Add inference tests for ForwardDiff Dual problems

- Add @inferred tests to verify init returns concrete DualLinearCache type
- Update GenericLUFactorization test to expect DualLinearCache (optimization moved to solve-time)
- Add test for solve correctness with GenericLUFactorization

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Opus 4.5 <[email protected]>

Author: ChrisRackauckas

test/forwarddiff_overloads.jl

@@ -196,20 +196,37 @@ overload_x_p = solve!(cache, UMFPACKFactorization())
 backslash_x_p = A \ b
 @test ≈(overload_x_p, backslash_x_p, rtol = 1e-9)
 
-# Test that GenericLU doesn't create a DualLinearCache
+# Test type inference for init with ForwardDiff Dual numbers
+# This ensures init returns a concrete type (not a Union) for type stability
 A, b = h([ForwardDiff.Dual(5.0, 1.0, 0.0), ForwardDiff.Dual(5.0, 0.0, 1.0)])
 
 prob = LinearProblem(A, b)
-@test init(prob, GenericLUFactorization()) isa LinearSolve.LinearCache
 
-@test init(prob) isa LinearSolve.LinearCache
+# GenericLUFactorization now returns DualLinearCache for type stability
+# (the optimization for GenericLU happens at solve-time instead of init-time)
+@test init(prob, GenericLUFactorization()) isa LinearSolve.LinearSolveForwardDiffExt.DualLinearCache
 
-# Test that SparspakFactorization doesn't create a DualLinearCache
+# Test inference with explicit algorithm
+@test (@inferred init(prob, LUFactorization())) isa LinearSolve.LinearSolveForwardDiffExt.DualLinearCache
+@test (@inferred init(prob, GenericLUFactorization())) isa LinearSolve.LinearSolveForwardDiffExt.DualLinearCache
+
+# Test inference with default algorithm (nothing) - this was the main bug
+# Previously returned Union{LinearCache, DualLinearCache} due to runtime conditional
+@test (@inferred init(prob, nothing)) isa LinearSolve.LinearSolveForwardDiffExt.DualLinearCache
+
+# Test that SparspakFactorization still opts out (sparse solvers can't handle Duals the same way)
 A, b = h([ForwardDiff.Dual(5.0, 1.0, 0.0), ForwardDiff.Dual(5.0, 0.0, 1.0)])
 
 prob = LinearProblem(sparse(A), b)
 @test init(prob, SparspakFactorization()) isa LinearSolve.LinearCache
 
+# Test that solve still works correctly with GenericLUFactorization
+A, b = h([ForwardDiff.Dual(5.0, 1.0, 0.0), ForwardDiff.Dual(5.0, 0.0, 1.0)])
+prob = LinearProblem(A, b)
+sol_generic = solve(prob, GenericLUFactorization())
+backslash_result = A \ b
+@test ≈(sol_generic.u, backslash_result, rtol = 1e-9)
+
 # Test ComponentArray with ForwardDiff (Issue SciML/DifferentialEquations.jl#1110)
 # This tests that ArrayInterface.restructure preserves ComponentArray structure