Notation and shapes

All observation vectors in this companion are rows. A matrix direction, such as a column of $W$ , is a column. The shapes below summarize the definitions introduced in the chapters. The letter $n$ counts fitted people; $N$ counts reviewed training articles for the proposed calibration fit.

Symbol	Shape	Meaning
$x_a,\mu$	$1\times d$	Article embedding and its fitted reference mean
$V$	$d\times K$	Retained raw news directions
$R,D$	$K\times K$	Naming rotation and named-axis scale matrix
$c_a,y_a,z_a$	$1\times K$	Raw, named, and standardized article coordinates
$q_g,m_{pg}$	$1\times K$	Cell background and person-cell mean
$r_p,b_p$	$1\times K$	Balanced contrast and its unit direction
$B,H$	$n\times K$	Unit person rows and their centered rows
$\bar b$	$1\times K$	Mean unit person profile in the fitted population
$C_P$	$K\times K$	Equal-person covariance
$W$	$K\times r$	Retained people-pattern loadings
$u_p$	$1\times r$	A person’s retained people coordinates
$WW^{\mathsf T}$	$K\times K$	Rank- $r$ projector in news-coordinate geometry
$Z,Y$	$N\times K$ , $N\times m$	Article inputs and reviewed concept targets
$A,\beta$	$K\times m$ , $m\times1$	Proposed concept slopes and intercepts
$W^{\mathsf T}A$	$r\times m$	Algebraic slopes from people directions to concept scores

Here $d=384$ , $K=60$ , and $r=24$ in the published models described by the paper; the main fictional example uses $K=3$ and $r=2$ . The bare $r$ is a retained dimension count; the indexed $r_p$ is a person’s contrast row. The hypothetical number of ontology targets, $m$ , is a modeling choice rather than a statement that every ontology concept has a trained classifier.

In the thin SVD $H=U\Sigma Q^{\mathsf T}$ , $k_*=\min(n,K)$ . The matrix $U$ is $n\times k_*$ , $\Sigma$ is $k_*\times k_*$ , and $Q$ is $K\times k_*$ . Selecting the first $r$ columns of $Q$ with the fitted orientation produces $W$ ; the corresponding columns of $U$ and diagonal block of $\Sigma$ produce the score matrix $U_r\Sigma_r$ . Coordinate sums $S_{pgw}$ and concept-score rows $s$ are different objects.

Time-window subscripts extend the same shapes: $b_{pw},q_{gw},m_{pgw}$ have $K$ coordinates, while $u_{pw}$ has $r$ . The displacement $\Delta_p$ , discarded squared length $Q_p$ , attention share $\alpha_a$ , and ranking score are scalars. In the alignment example, $X,Y$ are $M\times k$ anchor tables and $Q_{\mathrm{align}}$ is the $k\times k$ orthogonal map; that use of $Y$ is local to alignment.