What does model complexity mathematically represent/relates in the bias–variance trade-off? (uisng simple math)

I am reviewing my core AI knowledge on the bias–variance trade-off ↑↓ and trying to reconcile its mathematical definition with common explanations based on training and test errors. (at the end, cause fitting problem, either overfitting or underfitting) In this article , bias is described informally as training error and variance as test error !? Aspect Bias Variance Definition Difference between the average model prediction and the correct value we are trying to predict. Variability of model predictions for a given data point. High values The model pays little attention to training data and oversimplifies the relationship. The model pays too much attention to training data and may fail to generalize to unseen data. Simple definition (from article) "Resulted Error from Training Data!" "Resulted Error from Test Data!" Typical problem Underfitting Overfitting The article's simple definitions are misleading. Bias is not equivalent to training error, and variance is not equivalent to testing error. Right? However, my understanding from the mathematical definitions is that bias measures systematic prediction error, while variance measures the variability of predictions across different training datasets. Mathematical definitions Bias measures the systematic difference between the expected model prediction and the true underlying function: $$ \operatorname{Bias}(x) = \mathbb{E}_D[\hat{f}_D(x)] - f(x) $$ Variance measures how much predictions change across independently generated t…