Positive-Definite Quadratic Feature Pair

Integrate this into an MLP block after a hidden vector h in R^n, using two symmetric quadratic heads q_A(h)=h^T A h and q_B(h)=h^T B h. Parameterize A=(U+U^T)/2 and B=(V+V^T)/2, and add a scalar pencil angle theta with lambda=cos(theta), mu=sin(theta). At every training step: (1) compute q_A and q_B for the minibatch; (2) form P=cos(theta)A+sin(theta)B; (3) estimate gamma=lambda_min(P) using an exact symmetric eigendecomposition for small n or 3-5 Lanczos or power iterations for larger n; (4) add alpha*softplus(m-gamma) to the task loss, where m is a target positive margin; (5) backpropagate through the eigensolver or use the Rayleigh-quotient estimate gamma_hat=v^T P v with v updated by normalized power iteration on -P. Optionally make the guarantee exact by parameterizing P=LL^T+epsilon I and setting A=P cos(theta)-C sin(theta), B=P sin(theta)+C cos(theta), where C is an unconstrained symmetric matrix; this makes the chosen pencil positive definite by construction. The mathematics supplies the certificate gamma>0; empirical quantities are the hidden-vector norm distribution, the frequency of simultaneous near-zero outputs, and downstream conditioning. First run a cheap ablation on MNIST or CIFAR-10 with a 64-dimensional MLP, comparing two unconstrained quadratic heads against the pencil-regularized version at matched parameter count and FLOPs. Measure training stability, the minimum observed sqrt(q_A(h)^2+q_B(h)^2), gradient norms, and test accuracy. Success is a substantial reduction in near-simultaneous head collapse without slower loss descent or a meaningful parameter or FLOP increase.