modeling: analytic Jacobian drops tied parameters' columns instead of chaining them into the free parameter, so LM/TRF fits with tied params stop off the minimum #20664

Where. astropy/modeling/fitting.py, _wrap_deriv (main, ~L1221-1251). When model.has_tied, the full analytic derivative model.fit_deriv(...) is computed for every parameter, then the columns of fixed and tied parameters are deleted:

tied = list(np.where([par.tied is not False for par in pars], True, tied))
fix_and_tie = np.logical_or(fixed, tied)
ind = np.logical_not(fix_and_tie)
residues = full_deriv[np.nonzero(ind)]

Deleting the column is correct for a fixed parameter. For a tied parameter (p_t = f(p_free)), it drops the term ∂y/∂p_t · ∂f/∂p_free from the derivative with respect to the free parameter. So the Jacobian the optimiser sees is wrong whenever a tie depends on a free parameter, and LM/TRF converge to a point that is not the least-squares minimum. Using the numerical Jacobian (estimate_jacobian=True) avoids it, because the tie is then applied before differencing.

Measured impact (specutils fit_lines, whose default is TRFLSQFitter). The model is [O III] λ4960 + λ5007 as two Gaussian1D with the 4960 amplitude, mean and stddev tied to 5007, plus a fixed Const1D continuum. It was refit on 6,812 DESI DR1 spectra against DESI's own scipy.optimize.curve_fit fit of the same pixels, model, start and bounds:

• default fitter: off the true χ² minimum in 6,608 of 6,812 fits, line flux off by a median 1.9% (p90 9.6%);
• TRFLSQFitter() with estimate_jacobian=True: on DESI's minimum.

Untied lines (Hα, Hβ and so on) agreed with DESI on the same spectra.

Minimal reproduction to try: two Gaussian1D with the second's amplitude tied as lambda m: m.amplitude_0 / 2.98 (and mean/stddev tied similarly), plus a fixed Const1D, on synthetic noisy data. Compare TRFLSQFitter()(...) with TRFLSQFitter()(..., estimate_jacobian=True) and with curve_fit on the equivalent function.

Fix direction: for tied parameters, either fall back to the numerical Jacobian, or apply the chain rule through the tie (numerically differencing the tie function if it is a lambda).

(Found while building https://github.com/scttfrdmn/scientific-codes-cookbook/tree/main/measurements/specutils-real)