docs(examples): use gauge-vertex overload everywhere (U6)

Finding-U6 from doc/reviewer/usability-audit-2026-05-31.md.

The Finding-D fix (external-audit-2026-05-30) added a clean one-call
gauge-vertex overload:
  assign_euclidean_vertex_dof_indices(mesh, maps, gauge_vertex)

But all user-facing code still showed the old verbose manual loop:
  auto vit = mesh.vertices().begin();
  maps.v_idx[*vit++] = -1;
  int idx = 0;
  for (; vit != mesh.vertices().end(); ++vit) maps.v_idx[*vit] = idx++;

Replaced in three places:
  README.md 'Minimal usage' code block
  example_euclidean.cpp Step 3
  example_layout.cpp  pin_first() helper

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

README.md

@@ -109,14 +109,12 @@ ConformalMesh mesh = load_mesh("input.off");
 EuclideanMaps maps = setup_euclidean_maps(mesh);
 compute_euclidean_lambda0_from_mesh(mesh, maps);
 
-// Assign DOFs — pin first vertex (gauge fix)
-auto vit = mesh.vertices().begin();
-maps.v_idx[*vit++] = -1;
-int idx = 0;
-for (; vit != mesh.vertices().end(); ++vit) maps.v_idx[*vit] = idx++;
+// Assign DOFs — pin first vertex as gauge fix, index the rest 0..n-1
+auto gauge = *mesh.vertices().begin();
+int n = assign_euclidean_vertex_dof_indices(mesh, maps, gauge);
 
 // Natural equilibrium target: x* = 0 by construction
-std::vector<double> x0(idx, 0.0);
+std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
 auto G0 = euclidean_gradient(mesh, x0, maps);
 for (auto v : mesh.vertices())
     if (maps.v_idx[v] >= 0) maps.theta_v[v] -= G0[maps.v_idx[v]];

code/examples/example_euclidean.cpp

@@ -59,13 +59,10 @@ int main(int argc, char* argv[])
     compute_euclidean_lambda0_from_mesh(mesh, maps);
 
     // ── Step 3: pin the first vertex (gauge fix) ──────────────────────────
-    auto vit = mesh.vertices().begin();
-    Vertex_index v_pinned = *vit++;
-    maps.v_idx[v_pinned] = -1;   // pinned: u[v_pinned] = 0 (fixed)
-    int idx = 0;
-    for (; vit != mesh.vertices().end(); ++vit)
-        maps.v_idx[*vit] = idx++;
-    const int n = idx;
+    // The gauge-vertex overload pins the chosen vertex (v_idx = -1) and
+    // assigns sequential indices 0..n-1 to the rest in a single call.
+    Vertex_index v_pinned = *mesh.vertices().begin();
+    const int n = assign_euclidean_vertex_dof_indices(mesh, maps, v_pinned);
 
     std::cout << "[example_euclidean] DOFs: " << n << " (1 vertex pinned).\n";
 

code/examples/example_layout.cpp

@@ -54,14 +54,13 @@ static void set_natural_theta(ConformalMesh& mesh, EuclideanMaps& maps, int n)
 }
 
 // ── Helper: pin vertex 0, assign DOF indices 0..n-1 to the rest ──────────────
+// Uses the gauge-vertex overload introduced in external-audit Finding-D:
+// assign_euclidean_vertex_dof_indices(mesh, maps, gauge) pins the chosen
+// vertex and assigns sequential indices in a single pass.
 static int pin_first(ConformalMesh& mesh, EuclideanMaps& maps)
 {
-    auto vit = mesh.vertices().begin();
-    maps.v_idx[*vit++] = -1;   // pinned
-    int idx = 0;
-    for (; vit != mesh.vertices().end(); ++vit)
-        maps.v_idx[*vit] = idx++;
-    return idx;
+    return assign_euclidean_vertex_dof_indices(
+        mesh, maps, *mesh.vertices().begin());
 }
 
 // ── Main ─────────────────────────────────────────────────────────────────────

doc/reviewer/usability-audit-2026-05-31.md

@@ -541,7 +541,7 @@ exposed via the CGAL public API, but a CLI user cannot reach them.
 | U3 | `contracts.md:16` | Doc error | 🟡 Medium | ✅ Fixed 2026-05-31 |
 | U4 | `README.md:17` | Stale | 🟡 Medium | ✅ Fixed 2026-05-31 |
 | U5 | `Discrete_conformal_map.h:14` | Stale | 🟡 Medium | ✅ Fixed 2026-05-31 |
-| U6 | README + 2 examples | Stale | 🟡 Medium | 🟡 Open |
+| U6 | README + 2 examples | Stale | 🟡 Medium | ✅ Fixed 2026-05-31 |
 | U7 | `getting-started.md` | Missing | 🟠 Minor | ✅ Fixed 2026-05-31 |
 | U8 | `README.md` | Missing | 🟠 Minor | ✅ Fixed 2026-05-31 |
 | U9 | `layout.hpp` + example | Missing | 🟠 Minor | 🟡 Open |