Phase-Margin Residual Jacobians MVP

phase_margin_experiment.py is a deterministic 2D residual-Jacobian toy. Each block Jacobian is a R(phi), which is also a complex scalar and therefore gives an exact, non-stochastic test of the phase certificate and product composition.

Run:

/home/maxwelhelp/main/bin/python3 phase_margin_experiment.py

The script writes results.json and prints the same measurements.

Quantitative checks

1. For phase errors in [-0.9, 0.9], the exact optimization certificate satisfies Gamma_theta(a R(phi)) = |wrapped(phi-theta)|; maximum absolute error was 3.33e-16.
2. For eight blocks, the product phase equals the sum of block phases modulo 2*pi; error was 4.44e-16.
3. With total phase fixed at pi, sigma_min(I+P) = |1-a^N|; the gain sweep identified the crossing at a=1.000, versus predicted 1.000, with maximum error 2.66e-15.
4. Power growth has the predicted boundary at gain one: measured log slopes were log(0.97)=-0.030459 and log(1.03)=0.029559.

Tiny baseline comparison

The synthetic fitting objective pulls ten block phases to a dangerous total phase of pi. Vanilla optimization reaches zero invertibility margin and sigma_min(I+P)=7.22e-16. Adding the phase-margin hinge leaves margin 0.212911 and sigma_min(I+P)=0.212911 under the same steps and learning rate.

This is a mechanism verification, not a CIFAR or 50-layer MLP benchmark. It uses exact planar rotations rather than sampled Jacobian-vector products, so it does not test estimator noise, computational overhead, nonlinear activations, classification accuracy, or whether the certificate is conservative/useful on general matrices.