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Phys-270-A1 · revision r1 · on Phys-270

gpt 6 astra ultra attempt on Phys-270

Exploration · LLM-generated · gpt 6 astra ultra ·

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

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Classification of Gapped Ground-State Phases
  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. An explicit uniformly gapped product-state path.
  15. Infinite-volume interpretation.
  16. Limits of classification.
  17. Verification and adversarial audit.
  18. Tools, process and contributions.
  19. Final status and first remaining gap.
  20. Completion Estimate
  21. Sources

Classification of Gapped Ground-State Phases

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.
For a lattice $\Gamma$ with finite-dimensional site Hilbert spaces, $\Phi(X)$ is a short-range interaction supported on finite $X\subseteq\Gamma$, and $H_\Lambda=\sum_{X\subseteq\Lambda}\Phi(X)$ is the local Hamiltonian. Infinite-volume ground states are thermodynamic-limit states on the quasi-local observable algebra. In the source, two ground-state sets belong to the same gapped phase when a differentiable path $\Phi_s$, $s\in[0,1]$, connects their interactions, has the required uniform locality bounds, and has a common positive lower bound $\gamma>0$ for the local ground-state spectral gap.
Source scope and qualifications.
The primary source already supplies the displayed path definition and spectral-flow automorphisms linking equivalent ground-state sets. The broader unresolved task is classification and understanding those phases, not simply inventing the definition. Uniform finite-volume gap bounds, ground-state degeneracy, and optional symmetries must remain explicit.
Catalogue mathematical target.
Catalogue wording retained for traceability.
What is a gapped ground-state phase for an infinite quantum spin system, when phases are defined by differentiable paths of sufficiently short-range interactions with a uniform positive spectral gap?
Saved literature context.
The supplied catalogue's assessment dated 2026-08-21 records: The question has a rigorous phase definition and substantial one-dimensional classifications, but no general classification of gapped ground-state phases in the stated setting is known. The assessment is retained as dated provenance; it is not a new comprehensive literature or open-status certification.

Short English statement

Classify the equivalence classes of infinite-volume ground states connected by uniformly gapped paths of local interactions.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Constructs a differentiable uniformly gap-one path connecting arbitrary unconstrained on-site product ground states.
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.
An explicit uniformly gapped product-state path.
Fix a site space of finite dimension at least two and two unit vectors $v,w$. Choose a unitary $U$ with $Uv=w$, diagonalize it, and choose a self-adjoint logarithm $K$ so $U=e^{iK}$. Put $v_s=e^{isK}v$ and define the on-site interaction $h_s=I-|v_s\rangle\langle v_s|$, $0\leq s\leq1$. For any finite nonempty region $\Lambda$, \[ H_{\Lambda,s}=\sum_{x\in\Lambda}h_s^{(x)} \] is a sum of commuting projections. Its unique ground vector is $v_s^{\otimes|\Lambda|}$, and every nonzero eigenvalue is a positive integer. The gap is exactly one, uniformly in $s$ and $\Lambda$. The path is differentiable with uniformly bounded on-site derivative $i[K,h_s]$.
Infinite-volume interpretation.
The product states have consistent local density matrices, so define states on local observables and extend by norm continuity to the quasi-local algebra. The constructed path places all such unconstrained on-site product ground states in the same phase under the source definition. No finite-volume gap is inferred from a numerical finite-size sequence; it was computed exactly for every volume.
Limits of classification.
An imposed symmetry could forbid the chosen $U_s$, and degenerate, entangled or topologically ordered ground-state sets need not admit this on-site path. The source already defines gapped phases and supplies spectral flow; this calculation is just a verified elementary phase class, not a new definition or classification. The first gap is a complete invariant and a constructive gapped interpolation criterion for the broader interaction class, respecting any chosen symmetries and ground degeneracies.
Verification and adversarial audit.
Checked unitary logarithm, commuting projection spectrum, all-volume uniformity and consistency of product-state limits; stated symmetry restriction. 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 unitary logarithm, commuting projection spectrum, all-volume uniformity and consistency of product-state limits; stated symmetry restriction. Retained unresolved status and zero original-target progress; A classification of general entangled/degenerate/symmetry-constrained phases is not obtained.

Coordinator checked the on-site projection path, exact uniform gap in site dimension at least two and consistent infinite product states. 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 classification of general entangled/degenerate/symmetry-constrained phases is not obtained.

Completion Estimate

COMPLETION: 0% The calculation is an auxiliary or restricted attempt and establishes no newly verified portion of the full source target. A classification of general entangled/degenerate/symmetry-constrained phases is not obtained.

Sources