feat(thermal): certify shell geometry and local heat operators

src/fesa/elements/thermal_shell_geometry.h

@@ -1,10 +1,59 @@
 #ifndef FESA_ELEMENTS_THERMAL_SHELL_GEOMETRY_H_
 #define FESA_ELEMENTS_THERMAL_SHELL_GEOMETRY_H_
+
 #include <array>
+
 #include "fesa/core/status.h"
 #include "fesa/math/vector3.h"
+#include "thermal_interval.h"
+
 namespace fesa {
-/// @brief Validates the finite, oriented Q4 geometry used by the thermal kernel.
-Status ValidateThermalShellGeometry(const std::array<Vector3,4>& coordinates);
-}
-#endif
+struct ModelDefinition;
+namespace thermal_internal {
+
+/// @brief Holds translated, power-of-two scaled tangents and their enclosures.
+struct Geometry {
+  Vector3 b;
+  Vector3 c;
+  Vector3 d;
+  IntervalVector bi;
+  IntervalVector ci;
+  IntervalVector di;
+  int length_exponent = 0;
+  double scaled_jacobian_lower = 0;
+};
+
+/// @brief Holds actual-point product metric, gradients and orthonormal columns.
+struct SurfacePoint {
+  std::array<double, 4> shape{};
+  std::array<Interval, 4> shape_interval{};
+  std::array<Vector3, 4> gradient{};
+  std::array<IntervalVector, 4> gradient_interval{};
+  std::array<Vector3, 3> axes{};
+  double jacobian = 0;
+  Interval jacobian_interval;
+};
+
+/// @brief Certifies whole-patch rank/injectivity from fixed candidate
+/// directions.
+Result<Geometry> BuildGeometry(const std::array<Vector3, 4>& coordinates,
+                               const SourceEntityId& source,
+                               const SourceLocation& location);
+/// @brief Checks finite metric inverse and gradients at one used surface point.
+Result<SurfacePoint> EvaluateSurface(const Geometry& geometry, double xi,
+                                     double eta, Interval xi_interval,
+                                     Interval eta_interval,
+                                     const SourceEntityId& source,
+                                     const SourceLocation& location);
+/// @brief Builds a geometry failure retaining original source element identity.
+Status GeometryFailure(const SourceEntityId& source,
+                       const SourceLocation& location, const char* code,
+                       const char* message);
+}  // namespace thermal_internal
+
+/// @brief Validates DS4 whole-patch geometry and source-only manifold topology.
+/// @pre Node indices and section assignments have been checked by Domain.
+Status ValidateThermalShellMesh(const ModelDefinition& definition);
+
+}  // namespace fesa
+#endif  // FESA_ELEMENTS_THERMAL_SHELL_GEOMETRY_H_