[SteveBronder]

Summary

The adjoint cosine ODE tests fail on Windows x64 with incorrect gradients and on Windows ARM64 with CVodeB error -3 in the CMake CI run. CVODES reuses its quadrature derivative output buffer, but accumulate_adjoints adds to that buffer with +=, retaining stale or uninitialized values between callbacks.

Clear the output before accumulating derivatives. Add an analytic value/gradient regression covering scalar and vector parameters, repeated solves, and repeated reverse passes. This is a solver correctness fix reviewed separately from the CMake infrastructure; existing tolerances stay unchanged.

Tests

• Observed failures before the fix in the existing adjoint tests and the new regression when allocator poisoning fills reused memory with nonzero bytes (MALLOC_PERTURB_).
• After the fix, all 56 cosine ODE cases pass ten shuffled repetitions with allocator poisoning; the 12 adjoint/regression cases pass ten repetitions with a second poison value.
• ARM64 Linux under QEMU: the regression fails before the fix and passes three repetitions after it.
• Windows x64 and ARM64 LLVM-Mingw cross-compilation passes with -Wall -Werror; native Windows runtime validation remains for CI.
• Dependency checks, cpplint, and git diff --check pass.

Side Effects

Adjoint quadrature derivatives no longer depend on previous output-buffer contents. Public APIs and solver tolerances are unchanged.

Release notes

Fix incorrect adjoint ODE parameter gradients and intermittent CVODES backward-integration failures caused by stale quadrature output.

Checklist

• Copyright holder: Simons Foundation
• New value and gradient regression added
• Focused validation passes as described above
• Complete CI matrix passes
• C++ idioms and Doxygen documentation have completed review

Related PRs

• 1/7: [CMake 1/7] Fix test compilation with strict compiler warnings
• 2/7: [CMake 2/7] Share OpenCL kernel-cache registration across translation units
• 3/7: [CMake 3/7] Isolate TBB construction tests from process-wide initialization
• 4/7: [CMake 4/7] Make probability test fixtures safe for grouped linking
• 5/7: [CMake 5/7] Make unit tests safe for grouped execution
• 6/7: [CMake 6/7] Initialize CVODES adjoint quadrature derivative buffers (this PR)
• 7/7: [CMake 7/7] Add CMake builds and migrate the test runner and CI

Native stack: https://github.com/stan-dev/math/stack/3509