1\section{Robustness, alternative specifications and external validation}\label{sec:robust}23\subsection{Preregistered and referee-requested variants}4Table~\ref{tab:h1} (lower panel) and Figure~\ref{fig:robust} recompute the H1 statistics under the classifier labelling on the preregistered variants and on three added after refereeing: the platform-label-only region assignment (1{,}214 stores), the point-in-polygon-first assignment (1{,}331), stores found through province-wide sources only --- certification registries, sector associations, marketplaces and search sweeps, excluding every regional directory (1{,}012), stores found through certification and association sources (1{,}068; this variant, advertised in a first draft as the answer to directory-driven discovery, includes the regional directories and is kept for completeness only), excluding Montréal (1{,}094), excluding the one per cent largest stores (1{,}318), eight macro-categories, dropping ``other'' (1{,}306), excluding resellers and collective storefronts (1{,}306), dropping the 124 stores whose platform label disagrees with the polygon containing their FSA centroid (1{,}207), and dropping the 342 stores located in FSAs that straddle a regional boundary (924). The merchant-equal result holds in all of them ($V$ between 0.106 and 0.123; permutation $p\le0.005$ in every variant, 1{,}000 permutations) and the trimmed-listing result is never significant ($p$ between 0.07 and 0.94). The same is true under the rule labelling ($p\le0.005$ merchant-equal, $p\ge0.19$ trimmed). The modifiable areal unit problem is addressed directly: at the level of the 56 MRCs with at least five geocoded stores the merchant-equal dependence is $V=0.185$ against a null of 0.156 ($p<0.001$). A hundred reassignments of categories drawn from the classifier's audit confusion matrix keep the merchant-equal $V$ between 0.099 and 0.112 and reject the null in every draw (82\% of draws with the rule-label confusion matrix). The Gini of catalogue sizes stays between 0.76 (without the largest one per cent) and 0.87 (without Montréal). The regional diversity ordering, by contrast, is labelling-sensitive beyond its extremes (Section~\ref{sec:results_h2}): we report it as an interval statement, not a league table.56\begin{figure}[htbp]\centering7\includegraphics[width=0.85\textwidth]{\FM{robustness_h1.pdf}}8\caption{H1 across sample and category variants, classifier labelling: observed Cramér's $V$ against the mean of the merchant-reshuffle null under merchant-equal (M) and trimmed-listing (T) weights. Source: Fabri-Ka research snapshot (2026-11-01).}\label{fig:robust}9\end{figure}1011\subsection{Price dispersion}12The channel effect is the single most stable estimate in the paper ($-0.10$ to $-0.12$ across specifications, wild-bootstrap $p\le0.017$), and it is estimated within match group, so false matches attenuate rather than inflate it. The geographic joint test, by contrast, is fragile in three documented ways: it depends on the bootstrap implementation (a disturbance not re-projected off the group fixed effects gave $p=0.005$; the correct procedure gives 0.098), it is dominated by one region with one large reseller (Outaouais, 65 of 164 Wald points), and it weakens further when fuzzy matches are added ($p=0.41$). Region effects for regions with one to three matched stores are individual merchants' pricing. We therefore make no claim about regional price levels.1314\subsection{Holdouts}15Leave-one-region-out recomputation of the H2 correlations never moves the rarefied-diversity--population correlation outside $[-0.3, 0.3]$ under either labelling; the H7 leave-one-region-out predictive correlation is the V3 statistic reported above; the H5 leave-one-region-out Wald decomposition is in the text. The H4 temporal holdout (density from weeks 36--39, entries in 40--44) is built into the design. Duplicate products cannot leak between training and test in the classifier because the audit sample's identifiers were removed from the training sample before annotation, although 92\% of audit \emph{stores} also appear in the training set (no merchant holdout); the embedding model saw all listings but was never fitted to labels.1617\subsection{External validation and selection}18Fabri-Ka's store counts per region correlate at 0.91 with ISQ business locations, 0.89 with retail locations, 0.93 with manufacturing locations and 0.84 with population (Spearman, 17 regions; Table~\ref{tab:external}), so the aggregator reproduces the size ordering of regional economies. Coverage is nonetheless uneven: from 1.8 stores per thousand business locations in Laval and 3.4 in Montréal to 9.0 in Estrie and 11.1 in Saguenay--Lac-Saint-Jean, a six-fold range, with the regions of high coverage being exactly those with a connected regional agri-food directory (Zone boréale, Créateurs de saveurs, Gaspésie gourmande). The 226 stores of the assortment sample without a recoverable location come predominantly from a gift directory and from search sweeps, i.e.\ from the sources whose records carry no address. Discovery source and e-commerce platform mixes differ by region (Shopify dominates in Montréal and Laval, WooCommerce in the agri-food regions), and category mix differs by platform, so a platform-connector failure would be region-correlated; the province-wide-sources, trimmed and no-Montréal variants are the operational answer to this threat and do not change the qualitative results. We have no regional measure of e-commerce adoption to weight by, and we do not attempt selection weights: the coverage ratios are reported so that readers can see where the merchant population is thinly or thickly observed.19