Loss Functions in v0.4¶

ZeroProofML v0.4 combines multiple losses to stabilise training near singularities and to distinguish projective orientations.

Implicit Loss¶

Cross-product form that avoids direct division: [ E = (P \cdot Y_d - Q \cdot Y_n)^2, \quad L = \operatorname{mean}\left( \frac{E}{\operatorname{sg}(Q^2 Y_d^2 + P^2 Y_n^2) + \gamma} \right) ] Handles poles by keeping gradients defined even when Q → 0.

Margin Loss¶

Encourages denominators to stay away from zero during training: [ L_{\text{margin}} = \operatorname{mean}(\max(0, \tau_{train} - |Q|)^2) ] Optionally masked to finite paths.

Sign Consistency Loss (Critical)¶

Projective cosine similarity to disambiguate +∞ vs −∞ by aligning (P, Q) with target tuples (Y_n, Y_d).

Coverage & Rejection¶

• Coverage metric: fraction of outputs that are finite (non-⊥).
• Rejection loss: penalises coverage below a target threshold, typically paired with adaptive sampling.

Combined Training Objective¶

SCMTrainingLoss mixes the components: [ L = L_{fit} + \lambda_{margin} L_{margin} + \lambda_{sign} L_{sign} + \lambda_{rej} L_{rej} ] Default hyperparameters: γ=1e-9, τ_{train}=1e-4, ε_{sing}=1e-3, λ_{margin}=0.1, λ_{sign}=1.0.