Multi-race stitching

Several races sharing some runners. Each line is one race in the format id:dividend, id:dividend, .... We fit a single global ability $\theta_h$ per item plus a per-race bias $b_r$ two ways:

• Curve-based Gauss–Newton (primary). The proper method — matches prices in probability space, alternating ridge updates for $(b_r)$ and $(\theta_h)$ on the cached ability$\to$price curves. See the rating-systems page.
• Relative-then-LS (comparison). The fast baseline — invert each race independently, centre by median, then slope-weighted-average per item. Doesn't revisit the prices, so it tends to shrink towards the mean when items are sparsely connected.

The two methods agree closely on well-connected items and drift on items that appear in few races. The GN bars are the proper answer; the LS bars give you a feel for how much the cheap stitch costs you when coverage is thin. Watch the "max GN residual" in the status line — that's the largest price error the GN fit leaves behind, in probability units.