Planckian fit is exactly non-identifiable: x0 and sigma cannot be determined separately

[PierreRaybaut]

Summary

PlanckianFitComputer.evaluate() is invariant under the exact reparameterization

$$x_0 \to k,x_0, \qquad \sigma \to k,\sigma, \qquad \mathrm{amp} \to \mathrm{amp}/k^5$$

The curve is bit-identical (verified: max|diff| = 0.0 for k = 2). Only the combinations x0/sigma and amp·x0⁵ are observable, so the individual values of x0, sigma and amp returned to the user are meaningless — they depend solely on where the optimiser happened to start.

Secondary defect

The docstring and the DataLab GUI label call x0 the peak wavelength (Wien's displacement law). This is false unless sigma == 1:

sigma	argmax for x0 = 1
0.5	2.0165
1.0	1.0060
2.0	0.5008

The maximum sits at approximately 1.007·x0/sigma.

Proposed resolution

Two options, to be decided before implementation:

1. Reparameterize to three genuinely independent parameters, e.g. (peak_amplitude, peak_position, y0) with the Planck shape fully determined by peak_position — removes the degeneracy and makes the reported peak position correct by construction. Breaking change for stored fit_params.
2. Keep the model and document the degeneracy explicitly, renaming x0 so it no longer claims to be the peak wavelength.

The branch fix/fitting-audit implements option 2 (documentation and label only) so the currently misleading claim disappears immediately; option 1 remains open.

[PierreRaybaut]

Status after fix/fitting-audit

The secondary defect is now fixed everywhere; the degeneracy itself is not.

Done:

• bc4db39 (Sigima) — .. warning:: on the degeneracy in the PlanckianFitComputer class docstring, x0 no longer documented as the peak wavelength in evaluate, plus two regression tests (test_planckian_model_is_scale_degenerate, test_planckian_peak_is_not_at_x0).
• 7489d6ca (DataLab) — note on the over-parameterization added to doc/features/signal/menu_processing.rst, with the French translation.
• 1e5f6e0a (DataLab) — the GUI label for x0 in the interactive fit dialog changed from Peak wavelength to Scale factor, matching the wording already used in the documentation. Since non-peak interactive fits key their metadata on the translated UI label (Non-peak interactive fits store metadata keyed by translated UI labels DataLab#355), this also changes the stored key for that parameter.

What remains: the design decision

Only option 2 of the issue body (document and relabel) has been implemented. The model is still over-parameterized: amp, x0 and sigma are three free parameters for two observable degrees of freedom, so the optimiser drifts along the degenerate direction and the individual values returned to the user depend solely on the starting point. The bounds in compute_bounds (x0 in [min(x), max(x)], sigma in [0.1, 5.0]) are arbitrary and do nothing to constrain that direction.

Three candidate resolutions, in increasing order of cost:

1. Freeze sigma = 1.0 and remove it from the free parameters. This removes the degeneracy exactly: x0 becomes a genuine scale and the maximum sits at 1.007 * x0. Much lighter than a full reparameterization, but still a model change, and it removes a parameter users may currently be adjusting.
2. Reparameterize to three independent quantities, e.g. (peak_amplitude, peak_position, y0), with the Planck shape fully determined by peak_position. This makes the reported peak position correct by construction.
3. Keep the model as is and consider the documentation sufficient — acceptable only if the returned amp/x0/sigma are understood to be non-interpretable individually.

Migration cost for options 1 and 2

FIT_PARAMS_VERSION is currently 2 and validate_fit_params rejects any other value. Changing the number or the meaning of the planckian parameters therefore requires a bump to version 3, a converter for stored fit_params, and a synchronised update of FIT_TYPE_MAPPING, the DataLab fit dialog and DataLab-Web.