Fix cache reuse safety in non-allocating finite_difference_jacobian

[ChrisRackauckas-Claude]

Summary

• Fixes Fix cache updates #213. The non-allocating cached finite_difference_jacobian was reading the unperturbed components of x from cache.x1 (which can be uninitialized when the cache is built via similar(x), or stale when the cache is reused at a different x) instead of from the input x. This produced junk Jacobians in :central mode — most visibly via DifferentiationInterface.jl (see FiniteDiff with :central includes junk DifferentiationInterface.jl#983).
• The fix has the :central dense and sparse paths perturb around vecx (the input) directly. The in-place ! version was already safe via copyto!(x1, x). A defensive copyto!(cache.x1, x) is added to the non-allocating entry point too (guarded for immutability), so the cache is observably consistent with x after the call.
• Audited the broader caveat from @gdalle that the issue might be wider: GradientCache, HessianCache, JVPCache already update their scratch fields from x correctly. DerivativeCache has no x cache. New tests pin all of those down to prevent regressions.

Reproducer (from JuliaDiff/DifferentiationInterface.jl#983)

julia> using DifferentiationInterface, FiniteDiff

julia> foo(x) = [2x[1], 3x[2], 4x[1]];

julia> jacobian(foo, AutoFiniteDiff(fdjtype=Val(:central)), zeros(2))
# before:
# 3×2 Matrix{Float64}:
#  2.0           3.94723e-309
#  6.73119e-317  3.0
#  4.0           7.89446e-309
# after:
# 3×2 Matrix{Float64}:
#  2.0  0.0
#  0.0  3.0
#  4.0  0.0

Test plan

• New test/cache_reuse_tests.jl (27 tests) covers:
  • JacobianCache reuse, out-of-place + in-place, :forward/:central/:complex, dense + sparse, including: fresh cache reused at a new x, explicitly-poisoned (fill!(.., 1e10)) cache, and a direct DI-style similar(x) reproduction of #983.
  • GradientCache, JVPCache, HessianCache reuse with poisoned scratch buffers (these were already safe but lacked regression coverage).
  • In-place :central sparse path: verifies x is not mutated across the call.
• Wired the new file into the Core test group in test/runtests.jl.
• Full Core suite passes locally: Standard 219 + Color 39 + OutOfPlace 25 + new CacheReuse 27 = 310 passing (plus 1 pre-existing @test_broken).
• CI green.

🤖 Generated with Claude Code

[ChrisRackauckas-Claude]

CI status:

Check	Result
test (Core, 1)	✅ pass — includes the new 27-test cache-reuse safety set
build	✅ pass
OrdinaryDiffEq.jl/InterfaceII/1	✅ pass
codecov/patch, codecov/project	✅ pass
test (Downstream, 1)	❌ fail — pre-existing on master, unrelated to this PR

The Downstream failure is UndefVarError: Rodas4P not defined in test/downstream/ordinarydiffeq_tridiagonal_solve.jl, caused by an upstream rename in OrdinaryDiffEq. Same failure on master in run 24500721998 (2026-04-16, before this PR). Worth fixing separately by either re-importing Rodas4P from its new location or replacing it with whatever solver superseded it.