Pfaffian-horizontal decoder

Integrate this into a decoder or implicit neural representation, not into the encoder. Let the decoder take latent coordinates \(z\in\mathbb{R}^m\) and output \(y=F_\theta(z)\in\mathbb{R}^N\). Provide a differentiable one-form \(\lambda(y)\in\mathbb{R}^N\), either analytically specified or represented by a small frozen network. For every minibatch, use automatic differentiation to compute the Jacobian columns \(J_i=\partial F_\theta(z)/\partial z_i\), then compute \(c_i=\lambda(F_\theta(z))^{\mathsf T}J_i\). Optimize \(\mathcal{L}_{\mathrm{task}}+\mu\sum_i c_i^2\), with \(\mu\) ramped from zero during the first 10% of training. A stronger variant forms \(q=\lambda/\max(\|\lambda\|_2^2,\epsilon)\) and the horizontal projection \(J_i^{\mathrm{hor}}=J_i-q(\lambda^{\mathsf T}J_i)\), then penalizes \(\|J_i-J_i^{\mathrm{hor}}\|^2\); do not overwrite autodiff Jacobians directly because projected columns may be non-integrable. Restrict minibatches to a small output neighborhood \(V\), matching the proposition's local assumption. If \(\lambda\) is not analytically known, estimate the integrability diagnostic by finite differences for \(d\lambda\) and report the norm of \(\lambda\wedge d\lambda\). First run a synthetic sphere-boundary-to-ball interpolation task with an analytically integrable form, comparing an equal-size MLP with and without the penalty. Measure reconstruction error, maximum and RMS Pfaffian violation, Jacobian spectral norm, and interpolation smoothness. Success is a 10-fold reduction in constraint violation with at most 5% reconstruction loss, or lower Jacobian variation at equal violation.