cyclic-peptide-holonomy

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.

The two strain laws

Two normalisations answer two different questions, and conflating them is the most likely source of a wrong exponent:

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.