feat: adopt lumped mass and reduced-coordinate shell HHT

Integrate Hex8 mass verification and fixed-null-basis shell dynamics with numerical and physics handoff evidence. Preserve interrupted reference and full-validation status; full integration validation is deferred until after performance optimization.

src/fesa/analysis/hht_dynamic_internal.h

@@ -2,6 +2,7 @@
 #define FESA_ANALYSIS_HHT_DYNAMIC_INTERNAL_H_
 
 #include <array>
+#include <optional>
 #include <vector>
 
 #include "fesa/analysis/dynamic_analysis_state.h"
@@ -28,6 +29,8 @@ struct RuntimeShell {
 struct DynamicAssembly {
   DynamicAcceptedFrameState frame;
   SparseMatrix mass;
+  SparseMatrix position_tangent;
+  SparseMatrix velocity_tangent;
   SparseMatrix internal_tangent;
   SparseMatrix effective_tangent;
   SparseMatrix drilling_metric;
@@ -38,6 +41,8 @@ struct DynamicAssembly {
   double angular_momentum_rate_scale;
   Vector hidden_second{0U};
   Vector hidden_second_scale{0U};
+  // Canonical scalar drilling-constraint derivative, indexed by global node.
+  std::optional<SparseMatrix> hidden_gradient;
 };
 
 /// @brief Builds checked shell evaluators in stable source order.
@@ -59,6 +64,20 @@ Result<DynamicAssembly> EvaluateAssembly(
     const HhtParameters& parameters, const ParallelFor& parallel_for,
     bool directional_third);
 
+/// @brief Evaluates an increment's initial iterate without committing history.
+/// @param frame New-time trial containing the last accepted coordinates.
+/// @param predictor Frozen HHT predictor used for both possible starting points.
+/// @param old_acceleration Last accepted physical coordinate acceleration.
+/// @return The valid initial assembly with smaller free ||D^T E||_infinity,
+/// preferring the predictor on a tie. Invalid predictor chart/geometry selects
+/// accepted coordinates; rate, mass and arithmetic failures remain failures.
+Result<DynamicAssembly> EvaluateNewtonStart(
+    const Domain& domain, const DofManager& dofs,
+    const std::vector<RuntimeShell>& shells, DynamicAcceptedFrameState frame,
+    const HhtPredictor& predictor, const Vector& old_acceleration,
+    const Vector& load_scale, double time_increment,
+    const HhtParameters& parameters, const ParallelFor& parallel_for);
+
 /// @brief Builds deterministic CSR structure from an ordered local list.
 Result<SparseMatrix> MakeSparse(std::size_t rows, std::size_t columns,
                                 std::vector<CooContribution> contributions);
@@ -77,6 +96,10 @@ std::vector<std::size_t> NullNodes(
     const std::vector<DynamicRotationKinematics>& rotations);
 
 /// @brief Solves the positive-mass and differentiated-null initial equations.
+///
+/// The returned candidate is finite and exact-zero at constrained DOFs. Its
+/// complete free-equation N2-R certificate requires reassembly with that
+/// acceleration and the caller's source-uncancelled residual scale.
 Result<Vector> InitializeAcceleration(const Domain& domain,
                                       const DofManager& dofs,
                                       const DynamicAssembly& initial,