perf(linear): share one eccentric-radius grid across an MGE basis (PyAutoArray#507) - #590

Summary

The PyAutoGalaxy half of PyAutoLabs/PyAutoArray#507 phase 2. Step 0 of that
issue measured the MGE operated mapping matrix at 0.209 s of a 0.633 s HST
rectangular numba CPU likelihood evaluation, and — this is the part that was not
guessable — 78 % of it is profile evaluation, not the PSF convolution, which
phase 1's batching had already reduced to 0.054 s.

A multi-Gaussian expansion is a LightProfileLinearObjFuncList of tens of
Gaussian profiles which share a centre and ell_comps and differ only in
sigma. Building its mapping matrix evaluated image_2d_from once per profile,
and every one of those calls redid the same two grid calculations: the
reference-frame transform (an arctan2, a sin and a cos per coordinate) and
the eccentric radii formed from it. Both dwarf the profile itself, which is one
exp. cProfile on the HST cell: the transform alone was 951 us of the 1.3 ms
each profile took.

LightProfileLinearObjFuncList._image_slim_list_from now groups the profiles by
(class, centre, ell_comps) and computes each group's transform and radii once,
then evaluates every profile's image_2d_via_radii_from against its group's
radii. Both the data grid and the blurring grid go through it, so the numpy
batched-convolution path in operated_mapping_matrix_override picks it up too.

Grouping rather than an all-or-nothing check is deliberate. The workspace's
canonical MGE recipe (features/multi_gaussian_expansion/modeling.py) stacks two
sets of 30 Gaussians which share a centre but carry their own ell_comps, and
GalaxiesToInversion lands both sets in a single func list — an all-or-nothing
test would fall back to 60 independent evaluations on exactly the model users are
told to write.

Bit-identical, not approximate. The hoisted quantities are produced by
calling the same methods, with the same inputs, that each profile would have
called on itself.

Measured on the HST breakdown cell, 60 linear Gaussians, OMP_NUM_THREADS=1:
60x image_2d_from(grid) 0.0867 -> 0.0168 s, 60x
image_2d_from(blurring_grid) 0.0346 -> 0.0092 s. Paired in the harness, the
MGE step falls 0.194 -> 0.084 s at HST rectangular (2.3x) and 0.080 -> 0.032 s at
euclid (2.5x); with the PyAutoArray half it takes the whole HST evaluation
0.6214 -> 0.3334 s, which is the issue's goal.

API Changes

None — internal changes only. Three new private members on
LightProfileLinearObjFuncList (_shared_eccentric_radii_index_groups,
_image_slim_list_from, _shared_eccentric_radii_image_slim_list_from) and one
private module function (_gaussian_image_2d_from). mapping_matrix and
operated_mapping_matrix_override keep their signatures and return
bit-identical values.

See full details below.

Test Plan

• pytest test_autogalaxy — 1149 passed (1144 before; +5).
• test__mapping_matrix__shared_geometry_matches_per_profile_loop — an MGE
  basis matches the per-profile loop bit-identically (np.array_equal)
  on both the data grid and the blurring grid, and through
  operated_mapping_matrix_override against the per-profile
  psf.convolved_image_from.
• ..._spherical_gaussians_share_radii_and_match_the_loop,
  ..._two_basis_mge_is_grouped_by_ell_comps (two groups plus a Sersic
  taking the loop), ..._mixed_geometry_falls_back (different centre /
  different ell_comps / different class),
  ..._non_gaussian_and_single_profile_take_the_loop.
• Control runs recorded — each of these deliberately wrong variants fails
  the new tests: dropping the transform; using elliptical instead of
  eccentric radii; transforming the un-over-sampled grid; removing the
  image_2d_from-identity membership test.
• Bit-identity on real data (HST, 60 Gaussians): the mapping matrix
  (60 x 15361), the blurring stack (60 x 5960) and the MGE operated mapping
  matrix all np.array_equal to the per-profile loop; the three pinned
  profiling log likelihoods unchanged to every recorded digit.

Why the membership test is image_2d_from identity

A group needs numpy, a Grid2D, two or more profiles, one class, an identical
centre and ell_comps, and — the load-bearing part — that the class's
image_2d_from is Gaussian.image_2d_from, tested by function-object
identity rather than isinstance. That is what makes "the transform and the
eccentric radii are the whole of the shared work" true:

• subclasses which inherit it (linear, operated, spherical Gaussians) match;
• GaussianMultipole, which overrides it to perturb the radius, does not;
• Sersic and its children, which also share a geometry in a Sersic basis, do
  not — they already avoid the polar transform via
  _eccentric_radii_grid_from_cartesian, a branch taken only when the grid
  handed to them is not pre-transformed, so hoisting a transformed grid into
  them would be both slower and a different floating-point expression.

Follow-up worth filing: giving Gaussian that same Cartesian shortcut would
remove the polar transform for every Gaussian evaluation, not just an MGE
basis — but it moves values, so it is a separate, wider-blast-radius task.

Merge order

Merge PyAutoLabs/PyAutoArray#508 before this PR. This repo's CI clones
PyAutoArray from source and checks out the same-named branch when it exists, so
this PR is currently testing against the PyAutoArray branch; once #508 merges and
that branch is deleted, it falls back to PyAutoArray main. No test here depends
on #508's new OverSampler behaviour, so this PR is green either way — the order
is about not leaving the paired measurement half-landed. The workspace PR
PyAutoLabs/autolens_profiling#190 merges last, behind both.

Heart

Heart is RED at ship time for a pre-existing reason unrelated to this task.
Verbatim from pyauto-heart readiness --json (2026-08-28T21:34:42Z, score 45):

• red_reasons: "release validation FAILED (stage integrate)"
• yellow_reasons: "workspace validation not passing (2 failed, cloud#33179766004: autolens_test scripts/imaging/rectangular_mge.py, autolens_test scripts/imaging/rectangular_mge_rtu.py)",
  "manifest drift: session-start hooks (generated) — 32 mismatch(es) vs PyAutoMind/repos.yaml"

Human authorisation to ship over it, given 2026-08-28, verbatim:

prm and then kick off phase 2, I authorize the heart RED thing

Generated by the PyAutoLabs agent workflow.