Neural project defense and a new debugging case (Day 40 extended assessment)

Constructed debugging case (synthetic data; this record states no results - run it yourself). A colleague trained a small network to predict a synthetic "late-booking share" from two synthetic features (days before departure and fare relative to the route median). They report that the network's logged MSE stops improving and stays far above the linear-regression baseline, and they propose increasing the hidden layer to 128 units and training ten times longer. Before agreeing, investigate.

import numpy as np
from sklearn.linear_model import LinearRegression

rng = np.random.default_rng(8)
n = 2000
days_out = rng.uniform(0, 90, n)          # synthetic: days before departure
fare_idx = rng.uniform(0.6, 1.6, n)       # synthetic: fare relative to route median
X = np.column_stack([days_out, fare_idx])
y = 0.9 * np.exp(-days_out / 25) - 0.25 * (fare_idx - 1.0) ** 2 + rng.normal(0, 0.03, n)
idx = rng.permutation(n)
tr, va = idx[:1500], idx[1500:]
mu, sd = X[tr].mean(0), X[tr].std(0)
Xs = (X - mu) / sd

def mse(pred, target):
    return float(np.mean((pred - target) ** 2))

W1 = rng.normal(0, 0.5, (2, 16)); b1 = np.zeros(16)
W2 = rng.normal(0, 0.5, (16, 1)); b2 = np.zeros(1)
y_tr = y[tr].reshape(-1, 1)
lr = 0.05
for epoch in range(301):
    h = np.maximum(0, Xs[tr] @ W1 + b1)
    pred = h @ W2 + b2
    g = 2 * (pred - y_tr) / len(tr)
    gW2, gb2 = h.T @ g, g.sum(0)
    gh = (g @ W2.T) * (h > 0)
    gW1, gb1 = Xs[tr].T @ gh, gh.sum(0)
    W1 -= lr * gW1; b1 -= lr * gb1; W2 -= lr * gW2; b2 -= lr * gb2
    if epoch % 50 == 0:
        pred_va = np.maximum(0, Xs[va] @ W1 + b1) @ W2 + b2
        print(f"epoch {epoch:3d}  train_mse {mse(pred, y[tr]):.4f}  val_mse {mse(pred_va, y[va]):.4f}")

lin = LinearRegression().fit(Xs[tr], y[tr])
print(f"linear baseline val_mse {mse(lin.predict(Xs[va]), y[va]):.4f}")

Deliver, in order:

1. Your first two hypotheses, written before running anything, and which check you will run first and why.
2. The fault: where it is, the printed shape or value that proves it, and why Python raised no error.
3. The corrected train and validation MSE from your own re-run, next to the baseline, and what the comparison now shows (single seed, synthetic data, validation only).
4. Your answer to the colleague's proposal: accept or reject, with the evidence.
5. One guard you would add to the training code so this class of fault fails loudly, and which of your own Week 7-8 checks would or would not have caught it.