Newton optimizer produces NaN parameters for a target with a flat direction, no error raised

make_negative_definite_and_solve divides by fabs(eigenvalues[i]) without checking for zeros (division at line 24). For a parameter absent from the target, the Hessian is zero and the step is 0/0:

// model whose log_prob ignores its single parameter, returns 0.5
stan::optimization::newton_step<flat_model, false>(model, params_r, params_i);
// f0 = 0.5, params_r[0] = -nan

The caller loop accepts NaN (while (f1 < f0) is false for NaN) and the service returns error_codes::OK without a finiteness check, so the run reports success with NaN parameters. Probably related to #3306.

Environment:

• Ubuntu 22.04.5 LTS (WSL2, kernel 6.18.33.2-microsoft-standard-WSL2), x86_64
• stan-dev/stan develop @ 8ad5c989c (stan::math develop @ d20b238b9c)
• g++ 11.4.0 (Ubuntu 22.04), -std=c++17

Start with make_negative_definite_and_solve in src/stan/optimization/newton.hpp and trace the caller loop and return path in src/stan/services/optimize/newton.hpp. Reproduce the flat_model case described in the issue, then inspect how the optimizer handles zero Hessian eigenvalues and non-finite parameters. Done means the flat-direction run no longer reports success with NaN parameters and has coverage for the failure or safe result.