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

The baseline bounds the market-wide response to isolated idiosyncratic
extremes at a few percent of the base rate. 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}

Market-wide breadth spreads any response across several 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 network differential is large relative to the market-wide estimates,
precisely estimated, and decays slowly with the horizon: five minutes after
an isolated $|z|\ge6$ move, the source's neighbors are
\NetDiffFive$\times10^{-4}$ more likely than other receivers to show a tail
exceedance, net of the same contrast for placebos, with a day-block bootstrap
standard error of \NetDiffFiveSE$\times10^{-4}$ and a ticker-cluster
standard error of \NetDiffFiveTickerSE$\times10^{-4}$; the differential
declines to \NetDiffFifteen$\times10^{-4}$ at fifteen minutes and
\NetDiffThirty$\times10^{-4}$ at thirty. Against a neighbor base rate of
about \NeighborBaseShare{}, a neighbor's exceedance probability rises by
roughly six percent of its unconditional level. The response is symmetric in
the sign of the shock (\NetDiffNeg$\times10^{-4}$ after negative and
\NetDiffPos$\times10^{-4}$ after positive moves). Idiosyncratic extremes do
propagate---to the connected few, not to the diffuse many, which is why
market-wide breadth cannot detect them.

\input{tables/table7_network}

Table~\ref{tab:network_components} decomposes the differential into its
four components. Over the full sample the contrast is driven by the real
arm: neighbors of a shocked stock show about $61\times10^{-4}$ more
exceedances than other receivers, while the placebo contrast is
$-2.7\times10^{-4}$. By decade, however, both arms move. In the 1990s,
neighbors show \NetRealNdNineties$\times10^{-4}$ more exceedances than
other receivers after a real event and \NetPlaceboNdNineties$\times10^{-4}$
more at placebo minutes; in the 2010s the real contrast is
\NetRealNdTens$\times10^{-4}$ and the placebo contrast is
\NetPlaceboNdTens$\times10^{-4}$. The placebo contrast is thus far from zero
within decades even though it averages to nearly zero over the sample: in the 1990s, the twenty most correlated names are simply the most
actively traded names, which show more exceedances in every minute; after
2000, the neighbors of a stock that is quiet at the placebo minute are
themselves quieter than the rest of the universe, as correlation implies.
The network differential is therefore legitimately a difference of two
contrasts, and only the real-minus-placebo column of
Table~\ref{tab:network_components} carries a propagation interpretation.
Panel~B of Table~\ref{tab:design_checks} adds that the differential survives
the restriction to minutes in which the source is the only $|z|\ge6$ event
(\NetDiffSingleton$\times10^{-4}$, \NetDiffSingletonSE), so it is not an
artifact of correlated pairs moving in the same minute and remaining active
five minutes later, although co-arrivals do account for about a fifth of the
pooled estimate.

\input{tables/table10_network_components}

The evolution across decades (Figure~\ref{fig:network}) is the paper's
sharpest dynamic fact and its most delicate one. The five-minute network
differential was \NetDiffFiveNineties$\times10^{-4}$ in 1992--1999 and
\NetDiffFiveOughts$\times10^{-4}$ in 2000--2009, then rises to
\NetDiffFiveTens$\times10^{-4}$ in 2010--2019 and
\NetDiffFiveTwenties$\times10^{-4}$ in 2020--2026, a threefold increase that
coincides with the era of algorithmic liquidity provision, correlation-aware
execution, and index intermediation, and that occurs while market-wide
spillover stays bounded near zero. Two readings are consistent with this
pattern and the data do not separate them. Under the first, transmission
along economic linkages became stronger and faster---basket arbitrage and
statistical-arbitrage strategies convert a move in one name into trades in
its peers within minutes, as \citet{bendavid2018etfs} and
\citet{foucault2016news} would predict. Under the second, transmission was
always present but the one-minute correlations from which we identify
neighbors were too noisy in the 1990s to find the true peers: with the Epps
effect of Figure~\ref{fig:epps}, the prior-month one-minute residual
correlation of a genuine industry pair in 1996 is barely distinguishable from
sampling noise, and a neighbor set contaminated by false peers dilutes the
contrast mechanically. The decline of the 1990s differential at $h=5$
relative to $h=30$ (Table~\ref{tab:network_mechanism}), the opposite of the
decay seen in later decades, is consistent with the second reading, since
slow comovement is what an Epps-era network would transmit. The honest
summary is that propagation along the measured network at least tripled, and
that transmission became sharper rather than broader; how much of the
increase is a change in the market and how much a change in the measurement
requires neighbor sets built on coarser-sampled or transaction-time
correlation estimators \citep{hayashi2005covariance}, an extension we leave
for future work.

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

\subsection{Cumulative responses}

Point-in-time breadth at $t+h$ misses a receiver that moves 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$\times10^{-4}$, standard error
\CumThirtySE$\times10^{-4}$, against a base rate near
\CumBaseFifteen\% at fifteen minutes). The market-wide bound is therefore not
an artifact of sampling the response at isolated instants; transient
market-wide cascades after isolated idiosyncratic extremes are absent as
well.

\subsection{Systemic co-jump minutes}

The isolation filters exclude minutes in which more than 2\% of the universe
moves beyond four robust standard deviations simultaneously. These minutes
are the natural home of the fragility hypothesis, and their time series
(Figure~\ref{fig:cojumps}) delivers two messages, each with a comparability
limit. First, the ordinary cross-section de-synchronized secularly: with
idiosyncratic tail activity two orders of magnitude rarer, minutes breaching
the fixed 2\% threshold fell from near-ubiquity in the sparse, wide-tick
1990s to about \CoJumpPerThousandRecent{} per 1,000, roughly one minute in
eight, in 2021--2025. Under the same standardization the early 1990s also
produce \SevereEarlyNineties{} severe minutes per 1,000 and a mean affected
share of \CoJumpSevEarlyNineties\% within co-jump minutes---both above
2008---which we read as discreteness rather than fragility: when most minute
returns are zero or one tick, a two-tick move in many names at once breaches
a four-MAD threshold without any economic synchronization. The fixed-threshold
count and the pre-2001 severity series are therefore not fragility measures,
and no claim in this paper compares synchronization across the
decimalization boundary.

Second, within the post-decimalization era, conditional severity moves
sharply 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 moving
together) rise from \SeverePreCovid{} per 1,000 in 2017--2019 to
\SevereCovid{} per 1,000 in 2020, with \SevereGFC{} per 1,000 in 2008
against fewer than 6 per 1,000 in 2006 and 2019: an increase of roughly an
order of magnitude in both crises. Combined with the
evidence of Figure~\ref{fig:extreme_minutes} that $|r|>5\%$ minutes rose by a
factor of \ExtremeCovidFactor{} in 2020, the pattern is consistent across
measures and with the collective-synchronization events documented by
\citet{calcagnile2018collective} on a shorter sample: modern stress arrives
as market-wide synchronized minutes, not as single-name events. Whether those
minutes are common news arrivals or endogenous synchronization of the kind
modelled by \citet{bormetti2015modelling} is not identified by bars alone.

\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 observable top-500 names exceed $|z|>4$
simultaneously (log scale). (b) Mean affected share within co-jump minutes
($>$2\% threshold). Pre-2001 levels are inflated by price discreteness and
are not comparable with later years.}
\label{fig:cojumps}
\end{figure}

Taken together, the mechanism evidence sharpens the baseline into a bounded
statement: idiosyncratic extremes propagate narrowly along the lagged
correlation network, that measured channel at least tripled in the
algorithmic era, and market-wide tail risk in the post-decimalization era is
the province of common-shock minutes whose severity in crises, not their
frequency in calm times, is the binding object.