Check reference diagnostic sets before real learners use them.

Checks notation, matrix reasoning, random variables, Bayes, distributions, convergence, and concentration before ML theory.

Checks LLN versus CLT, convergence modes, standard errors, and finite-sample caveats before estimation and asymptotics.

Checks the sub-Gaussian/Hoeffding bridge plus ERM, VC, and uniform-convergence prerequisites.

Checks exp/log identities, sigmoid/logit/tanh, softplus, safe softmax, and optimizer mechanics before neural-network training paths.

Checks optimization, stochastic gradients, likelihood, Fisher information, and estimation assumptions.

Checks surprisal, entropy, bit/nat units, bpb in language modeling, KL asymmetry, and the cross-entropy decomposition before classifier-training and LM-evaluation paths. Pairs IT foundations with the activation-function mechanics that the cross-entropy loss combines with.

Checks the differential calculus and geometry beneath gradient descent: directional derivatives, Jacobians, Hessians and symmetry, first- and second-order optimality conditions, Taylor quadratic models, eigenvalues as curvature, saddle classification, and the Newton step. Pairs the calculus foundations with optimizer mechanics so a learner who walks both knows what GD is walking on AND how stochastic gradients and step-size choices behave.

Checks the spine of estimator vocabulary and MLE properties: estimator-as-random-variable, bias and MSE decomposition, OLS as MLE under Gaussian noise, MLE as KL minimization (Goodfellow Ch 5.5), consistency, Gaussian-variance MLE bias-vs-consistency, asymptotic efficiency at the Cramer-Rao floor, and OLS as BLUE under Gauss-Markov. Pairs the depth set with the breadth statistics-estimation set so a learner who walks both has Bayesian/frequentist context AND clean property definitions.

Checks the constrained-optimization vocabulary: convex set vs convex function, Lipschitz vs L-smoothness, feasible set, projection onto a convex set, the descent lemma, Lagrangian, KKT conditions, weak vs strong duality with Slater, and projected gradient. Pairs the constrained-side set with the unconstrained-calculus set (gradient-based-optimization-foundations-v1) so a learner walks both halves of optimization: what GD walks on, and what changes when constraints arrive.

Checks the pre-ML-theory vocabulary: supervised vs unsupervised, parametric vs non-parametric, train vs generalization error, bias-variance, k-means, KNN distance concentration, SVM max-margin, the kernel trick, decision-tree splitting, and overfitting diagnosis. Pairs the fundamentals set with the learning-theory set so a learner who walks both has the basic ML vocabulary AND the formal generalization machinery.

Checks whether learners can separate iid sampling from adaptive diagnostic feedback and use martingale-style assumptions for retry/checkpoint loops.

Checks the discrete-math vocabulary every learner needs before reading proofs in any branch of math: subset vs member, functions and relations, injective/surjective/bijective, implication truth tables, quantifier negation, contrapositive vs contradiction, induction structure, Cantor diagonalization, equivalence relations, power-set cardinality, and Russell's paradox. Pairs the sets-and-logic depth set with math-foundations so a learner gets both proof vocabulary AND the broader notation literacy for matrices, calculus, and inequalities.

Checks the multivariable-calculus prerequisites before optimization theory: derivative as a limit, continuity vs differentiability, partials, MVT, chain rule via Jacobians, the Frechet derivative, implicit function theorem, extreme value theorem on compact sets, Lipschitz vs uniform vs pointwise continuity, and Schwarz commutativity. Pairs the depth set with the gradient-based-optimization set so a learner walks both the calculus and its application to GD/Newton.

Checks the measure-theory foundations underpinning modern probability: sigma-algebras, Borel sets, measurable functions, MCT/Fatou/DCT, almost-everywhere equality, Borel-Cantelli, Fubini, Lebesgue vs Riemann, and countable additivity. Pairs the measure-theory depth set with the probability-foundations breadth set so a learner promoting from undergrad probability into rigorous measure-theoretic probability has both sides.