Fit, don't train
Every kernel is a closed-form equation solved from quantum chemistry — no learnable weights, no gradient descent, no empirical correction tables. Here's the full method and the rigor gate every claim passes through.
Closed-form, not trained
Every kernel is ŷ = β₀ + Σⱼ βⱼ · z(fⱼ^pⱼ) solved by a single Cholesky solve (ridge λ=1e-3). No SGD, no learnable weights, no hyperparameter tuning. We say 'fit' and 'solve', never 'train'.
Quantum-chemical inputs only
The 9-dimensional feature vector comes entirely from one SCF per molecule (wB97M-V / def2-SVP / C-PCM, ε=80). No empirical group contributions, no fitted correction tables.
Quarter-integer exponents
Optimal exponents fall on the ¼-integer grid {…,−0.5,−0.25,0,0.25,0.5,0.75,1.0,…}. Finer grids add <0.002 r; continuous optimisation lands within 0.06 of the nearest ¼-integer with no gain.
Scaffold-blind is the real metric
In-sample r is a sanity check. The number that matters — and that we quote as commercial-grade — is the 25% Bemis–Murcko scaffold-blind held-out lift.
- fⱼ — quantum-chemical descriptor from one SCF (wB97M-V / def2-SVP / C-PCM, ε=80)
- pⱼ — exponent on the ¼-integer grid {−2.25 … +2.25}
- z() — z-scoring of the transformed feature
- βⱼ — solved by a single Cholesky solve, ridge λ = 1e-3
The 2-term search is a vmap’d best-of-17,766 over ordered (feature, exponent) pairs, with per-fold independent selection inside nested CV. No SGD, no hyperparameter tuning.
The 8-step rigor gate
Every kernel claim passes this gate before it ships. It was hardened across 12 audit rounds; several steps exist specifically because an earlier version of a kernel failed them.
Grouped-LOO by InChI-Key-14
Same molecule under different conditions never straddles train/test — catches cross-condition leakage that anchor-level LOO misses.
Bootstrap 95% CI on r & MAE
Exposes small-n volatility. For an A-grade ship the CI lower bound on r must sit at or above 0.50.
Permutation sanity floor
Shuffle the target, run the identical search, record the noise p95. A real winner must beat noise by ≥ +0.30 r.
Nested cross-validation
The exponent/pair search happens on the TRAINING fold only, scored on held-out chemistry. This is the decisive killer of winner's curse.
Pair-stability & plateau width
If ≥3 exponent pairs sit within ±0.01 r of the top, ship the FAMILY — not a single 'magic' pair (this is exactly why Bradley_Tm is a family kernel).
Calibration diagnostic
Fit y = α + β·ŷ on grouped-LOO predictions. If the slope ≈ 1 there is no compression to 'recover' — the low-lift verdict is honest signal, not an artefact.
Feature-addition gate
A new feature must: raise nested-CV r with a Δr CI excluding 0, enter the winning pair in ≥50% of folds, AND pass a held-out negative control. 'r went up' alone is not success.
25% scaffold-blind held-out
A Bemis–Murcko scaffold split (seed 2026) set aside up front, never seen by the fitter. Re-fit β on the 75%, predict the blind 25%. This is the commercial out-of-distribution estimate.
The Quarter-Integer Law
Across every kernel, the optimal exponents fall on the ¼-integer grid. Finer grids (⅛, 1/16) add <0.002 r; coarser grids (½) lose ~0.010 r; continuous optimisation lands within 0.06 of the nearest ¼-integer with no gain. The exponents appear to reflect integer-rational dimensional couplings in continuum solvation theory (Born, Onsager, scaled-particle).
| Property | g_polar exponent | Physical regime |
|---|---|---|
| Melting point (Tm) | 0.25 | Weak polar coupling to the lattice (¼ of solvation) |
| Hydration ΔG | 0.75 | Full solvation (Born + scaled-particle combined) |
| Surface tension (γ) | 1.50 | Interfacial cooperative regime (3/2) |
| Enthalpy of vaporisation (ΔH_vap) | 0.50 | Cohesion, cavity-dominated (the 'T₄ prophecy' was falsified) |
Full derivation in the internal PAPER_DRAFT_quarter_integer_law. All 21 published exponents sit on the ¼-integer grid.