\section{Mechanisms and pricing}
\label{sec:mechanisms}

\subsection{Where would one-minute information live? (H3)}

The pre-registered heterogeneity hypothesis holds that if fine-grained activity carries incremental information, it
should do so where information asymmetry and processing frictions are largest: small, illiquid and highly levered
firms. Table~\ref{tab:subgroups} and Figure~\ref{fig:heterogeneity} split the test period along these ex-ante terciles
(cut-points from training rows), by profitability, release timing, sector, bar density, and by whether the 20-day window
straddles the previous release. In no subgroup does the 1-minute set improve on the daily benchmark. The ordering is
nevertheless informative: the deterioration is smallest and statistically indistinguishable from zero for small,
illiquid and loss-making firms (\emph{e.g.}, small firms: within $\pm0.002$ of zero), and largest for large, liquid and
profitable firms, where the extra features add noise to an already accurate forecast. The sign of the H3 ordering is
therefore the one predicted by frictions, but the level is zero: the groups in which intraday activity would be most
informative are also those in which it is not informative \emph{enough} to help.

\begin{table}[H]
\centering\footnotesize
\caption{The 1-minute increment by pre-specified subgroup (LightGBM, test period, $h=20$)}
\label{tab:subgroups}
\adjustbox{max width=\textwidth}{\input{tables/tab07b_subgroups}}
\begin{minipage}{\textwidth}\vspace{0.3em}\footnotesize
\emph{Notes.} Terciles of log market capitalisation, Amihud illiquidity and debt-to-assets are formed on training rows.
``Window straddles previous release'': the 20-session feature window contains $A_{q-1}$. ``Dense bars'': at least
98\% of expected 1-minute bars present. 95\% block-bootstrap CIs over calendar quarters; 2020 is in the validation
period and therefore absent from the test subgroups.
\end{minipage}
\end{table}

\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/fig07_heterogeneity.pdf}
\caption{Change in test log loss from adding 1-minute families to the daily-and-accounting benchmark, by subgroup
(LightGBM). Negative values would indicate a gain.}
\label{fig:heterogeneity}
\end{figure}

The panel regressions of Table~\ref{tab:panel} use the out-of-fold Latent Fundamental Deterioration Signal from the
expanding-window refits. The daily-benchmark signal $\LFDS^{\mathrm{B2}}$ is a powerful predictor of the
realised surprise with firm and sector-by-quarter fixed effects ($\beta=\panSonlfdsTwoanddlfdslfdsTwo$,
$t=\pantSonlfdsTwoanddlfdslfdsTwo$, two-way clustered): a one-point increase in the predicted probability of an adverse
quarter is associated with a surprise lower by 7.6\% of assets. The \emph{incremental} 1-minute component
$\LFDS^{\mathrm{B5}}-\LFDS^{\mathrm{B2}}$ has a coefficient of \panSonlfdsTwoanddlfdsdlfds{}
($t=\pantSonlfdsTwoanddlfdsdlfds$), one sixth of the benchmark's and marginal at conventional levels. Its interactions
with ex-ante terciles point in the direction \emph{opposite} to H3: the incremental component is less informative for
the smallest firms ($t=\pantdlfdsxsizedlfdsxsizeZero$ on the bottom-tercile interaction) and for the least levered
firms ($t=\pantdlfdsxlevdlfdsxlevZero$), and does not vary with illiquidity. Whatever small amount of information the
1-minute features add is concentrated in large, levered firms, consistent with fine-grained activity reflecting the
trading of well-informed participants in liquid names rather than the slow processing of information in neglected ones;
in the forecasting exercise this amount is not large enough to survive the noise the features bring.

\begin{table}[H]
\centering\footnotesize
\caption{Panel regressions of the realised surprise on out-of-fold deterioration signals}
\label{tab:panel}
\adjustbox{max width=\textwidth}{\input{tables/tab06_panel}}
\begin{minipage}{\textwidth}\vspace{0.3em}\footnotesize
\emph{Notes.} Dependent variable: $S_{i,q}$ winsorised at 1/99\% (specifications ``S\_on''), or the binary adverse outcome
(``y\_on''). $\mathrm{lfds2}$, $\mathrm{lfds5}$: out-of-fold predicted probabilities from LB2 and LB5 (expanding-window
annual refits, 2016--2026); $\mathrm{d\_lfds}=\mathrm{lfds5}-\mathrm{lfds2}$; interactions with within-quarter terciles
(0 = bottom, 2 = top). Controls: ROA, leverage, log market cap, 20-day return and volatility, last and mean surprise.
Firm and sector$\times$calendar-quarter fixed effects; standard errors two-way clustered by firm and calendar quarter.
$^{*}$, $^{**}$, $^{***}$: 10, 5, 1\%.
\end{minipage}
\end{table}

\subsection{Is the signal in prices? (H5)}

If the deterioration probability were news to the market, firms with a high ex-ante probability should experience
negative abnormal returns between the cutoff and the day after the release. Appendix Table~\ref{app:tables} shows that
the market-adjusted return from $c+1$ to $A_q+1$ is unrelated to $\LFDS^{\mathrm{B5}}$
($\beta=\carcaronlfdsFivelfdsFive$, $t=\cartcaronlfdsFivelfdsFive$) and to the daily benchmark signal: by the cutoff,
twenty trading days before the release, the information that the models extract is already in prices. The incremental
1-minute component carries a positive coefficient ($\carcaronlfdsTwodlfdsdlfds$, $t=\cartcaronlfdsTwodlfdsdlfds$),
i.e., firms whose 1-minute features push the probability \emph{up} relative to daily data earn slightly \emph{higher}
announcement-window returns, the opposite of a deterioration signal; we read this as a partial reversal of recent
intraday selling pressure rather than as information, and note that it is one of several H5 tests.

The secondary portfolio exercise confirms the absence of economic value (Table~\ref{tab:economic},
Figure~\ref{fig:economic}). Month-end decile sorts on the out-of-fold calendar-design scores, with positions taken at the
close of the following session and held one month, produce long-minus-short (low minus high deterioration probability)
FF5+momentum alphas of \alphaLBFivegross\% per month ($t=\alphatLBFivegross$) gross and \alphaLBFivenet\% net of
20 basis points per side for the 1-minute scores, and \alphaLBTwogross\% ($t=\alphatLBTwogross$) for the daily
scores; mean returns \emph{rise} with the deterioration decile, as expected if high-probability firms are small,
volatile and compensated for risk. A forecastable accounting deterioration is therefore not a tradable anomaly in
this sample, at any sampling frequency.

\begin{table}[H]
\centering\footnotesize
\caption{Economic value: monthly decile portfolios on the ex-ante deterioration score, 2016--2026}
\label{tab:economic}
\adjustbox{max width=\textwidth}{\input{tables/tab11_economic}}
\begin{minipage}{\textwidth}\vspace{0.3em}\footnotesize
\emph{Notes.} Long decile 1 (lowest score), short decile 10; equal weights; entry at the close of the session after the
month-end cutoff, exit at the close of the session after the next cutoff; turnover is one-way; net returns deduct
20 bps per side. Alphas from a regression on FF5 and momentum factors compounded over the same holding windows, HAC
standard errors.
\end{minipage}
\end{table}

\begin{figure}[H]
\centering
\includegraphics[width=0.6\textwidth]{figures/fig09_economic.pdf}
\caption{Mean monthly return by decile of the 1-minute deterioration score.}
\label{fig:economic}
\end{figure}