ARTS2D: define a true reduced 2-V EDF marginal instead of a renormalized 3-D slice

[joglekara]

Problem

ARTS2D needs a reduced two-velocity EDF, but the spherical-harmonic path currently constructs 3-D-normalized radial/spherical components, samples them on the vz=0 plane, clips the result, and renormalizes that slice in 2-D (tsadar/core/modules/distribution_functions/spherical_harmonics.py:263-312).

A central slice is not the same as the marginal over the unobserved velocity component except for special distributions. Post-hoc 2-D normalization hides the dimensional mismatch while changing moments and introducing an artificial degeneracy between EDF shape and Te.

Directly measured normalized 2-V second moments from the current reconstruction are approximately:

• super-Gaussian m=2: 2.01;
• m=3: 2.19;
• m=5: 2.35.

The intended normalized 2-V thermal second moment should remain 2 across shape parameterizations if Te keeps the documented meaning. The 1-D DLM construction maintains its corresponding variance across these values.

Decision required

Choose and document one model:

1. ARTS2D directly parameterizes a physical reduced 2-V EDF on the velocity plane; or
2. it parameterizes a 3-D EDF and analytically/numerically marginalizes the unobserved component before the Radon projection.

Do not silently substitute a central slice for a marginal.

Acceptance criteria

• State the EDF dimensional convention, units, normalization, and relation between Te and velocity moments in code/docs/config schema.
• Add an analytic Maxwellian 3-D-to-2-D marginal test.
• Add high-accuracy numerical marginal tests for m != 2 and representative anisotropic modes.
• Density, first moments, and thermal second moments obey the chosen normalization without post-hoc shape-dependent temperature drift.
• Every 1-D projected EDF used by the susceptibility agrees with direct integration of the declared underlying distribution to a documented tolerance.
• Positivity is imposed by construction or a smooth parameterization rather than a hard gradient-killing clip.

Related: #124 and the separate spherical-harmonic trainability issue.

[joglekara]

The mechanical basis blocker is tracked in #136. That issue should make coefficients nonzero and differentiable; this issue should define the physical 2-V representation and normalization contract before the final basis API and reference fixtures are frozen.

[joglekara]

Design clarification: the physical VDF is 3-V, but the ARTS observable can be exactly 2-V

The electron VDF is physically a function (f_3(v_x,v_y,v_z)). This issue should not be interpreted as claiming that electrons live in a two-dimensional velocity space. The important distinction is between the physical state and the sufficient marginal observed by a particular scattering geometry.

For each scattering direction (\hat{\mathbf k}), longitudinal Thomson scattering depends on the one-dimensional plane marginal

[
F_{\hat k}(u)=\int f_3(\mathbf v),\delta!\left(u-\hat{\mathbf k}\cdot\mathbf v\right)d^3v.
]

The spectral numerator contains (F_{\hat k}(\omega/k)), while the electron susceptibility depends on its derivative and Hilbert/principal-value transform. It does not sample the value of the 3-V VDF at the single point ((\omega/k)\hat{\mathbf k}).

If every collected (\mathbf k) lies in one plane (P=(v_1,v_2)), define

[
f_2(v_1,v_2)=\int f_3(v_1,v_2,v_3),dv_3.
]

Then, exactly and without isotropy or separability,

[
F_{\hat k}(u)=\int f_2(v_1,v_2),
\delta(u-\hat k_1v_1-\hat k_2v_2),dv_1dv_2
\qquad (\hat{\mathbf k}\in P).
]

Therefore a native 2-V Cartesian marginal is not an approximation for coplanar ARTS views; it is the result of integrating the unobserved direction before applying the Radon transform. A full 3-V fit to purely coplanar data would add an exact out-of-plane nullspace. The inferred conditional structure in (v_3), out-of-plane drift, (P_{33}), and out-of-plane heat flux would come from the prior/model rather than the data.

These objects should remain distinct:

Representation	Meaning	Status for ARTS
(f_3(v_x,v_y,v_z))	Full physical VDF	Ground-truth object
(f_2(v_x,v_y)=\int f_3,dv_z)	Cartesian out-of-plane marginal	Exact sufficient object for coplanar (\mathbf k)
(G(v_\parallel,v_\perp))	Full 3-V VDF under axisymmetry/gyrotropy	Valid only with a justified symmetry axis and cylindrical Jacobian
(f_3(v_x,v_y,0))	Central slice	Generally not the measured object

Current TSADAR contract

The documentation already states that fe.dim: 2 is an x-y distribution integrated along z, where z is the probe-polarization direction (docs/source/defaults.rst:40). The form factor then performs one more line/Radon projection of that 2-D input for each k direction (tsadar/core/physics/form_factor.py:415-436). That contract is internally sensible.

The spherical-harmonic path violates it: it constructs 3-D-normalized (f_{\ell m}(v)Y_{\ell m}), evaluates those functions on a fixed 2-D plane, clips, and renormalizes in 2-D. That is a slice-like surrogate, not the declared marginal except in special cases such as an isotropic Gaussian where slice and marginal have the same normalized shape.

Proposed model split

1. Marginal2V — default data-level inversion. Parameterize a positive Cartesian marginal directly and report it explicitly as an in-plane marginalized EDF. It can determine density, in-plane drift, and in-plane moments, but should not be presented as a complete 3-V reconstruction.
2. Harmonic3V — physical hypothesis model. Define a genuine low-order spherical-harmonic VDF and analytically/numerically marginalize it, or directly compute its 1-D projection for every actual scattering vector. Never evaluate only (v_z=0). Fit only the harmonic combinations supported by the noise-weighted geometry/Jacobian; invisible components must remain fixed or prior-dominated.
3. Gyrotropic 2-V — optional symmetry model. If a known magnetic/physical axis justifies (G(v_\parallel,v_\perp)), implement its cylindrical measure and forward projection explicitly. Do not rotate it as a Cartesian image.

Before selecting the production model, propagate the actual finite-aperture ray distribution and check whether the collected k vectors are sufficiently coplanar. Genuine non-coplanar views, or rotated measurements of the same underlying plasma, can justify a low-order 3-V model; otherwise the 2-V marginal is the more identifiable primary result and any 3-V completion must be labeled model-dependent.

Useful theory references:

• Foo, Schaeffer, and Heuer, Recovering non-Maxwellian particle velocity distribution functions from collective Thomson-scattered spectra: https://doi.org/10.1063/5.0169393
• Williamson and Clarke, Construction of electron distribution functions from laser scattering spectra: https://doi.org/10.1017/S0022377800025794

Suggested additions to this issue's acceptance criteria:

• Verify coplanarity using the actual ray/aperture model, not only nominal scattering angles.
• Test that direct 3-D projection equals projection of the computed 2-D marginal for every in-plane view.
• Separate data-resolved in-plane moments from model-dependent out-of-plane moments in outputs and documentation.
• If Harmonic3V is retained, implement its marginal/projection operator directly and rank-test its observable coefficient combinations.

[almilder]

Building machinery to store and produce true 3V distributions is productive as it will allow 3V theory models and 3V code outputs to be ported directly. However the null-space problem of the third dimension means only a 2V-marginal should be presented to formfactor. And since a 2V-marginal can't be restored to a unique 3V distribution the 3V distributions only ever need to be use to produce an initial 2V-marginal. As for alternative 2V formalisms like v_perp/v_parallel these are generally excluded by construction. All the scattering vectors for a ray fan of differing angles and lengths which lie in a plane. That plane also contains the only unique static vector of the problem k-probe. For this reason the math probably must be done on a cartesian grid with a cardinal vector aligned to k-probe (we chose to align k-probe to x).