ill-conditioning can demand additional precision.
Beyond the harmonic case.
The formula handles multidimensional atom positions and makes the confinement convention explicit. It supplies no uniform thermodynamic limit for arbitrary sequences $K,b,c$, and no efficient procedure for generic nonlinear atomistic potentials. The source already has chain transfer-operator methods and closely repeats 30001115. The first gap is a controlled nonlinear approximation whose error per removed atom and computational cost remain bounded at the specified tolerance, with the retained-coordinate scaling fixed.
Verification and adversarial audit.
Checked square completion, determinant normalization, zero-mode divergence and finite bandwidth; separated arithmetic cost from numerical conditioning. Model self-audit only; no independent human proof checking. Prompts A and B were applied to this canonical record and its qualified source scope. Checked square completion, determinant normalization, zero-mode divergence and finite bandwidth; separated arithmetic cost from numerical conditioning. Retained unresolved status and zero original-target progress; A nonlinear error/complexity guarantee and a specified thermodynamic scaling are missing.

Coordinator checked the completed-square integral, zero-mode divergence and exact-arithmetic band elimination cost. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 was used for file preparation and structural checks; no numerical experiment or theorem prover was used. 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. A nonlinear error/complexity guarantee and a specified thermodynamic scaling are missing.

Completion Estimate

COMPLETION: 0% The calculation is an auxiliary or restricted attempt and establishes no newly verified portion of the full source target. A nonlinear error/complexity guarantee and a specified thermodynamic scaling are missing.

Sources