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
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.