Use separate cache flags for A and b in ForwardDiff extension (rebased #1054 + test)

[ChrisRackauckas-Claude]

Please ignore until reviewed by @ChrisRackauckas.

Finishes and supersedes #1054 (by @hersle). That PR's change is correct but its CI was red only because it was based on main from before #1053, so test/Core/resolve.jl errored on SuperLUDISTFactorization() (the alg wasn't yet in the resolve skip list). This branch is #1054 rebased cleanly onto current main (which has the #1053 fix), plus a test.

Contents

• Use separate cache flags for A and b in the ForwardDiff extension (original commit by @hersle, rebased). Splits the single rhs_cache_valid flag into independent A_partials_valid / b_partials_valid flags, so mutating only A or only b no longer forces the other side's partials list to be recomputed (e.g. in an ODE where A is fixed while b changes, and vice versa).
• New test DualLinearCache separate A/b partials validity in test/Core/forwarddiff_overloads.jl: asserts a one-sided A/b mutation invalidates only that side's flag, that solve! revalidates both, and that results stay correct across one-sided updates.

Verification (local, Julia 1.10)

• test/Core/forwarddiff_overloads.jl — all testsets pass, including the new one (12/12).
• test/AD/caching_allocation_tests.jl — all pass.
• Runic format check on changed files — clean.

Credit for the underlying change goes to @hersle (#1054); the original commit is preserved with authorship.

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