Define previously-undefined variables across all equations

[cnellington]

Summary

Scanned every displayed equation in the manuscript and ensured each symbol is introduced at least once in the surrounding prose. Where variables were missing or under-explained, added succinct inline glosses in the existing "where ..." style. Definitions already established earlier (for example $\theta_i$, $c_i$, $f$, $\ell$, $\lambda$) are reused rather than repeated.

What changed, by file

• content/03.historical.md
  • Least squares: $n$ observations, covariate $x_i$, outcome $y_i$.
  • Linear mixed model: design matrices $X$, $Z$; fixed effects $\beta$; random effects $u$ with covariance $G$; residual $\varepsilon$ with covariance $R$.
  • Bayesian hierarchy: $y_{ij}$, group parameters $\theta_j$, hyperparameters $\phi$.
  • GAM: intercept $\alpha$, smooths $f_j$ over $p$ predictors $x_{ij}$, residual $\varepsilon_i$.
  • Multi-task objective: per-task parameters $w^t$, loss $\ell$, $n_t$ samples, $T$ tasks, strength $\lambda$.
  • LinUCB: action set $\mathcal{A}$, reward parameters $\hat\theta_a$, design matrix $A_a$, exploration coefficient $\alpha$.
  • Autoregressive LM: token $x_t$, prefix $x_{<t}$, length $T$; fixed pretrained $\theta^\star$.
• content/04.explicit.md
  • Varying-coefficient model: coefficient functions $\beta_j(c_i)$, covariates $x_{ij}$, residual $\varepsilon_i$.
• content/06.bridge.md
  • Joint features: $\otimes$ named as the Kronecker product; dims $d_x$, $d_c$. Ridge estimator design $X$, labels $y$, penalty $\lambda$, identity $I$. Attention query/key/value projection matrices $Q$, $K$, $V$, with a note that $K$ here differs from the Gram matrix $K$ used just above.
• content/07.principles.md
  • Potential outcomes $Y(1)$, $Y(0)$.
• content/08.evaluation.md
  • Bias-variance bound: noise variance $\sigma^2$, smoothness/convexity constants $L$, $\mu$, smoothness exponent $\alpha$. Convergence bound: population objective $\mathcal{L}$, initialization $\theta^{(0)}$, minimizer $\theta^\star$, step size $\eta$.
• content/09.interpretation.md
  • Local $R^2$: black-box and surrogate predictions $h_i$, $g_i$; weighted mean $\bar h$; locality kernel $\kappa$.

Notes

• Edits are additive glosses only; no equations, results, or narrative were changed.
• One genuine notational clash surfaced and is now flagged inline: in the theoretical bridge, $K$ denotes both the Gram matrix (part A) and the attention key-projection matrix (part B).
• CI rebuilds HTML/PDF on merge to main.

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