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

The baseline null says that isolated idiosyncratic shocks do not raise
market-wide jump activity. This section asks where transmission lives if not
there: in the shocked stock's network neighborhood, in transient responses
between measurement points, and in the systemic co-jump minutes that the
idiosyncratic design deliberately sets aside.

\subsection{Network-concentrated propagation --- and its intensification}

Market-wide breadth spreads any response across five hundred receivers; a
transmission channel operating through economic or statistical linkages
concentrates it among a handful. For every event we therefore compare the
response of the source's twenty closest lagged-correlation neighbors with the
response of all other receivers, real minus placebo
(Table~\ref{tab:network_mechanism}).

The differential is large, precisely estimated, and horizon-decaying: five
minutes after an isolated $|z|\ge6$ shock, the source's neighbors are
\NetDiffFive$\times10^{-4}$ more likely than non-neighbors to exceed
$|z|>2$---relative to placebo---with a bootstrap standard error of
\NetDiffFiveSE$\times10^{-4}$ ($t\approx34$); the differential declines to
\NetDiffThirty$\times10^{-4}$ at thirty minutes. Against the $\sim700\times
10^{-4}$ base rate, a neighbor's jump probability rises by roughly nine
percent of its unconditional level. Idiosyncratic shocks do propagate---to the
connected few, not to the diffuse many, which is precisely why market-wide
breadth cannot detect them.

\input{tables/table7_network}

The evolution across decades is the paper's sharpest dynamic fact
(Figure~\ref{fig:network}): the five-minute neighbor differential was
\NetDiffFiveNineties$\times10^{-4}$ in 1992--1999 and
\NetDiffFiveOughts$\times10^{-4}$ in 2000--2009, then jumps to
\NetDiffFiveTens$\times10^{-4}$ in 2010--2019 and
\NetDiffFiveTwenties$\times10^{-4}$ in 2020--2026---a tripling concentrated
exactly in the era of algorithmic liquidity provision, correlation-aware
execution, and index intermediation. Propagation along the statistical network
became roughly three times stronger while market-wide spillover remained zero:
transmission grew \emph{sharper}, not broader.

\begin{figure}[t]
\centering
\includegraphics[width=.62\textwidth]{figF4_network.pdf}
\caption{\textbf{Network-concentrated propagation by decade.} Neighbor minus
non-neighbor $\Delta\Delta$ breadth at $h\in\{5,15,30\}$ with 95\% day-block
bootstrap intervals. Neighbors are the source's top-20 absolute-correlation
partners estimated on prior-month residuals.}
\label{fig:network}
\end{figure}

\subsection{Cumulative responses}

Point-in-time breadth at $t+h$ misses a receiver that jumps at $t+3$ and is
quiet at $t+15$. Cumulative breadth over $(t,t+h]$, differenced against its
mirrored pre-window and the placebo, remains null at the market level (at
thirty minutes: \CumThirty$\times10^{-4}$, SE \CumThirtySE$\times10^{-4}$).
The market-wide null is therefore not an artifact of sampling the response at
isolated instants; transient market-wide cascades after isolated idiosyncratic
shocks are absent as well.

\subsection{Systemic co-jump minutes: where fragility lives}

The isolation filters exclude minutes in which more than 2\% of the universe
jumps simultaneously. These minutes are the natural home of the fragility
hypothesis, and their time series (Figure~\ref{fig:cojumps}) delivers two
messages. First, the ordinary cross-section \emph{de-synchronized} secularly:
with idiosyncratic tail activity a hundred times rarer, minutes breaching the
fixed 2\% threshold fell from near-ubiquity in the sparse, wide-tick 1990s to
about one minute in eight today---a comparability warning we take seriously,
and the reason the fixed-threshold count is not itself a fragility measure.
Second, conditional severity moves the other way in stress: within co-jump
minutes, the mean share of names beyond $4\sigma$ averages
\CoJumpSevCalm\% in calm recent years but reaches \CoJumpSevGFC\% in 2008 and
\CoJumpSevCovid\% in 2020, and severe-synchronization minutes (more than 10\%
of the universe jumping together) spike from \SeverePreCovid{} per 1,000 in
2017--2019 to \SevereCovid{} per 1,000 in 2020. Combined with the
figE5 evidence that $|r|>5\%$ minutes rose by a factor of
\ExtremeCovidFactor{} in 2020, the pattern is consistent across measures:
modern stress arrives as market-wide synchronized minutes, not as
single-name events.

\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figF5_cojumps.pdf}
\caption{\textbf{Systemic co-jump minutes, 1992--2026.} (a) Minutes per 1,000
in which more than 10\% of eligible top-500 names exceed $|z|>4$
simultaneously (log scale). (b) Mean affected share within co-jump minutes
($>$2\% threshold).}
\label{fig:cojumps}
\end{figure}

Taken together, the mechanism evidence resolves the baseline null into a
sharper statement: idiosyncratic shocks propagate narrowly along the lagged
correlation network, that channel strengthened threefold in the algorithmic
era, and market-wide fragility is the province of common-shock minutes whose
severity---not frequency---is the binding object.