Improve backbone prediction accuracy

[SabariKumar]

Background

Issue #22 decomposed the all-atom CREST-coverage gap for inverted macrocyclic peptides and
isolated the sole remaining limiter as backbone-dihedral prediction accuracy. Everything else
is handled: side-chain χ is fine (0.11 Å on a true backbone), and constrained-DG embedding slop is
removed by a torsion-constrained FF build (0.47 Å at true dihedrals). But with predicted
dihedrals the seed lands ~0.77–1.75 Å all-atom from the dominant, above the 0.75 Å match.

The current backbone model (per-residue transformer, 24-bin φ/ψ classification + ω cis/trans;
peptide_all_ok ≈ 0.43, ~75% of dihedrals within-1-bin) mispredicts ~1 backbone dihedral per
peptide, and top-K sampling cannot recover it — 0/64 samples cover all backbone dihedrals
within-1-bin of the true dominant. So the fix is a genuinely more accurate/better-calibrated
model, not more sampling.

Proposed approach

• Frame/screw (Plücker) representation (see Plucker Coordinate Embedding #24). Predict SE(3) residue frames / screw
  coordinates instead of chained binned dihedrals, so prediction errors stay local (not
  propagating along the chain into large RMSD) and are continuous (sub-bin precision). This is
  the most promising lever and is developed jointly with the analytical-closure construction in
  Plucker Coordinate Embedding #24.
• Continuous regression (sin/cos or von Mises over φ/ψ) instead of 24-bin classification, if a
  full frame representation is deferred.
• More capacity / data / features — the Backbone Prediction Seeding #23 sweep showed capacity helps (peptide_all_ok
  0.36→0.43); revisit with the above.
• Distribution / top-K backbone targets (mirroring the Side Chain Rotamer Sampling #22 soft-χ work) to handle backbone
  multimodality — but note top-K sampling alone was insufficient (0/64), so this must come with
  better calibration, not just breadth.
• Compose with the validated torsion-constrained build and the lever-2 soft-restraint
  MACE-relax (0.77 Å) — a more accurate prediction + soft restraint should clear 0.75.

Key questions / unknowns

• Can the model get all backbone dihedrals within tolerance (currently ~1 mispredicted)?
• Does a local-error frame/screw representation (Plucker Coordinate Embedding #24) reduce the RMSD impact of the residual
  prediction error, vs binned/chained dihedrals?
• Does regression beat classification for the sub-bin precision the ≤0.75 Å all-atom match needs?
• End-to-end target: lift all-atom cov_bw_ceil on inverted peptides above the current 0.000
  (backbone-fold coverage already works; all-atom is the goal).

Relationship to other work

• Upstream: Side Chain Rotamer Sampling #22 — the decomposition, the torsion-constrained build, the soft-restraint relax,
  and the coverage harness (validate_seeding_coverage.py / aggregate_seeding_coverage.py).
• Tightly linked to Plucker Coordinate Embedding #24 (Plücker / analytical closure + frame/screw prediction): the
  frame/screw representation proposed there is a primary approach for this issue, and analytical
  closure gives the precise construction that lets prediction accuracy be the sole lever. Pursue
  Plucker Coordinate Embedding #24 first; if the representation change alone doesn't clear 0.75, this issue carries the broader
  model-improvement effort (regression, capacity, data, calibration).