\section{Economic framework and hypotheses}
\label{sec:theory}

\subsection{Anticipation versus revelation}

Let $X_{i,q}$ denote an accounting flow of firm $i$ for fiscal quarter $q$ (operating income in our primary
specification), revealed to the public at the earnings-release date $A_{i,q}$ and structured in the SEC filing of date
$F_{i,q}\ge A_{i,q}$. A forecaster standing at a trading day $c<A_{i,q}$ holds an information set $\mathcal{I}(c)$ that
contains, at most, prices and volumes up to the close of $c$ and the accounting statements that were \emph{available} at
$c$, i.e., quarters $j$ with $F_{i,j}+1\le c$. The quantity of interest is the conditional distribution of the surprise
\begin{equation}
  S_{i,q} \;=\; \frac{X_{i,q}-\mathbb{E}\!\left[X_{i,q}\mid \mathcal{A}(c)\right]}{TA_{i,j^*(c)}},
  \label{eq:surprise}
\end{equation}
where $\mathcal{A}(c)\subset\mathcal{I}(c)$ is the accounting information alone, $j^*(c)$ is the latest available quarter,
and the expectation is a seasonal random walk with drift \citep{foster1977}, the classic honest benchmark when no
point-in-time consensus exists. Semi-strong efficiency \citep{fama1970} implies that prices at $c$ impound all public
information, including $\mathcal{A}(c)$; the ``prices lead earnings'' literature \citep{beaver1980, kothari1992,
collins1994} documents that daily and annual returns indeed forecast accounting changes. None of this is the question
we ask. Our question is whether the \emph{resolution} at which the market is observed carries incremental information,
holding the information set, the universe, the calendar coverage, the horizon and the model capacity fixed.

\subsection{Why resolution could matter, and why it might not}

Three mechanisms make fine-grained activity informative beyond daily aggregates. First, in sequential-trade models
\citep{kyle1985, easley1987}, informed traders split orders and trade when liquidity is deep, so the \emph{composition} of
a day's volume and return (intraday autocorrelation, concentration of variance in a few minutes, volume on down-bars) can
reveal private information that the daily net return hides. Second, realized measures built from fine bars separate the
continuous and jump components of variation \citep{barndorff2004, andersen2003} and their sign
\citep{barndorff2010semivariance, patton2015}; negative semivariance and signed jumps have been linked to the
cross-section of expected returns \citep{bollerslev2020good} and may proxy for the arrival of adverse firm-specific news.
Third, limited attention \citep{hirshleifer2003, hirshleifer2009, dellavigna2009} predicts that the incorporation of
public but costly-to-process information is gradual, so that the time path of trading within the day (opening and closing
concentration, overnight versus intraday returns) carries information about ongoing processing.

Against these channels stand two forces. The Epps effect and microstructure noise contaminate 1-minute returns
\citep{hansen2006, barndorff2008}, so that the finest sampling is not the most informative for variance measurement
\citep{liu2015}; and whatever information intraday activity contains about fundamentals may be fully summarised by the
daily close, in which case the frequency premium is zero. Algorithmic trading may further have reduced the amount of
pre-announcement information acquisition \citep{weller2018}. The sign and size of the frequency premium are therefore
empirical questions, and a precise null is as informative as a positive estimate.

\subsection{Hypotheses}

All hypotheses, endpoints and decision rules were frozen before the test period was opened (\texttt{docs/RESEARCH\_DESIGN.md}
in the replication package). Let $\mathcal{L}(M)$ denote the out-of-sample log loss of model $M$ for the binary adverse
outcome $D_{i,q}=\mathbb{1}[S_{i,q}\le -0.5\%]$, and let B1--B5 be the nested information sets of Table~\ref{tab:infosets}.

\begin{description}[leftmargin=1.5em]
\item[H1 (incremental information).] $\mathcal{L}(\mathrm{B5})<\mathcal{L}(\mathrm{B2})$ at the 20-trading-day horizon:
  one-minute features improve forecasts beyond point-in-time fundamentals and daily OHLCV. Null: no improvement.
\item[H2 (temporal resolution).] The gain of B5 over B2 exceeds the gains of B4 (5-minute) and B3 (30-minute) over B2,
  with the same feature families and the same number of inputs. Null: the gains are equal.
\item[H3 (heterogeneity).] The B5$-$B2 gain is larger for small, illiquid and highly levered firms, the groups in which
  information asymmetry and processing costs are highest. Null: homogeneous gain.
\item[H4 (timing).] The gain increases as the cutoff approaches the release ($h=40\to1$) and is absent at placebo cutoffs.
  Null: no gradient.
\item[H5 (pricing).] The ex-ante deterioration probability predicts the announcement-window abnormal return and
  post-cutoff factor-adjusted returns. Null: no return predictability after risk adjustment and costs.
\end{description}

A positive H1 does not identify \emph{who} trades on what: observational prediction is not causal evidence of informed
trading, and we do not have quotes, trades or trader identities \citep{gregoire2022, hendershott2015}. The paper's claims
are confined to the forecasting content of anonymous bars and to the frequency at which that content appears.