An exact small-$n$ computation of the minimum 2-coloring discrepancy of $K_n^{(3)}$

For an integer $r\ge 2$ and an order $n\equiv 1,3(\mathrm{m}\mathrm{o}\mathrm{d}6)$, write ${\delta }_r(n)$ for the minimum, over all $r$-colourings $\chi :\left(\genfrac{}{}{0px}{}{[n]}{3}\right)\to [r]$, of ${\max }_{\mathcal{S}}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}(\mathcal{S},\chi )$, where the maximum is over labelled Steiner triple systems $\mathcal{S}$ of order $n$ and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}(\mathcal{S},\chi )={\max }_c\mathrm{\mid }\mathrm{\#}\{T\in \mathcal{S}:\chi (T)=c\}-\mathrm{\mid }\mathcal{S}\mathrm{\mid }\mathrm{/}r\mathrm{\mid }$. Following Gishboliner, Glock, and Sgueglia \cite{GishbolinerGlockSgueglia2025}, the bulk of the recent work on this quantity has been on lower bounds for $r\ge 3$ (proving ${\delta }_r(n)=\mathrm{\Omega }(n^2)$) and on structural characterisation of the low-discrepancy 2-colourings. We give three small computational contributions in the small-$n$ regime $n\in \{7,9,13,15,19,21\}$: An exact value of ${\delta }_2(n)$ for each such $n$, matching the formula ${\delta }_2(n)={\min }_{x\in [0,n]\cap \mathbb{Z}}\mathrm{\mid }x(n-x)\mathrm{/}2-n(n-1)\mathrm{/}12\mathrm{\mid }$ obtained by optimising the GGS Example 1.1 family. Rigorous for $n\in \{7,9\}$ via exhaustive search over labelled STSs ($30$ resp. $840$ systems) and over all $2$-colourings; computational for $n\in \{13,15,19,21\}$ by simulated-annealing search; A wide near-optimal basin: at $n=9$, every two-colour-flip neighbour of the optimal Example~1.1 colouring that maintains discrepancy $1.0$ exists; about $34\mathrm{\%}$ of two-flip perturbations preserve optimality; Random-colouring statistics for $r\in \{2,3,4\}$: $⟨{\max }_{\mathcal{S}}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}⟩$ grows linearly in $n$, in agreement with a heuristic Gaussian estimate $n\mathrm{/}\sqrt{6r}\cdot \sqrt{2\log K}$ over $K$ sampled labellings; the typical-case discrepancy is far below the GGS worst-case $\mathrm{\Omega }(n^2)$. We additionally state a conjectural exact formula for ${\delta }_2(n)$ that holds for every $n\equiv 1,3(\mathrm{m}\mathrm{o}\mathrm{d}6)$.