perf(solvers): scope default one-equation calls to one thread

src/fesa/math/dense_blas_internal.cpp

@@ -8,6 +8,24 @@
 
 namespace fesa::dense_blas_internal {
 
+ScopedSingleEquationThreads::ScopedSingleEquationThreads(
+    const std::size_t equation_count) noexcept {
+  // The measured 1x1 PARDISO case has backend/thread overhead and an
+  // intermittently failing default-thread lifecycle. Broader sizes are not
+  // inferred from this auxiliary evidence.
+  if (equation_count != 1U) return;
+  const int previous = mkl_set_num_threads_local(1);
+  if (previous == 0) {
+    restore_default_ = true;
+  } else {
+    static_cast<void>(mkl_set_num_threads_local(previous));
+  }
+}
+
+ScopedSingleEquationThreads::~ScopedSingleEquationThreads() {
+  if (restore_default_) static_cast<void>(mkl_set_num_threads_local(0));
+}
+
 bool CholeskyLowerAndInverse(const Matrix& matrix, Matrix& lower,
                             Matrix& inverse_lower) {
   if (matrix.Rows() != matrix.Columns())

src/fesa/math/dense_blas_internal.h

@@ -11,6 +11,21 @@
 
 namespace fesa::dense_blas_internal {
 
+/// @brief Limits only default-policy one-equation backend calls to one thread.
+/// @note Existing caller TLS overrides are retained. Destruction restores the
+/// default request; process-wide MKL/OpenMP settings are never changed.
+class ScopedSingleEquationThreads {
+ public:
+  explicit ScopedSingleEquationThreads(std::size_t equation_count) noexcept;
+  ScopedSingleEquationThreads(const ScopedSingleEquationThreads&) = delete;
+  ScopedSingleEquationThreads& operator=(const ScopedSingleEquationThreads&) =
+      delete;
+  ~ScopedSingleEquationThreads();
+
+ private:
+  bool restore_default_{false};
+};
+
 /// @brief Converts a dense storage length to the private MKL integer type.
 /// @return The converted length or an uncategorized overflow failure.
 Result<MKL_INT> ToMklSize(std::size_t size);

src/fesa/solvers/linear/mkl_pardiso_solver.cpp

@@ -14,6 +14,7 @@
 
 #include "fesa/math/sparse_matrix.h"
 #include "fesa/math/vector.h"
+#include "math/dense_blas_internal.h"
 
 namespace fesa {
 namespace {
@@ -286,6 +287,8 @@ class MklPardisoSolver::Impl {
   /// @brief Calls PARDISO with retained private arrays and phase state.
   void CallPardiso(const MKL_INT phase, double* rhs, double* solution,
                    MKL_INT& error) {
+    const dense_blas_internal::ScopedSingleEquationThreads thread_scope{
+        static_cast<std::size_t>(equation_count_)};
     pardiso(pt_.data(), &max_factorizations_, &matrix_number_, &mtype_, &phase,
             &equation_count_, values_.data(), row_offsets_.data(),
             column_indices_.data(), permutation_.data(), &rhs_count_,

src/fesa/solvers/linear/mkl_pardiso_tangent_solver.cpp

@@ -13,6 +13,7 @@
 
 #include "fesa/math/sparse_matrix.h"
 #include "fesa/math/vector.h"
+#include "math/dense_blas_internal.h"
 
 namespace fesa {
 namespace {
@@ -230,6 +231,8 @@ class MklPardisoTangentSolver::Impl {
   /// @brief Calls PARDISO with retained private arrays and phase state.
   void CallPardiso(const MKL_INT phase, double* rhs, double* correction,
                    MKL_INT& error) {
+    const dense_blas_internal::ScopedSingleEquationThreads thread_scope{
+        static_cast<std::size_t>(equation_count_)};
     pardiso(pt_.data(), &max_factorizations_, &matrix_number_, &mtype_, &phase,
             &equation_count_, values_.data(), row_offsets_.data(),
             column_indices_.data(), permutation_.data(), &rhs_count_,