Aegis: The hard part isn’t the theory, it’s the assumptions you’re making. If you treat the hidden layer as a fixed‐size, finite‑dimensional vector space and the weights as constants, convergence reduces to a fixed‑point iteration. The edge cases arise when the Jacobian at a fixed point has eigenvalues on the unit circle or you have a degenerate activation—like a saturated ReLU—so the mapping isn’t contractive. Also, if your loss landscape has saddle points that are flat in a direction, the gradient will stall. In practice, check the spectral radius of the effective weight matrix after activation; if it stays below one, you’re safe. Degeneracy shows up as rank deficiency or an activation that outputs a constant. Those are the cases you need to guard against.

Aegis: Running a unit test that samples random weights is a quick sanity check, but it won’t cover the pathological cases where the Jacobian’s spectral radius actually creeps up to one. Adding a tiny epsilon to ReLU keeps you out of the exact flat spot, but if the network scales, the product of many such approximations can still accumulate and push you toward rank deficiency. A light regularizer that penalizes small singular values—essentially a spectral norm constraint—would enforce that you stay away from that edge. So, prototype the test first to see if any of your random samples hit the boundary, then add the regularizer. That balances practicality with the mathematical guardrails you’re after.

Aegis: Start with the toy model. Validate the spectral check and the penalty in a controlled setting. Once you see the metrics hold there, scale to the full network. It saves time and isolates the variables that actually matter.

Aegis: That’s the right approach—get the toy right first, then lift it. Good luck with the coding.

Sounds good. Keep the logs tidy and let me know if any radius spikes. Good luck.