% Scryglass formulas
% Where used: Ratings (Dual Elo, Trust, league-aware), Match checklist (Elo→WR, Draft Score), Void-grubs article (contest bar)

\section{Dual Elo soft rating (league-aware)}
\label{eq:soft-mu}
\[
\mu_{\mathrm{aware}}
=
\mu - \max(0,\sigma - \sigma_{\min})
\]
Team \(\sigma_{\min}=25\). Player \(\sigma_{\min}=28\).
Used on Ratings ladders and profiles as \textbf{League-aware} rating.

\section{Elo to win probability}
\label{eq:elo-wr}
\[
\operatorname{logit}(p)
=
a + b\cdot\frac{\mu_i-\mu_j}{400},
\qquad
p=\sigma(\operatorname{logit})
\]
Calibration coefficients from \texttt{models/elo\_wr\_calibration.json}.
Used for Dual Elo pre-match favorites on match boards.

\section{Draft Score (league temperature)}
\label{eq:draft}
\[
p_{\mathrm{blue}}
=
\tfrac12 + \Bigl(\sigma(t\cdot e)- \tfrac12\Bigr)\cdot c
\]
where \(e\) is draft win-edge (champ logits + role pairs), \(t\) is league/tier temperature,
and \(c\) is confidence shrink.
Elo-controlled bump:
\[
\Delta\mathrm{WR}_{\mathrm{pp}}
\approx
100\cdot \kappa\cdot e\cdot c
\]
with \(\kappa\) from residual ridge in \texttt{draft\_wr\_calibration.json}.

\section{Void grubs contest bar (article)}
\label{eq:grubs}
At even gold, leave for the two-wave farm reference until fight-win probability reaches
the opportunity-cost break-even (≈58.9\% in the published reference). Gold@10→map-win uses a
side-neutral associational logit; see the article pack JSON for fitted coefficients.
