What can a finite system's energies reveal about its connectivity and physical symmetry?
This research note separates information fixed by eigenvalues from information that requires a physical basis, local probes, or controlled perturbations. It combines primary-source synthesis, explicit proofs, and reproducible exact examples. It makes no claim of certified original novelty or external peer review.
The central results are:
- Energies alone generally identify neither a site's neighbors nor the system's physical symmetry.
- Connected unweighted graphs can share every site's local spectral measure yet have different automorphism groups.
- Even complete boundary response and visibility of every mode can leave a positive weighted path indistinguishable from a cycle.
- Extra information works under explicit assumptions: an endpoint measure reconstructs a positive Jacobi path; calibrated one-site and two-site shifts reconstruct arbitrary Hermitian coupling magnitudes.
| Resource | Purpose |
|---|---|
| Research note | Main argument, guarantees, limitations, and a precise open problem |
| Derivations | Complete elementary arguments and counterexamples |
| Reproduction guide | Exact commands, inputs, dependencies, and saved results |
| Evidence records | Primary-source locators, support levels, and research decisions |
| Mathematical review / source review | Independent agent review, findings, and limitations |
From the repository root, with Python 3.9 or later:
python analysis/verify_examples.py
python analysis/verify_boundary.py
python analysis/verify_review.py
python tools/validate_research.pyThe mathematical checks use only the Python standard library, exact integer/rational arithmetic, and deterministic inputs. Figure regeneration is optional and uses Pillow; see the reproduction guide. The review is independent agent scrutiny, not journal peer review. No experiment was performed.
Read AGENTS.md first, then the human-owned PROJECT.md and current STATE.md. Reconcile the checkpoint with saved evidence before acting. The unresolved optimization concerns smaller measurement families and their noise margins for a stated model and control class; the present sufficient protocol does not establish optimality.
The reusable continuation prompt is in .claude/commands/loop.md. Earlier template material is preserved in a verified archive, outside the active research presentation.
This project is already initialized. For a new project, the canonical setup prompt retained for compatibility with AGENTS.md is:
New-project setup prompt
Read AGENTS.md and set up this new research project from our preceding
discussion. Write PROJECT.md with the objective, relevant context, constraints,
supplied resources, desired outputs, stopping conditions, and important
unknowns. Initialize STATE.md with inferred practical success criteria, labeled
assumptions, the key uncertainty, and a concrete next action. Preserve supplied
originals and distinguish human-provided information from agent inference. Ask
only for essential inaccessible information or consequential choices that depend
on my priorities; do not ask me for discoverable background or a research plan.
Keep setup compact and stop with the workspace ready for the research loop.
Licensed under MIT. Source papers retain their own rights; they are cited, not redistributed here.