\section{Results}
\label{sec:results}

\subsection{The transformation of the intraday market, 1992--2026}

Figures~\ref{fig:seasonality_decade} through \ref{fig:epps} document the
environment in which propagation must be measured, on liquidity-screened
universes with strictly lagged membership.

Three secular changes stand out. First, ordinary minute-level variability
collapsed. Mean absolute one-minute returns on the top-1000 universe fell
decade by decade at every minute of the trading day
(Figure~\ref{fig:seasonality_decade}a), and the frequency of one-minute
returns exceeding 1\% fell from about \ExtremeGtOneNineties{} per 10,000 in
the 1990s to \ExtremeGtOneTens{} per 10,000 in the 2010s---a factor of roughly
\ExtremeDropFactor{} (Figure~\ref{fig:extreme_minutes}). Part of the 1990s
level reflects bid--ask bounce under eighth and sixteenth ticks rather than
true volatility; the decade-by-decade decline survives range-based measures
and coarser sampling, and we flag the tick-regime caveat wherever it binds.

Second, the collapse was not monotone in the tails. The frequency of
$|r|>5\%$ minutes rose from \ExtremeGtFivePreCovid{} per 10,000 in 2017--2019
to \ExtremeGtFiveCovid{} in 2020---a factor of \ExtremeCovidFactor---and 2008
shows the same signature (Figure~\ref{fig:extreme_minutes}). Extremes became
rarer but, when the market is stressed, arrive in synchronized bursts; we
return to this in Section~\ref{sec:mechanisms}.

Third, cross-sectional information transmission accelerated dramatically. In
1996 the mean pairwise correlation of one-minute returns among the hundred
largest stocks was \EppsNinetySixOne{}, against \EppsNinetySixThirty{} at
thirty minutes: comovement existed but took tens of minutes to materialize,
the classical Epps effect \citep{epps1979comovements}. By 2015 the one-minute
correlation was \EppsFifteenOne{} against \EppsFifteenThirty{} at thirty
minutes, and 2024 shows the same near-flat profile
(Figure~\ref{fig:epps}). The Epps effect has essentially disappeared from
large-cap U.S. equities: common information is now impounded across the
cross-section within a minute.

\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figE3_seasonality.pdf}
\caption{\textbf{Intraday seasonality by decade.} Minute-of-day means during
regular hours, lagged top-1000 dollar-volume universe, split-window and
flagged bars excluded: (a) absolute one-minute log return; (b) share of the
stock-day's volume; (c) high--low log range.}
\label{fig:seasonality_decade}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=.72\textwidth]{figE5_extreme_minutes.pdf}
\caption{\textbf{Extreme one-minute returns.} Annual frequency (log scale) of
one-minute absolute log returns exceeding 1\%, 2\%, and 5\%, lagged top-1000
universe, regular hours. Pre-2001 levels partly reflect eighth/sixteenth tick
bid--ask bounce.}
\label{fig:extreme_minutes}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=.62\textwidth]{figE8_epps.pdf}
\caption{\textbf{The disappearance of the Epps effect.} Mean pairwise return
correlation among the top-100 lagged dollar-volume stocks at 1-, 5-, 15-, and
30-minute sampling, previous-tick synchronization within day, for 1996, 2005,
2015, and 2024.}
\label{fig:epps}
\end{figure}

\subsection{Baseline: idiosyncratic shocks do not propagate}

Table~\ref{tab:baseline_propagation} reports the paper's baseline estimates of
equation~\eqref{eq:dd} for \NEventsReal{} isolated $|z|\ge6$ source events and
\NEventsPlacebo{} matched placebos. The abnormal breadth response is
economically zero at every horizon: at $h=5$ minutes the point estimate is
\DDBreadthFive$\times10^{-4}$ (bootstrap SE \DDBreadthFiveSE$\times10^{-4}$),
and at $h=30$ minutes \DDBreadthThirty$\times10^{-4}$
(SE \DDBreadthThirtySE$\times10^{-4}$). Against a placebo base rate of roughly
$7\times10^{-2}$, the largest estimate in absolute value implies that an
isolated six-sigma shock changes the probability that any given other security
jumps by less than half of one percent \emph{of the base rate}---with the
wrong sign. Abnormal intensity estimates are equally null.

\input{tables/table4_baseline_propagation}

Three features of Table~\ref{tab:baseline_propagation} deserve emphasis.
First, this is a powered null: standard errors are small enough to detect a
breadth response of $3\times10^{-4}$---one additional responding security per
seventeen events---at conventional levels. Second, the pre-event falsification
passes: real-event outcomes at $t-5$ equal placebo levels, so the null is not
produced by anticipation contaminating the baseline. Third, the null is
uniform: Figure~\ref{fig:irf_decade} shows the same flat response profile in
each of the four decades, so absorption is not a recent acquisition of
electronic markets---though the composition of what is being absorbed changed
radically, as the descriptive results show.

\begin{figure}[t]
\centering
\includegraphics[width=.6\textwidth]{figF1_irf_decade.pdf}
\caption{\textbf{Abnormal propagation breadth by decade.} Point estimates of
$\Delta\Delta$ breadth (equation~\eqref{eq:dd}) with 95\% day-block bootstrap
bands, by horizon and decade.}
\label{fig:irf_decade}
\end{figure}

Conditioning does not resurrect average propagation
(Figure~\ref{fig:gradient_state}): estimates by shock sign and by calm versus
stressed market states are all within two standard errors of zero. The only
suggestive pattern is monotonicity in magnitude---the largest shocks
($12\sigma$--$20\sigma$) carry a positive but insignificant point estimate of
\DDGradLargest$\times10^{-4}$ (SE \DDGradLargestSE$\times10^{-4}$)---which
motivates the concentrated-propagation probes of the next section.

\begin{figure}[t]
\centering
\includegraphics[width=.66\textwidth]{figF3_gradient_state.pdf}
\caption{\textbf{Shock size, sign, and market state.} $\Delta\Delta$ breadth
at $h=15$ by source-shock magnitude bins, shock sign, and prior-30-minute
market volatility state, with 95\% bootstrap intervals.}
\label{fig:gradient_state}
\end{figure}