% ============================================================================
% Appendix C — Mathematical derivations
% Written by Agent L (Appendices B/C instance), 2026-11-03. Derivations and
% proofs referenced by Part IV (sections/part04_methodology.tex); notation
% follows Part IV. Each derivation is checked against the implementation in
% econometrics/structural_change.py, bibliometrics/03_citation_impact.py,
% bibliometrics/04_diversity_cagr.py, knowledge_graph/02_category_flows.py,
% taxonomy/classify.py, taxonomy/validate_sample.py and
% experiments/emerging_detection.py; where the code and the usual textbook
% convention differ, the code's convention is stated and the difference noted.
% ============================================================================

\section{Mathematical derivations}
\label{sec:appC}

Part~IV defines the statistics of this study and states their properties;
this appendix supplies the derivations behind those statements, in the order
in which Part~IV uses them. Each subsection fixes the object as implemented,
derives the property that the main text relies on, and records any point at
which the implementation departs from the convention of the cited source.
The derivations are elementary and self-contained; their purpose is not
novelty but the assurance that the quantities tabulated in Part~IX are what
the text says they are. Throughout, $n$ is the length of a monthly series,
$y_t = \log(N_t + 1)$ its logarithm, and $\mathbf{1}[\cdot]$ the indicator
function.

\subsection{The interrupted time series as a linear regression}
\label{sec:appC-its}

\paragraph{Design matrix.}
Equation~\eqref{eq:its} of Part~IV is the linear model $y = X\beta +
\varepsilon$ with $y \in \R^n$ and $X \in \R^{n \times 15}$ whose columns
are, for $t = 0, \dots, n-1$ and event index $t_e$,
\begin{equation}
  x_t \;=\; \bigl(1,\; t,\; \mathbf{1}[t \ge t_e],\; (t - t_e)\,\mathbf{1}[t \ge t_e],\;
  \mathbf{1}[\mathrm{month}(t) = 2],\; \dots,\; \mathbf{1}[\mathrm{month}(t) = 12]\bigr)^{\top},
  \label{eq:appC-design}
\end{equation}
exactly the data frame built by \texttt{\_design()} in
\texttt{econometrics/structural\_change.py} (columns \texttt{const},
\texttt{t}, \texttt{post}, \texttt{t\_post}, \texttt{m2}, \dots,
\texttt{m12}). The post-event trend is centred at $t_e$ rather than at zero
so that $\beta_2$ is the level change \emph{at} the event month and not the
extrapolated intercept difference at $t = 0$; with an uncentred post-trend
the two parametrisations are related by $\beta_2^{\mathrm{unc}} = \beta_2 -
\beta_3 t_e$, and only the centred one has the interpretation used in the
text. The eleven month dummies are linearly independent of the constant
because January is omitted; the full-column-rank condition requires at least
one observation in every month of the year, satisfied by any series of
twelve months or more, and at least two post-event months with $t > t_e$
(with a single one the centred post-trend column is identically zero and
$\beta_2$ and $\beta_3$ are not separately identified), guaranteed by the
driver's rule that twelve months follow every event.
The ordinary least-squares estimator is $\hat\beta = (X^{\top}X)^{-1}
X^{\top}y$ and the residuals $\hat\varepsilon = y - X\hat\beta$.

\paragraph{The HAC covariance estimator.}
Under the assumption that $\varepsilon_t$ is a stationary, serially
correlated and possibly heteroskedastic sequence with
$\E[x_t \varepsilon_t] = 0$, the asymptotic covariance of $\sqrt{n}(\hat\beta
- \beta)$ is $Q^{-1}\Omega Q^{-1}$ with $Q = \lim n^{-1}X^{\top}X$ and
$\Omega = \sum_{\ell=-\infty}^{\infty} \Gamma_\ell$, where
$\Gamma_\ell = \E[x_t \varepsilon_t \varepsilon_{t-\ell} x_{t-\ell}^{\top}]$
is the autocovariance of the score at lag $\ell$. The estimator of
\citet{newey1987hac} replaces the infinite sum by a kernel-weighted finite
one,
\begin{equation}
  \hat\Omega_L \;=\; \hat\Gamma_0 + \sum_{\ell=1}^{L} w_\ell\,\bigl(\hat\Gamma_\ell + \hat\Gamma_\ell^{\top}\bigr),
  \qquad
  \hat\Gamma_\ell = \frac{1}{n}\sum_{t=\ell}^{n-1} \hat\varepsilon_t \hat\varepsilon_{t-\ell}\, x_t x_{t-\ell}^{\top},
  \qquad
  w_\ell = 1 - \frac{\ell}{L+1},
  \label{eq:appC-nw}
\end{equation}
and the variance estimate is $\hat V = n\,(X^{\top}X)^{-1} \hat\Omega_L
(X^{\top}X)^{-1}$, which is the expression of Part~IV once the factors of
$n$ are absorbed into the un-normalised $\hat\Gamma_\ell$. The Bartlett
weights $w_\ell$ are what make $\hat\Omega_L$ positive semi-definite for every
sample: $\hat\Omega_L$ can be written as $\frac{1}{n(L+1)}\sum_{t}
\bigl(\sum_{s=t-L}^{t} \hat u_s\bigr)\bigl(\sum_{s=t-L}^{t} \hat u_s\bigr)^{\top}$
up to end effects, with $\hat u_s = x_s \hat\varepsilon_s$, a sum of outer
products, whereas the truncated sum with $w_\ell \equiv 1$ is not guaranteed
to be positive semi-definite. The implementation calls
\texttt{statsmodels} with \texttt{cov\_type="HAC"} and \texttt{maxlags=6}, so
$L = 6$ and $w_\ell = 1 - \ell/7$ as stated in Part~IV, without the
small-sample degrees-of-freedom correction and with $p$-values from the
normal approximation to the $t$-ratio, which is the library's default for
robust covariances. With $n$ of the order of one hundred and fifteen
columns in $X$, the constant included, the difference between the normal and the $t_{n-15}$ reference
distribution is immaterial at the magnitudes of the $t$-statistics reported;
what matters is the placebo rank of the next paragraph, which does not
depend on any reference distribution.

\paragraph{The placebo $p$-value as a rank statistic.}
Let $\mathcal{T}' = \{t' : 24 \le t' \le n - 13\}$ be the candidate event
indices (the implementation's \texttt{min\_pre = 24} and
\texttt{min\_post = 12}) and, for each $t' \in \mathcal{T}'$, let
$t^{(2)}_{t'} = \hat\beta_2(t') / \widehat{\mathrm{se}}(\hat\beta_2(t'))$ be
the HAC $t$-ratio of the level change when the model
\eqref{eq:appC-design} is fitted with $t_e = t'$. The placebo value of the
true event $t_e \in \mathcal{T}'$ is
\begin{equation}
  p^{\mathrm{plac}} \;=\; \frac{1}{|\mathcal{T}'|}\sum_{t' \in \mathcal{T}'}
  \mathbf{1}\bigl[\,|t^{(2)}_{t'}| \ge |t^{(2)}_{t_e}|\,\bigr]
  \;=\; \frac{\mathrm{rank}\bigl(|t^{(2)}_{t_e}|\bigr)}{|\mathcal{T}'|},
\end{equation}
where $\mathrm{rank}$ is the descending rank of the true event's absolute
$t$-ratio among the $|\mathcal{T}'|$ values, ties counted as not better
(\texttt{>=}). Three properties follow directly. Because $t_e \in
\mathcal{T}'$, the sum contains the term $t' = t_e$, so $p^{\mathrm{plac}}
\ge 1/|\mathcal{T}'|$, which for the default window ($n = 101$, hence
$|\mathcal{T}'| = 65$) is about $0.015$: a placebo rank cannot be smaller,
however large the true statistic, and a reported value of $1/|\mathcal{T}'|$
means ``largest of all candidate dates'', not ``significant at that level''.
Because the statistic is a rank, it is invariant to any monotone
transformation of the $|t|$-ratios and in particular does not depend on the
HAC bandwidth through anything but the ordering it induces. And if the
candidate dates were exchangeable under a null hypothesis of no break
anywhere, $p^{\mathrm{plac}}$ would be a valid permutation-type $p$-value
with $P(p^{\mathrm{plac}} \le \alpha) \le \alpha$; they are not exchangeable,
since all $|\mathcal{T}'|$ regressions share the same $n$ observations and
adjacent candidates give nearly identical fits, so the value is reported as
a rank and read as such. The same construction gives the placebo rank of
the slope change from $t^{(3)}_{t'}$. The driver additionally computes the
rank restricted to the baseline candidates $\{t' < t_e\}$, which answers the
protocol's original question (is the reference date special relative to the
pre-period?) and is reported next to the full-range rank.

\subsection{Bai--Perron multiple breaks by dynamic programming}
\label{sec:appC-bp}

\paragraph{Segment costs and additivity.}
For a segment $[i, j)$ of the series with $j - i \ge h$, define
\begin{equation}
  S(i, j) \;=\; \min_{a, b \in \R}\ \sum_{t=i}^{j-1} (y_t - a - b\,t)^2,
  \label{eq:appC-segcost}
\end{equation}
the residual sum of squares of a linear trend fitted to the segment alone
(the implementation solves the two-parameter least-squares problem for each
segment with \texttt{numpy.linalg.lstsq}; with \texttt{trend=False} the
regressor $t$ is dropped and $S(i,j)$ is the centred sum of squares). The
model with $m$ breaks $0 < \tau_1 < \dots < \tau_m < n$ assigns its own
$(a, b)$ to each of the $m+1$ segments $[\tau_{k-1}, \tau_k)$ with
$\tau_0 = 0$ and $\tau_{m+1} = n$. Because the parameters of different
segments are distinct and each observation enters exactly one segment, the
residual sum of squares of the whole model separates,
\begin{equation}
  \mathrm{SSR}(\tau_1, \dots, \tau_m) \;=\; \sum_{k=1}^{m+1} S(\tau_{k-1}, \tau_k),
  \label{eq:appC-additive}
\end{equation}
and the global minimisation over $\tau$ decomposes: the optimal partition of
the first $j$ observations into $m+1$ segments contains, as its first $m$
segments, an optimal partition of the first $\tau_m$ observations into $m$
segments. This is the principle of optimality, and it is the only property of
the cost that the dynamic programme uses; it holds for any cost that is a
sum over segments of a function of the segment alone, which is why the same
recursion serves the piecewise-constant model of Section~\ref{sec:appC-pelt}.
It fails for a model with parameters shared across segments (a common slope,
or the seasonal dummies of the ITS), which is why the segmented regression
carries no seasonal terms and the series is seasonally adjusted beforehand
by subtracting the baseline-period month means
(\texttt{run\_structural\_change.py}).

\paragraph{The recursion.}
Let $C_m(j)$ be the minimum of \eqref{eq:appC-additive} over partitions of
observations $0, \dots, j-1$ into $m+1$ admissible segments. Then
\begin{equation}
  C_0(j) = S(0, j), \qquad
  C_m(j) = \min_{\,mh \le i \le j-h} \bigl\{ C_{m-1}(i) + S(i, j) \bigr\},
  \qquad j \ge (m+1)h,
  \label{eq:appC-dp}
\end{equation}
the lower limit of $i$ ensuring that $m$ segments of length at least $h$
fit before the last break and the upper limit that the last segment has
length at least $h$. The argument minimiser $i^{\ast}_m(j)$ is stored, and
the breaks of the $m$-break solution for the full series are recovered
backward: $\tau_m = i^{\ast}_m(n)$, $\tau_{m-1} = i^{\ast}_{m-1}(\tau_m)$,
and so on, which is what the dictionary \texttt{B[m][j]} of the
implementation accumulates forward instead. This is the dynamic programme
that \citet{bai2003computation} propose for the global minimiser of
\citet{bai1998breaks}; the earlier literature used it for the
piecewise-constant case, and its correctness requires nothing beyond
\eqref{eq:appC-additive} and the induction on $m$: if $C_{m-1}(i)$ is the
true minimum for every $i$, then any partition of $[0, j)$ into $m+1$
segments has a last segment $[i, j)$ and a prefix partition of cost at least
$C_{m-1}(i)$, so its cost is at least the right-hand side of
\eqref{eq:appC-dp}, which is attained.

\paragraph{Complexity.}
The table of segment costs has one entry per admissible pair $(i, j)$, that
is $\binom{n-h+1}{2} + O(n)$ entries, of order $n^2/2$. The recursion
\eqref{eq:appC-dp} evaluates, for each $m \le M$ and each $j$, a minimum over
at most $n$ candidates, so its cost is $O(n^2 M)$ operations; the
back-tracking costs $O(M)$. The implementation fills the cost table by
solving each segment's least-squares problem independently, at a cost
proportional to the segment length, so the table costs $O(n^3)$ rather than
the $O(n^2)$ obtainable by updating running sums as in
\citet{bai2003computation}; for $n$ of the order of one hundred, as here,
the difference is a fraction of a second and the simpler code was preferred
for its transparency. Trimming enters only through the admissibility
condition $j - i \ge h$; with $h = \max\{8, \lfloor 0.15\,n \rfloor\}$ and
$M = 3$, the default window of $n = 101$ months gives $h = 15$ and a
feasible region $(m+1)h \le n$ for every $m \le 3$, so all four candidate
models exist.

\paragraph{Information criteria as implemented.}
For $m = 0, \dots, M$ let $\mathrm{SSR}_m = C_m(n)$ and let $p_m = k(m+1) +
m$ be the parameter count, with $k = 2$ for the trend model ($k = 1$ for the
mean model): one intercept and one slope per segment, plus the $m$ break
positions, counted as parameters by convention. The two criteria computed by
the code are
\begin{align}
  \mathrm{BIC}(m) &= n \log\frac{\mathrm{SSR}_m}{n} + p_m \log n, \label{eq:appC-bic}\\
  \mathrm{LWZ}(m) &= n \log\frac{\mathrm{SSR}_m}{n - p_m} + p_m\, c_0\, (\log n)^{2.1}, \qquad c_0 = 0.299, \label{eq:appC-lwz}
\end{align}
and the selected number of breaks is $\hat m = \arg\min_m$ of each. The BIC
form \eqref{eq:appC-bic} is the Gaussian log-likelihood, maximised over the
error variance, in which $-2\log\hat L = n\log(\mathrm{SSR}/n) + \text{const}$,
plus the Schwarz penalty $p \log n$ \citep{schwarz1978bic}; the two
differences from a textbook BIC are that the break positions are counted
among the parameters and that the constant and the segment variances are
treated as common, both as in \citet{bai2003computation}. The LWZ form
\eqref{eq:appC-lwz} is the modified Schwarz criterion of
\citet{liu1997segmented}, whose penalty grows as $(\log n)^{2+\delta_0}$ with
$\delta_0 = 0.1$ and whose variance estimate divides by $n - p_m$; the
constant $c_0 = 0.299$ is the value they recommend and the one used in the
simulations of \citet{bai2003computation}, which find that LWZ selects fewer
spurious breaks than BIC under serial correlation at the cost of some
power. The comparison across $m$ is legitimate because every $\mathrm{SSR}_m$
is a global minimum over the same observations and the same admissible
region; a sequential procedure that fixed earlier breaks before adding the
next would not have this property and would need the sup-$F$ machinery that
is not implemented. Neither criterion yields a confidence set for the break
dates; the text reports the agreement of the two selections and the
sensitivity to the truncation month instead.

\subsection{PELT: cost, penalty and pruning}
\label{sec:appC-pelt}

\paragraph{Objective.}
With the $\ell_2$ cost $\mathcal{C}(y_{i:j}) = \sum_{t=i}^{j-1} (y_t -
\bar y_{i:j})^2$ for a segment $[i,j)$ and a penalty $\lambda > 0$ per
changepoint, the problem of equation~\eqref{eq:pelt} of Part~IV is
\begin{equation}
  F(n) \;=\; \min_{m,\ \tau_1 < \dots < \tau_m}\ \Bigl\{ \sum_{k=1}^{m+1} \mathcal{C}(y_{\tau_{k-1}:\tau_k}) + \lambda m \Bigr\},
  \label{eq:appC-peltobj}
\end{equation}
and, by the same additivity argument as in Section~\ref{sec:appC-bp}, it
satisfies the optimal-partitioning recursion of \citet{jackson2005optimal},
\begin{equation}
  F(0) = -\lambda, \qquad F(j) \;=\; \min_{0 \le i < j}\ \bigl\{ F(i) + \mathcal{C}(y_{i:j}) + \lambda \bigr\},
  \label{eq:appC-op}
\end{equation}
where the number of breaks is not fixed in advance but chosen by the
penalty; the convention $F(0) = -\lambda$ cancels the penalty charged to the
first segment. Solving \eqref{eq:appC-op} for all $j$ costs $O(n^2)$ cost
evaluations.

\paragraph{The pruning condition.}
\citet{killick2012pelt} observe that if there exists a constant $K$ such that
for all $i < j < l$
\begin{equation}
  \mathcal{C}(y_{i:j}) + \mathcal{C}(y_{j:l}) + K \;\le\; \mathcal{C}(y_{i:l}),
  \label{eq:appC-Kcond}
\end{equation}
then any candidate $i$ for which
\begin{equation}
  F(i) + \mathcal{C}(y_{i:j}) + K \;\ge\; F(j)
  \label{eq:appC-prune}
\end{equation}
can never be the last changepoint before any later $l > j$ and may be
discarded from the candidate set. The proof is two lines. For $l > j$, by
\eqref{eq:appC-Kcond} and \eqref{eq:appC-prune},
$F(i) + \mathcal{C}(y_{i:l}) + \lambda \ge F(i) + \mathcal{C}(y_{i:j}) +
\mathcal{C}(y_{j:l}) + K + \lambda \ge F(j) + \mathcal{C}(y_{j:l}) +
\lambda$, so the partition that places a changepoint at $j$ after the
optimal prefix of $[0,j)$ is at least as good as any that ends its last
segment at $i$; hence $i$ never needs to be examined again. For the
$\ell_2$ cost, \eqref{eq:appC-Kcond} holds with $K = 0$, because adding a
split point to a segment can only reduce its residual sum of squares (the
split model nests the unsplit one). The recursion with pruning examines, at
each $j$, only the candidates not yet discarded; under the regularity
conditions of \citeauthor{killick2012pelt} (changepoints whose number grows
linearly with $n$) the expected number of surviving candidates is bounded and
the total cost is linear in $n$, whence the name. For a short series with
few changes, as here, the pruning saves little and the method is simply an
exact solution of \eqref{eq:appC-peltobj}; the point of using it is
exactness at a given penalty, not speed. The implementation
(\texttt{ruptures.Pelt}, \citealp{truong2020ruptures}) is called with
\texttt{model="l2"}, \texttt{min\_size=6} and \texttt{jump=1}, the last two
restricting segments to at least six months and allowing a changepoint at
every index; the library returns the segment end points including $n$, which
the wrapper drops.

\paragraph{The penalty $2\sigma^2 \log n$.}
Suppose the observations are independent Gaussian with piecewise-constant
mean and a common variance $\sigma^2$. Twice the negative log-likelihood of
a partition, up to a constant, is $\sigma^{-2}\sum_k \mathcal{C}(y_{\tau_{k-1}:\tau_k})$.
Each additional changepoint adds two parameters, a segment mean and a
position, so the Schwarz criterion in this scale adds $2\log n$ per
changepoint. Multiplying through by $\sigma^2$ to express the criterion in
the units of the cost gives
\begin{equation}
  \sum_k \mathcal{C}(y_{\tau_{k-1}:\tau_k}) + 2\sigma^2 \log n \cdot m,
\end{equation}
which is \eqref{eq:appC-peltobj} with $\lambda = 2\sigma^2\log n$. The
implementation replaces $\sigma^2$ by the sample variance of the whole
series, $\widehat{\mathrm{Var}}(y) = n^{-1}\sum_t (y_t - \bar y)^2$
(\texttt{numpy.var}, population form). This is the conservative choice
discussed in Part~IV: in a series with a trend or with genuine level
shifts, the overall variance exceeds the within-segment variance, so the
penalty is larger than the within-segment Schwarz penalty and fewer
changepoints are declared. \citet{yao1988schwarz} proves that the Schwarz
criterion, with a known or consistently estimated $\sigma^2$, selects the
true number of changes with probability tending to one in the
piecewise-constant Gaussian model; the result does not extend to a series
whose mean is a trend, for which a piecewise-constant fit must spend
changepoints to track the slope, and the PELT dates of this study are read
with that in mind.

\subsection{Bayesian online changepoint detection}
\label{sec:appC-bocpd}

\paragraph{Conjugate updates.}
Within a run the observations are modelled as
$y \mid \mu, \sigma^2 \sim \mathcal{N}(\mu, \sigma^2)$ with the
Normal--Inverse-Gamma prior
$\mu \mid \sigma^2 \sim \mathcal{N}(\mu_0, \sigma^2/\kappa_0)$,
$\sigma^2 \sim \mathrm{IG}(\alpha_0, \beta_0)$, whose joint density is
proportional to
\begin{equation}
  (\sigma^2)^{-\alpha_0 - 3/2} \exp\Bigl\{ -\frac{\beta_0}{\sigma^2} - \frac{\kappa_0 (\mu - \mu_0)^2}{2\sigma^2} \Bigr\}.
\end{equation}
Multiplying by the likelihood of one observation $y$, proportional to
$(\sigma^2)^{-1/2}\exp\{-(y-\mu)^2/(2\sigma^2)\}$, and completing the square
in $\mu$,
\begin{equation}
  \kappa_0 (\mu - \mu_0)^2 + (y - \mu)^2
  \;=\; (\kappa_0 + 1)\Bigl(\mu - \frac{\kappa_0\mu_0 + y}{\kappa_0 + 1}\Bigr)^2 + \frac{\kappa_0 (y - \mu_0)^2}{\kappa_0 + 1},
\end{equation}
shows that the posterior is again Normal--Inverse-Gamma with
\begin{equation}
  \mu_1 = \frac{\kappa_0 \mu_0 + y}{\kappa_0 + 1}, \quad
  \kappa_1 = \kappa_0 + 1, \quad
  \alpha_1 = \alpha_0 + \tfrac12, \quad
  \beta_1 = \beta_0 + \frac{\kappa_0 (y - \mu_0)^2}{2(\kappa_0 + 1)},
  \label{eq:appC-nigupdate}
\end{equation}
which are the four one-step updates of the implementation
(\texttt{mu\_new}, \texttt{kappa\_new}, \texttt{alpha\_new},
\texttt{beta\_new}). Applying \eqref{eq:appC-nigupdate} $r$ times and
collecting terms gives the batch form quoted in Part~IV, with
$\kappa_r = \kappa_0 + r$, $\mu_r = (\kappa_0\mu_0 + r\bar y)/\kappa_r$,
$\alpha_r = \alpha_0 + r/2$ and
$\beta_r = \beta_0 + \tfrac12\sum_{i \le r}(y_i - \bar y)^2 + \kappa_0 r (\bar y - \mu_0)^2 / (2\kappa_r)$;
the equality of the two forms is a matter of algebra (induction on $r$) and
is the reason the online recursion can be trusted to reproduce the batch
posterior.

\paragraph{Student-$t$ predictive.}
The predictive density of the next observation integrates the Gaussian
likelihood against the current Normal--Inverse-Gamma posterior. Integrating
$\mu$ first gives $y \mid \sigma^2 \sim \mathcal{N}(\mu_r, \sigma^2 (1 +
1/\kappa_r))$; integrating $\sigma^2 \sim \mathrm{IG}(\alpha_r, \beta_r)$
against this scale mixture of normals gives
\begin{equation}
  p(y \mid r) \;=\; \int \mathcal{N}\Bigl(y;\ \mu_r,\ \sigma^2\frac{\kappa_r + 1}{\kappa_r}\Bigr)\,\mathrm{IG}(\sigma^2; \alpha_r, \beta_r)\, d\sigma^2
  \;=\; t_{2\alpha_r}\Bigl( y;\ \mu_r,\ \frac{\beta_r(\kappa_r + 1)}{\alpha_r \kappa_r} \Bigr),
  \label{eq:appC-student}
\end{equation}
a Student-$t$ density with $2\alpha_r$ degrees of freedom, location $\mu_r$
and squared scale $\beta_r(\kappa_r+1)/(\alpha_r\kappa_r)$: the standard
result that a normal with inverse-gamma mixing on the variance is a
$t$ with degrees of freedom twice the shape and scale equal to the
rate over the shape times the normal's variance factor. The code evaluates
\eqref{eq:appC-student} through \texttt{scipy.stats.t.pdf} with
\texttt{df = 2*alpha} and \texttt{scale = sqrt(beta*(kappa+1)/(alpha*kappa))},
vectorised over the run lengths.

\paragraph{Run-length recursion and the mass on $r = 0$.}
Let $R_t(r) = P(r_t = r \mid y_{1:t})$ be the filtered run-length posterior
after $t$ observations, with $R_0(0) = 1$, and let $\pi_t(r) = p(y_{t+1} \mid
r_t = r, y_{1:t})$ be the predictive \eqref{eq:appC-student} evaluated with
the hyperparameters of the run of length $r$ ending at $t$. With a constant
hazard $H$ (a geometric prior on run lengths), the joint recursion of
\citet{adams2007bocpd} is, for the growth and the change moves,
\begin{equation}
  \tilde R_{t+1}(r+1) = R_t(r)\,\pi_t(r)\,(1 - H), \qquad
  \tilde R_{t+1}(0) = \sum_{r=0}^{t} R_t(r)\,\pi_t(r)\, H,
  \label{eq:appC-rl}
\end{equation}
followed by normalisation, $R_{t+1}(r) = \tilde R_{t+1}(r) / \sum_{r'}
\tilde R_{t+1}(r')$. Write $Z_t = \sum_{r} R_t(r)\pi_t(r)$ for the marginal
predictive likelihood of $y_{t+1}$. Then $\sum_{r \ge 1}\tilde R_{t+1}(r) =
(1-H) Z_t$ and $\tilde R_{t+1}(0) = H Z_t$, so the normaliser is $Z_t$ and
\begin{equation}
  R_{t+1}(0) \;=\; \frac{H Z_t}{H Z_t + (1-H) Z_t} \;=\; H \qquad \text{for every } t.
  \label{eq:appC-r0}
\end{equation}
The filtered probability that a changepoint occurs \emph{at} the current
step therefore equals the prior hazard identically, whatever the data: the
run-length posterior carries information about where the last change was
only through the relative masses at $r \ge 1$, which are reweighted by the
predictive likelihoods, and not through the mass at $r = 0$, which the
constant hazard fixes. This is why the detection statistics of Part~IV are
not $R_t(0)$. The first, $P(r_t \le 3 \mid y_{1:t}) = \sum_{r=0}^{3} R_t(r)$,
is $H$ plus the mass on runs of length one to three, that is, on changes
that occurred one to three months earlier and have since accumulated
evidence; its value exceeds $H$ only when the recent observations are
better explained by a fresh run than by the continuation of the old one. The
second is the maximum-a-posteriori run length $\hat r_t = \arg\max_r R_t(r)$,
whose resets from a value above twelve to a value at most two are the
online detector's changepoints. The implementation keeps the full
$(n+1)\times(n+1)$ matrix $R$, adds $10^{-300}$ to the normaliser against
underflow, and initialises $\mu_0$ and $\beta_0$ from the first six
observations with $\kappa_0 = \alpha_0 = 1$; the prior mean run length
$1/H = 60$ months is the only tuning constant. The recursion is the filtered
one; the smoothed posterior of \citet{fearnhead2006exact} would require a
backward pass and answers the retrospective question that the offline
methods of Sections~\ref{sec:appC-bp} and~\ref{sec:appC-pelt} already
address.

\subsection{The Gini coefficient: rank form and Lorenz form}
\label{sec:appC-gini}

Let $c_{(1)} \le c_{(2)} \le \dots \le c_{(n)}$ be the citation counts of
the $n$ works of a publication year sorted in ascending order, $S =
\sum_i c_{(i)} > 0$ their total, and $L_i = \sum_{j \le i} c_{(j)} / S$ the
cumulative share held by the $i$ least-cited works, with $L_0 = 0$ and
$L_n = 1$. The Lorenz curve \citep{lorenz1905concentration} is the polygon
through the points $(i/n, L_i)$, and the Gini coefficient is defined as one
minus twice the area $A$ under it \citep{ceriani2012gini}. By the
trapezoidal rule, which is exact for a polygon,
\begin{equation}
  A \;=\; \sum_{i=1}^{n} \frac{1}{n}\cdot\frac{L_{i-1} + L_i}{2}
    \;=\; \frac{1}{2n}\Bigl( 2\sum_{i=1}^{n} L_i - L_n \Bigr)
    \;=\; \frac{1}{n}\sum_{i=1}^{n} L_i - \frac{1}{2n}.
\end{equation}
Exchanging the order of summation in $\sum_i L_i$,
\begin{equation}
  \sum_{i=1}^{n} L_i \;=\; \frac{1}{S}\sum_{i=1}^{n}\sum_{j=1}^{i} c_{(j)}
    \;=\; \frac{1}{S}\sum_{j=1}^{n} (n - j + 1)\, c_{(j)}
    \;=\; (n+1) - \frac{1}{S}\sum_{j=1}^{n} j\, c_{(j)},
\end{equation}
so that
\begin{equation}
  G \;=\; 1 - 2A
    \;=\; 1 - \frac{2(n+1)}{n} + \frac{2}{nS}\sum_{j=1}^{n} j\, c_{(j)} + \frac{1}{n}
    \;=\; \frac{2\sum_{j=1}^{n} j\, c_{(j)}}{n \sum_{j} c_{(j)}} - \frac{n+1}{n},
  \label{eq:appC-gini}
\end{equation}
which is equation~\eqref{eq:gini} of Part~IV and the expression that
\texttt{bibliometrics/03\_citation\_impact.py} evaluates in SQL, the rank
$j$ being the window function \texttt{row\_number()} partitioned by year and
ordered by the citation count and then by the work identifier.
The secondary ordering by work identifier is needed only to make the rank a
permutation when counts tie; since tied values receive consecutive ranks and
contribute $c \cdot (j + j')$ to the numerator whichever of the two tied
works gets the smaller rank, the value of $G$ does not depend on how ties are
broken. The bounds follow from \eqref{eq:appC-gini}: $G = 0$ when all
$c_{(j)}$ are equal (then $\sum_j j c_{(j)} = c\,n(n+1)/2$ and the two terms
cancel), and $G = 1 - 1/n$ when a single work holds all citations (then
$\sum_j j c_{(j)} = nS$), so the maximum attainable value is $1 - 1/n$ rather
than one and the coefficient is not normalised by that factor. Uncited works
enter with $c_{(j)} = 0$ at the lowest ranks; adding $u$ uncited works to a
year leaves $\sum_j j c_{(j)}$ shifted by $uS$ and $n$ increased by $u$, and
one checks from \eqref{eq:appC-gini} that $G$ increases, which is the
direction stated in Part~IV and the reason the uncited share accompanies the
coefficient in the tables.

\subsection{Rao--Stirling and Leinster--Cobbold at order two}
\label{sec:appC-diversity}

Let $p \in \R^K$ be a probability vector over categories, $\delta_{ij} \in
[0,1]$ a dissimilarity with $\delta_{ii} = 0$, and $Z = (z_{ij})$ the
similarity matrix $z_{ij} = 1 - \delta_{ij}$, so that $z_{ii} = 1$. The
Rao--Stirling index \citep{rao1982diversity, stirling2007diversity} is
$\Delta_{\mathrm{RS}}(p) = \sum_{i \ne j} p_i p_j \delta_{ij} = p^{\top}\Delta p$
with $\Delta = (\delta_{ij})$, the diagonal contributing nothing. The
Leinster--Cobbold diversity of order $q$ \citep{leinster2012diversity} is
${}^{q}D^{Z}(p) = \bigl(\sum_{i : p_i > 0} p_i (Zp)_i^{q-1}\bigr)^{1/(1-q)}$;
at $q = 2$ the exponent $1/(1-q) = -1$ and the inner power is one, so
\begin{equation}
  {}^{2}D^{Z}(p) \;=\; \Bigl(\sum_i p_i (Zp)_i\Bigr)^{-1} \;=\; \frac{1}{p^{\top} Z p}.
\end{equation}
Since $Z = \mathbf{1}\mathbf{1}^{\top} - \Delta$ entrywise (using $z_{ii} = 1
= 1 - \delta_{ii}$ on the diagonal as well), and $p^{\top}\mathbf{1}\mathbf{1}^{\top}p
= (\sum_i p_i)^2 = 1$,
\begin{equation}
  p^{\top} Z p \;=\; 1 - p^{\top}\Delta p \;=\; 1 - \Delta_{\mathrm{RS}}(p),
  \qquad\text{hence}\qquad
  {}^{2}D^{Z}(p) \;=\; \frac{1}{1 - \Delta_{\mathrm{RS}}(p)},
  \label{eq:appC-lc2}
\end{equation}
the identity quoted in Part~IV\@. It shows that at order two the two
families carry the same information: ${}^{2}D^{Z}$ is a strictly increasing
function of $\Delta_{\mathrm{RS}}$ on $[0, 1)$, and the ordering of years or
categories by either index is the same. They differ in scale and in
behaviour near saturation. $\Delta_{\mathrm{RS}}$ is bounded by the mean
dissimilarity and compresses differences among very even distributions;
${}^{2}D^{Z}$ is an effective number of categories, equal to $K$ for the
uniform distribution over $K$ mutually dissimilar categories
($\Delta_{\mathrm{RS}} = 1 - 1/K$) and to one for a distribution concentrated
on a single category ($\Delta_{\mathrm{RS}} = 0$), and it expands the
compressed end. The identity requires only $z_{ii} = 1$ and $z_{ij} = 1 -
\delta_{ij}$; it does not require $\delta$ to be a metric, and the
dissimilarity used, one minus a cosine similarity of category centroids,
is not one. The script \texttt{bibliometrics/04\_diversity\_cagr.py}
computes both quantities from the yearly category mix exactly as
\eqref{eq:appC-lc2} prescribes, $\Delta_{\mathrm{RS}} = p^{\top}(1 - Z)p$
with the diagonal of $Z$ set to one and ${}^{2}D^{Z} = 1/(p^{\top}Zp)$, and
attaches a percentile bootstrap interval to $\Delta_{\mathrm{RS}}$ by
multinomial resampling of the works over categories ($B = 500$, seed 44),
which is the sampling model under which the category counts of a year are
a multinomial draw of fixed size; the interval of ${}^{2}D^{Z}$ follows by
the monotone transformation.

\subsection{Modularity of a directed weighted graph at a fixed partition}
\label{sec:appC-modularity}

Let $W \in \R_{\ge 0}^{K \times K}$ be the flow matrix of a citing year,
$W_{ij}$ the number of citations from category $i$ to category $j$, with
$m = \sum_{ij} W_{ij}$, out-strength $k^{\mathrm{out}}_i = \sum_j W_{ij}$ and
in-strength $k^{\mathrm{in}}_j = \sum_i W_{ij}$, and let $g : \{1, \dots, K\}
\to \mathcal{G}$ be a given assignment of categories to communities. The
statistic implemented by \texttt{modularity()} in
\texttt{knowledge\_graph/02\_category\_flows.py} is
\begin{equation}
  Q(W, g) \;=\; \frac{1}{m}\sum_{i,j}\Bigl( W_{ij} - \frac{k^{\mathrm{out}}_i k^{\mathrm{in}}_j}{m} \Bigr)\mathbf{1}[g_i = g_j],
  \label{eq:appC-Q}
\end{equation}
the directed form of \citet{leicht2008directed}, which reduces to the
Newman--Girvan modularity \citep{newman2004community} when $W$ is symmetric
(then $k^{\mathrm{out}} = k^{\mathrm{in}} = k$ and $m = 2|E|$ for an
unweighted graph). Three facts are used in Part~IV\@. First, the null term
is the expected weight of the edge $i \to j$ under random rewiring that
preserves every node's out- and in-strength in expectation: a configuration
model in which each of the $m$ units of weight leaves node $i$ with
probability $k^{\mathrm{out}}_i/m$ and enters node $j$ with probability
$k^{\mathrm{in}}_j/m$ independently gives expected weight
$m \cdot (k^{\mathrm{out}}_i/m)(k^{\mathrm{in}}_j/m) = k^{\mathrm{out}}_i
k^{\mathrm{in}}_j/m$, and summing over $j$ recovers $k^{\mathrm{out}}_i$, so
the null model has the observed strengths. Second, $Q$ vanishes when all
nodes are in one community: then the indicator is identically one and
$\sum_{ij} W_{ij} = m = \sum_{ij} k^{\mathrm{out}}_i k^{\mathrm{in}}_j/m$.
Third, writing $e_{gg} = \sum_{i,j : g_i = g_j = g} W_{ij}/m$ for the fraction
of weight inside community $g$, $a^{\mathrm{out}}_g = \sum_{i : g_i = g}
k^{\mathrm{out}}_i/m$ and $a^{\mathrm{in}}_g$ likewise,
\begin{equation}
  Q \;=\; \sum_{g \in \mathcal{G}} \bigl( e_{gg} - a^{\mathrm{out}}_g\, a^{\mathrm{in}}_g \bigr),
  \label{eq:appC-Qcomm}
\end{equation}
so that $Q \le \sum_g e_{gg} \le 1$ and $Q \ge -\sum_g a^{\mathrm{out}}_g
a^{\mathrm{in}}_g \ge -1$; the value is positive when more weight stays
inside communities than the strength-preserving null predicts. The
self-share reported alongside, $\operatorname{tr}(W)/m = \sum_i e_{ii}$ at
the category partition, is the first term of \eqref{eq:appC-Qcomm} alone and
differs from $Q_{\mathrm{cat}}$ by the correction $\sum_i a^{\mathrm{out}}_i
a^{\mathrm{in}}_i$, which grows with the concentration of flows on a few
large categories. For the area partition, \eqref{eq:appC-Qcomm} is summed
over the eight areas with $e_{\alpha\alpha}$ the within-area share, which is
one minus the cross-area share $X_{\alpha,t}$ of Part~IV weighted by the
area's out-strength; $Q_{\mathrm{area}}$ is therefore a strength-weighted,
null-corrected average of the within-area shares, and a fall in
$Q_{\mathrm{area}}$ can arise from a rise in boundary crossing or from a
shift of out-strength toward areas whose $a^{\mathrm{out}}_\alpha
a^{\mathrm{in}}_\alpha$ is large. Because the partition is fixed rather than
optimised, none of the resolution-limit results for maximised modularity
\citep{fortunato2007resolution} applies; $Q$ is here a descriptive index and
its comparison across years is meaningful only because the partition and
the category definitions are the same in every year.

\subsection{Krippendorff's \texorpdfstring{$\alpha$}{alpha} with the MASI distance}
\label{sec:appC-alpha}

\paragraph{General form.}
Krippendorff's $\alpha$ \citep{hayes2007krippendorff, artstein2008agreement}
is defined for any difference function $d(v, v') \ge 0$ on the values that
coders may assign, with $d(v, v) = 0$, as
\begin{equation}
  \alpha \;=\; 1 - \frac{D_o}{D_e},
  \label{eq:appC-alpha}
\end{equation}
where $D_o$ is the mean difference between the values assigned to the same
unit by different coders and $D_e$ the mean difference between all pairs of
values in the pooled data, irrespective of unit. With two coders $A$ and $B$
rating each of $n$ units, the pooled set contains $N = 2n$ values and
\begin{equation}
  D_o \;=\; \frac{1}{n}\sum_{u=1}^{n} d\bigl(a_u, b_u\bigr), \qquad
  D_e \;=\; \frac{1}{N(N-1)}\sum_{c \ne c'} d\bigl(v_c, v_{c'}\bigr)
        \;=\; \frac{1}{\binom{N}{2}}\sum_{c < c'} d\bigl(v_c, v_{c'}\bigr),
  \label{eq:appC-DoDe}
\end{equation}
the second equality using the symmetry of $d$. The coefficient is one when
the coders agree on every unit ($D_o = 0$), zero when the within-unit
differences are what chance pairing of the pooled values would produce
($D_o = D_e$), and negative when disagreement is systematic. Unlike Cohen's
$\kappa$, which compares observed agreement with the product of the two
coders' marginals, $\alpha$ pools the two coders' values into one
distribution for its chance term, so it does not reward coders for having
different marginal distributions; this and its accommodation of arbitrary
difference functions are the reasons it is the recommended coefficient for
set-valued annotation \citep{artstein2008agreement}.

\paragraph{The MASI distance.}
For set-valued labels $x, y \subseteq \mathcal{K}$, \citet{passonneau2006masi}
defines the Measuring Agreement on Set-valued Items distance as
\begin{gather}
  d_{\mathrm{MASI}}(x, y) \;=\; 1 - J(x, y)\, M(x, y), \qquad
  J(x, y) = \frac{|x \cap y|}{|x \cup y|}, \nonumber\\
  M(x, y) =
  \begin{cases}
    1    & x = y,\\
    2/3  & x \subsetneq y \text{ or } y \subsetneq x,\\
    1/3  & x \cap y \ne \emptyset \text{ and neither contains the other},\\
    0    & x \cap y = \emptyset ,
  \end{cases}
  \label{eq:appC-masi}
\end{gather}
the Jaccard similarity attenuated by a monotonicity factor that
distinguishes nesting from mere overlap. The distance is zero only for
identical non-empty sets, equals one for disjoint sets, and treats the pair
(empty, empty) as identical ($d = 0$) by the convention of the
implementation, which returns zero before computing $J$ for two empty sets;
the pair (empty, non-empty) has $J = 0$ and $d = 1$.

\paragraph{What the code computes.}
\texttt{taxonomy/validate\_sample.py} forms, for the two raters, the
observed term as the mean of $d_{\mathrm{MASI}}^2$ over the $n$ rated works
and the expected term as the mean of $d_{\mathrm{MASI}}^2$ over all
unordered pairs of the $2n$ pooled label sets (exhaustively when
$2n \le 1200$, otherwise over 200{,}000 random ordered pairs with $i \ne j$,
seed 44), and reports $1 - D_o/D_e$. Two remarks make this precise. The code
uses the constants $0.67$ and $0.33$ for the monotonicity factor rather than
$2/3$ and $1/3$, a rounding of the published definition that changes
distances by at most $0.0034$. More consequentially, it \emph{squares} the
MASI distance before averaging, that is, it applies
\eqref{eq:appC-alpha}--\eqref{eq:appC-DoDe} with $d = d_{\mathrm{MASI}}^2$.
Krippendorff's interval metric is the squared difference, and the code's
comment describes the computation as the ``interval form on the MASI
distance''; \citeauthor{passonneau2006masi}, in contrast, plugs
$d_{\mathrm{MASI}}$ itself into the $\alpha$ formula, and the agreement
packages that implement MASI follow her. The two conventions yield different
numbers from the same ratings---squaring compresses small disagreements and
so, as a rule, raises $\alpha$ when most disagreements are partial---and the
value reported in Part~IX is the squared-distance version, labelled as such.
Both are legitimate instances of \eqref{eq:appC-alpha}; what is not
legitimate is to compare the reported value with published MASI-$\alpha$
figures without this caveat. The mean Jaccard index reported next to it is
$n^{-1}\sum_u J(a_u, b_u)$ with $0/0 = 0$ and is a raw agreement rate with
no chance correction. Both statistics are computed on the full rated sample
and not on the 200-work subsample reserved for the principal investigator,
whose verdicts will add a third coder to \eqref{eq:appC-DoDe} once
checkpoint H-2 is signed.

\subsection{The multi-label score rule and the tie-break as a total order}
\label{sec:appC-score}

\paragraph{Score.}
For a work $w$ and a category $k$ the implementation
(\texttt{taxonomy/classify.py}, \texttt{score\_row}) computes five
components, one per signal type $\sigma \in \{\mathrm{T}, \mathrm{F},
\mathrm{X}, \mathrm{D}, \mathrm{kw}\}$,
\begin{align}
  s^{\sigma}_k(w) &\;=\; \min\Bigl\{ 1,\ \sum_{x \in \sigma(w)} \omega^{\sigma}_k(x) \Bigr\}
  \qquad (\sigma \ne \mathrm{kw}), \\
  s^{\mathrm{kw}}_k(w) &\;=\; \min\Bigl\{ 1,\ \sum_{(r, \omega, \phi) \in R_k} \omega\, \mathbf{1}[r \text{ matches } \phi(w)] \Bigr\},
\end{align}
where $\sigma(w)$ is the \emph{set} of the work's topic, subfield, arXiv
category or DBLP stream identifiers (Python \texttt{set()}, so a repeated
identifier contributes once), and the keyword loop stops at the first rule
that brings the running sum to one, which is equivalent to the capped sum.
The score is $s_k(w) = \sum_\sigma s^{\sigma}_k(w) \in [0, 5]$; a category
with $s_k(w) \ge \theta = 1$ is then vetoed if any exclusion pattern of
$Z_k$ matches the concatenation of title and abstract, in which case its
score is set to zero and it is not assigned. The set of assigned categories
is $C(w) = \{k : s_k(w) \ge \theta,\ \text{not vetoed}\}$, and the stored
score of an assigned category is $s_k(w)$ rounded to three decimals. Two
elementary properties justify the reading given in Part~IV\@. The cap makes
each component a bounded, monotone, subadditive function of the matched
weights: adding a signal never lowers a component, and no number of signals
of one type can contribute more than one. And because the threshold equals
the cap, a single weight-one signal of any type suffices for assignment
while signals of weight below one require corroboration from a different
type; formally, $s_k(w) \ge 1$ holds if and only if either some component
reaches one or the sum of the (uncapped) components across at least two
types reaches one.

\paragraph{Primary category as the minimum of a total order.}
Let $\mathcal{P}(w) = \bigcup_{k \in C(w)} \mathrm{parents}(k)$ be the set
of categories declared parents of at least one assigned category (the
\texttt{child\_of} relation of the ontology). The implementation sorts the
assigned categories by the key
\begin{equation}
  \phi(k) \;=\; \bigl( -s_k(w),\ \mathbf{1}[k \in \mathcal{P}(w)],\ \mathrm{number}(k) \bigr) \in \R \times \{0,1\} \times \mathbb{N}
  \label{eq:appC-key}
\end{equation}
in lexicographic order and takes the first element; this is
$\kappa^{\ast}(w) = \arg\min_{k \in C(w)} \phi(k)$ and coincides with the
lexicographic arg-max of Part~IV (the sign of the first component reversed).
The lexicographic order on $\R \times \{0,1\} \times \mathbb{N}$ is a total
order, and $\phi$ is injective on $\mathcal{K}$ because the ontology numbers
are distinct, so the restriction of $\phi$ to $C(w)$ has a unique minimum
whenever $C(w) \ne \emptyset$: the primary category is well defined and does
not depend on the order in which categories are scanned. The second
component implements ``child before parent'': if $k$ and $k'$ are both
assigned with equal scores and $k'$ is a declared parent of $k$, then
$k' \in \mathcal{P}(w)$ while $k$ may not be, and $k$ precedes $k'$. The rule
is not transitive in the sense of a hierarchy, however. Membership in
$\mathcal{P}(w)$ depends only on whether $k$ is the parent of \emph{some}
assigned category, not of the specific competitor; a category that is the
parent of a low-scoring assigned child is demoted in the tie against an
unrelated category of equal score. This is a property of the key as
implemented, stated here so that the primary counts of Part~IX are read with
it; it affects only exact ties in a score rounded to three decimals, which
are frequent precisely among the categories whose signals are weight-one
keywords. The third component, the ontology number, is arbitrary and makes
the order total; no claim is attached to it.

\subsection{The emerging-topic label and the absence of look-ahead}
\label{sec:appC-lookahead}

\paragraph{Filtrations.}
Index months by integers $t = 0, 1, \dots, M$ (January 2018 to the last
complete month) and let the panel of \texttt{experiments/emerging\_detection.py}
be the pair of arrays $n_{k,t}$ (eligible month-precision works of topic $k$
with first month $t$) and $c_{k,t}$ (internal citations received by works of
topic $k$ from works with first month $t$). Define the filtration
\begin{equation}
  \mathcal{F}_T \;=\; \sigma\bigl( n_{k,t},\ c_{k,t} :\ k \in \mathcal{K}_{\mathrm{top}},\ t \le T \bigr),
  \label{eq:appC-filtration}
\end{equation}
the information contained in the panel columns up to and including month
$T$. A statistic is \emph{available at} $T$ if it is
$\mathcal{F}_T$-measurable. Each of the six features of Part~IV is a
function of sums of panel entries over windows $(a, b]$ with $b \le T$:
$g^{12}_{k,T}$ uses $(T-24, T]$, $\mathrm{acc}_{k,T}$ uses $(T-36, T]$,
$z_{k,T}$ uses $[T-27, T]$, $\mathrm{cit}_{k,T}$ uses $(T-24, T]$ of $c$,
$\mathrm{share}_{k,T}$ uses $(T-12, T]$ of every topic, and
$\mathrm{age}_{k,T}$ uses the cumulative sum up to $T$. All are
$\mathcal{F}_T$-measurable by construction. The implementation enforces this
by index arithmetic (\texttt{win(a, lo, hi)} slices columns \texttt{lo+1}
to \texttt{hi} inclusive with \texttt{hi = T\_idx}), not by convention.

\paragraph{Label.}
The label $\ell_{k,T,h}$ of equation~\eqref{eq:label} is a function of
$V_k(T-h, T]$, $V_k(T, T+h]$, $V_k(T-12, T]$ and of the within-cutoff decile
of $g^{\mathrm{fut}}$ over eligible topics; it is therefore
$\mathcal{F}_{T+h}$-measurable and, because it depends on columns strictly
after $T$, not $\mathcal{F}_T$-measurable. The feature--label pair at cutoff
$T$ thus has the structure of a prediction problem: predict an
$\mathcal{F}_{T+h}$-measurable outcome from an $\mathcal{F}_T$-measurable
input. The eligibility condition $V_k(T-12, T] \ge 20$ is
$\mathcal{F}_T$-measurable, so conditioning on it does not leak the outcome.

\paragraph{Separation of training and test.}
Let $\mathcal{T}_{\mathrm{tr}}$ and $\mathcal{T}_{\mathrm{te}}$ be the
training and test cutoffs. Every quantity used to fit a model---features,
labels and the standardisation constants of the scaler, which is fitted on
training rows only---is $\mathcal{F}_{T^{\mathrm{tr}}_{\max} + h}$-measurable
with $T^{\mathrm{tr}}_{\max} = \max \mathcal{T}_{\mathrm{tr}}$. The
implementation takes as candidate cutoffs every third month of the panel,
which begins in January 2018 (January, April, July and October of each year),
and admits as training cutoffs those from December 2019 up to December 2021
for $h = 12$ and up to December 2020 for $h = 24$; the realised training
cutoffs therefore run from January 2020 to October 2021 ($h = 12$) or to
October 2020 ($h = 24$), so that $T^{\mathrm{tr}}_{\max} + h$ is October
2022 in both cases, and the first test cutoff is January 2023 (the last is
October 2024 at most, subject to the censoring condition below). The realised
lists are written to \texttt{results/emerging/metrics\_h12.json} and
\texttt{metrics\_h24.json} under \texttt{train\_cutoffs} and
\texttt{test\_cutoffs}.
Hence
\begin{equation}
  T^{\mathrm{tr}}_{\max} + h \;<\; \min \mathcal{T}_{\mathrm{te}},
  \label{eq:appC-gap}
\end{equation}
and the fitted predictor is measurable with respect to a $\sigma$-field
that precedes every test feature, let alone every test label, which closes
at $\max \mathcal{T}_{\mathrm{te}} + h \le M$. Condition \eqref{eq:appC-gap}
is the formal content of ``out of time'' in the sense of
\citet{bergmeir2012crossval}; it is stricter than the plan's wording, which
allowed training cutoffs through 2022; the code's rule is the one that
guarantees \eqref{eq:appC-gap}, and Part~IV describes the same rule. The test cutoffs
are further restricted to those with $T + h \le M$, so that no test label is
right-censored.

\paragraph{What measurability does not cover.}
The filtration \eqref{eq:appC-filtration} is generated by the panel, and the
panel is built once, in 2026, from a corpus whose topic assignments,
eligibility flags, linkage and month-precision filter were all fixed with
knowledge of the whole period. Formally, the random objects $n_{k,t}$
themselves are not $\mathcal{F}_t$-measurable functionals of the literature
as it stood at $t$: they are functionals of the literature as OpenAlex
indexed and classified it by October 2026. The statement proved above is
therefore conditional: \emph{given} the 2026 panel, no feature at $T$ uses a
column after $T$ and no training quantity uses a column at or after the
first test cutoff. It is not the statement that an observer in 2021 could
have computed the training set, which would require a topic vocabulary and a
corpus frozen in 2021. Part~IV draws the distinction and names the
cluster-based extension that would narrow it; the derivation here makes
precise which half of the claim is proved and which is assumed.