[ChrisRackauckas-Claude]

Please ignore this PR until it has been reviewed by @ChrisRackauckas.

Why

SciMLTesting.run_qa runs ExplicitImports' checks on the package module.
ExplicitImports does know about package extensions — it reads the [extensions]
table out of Project.toml — but it skips any extension whose module does not
exist, and an extension module only comes into existence once its trigger
(weak)dependency is loaded:

ext_mod = Base.get_extension(mod, Symbol(ext))
ext_mod === nothing && continue

test/qa/Project.toml did not depend on any of this package's weakdeps, so none
of the five extensions were being scanned by QA at all. Verified on the base
branch (main), from the unmodified QA environment:

BandedMatricesExt => nothing
BlockBandedMatricesExt => nothing
FastAlmostBandedMatricesExt => nothing
SpecialMatricesExt => nothing
ToeplitzMatricesExt => nothing
SCANNED: SparseMatrixIdentification

This was a real gap, not an incidental one — no test dependency was pulling any of
the weakdeps in transitively.

What changed

• test/qa/Project.toml: added BandedMatrices, BlockBandedMatrices,
  FastAlmostBandedMatrices, SpecialMatrices, ToeplitzMatrices under [deps]
  (same UUIDs as the root [weakdeps]) with [compat] entries mirroring the root
  Project.toml bounds.
• test/qa/qa.jl: using all five so the extension modules exist by the time
  run_qa runs.

Same probe on this branch:

BandedMatricesExt => BandedMatricesExt
BlockBandedMatricesExt => BlockBandedMatricesExt
FastAlmostBandedMatricesExt => FastAlmostBandedMatricesExt
SpecialMatricesExt => SpecialMatricesExt
ToeplitzMatricesExt => ToeplitzMatricesExt
SemiseparableMatricesExt => nothing
SCANNED: FastAlmostBandedMatricesExt
SCANNED: SparseMatrixIdentification
SCANNED: BlockBandedMatricesExt
SCANNED: BandedMatricesExt
SCANNED: SpecialMatricesExt
SCANNED: ToeplitzMatricesExt

Coverage

All 5 declared extensions are now scanned. Nothing was left out for
hardware/system-library reasons — every weakdep here is pure Julia.

SemiseparableMatrices is declared under [weakdeps] and [compat] in the root
Project.toml but has no [extensions] entry and no ext/ file, so there is
no extension module for ExplicitImports to scan. It is a dangling weakdep; adding
it to the QA environment would achieve nothing, so it was left out. Worth removing
from [weakdeps]/[compat] (or growing the extension it was presumably meant to
trigger) in a separate PR.

Findings the new coverage produced, and how each was handled

Fixed in the extension sources (no_implicit_imports)

Every extension did using SomePackage and then used names from it implicitly.
Replaced with explicit imports:

file	before	after
ext/BandedMatricesExt.jl	using SparseMatrixIdentification, using BandedMatrices	using SparseMatrixIdentification: SparseMatrixIdentification, using BandedMatrices: BandedMatrix
ext/BlockBandedMatricesExt.jl	+ using BlockBandedMatrices, using BandedMatrices	using BlockBandedMatrices: BlockBandedMatrix (the BandedMatrices using was dropped — nothing from it was used; it remains a declared trigger in [extensions])
ext/FastAlmostBandedMatricesExt.jl	+ using FastAlmostBandedMatrices, using BandedMatrices	using FastAlmostBandedMatrices: AlmostBandedMatrix, using BandedMatrices: BandedMatrix
ext/SpecialMatricesExt.jl	+ using SpecialMatrices	using SpecialMatrices: Cauchy, Hilbert, Strang, Vandermonde
ext/ToeplitzMatricesExt.jl	+ using ToeplitzMatrices	using ToeplitzMatrices: Toeplitz

No behavior change; GROUP=Core still passes (see below).

Ignored, with justification (all_qualified_accesses_are_public)

18 findings, and all 18 have SparseMatrixIdentification itself as the owner —
they are the extensions reaching into their own parent package, which is the only
way an extension can hook into it. One ignore group was added in test/qa/qa.jl
with a comment naming the owner and the reason:

• try_almostbanded, try_banded, try_blockbanded, try_special_matrices,
  try_toeplitz — the stub functions each extension adds methods to.
• _bandedmatrices_loaded, _blockbandedmatrices_loaded,
  _fastalmostbandedmatrices_loaded, _specialmatrices_loaded,
  _toeplitzmatrices_loaded — the Ref{Bool} flags each extension's __init__
  flips.
• detect_block_size, is_almost_banded, is_banded, is_cauchy, is_hilbert,
  is_strang, is_toeplitz, is_vandermonde — the structure-detection predicates
  the extensions dispatch on.

None of these are declared public/exported, and there is no public spelling to
use instead, so there is nothing to "fix" in the extension source. No check was
disabled and no @test_broken was added.

Follow-up worth considering (deliberately not done here to keep this PR
reviewable): the five try_* functions are already documented in
docs/src/api.md under "Extension Point Functions", i.e. they are de-facto public
API that has never been declared public. Declaring them public (a minor
version bump) would let five of the eighteen ignores be dropped.

Local results

GROUP=QA julia --project=. -e 'using Pkg; Pkg.test()' on Julia 1.12:

Test Summary: | Pass  Total     Time
QA/qa.jl      |   21     21  1m15.2s
     Testing SparseMatrixIdentification tests passed

Before this PR, the same command on this branch's test/qa content but without
the source fixes errored with no_implicit_imports and
all_qualified_accesses_are_public — i.e. the newly-scanned extensions did
produce real findings, which is the point.

GROUP=Core julia --project=. -e 'using Pkg; Pkg.test()' also passes with the
extension source changes (Testing SparseMatrixIdentification tests passed, all
12 test files green).

test/qa/Manifest.toml is a local build artifact and is not committed.