[ParadaCarleton]

PALC clamps the parameter at the ends of [p_min, p_max], but both clamp only p. The state x stays where the unclamped step put it, so the point the solver works with no longer lies on the tangent line. This means the correction can converge onto a different branch, or evaluate F outside its domain. The error occurs in two separate places:

1. The predictor (corrector!, Palc.jl:160-162). When the predictor (x_pred, p_pred) = (x, p) + ds·τ lands past a bound, corrector! sets p_pred to the bound and runs the Natural corrector at that parameter, starting from x_pred. But x_pred belongs to the overshot parameter, so the fixed-parameter Newton starts a full step's worth of x beyond where the curve meets the bound. If another branch is closer to that point, it converges there.

Example: F(x, λ) = (x − λ)(x − λ − 3), whose two branches are x = λ and x = λ + 3. Start at (0, 0) with ds = 4 and p_max = 1. The tangent is (1, 1)/√2, so the predictor lands at about (2.83, 2.83). p is clamped to 1, but x stays at 2.83. At λ = 1 the roots are x = 1 and x = 4, and 2.83 is closer to 4, so the continuation ends at x = 4, on the other branch.

The fix moves the predictor to where its line meets the bound, x + (bound − p)/(p_pred − p)·(x_pred − x), whenever the current point lies inside (p_min, p_max). In the example this is x = 1, the right root. The new test test/continuation/palc_bound_predictor.jl runs this problem with Secant and Bordered tangents. We first hit this on a 3-player logit equilibrium branch that folds back just past the target parameter; the clamped step landed on the folded-back segment.

2. The corrector iterates (newton_palc). Each Newton iterate sets p = clamp(p − up, p_min, p_max) and keeps x − u. An iterate that wants to leave the interval is again moved in p only, so the next residual is evaluated at a point that no Newton step produced. When F is undefined past the bound, the line search can also evaluate it there and throw a DomainError. Here the corrector stops when an iterate leaves [p_min, p_max], so PALC rejects the step and shrinks ds as for any failed corrector. The test test/continuation/palc_bounds.jl covers this case.

Both changes only affect steps that reach a bound; every step strictly inside the interval runs as before.

https://claude.ai/code/session_015sE9Latuiqsp6zSHQ1cshz