Add Schwinger hydrogen internal-level task

[AlienTu]

Motivation

This task represents a theoretical and computational quantum-field-theory workflow: determine whether a completely screened static unit charge in one-flavour massive QED in 1+1 dimensions supports a localized neutral internal excitation, and extract the weak-mass asymptotics of its squared-frequency binding gap. The work combines field-theory normalization, a nonlinear screened background, fluctuation spectral analysis, weak one-dimensional binding, and controlled asymptotic extrapolation. Researchers working on low-dimensional gauge theories, bosonization, semiclassical solitons, and spectral problems perform this kind of calculation.

Task history

This task comes from the author's own research. The project was explored over approximately one academic semester in Fall 2025, including several formulations and numerical approaches before the physical problem and a stable solution method became clear.

Report	Minimum	Value
Project time scale ? start and end date	2 weeks	approximately 2025-09-01 to 2026-01-15
Actual working hours spent exploring the task	40 hours	approximately 160 hours over one academic semester
Estimated hours for a first-year PhD to reproduce the results	10 hours	approximately 160 hours / one academic semester

Field	Value
What the original work was	Establishing whether the 1+1-dimensional hydrogen analogue has an internal bound excitation at small fermion mass, identifying a numerically stable observable, and determining its weak-mass coefficients.
Did an LLM agent help, and where?	Yes. The author supplied the physical problem, conventions, scientific guidance, and validation judgments. AI agents assisted repeatedly with implementing and checking background solvers, fluctuation eigensolvers, perturbative cross-checks, benchmark packaging, and submission preflight under the author's direction.
Your background	ECE PhD student at Princeton University working in theoretical physics.

Task

Field	Value
Task ID	schwinger-hydrogen-internal-levels
Physics area	Quantum field theory; massive Schwinger model; weakly bound spectral modes
Difficulty	Hard
Source and provenance	Self-contained task-authored model and code. Coleman (1976), DOI 10.1016/0003-4916(76)90280-3, supplies the massive-Schwinger-model scalar conventions; Simon (1976), DOI 10.1016/0003-4916(76)90038-5, supplies the weak one-dimensional bound-state expansion. Reviewer-only provenance is included under oracle/provenance/manifest.json.
Mentor skills	background-fluctuation-spectrum with a reusable conventions and numerical-check reference

The agent reports whether the localized state exists and the coefficients in

Delta_B / mu^2 = d1 (m/e) + d2 (m/e)^2 + d3 (m/e)^3 + o((m/e)^3).

The held-out result is generated computationally from a nonlinear screened-background solve, the bottom of the linear fluctuation spectrum, large-box and grid checks, and an independent perturbative coefficient integration. The verifier grades only the four scientific outputs and does not prescribe bosonization or any numerical method in the prompt.

Checklist

• task.md prompt body is outcome-focused
• Oracle derives the answer through computation rather than echoing constants
• Metadata follows taxonomy.yaml
• bench tasks check tasks/schwinger-hydrogen-internal-levels passes
• Verifier checks outcomes rather than implementation or skill usage
• Mentor skill is included and contains no final answer or verifier tolerance
• Dockerfile does not bake the mentor skill, oracle, or verifier into the agent image
• Scientific provenance is documented
• Repository, taxonomy, skill-frontmatter, Ruff, documentation, and website-build checks pass locally
• Containerized bench eval run --agent oracle --sandbox docker log attached
• Standardized repeated no-skill and with-skill trials on the final commit attached
• Exact author degree/status wording confirmed

Local test results

Current task commit: d30dbfb

Task digest before this PR was opened: sha256:ee17827e0b9398823ea50ff3ea3bbd12190f31de8b53ab3a19bda7e7d5479659.

The oracle was executed directly in the pinned Python scientific environment and produced:

{
  "bound_state_exists": true,
  "d1": 0.0,
  "d2": 108.3,
  "d3": -2510.4
}

All five verifier tests passed. The direct oracle run took about five seconds and the verifier about 0.05 seconds on the development machine. A formal Docker-backed BenchFlow reward log remains to be attached.

Agent	Model	Reasoning	No skill (primary)	With skills (control)	Time
Oracle, direct local execution	human-authored solver	n/a	n/a	5/5 verifier tests passed	about 5 s + 0.05 s verification
AI-assisted development trials	several development agents/settings; exact identifiers were not consistently logged	author-guided	not measured	multiple successful guided reproductions; exact N not recorded	spread across development
Formal final-commit matrix	pending	pending	pending	pending	pending

The successful author-guided AI reproductions are useful development evidence: the bundled mentor skill codifies essentially the same background-field and fluctuation-spectrum guidance. They are not represented as a standardized BenchFlow pass rate because exact trial counts, final-commit identity, model identifiers, and matched no-skill controls were not recorded.

Failure analysis

No infrastructure or verifier failure was observed in direct local execution. During development, the main scientific numerical issue was that extracting d3 from finite-box eigenvalue differences is substantially more sensitive than extracting d2. A coarse fit initially rounded d3 incorrectly. This was corrected by finer spectral checks, continuous-tail matching, and an independent second-order weak-binding integration, which agree on d3 = -2510.4268....

Formal no-skill and with-skill trajectory failure analysis remains pending the final matched trial matrix.

What we learned building it

The squared-frequency gap is much more stable than subtracting frequencies after taking square roots. Very shallow bound modes require a box set by the bound-state decay length, not the much shorter screened-background length. The lowest finite-box continuum state must also be tracked: it approaches the infinite-volume threshold from above, whereas the internal mode remains localized below it.

The weak-mass leading coefficient is stable under the direct spectral calculation, but the cubic coefficient requires either substantially higher precision or an independent perturbative calculation to control the last reported digit.

Artifacts

• Task package, oracle, verifier, mentor skill, and provenance manifest are committed in this PR.
• The oracle writes the result, spectral curve, fit-window diagnostics, box/resolution diagnostics, and a human-readable report.
• Formal BenchFlow job directories and matched agent trajectories will be attached before the PR is marked ready for review.

Credit

Role	GitHub handle(s)
Author	@AlienTu
Reviewers	pending

[vercel]

@AlienTu is attempting to deploy a commit to the Benchmarkthing Team on Vercel.

A member of the Team first needs to authorize it.

[bingran-you]

@AlienTu Thanks for contributing! Will review it soon.

[bingran-you]

@AlienTu Could you attach the local agent experiment results and traj files (jsonl format) zip with BenchFlow CLI (codex gpt5.6sol or claude code opus5)?