| --- |
| license: mit |
| tags: |
| - computational-chemistry |
| - cyclic-peptides |
| - applied-topology |
| - sheaf-theory |
| - structural-biology |
| --- |
| |
| # Ring Closure as Holonomy |
|
|
| Cyclic peptide backbones modelled as cellular sheaves over a cycle graph. The dihedral |
| mismatch accumulated on ring closure enters the sheaf Laplacian exactly where a magnetic |
| flux enters a tight-binding ring, which puts the frustration spectrum in closed form. |
|
|
| ``` |
| lambda_k^{+/-} = 2 - 2 cos( (2 pi k +/- theta) / N ) |
| ``` |
|
|
| Verified against numerical diagonalisation to 4e-15. |
|
|
| ## Status |
|
|
| The mathematics is verified and reproducible. The chemistry interpretation is **partially |
| supported and still open**. Read `FINDINGS.md` before citing anything here. |
|
|
| | Claim | Status | |
| |---|---| |
| | Closed-form spectrum | Verified, 4e-15 | |
| | Flux / Aharonov-Bohm correspondence | Verified, unitary equivalence | |
| | Strain laws (spectral gap and total) | Verified, exact for all N | |
| | Betti invariance under stiffness | Verified | |
| | theta extractable from coordinates | Machinery validated; identification with the model's theta **unresolved** | |
| | Dilution with ring size | Direction confirmed (p = 9e-38); exponent undetermined | |
|
|
| ## The two strain laws |
|
|
| Two normalisations answer two different questions, and conflating them is the most likely |
| source of a wrong exponent: |
|
|
| | quantity | normalisation | closed form | limit | |
| |---|---|---|---| |
| | spectral gap | unit-norm section | `2 - 2cos(theta/N)` | `theta^2 / N^2` | |
| | total strain | unit per-residue amplitude | `2N(1 - cos(theta/N))` | `theta^2 / N` | |
|
|
| `E_tot = N * lambda_min` exactly. The second is the Kirchhoff elastic-rod law, recovered |
| without assuming elasticity anywhere. |
|
|
| ## Quick start |
|
|
| ```bash |
| pip install -r requirements.txt |
| python verify.py # theory checks, ~10 s, no network |
| python scaling.py # both strain laws, N = 4..256 |
| python run_p1.py # downloads 8 CCDC structures, measures theta |
| ``` |
|
|
| `holonomy_extract.py` is the piece most likely to be useful standalone: correct NeRF |
| backbone construction, CIF and PDB readers, Bishop-frame holonomy, a Gauss-Bonnet |
| cross-check, and the loop-closure Jacobian. |
|
|
| ## What this is not |
|
|
| Not a structure predictor and not a competitor to conformational sampling. It answers a |
| prior question: given a backbone's frame mismatch, can the ring close without strain, and |
| if not how much is irreducible. Applies to macrocycles only, since an open backbone is a |
| tree and carries no holonomy. |
|
|
| ## Caveats before reuse |
|
|
| - Author line and affiliation in the manuscript are placeholders. |
| - Two 2026 arXiv references in the related-work section came from a literature pass, not |
| from direct reading. Verify before citing. |
| - The mathematics stands on its own; the biological identification does not yet. |
|
|
