\section{Ablations and falsification}
\label{sec:ablations}

Every variant in Table~\ref{tab:ablations_crps_scaled} is a Small model trained for \num{4000} steps with the same
seed and compared, on the 2024 development test, with a reference trained under exactly the same budget
(\textsc{mrpt-s}, 4k); two further seeds of the reference give the noise floor. Figure~\ref{fig:ablations_forest}
shows the relative CRPS changes at sixty minutes with their day-cluster intervals.

\input{tables/ablations_crps_scaled}

\paragraph{Resolutions (H1 again).} At this shorter budget removing the 5- and 30-minute branches costs
$+0.12$ to $+0.16$\,\% of CRPS, removing only the 30-minute branch $+0.10$\,\%; the seed-to-seed spread of the
reference is $-0.03$ to $+0.16$\,\%. Multi-resolution context is therefore a real but marginal regulariser at small
budgets and, as Section~\ref{sec:results} showed, nothing at all at the full budget and out of sample.

\paragraph{What the model actually uses.} The most informative ablations are the destructive ones. A model that
sees \emph{only} the calendar and missingness flags---no returns, no ranges, no volumes, just the trailing scale
$\sigma_t$ in its global token and the time of day---is worse than the full model by only $+0.20$ to $+0.29$\,\%.
Returns-only inputs cost $+0.19$ to $+0.28$\,\%; dropping relative volume $+0.08$ to $+0.14$\,\%; shuffling the order
of the 390 one-minute bars at training and evaluation time $+0.13$ to $+0.21$\,\%. In other words, of the
$\approx 4.3$\,\% improvement over the random walk at sixty minutes, roughly nine tenths come from the trailing
volatility level and the intraday seasonal, which every baseline also has, and at most three tenths of a percent
from any minute-level \emph{structure}---order, shape of bars, volume---that a sequence model can extract. The
falsification test passes in the sense intended: shuffling does hurt, so the model reads order; it just does not
have much to read.

\paragraph{Fusion, head, regime (H3), identity.} Cross-attention fusion of separate per-scale encoders is slightly
worse than the shared backbone ($+0.05$\,\%). The Gaussian head is \IfFileExists{tables/ablations_gaussian_note.tex}{\input{tables/ablations_gaussian_note}\unskip.}{worse than the Student-$t$ head (see Table~\ref{tab:ablations_crps_scaled}).}
Removing the regime embedding changes nothing ($+0.02$ to $+0.03$\,\%, Holm $p>0.3$ at 30 and 60 minutes), and
coverage on stress days is identical with and without it (0.57/0.85/0.97 at $H=60$): Hypothesis~\ref{h3} is
rejected---the global token already carries the regime statistics, and conditioning the decoder on them again
adds nothing. A learned eight-code regime codebook improves CRPS by $0.03$--$0.15$\,\% and lowers central coverage
toward nominal (0.56 at 50\,\%), the only architectural change with a consistent, if small, benefit. A ticker-identity
embedding hurts ($+0.05$ to $+0.31$\,\%): instrument identity over-fits at this scale, which is also why the core model
has none.

\paragraph{Joint paths (H4).} The copula head trained on the probability-integral transforms of the Stage-A
marginals converges to a copula log-likelihood of $-2.5\times10^{-5}$ per dimension, i.e.\ to independence, and
its loadings stay near zero. Correlated sampling changes the energy score by $-0.003$\,\%, the variogram score by
$-0.002$\,\%, the CRPS of maximum drawdown by $-0.01$\,\% and of realised variance by $+0.10$\,\%, all with intervals
containing zero (Table~\ref{tab:path_scores}); the CRPS of the cumulative return is $+0.08$\,\% worse at sixty
minutes. Hypothesis~\ref{h4} is rejected: once the marginals are conditioned on the recent history, the next sixty
minute returns are, to the resolution of these scores, conditionally independent, and a dependence-aware decoder
has nothing to model. We did not pursue diffusion decoders.

\begin{figure}[htbp]\centering
\includegraphics[width=\textwidth]{figures/fig_ablations_forest.pdf}
\caption{Ablations at the Small scale (4 000 steps each): relative CRPS change at $H=60$ versus the random walk,
with 95\,\% day-cluster bootstrap intervals; the equal-budget reference and its two extra seeds are included.}
\label{fig:ablations_forest}
\end{figure}

\begin{table}[htbp]\centering\small
\caption{Joint paths versus independent sampling of the same Student-$t$ marginals (MRPT-S + copula, 2024 development test, \num{986468} origins, 60-minute paths, 200 draws; day-cluster bootstrap 95\,\% intervals of the difference).}
\label{tab:path_scores}
\begin{tabular}{lccc}
\toprule
Score & Joint (copula) & Independent & Joint $-$ independent \\
\midrule
Energy score & 3.490e-3 & 3.490e-3 & $-9.2\times10^{-8}$ [$-2.3\times10^{-7}$, $+5.0\times10^{-8}$] \\
Variogram score ($p=0.5$) & 1.5113e-4 & 1.5114e-4 & $-3.8\times10^{-9}$ [$-1.0\times10^{-8}$, $+2.2\times10^{-9}$] \\
CRPS, maximum drawdown & 1.4632e-3 & 1.4633e-3 & $-1.8\times10^{-7}$ [$-6.2\times10^{-7}$, $+2.8\times10^{-7}$] \\
CRPS, realised variance & 1.1881e-5 & 1.1869e-5 & $+1.2\times10^{-8}$ [$-2.8\times10^{-9}$, $+3.0\times10^{-8}$] \\
\bottomrule
\end{tabular}
\end{table}