diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.md
index fd3b522..a270968 100644
--- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.md
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.md
@@ -10,16 +10,9 @@
 - tire-tree nodes: 4
 - tire-tree edges: 3
 
-| node | depth | faces | annular | up | singleton down | bite apexes | full medial tires |
+| node | depth | faces | annular cycles | annular | up | singleton down | bite apexes |
 |--:|--:|--:|--:|--:|--:|--:|--:|
-| T0 | 2 | 23 | 23 | 10 | 13 | 0 | 1 |
-| T0.0 |  |  | | 10 | 13 | - | `UDDUDUUDDUDUDUDUDDDUUDD` |
-| T1 | 1 | 15 | 15 | 5 | 10 | 0 | 1 |
-| T1.0 |  |  | | 5 | 10 | - | `UDDDUDDUDUDDDUD` |
-| T2 | 3 | 14 | 14 | 11 | 0 | 0 | 4 |
-| T2.0 |  |  | | 3 | 0 | - | `UUU` |
-| T2.1 |  |  | | 3 | 0 | - | `UUU` |
-| T2.2 |  |  | | 5 | 0 | - | `UUUUU` |
-| T2.3 |  |  | | 3 | 0 | - | `UUU` |
-| T3 | 3 | 5 | 5 | 5 | 0 | 0 | 1 |
-| T3.0 |  |  | | 5 | 0 | - | `UUUUU` |
+| T0 | 2 | 23 | 1 | 23 | 10 | 13 | 0 |
+| T1 | 1 | 15 | 1 | 15 | 5 | 10 | 0 |
+| T2 | 3 | 14 | 4 | 14 | 11 | 0 | 0 |
+| T3 | 3 | 5 | 1 | 5 | 5 | 0 | 0 |
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.pdf
index c2893ef..a1c5a0a 100644
Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.pdf differ
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.png
index 4fb8276..f24be12 100644
Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.png differ
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/draw_random_medial_tire_decompositions.py b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/draw_random_medial_tire_decompositions.py
index d48cc60..9491c30 100644
--- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/draw_random_medial_tire_decompositions.py
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/draw_random_medial_tire_decompositions.py
@@ -2,9 +2,8 @@
 
 The source graphs come from ``plantri -c5`` in graph6 format.  For each sampled
 30-vertex triangulation, this script chooses a random source vertex, builds the
-BFS depth-component tire tree, recognizes every full medial tire graph in the
-decomposition, and draws both the tire tree and the realized full medial tire
-graphs.
+BFS depth-component tire tree, and draws both the tire tree and the medial
+tread model for each depth component.
 """
 
 from __future__ import annotations
@@ -36,8 +35,11 @@ import networkx as nx
 if str(PAPER_DIR) not in sys.path:
     sys.path.insert(0, str(PAPER_DIR))
 
-from lib.medial_tire_decomposition import ekey, medial_tire_facemodel, recognise
-from lib.full_medial_tire_generator import FullMedialTireGraph
+from lib.medial_tire_decomposition import (
+    annular_cycle_components,
+    ekey,
+    medial_tire_facemodel,
+)
 
 
 @dataclass(frozen=True)
@@ -49,7 +51,8 @@ class TreadNode:
     up: frozenset
     down: frozenset
     bites: frozenset
-    tires: tuple[tuple[FullMedialTireGraph, dict], ...]
+    medial: nx.Graph
+    annular_cycles: tuple[tuple, ...]
 
 
 @dataclass(frozen=True)
@@ -228,8 +231,8 @@ def build_tire_tree(g: nx.Graph, source: int, augment: bool = True):
         if tread is None or len(tread["up"]) < 3:
             continue
         mt = medial_tire_facemodel(tread["tread_faces"])
-        tires = tuple(recognise(mt, tread))
-        if not tires:
+        annular_cycles = tuple(annular_cycle_components(mt, tread["annular"]))
+        if not annular_cycles:
             continue
         node = TreadNode(
             idx=len(nodes),
@@ -239,7 +242,8 @@ def build_tire_tree(g: nx.Graph, source: int, augment: bool = True):
             up=frozenset(tread["up"]),
             down=frozenset(tread["down"]),
             bites=frozenset(tread["bites"]),
-            tires=tires,
+            medial=mt,
+            annular_cycles=annular_cycles,
         )
         comp_to_node[comp_idx] = node.idx
         nodes.append(node)
@@ -343,7 +347,7 @@ def draw_tire_tree(ax, nodes: list[TreadNode], tree_edges):
         ax.text(
             x,
             y,
-            f"T{node.idx}\nd={node.depth}\n{len(node.tires)} tire(s)",
+            f"T{node.idx}\nd={node.depth}\n{len(node.annular_cycles)} cycle(s)",
             ha="center",
             va="center",
             fontsize=8,
@@ -373,58 +377,113 @@ def edge_midpoint_angle(i: int, n: int) -> float:
     return math.pi / 2 - 2 * math.pi * (i + 0.5) / n
 
 
-def draw_full_medial_tire(ax, graph: FullMedialTireGraph, title: str):
-    n = graph.n
-    ann = [vertex_xy(k, n, 1.0) for k in range(n)]
-    matched = graph.bite_edges
-    cyc_x = [p[0] for p in ann] + [ann[0][0]]
-    cyc_y = [p[1] for p in ann] + [ann[0][1]]
-    ax.plot(cyc_x, cyc_y, color="black", lw=1.3, zorder=2)
-    for i, tooth in enumerate(graph.tooth_word):
-        if tooth == "U":
-            r, color = 1.42, "#2563eb"
-        elif i not in matched:
-            r, color = 0.58, "#dc2626"
-        else:
-            continue
-        ang = edge_midpoint_angle(i, n)
-        apex = (r * math.cos(ang), r * math.sin(ang))
-        for corner in (ann[i], ann[(i + 1) % n]):
-            ax.plot([apex[0], corner[0]], [apex[1], corner[1]], color="#9ca3af", lw=0.5)
-        ax.scatter([apex[0]], [apex[1]], s=12, color=color, zorder=3)
-    for i, j in sorted(graph.bites):
-        corners = [ann[i], ann[(i + 1) % n], ann[j], ann[(j + 1) % n]]
-        apex = (0.82 * sum(p[0] for p in corners) / 4, 0.82 * sum(p[1] for p in corners) / 4)
+def draw_tread_model(ax, node: TreadNode):
+    cycle_count = len(node.annular_cycles)
+    offsets = [3.25 * (i - (cycle_count - 1) / 2) for i in range(cycle_count)]
+    apex_positions: dict[tuple, tuple[float, float]] = {}
+    apex_corners: dict[tuple, list[tuple[float, float]]] = defaultdict(list)
+    ann_positions: dict[tuple, tuple[float, float]] = {}
+
+    for cycle_idx, order in enumerate(node.annular_cycles):
+        n = len(order)
+        dx = offsets[cycle_idx]
+        ann = {
+            vertex: (dx + x, y)
+            for vertex, (x, y) in zip(order, [vertex_xy(k, n, 1.0) for k in range(n)])
+        }
+        ann_positions.update(ann)
+
+        cyc_x = [ann[v][0] for v in order] + [ann[order[0]][0]]
+        cyc_y = [ann[v][1] for v in order] + [ann[order[0]][1]]
+        ax.plot(cyc_x, cyc_y, color="black", lw=1.3, zorder=2)
+
+        for i, a in enumerate(order):
+            b = order[(i + 1) % n]
+            apexes = [
+                w for w in set(node.medial.neighbors(a)) & set(node.medial.neighbors(b))
+                if w not in node.annular
+            ]
+            for apex in apexes:
+                apex_corners[apex].extend([ann[a], ann[b]])
+                if apex in apex_positions:
+                    continue
+                angle = edge_midpoint_angle(i, n)
+                if apex in node.up:
+                    radius = 1.42
+                else:
+                    radius = 0.58
+                apex_positions[apex] = (
+                    dx + radius * math.cos(angle),
+                    radius * math.sin(angle),
+                )
+
+    for apex, corners in apex_corners.items():
+        if apex in node.bites and corners:
+            cx = sum(p[0] for p in corners) / len(corners)
+            cy = sum(p[1] for p in corners) / len(corners)
+            center_x = sum(offsets) / len(offsets) if offsets else 0.0
+            apex_positions[apex] = (
+                center_x + 0.82 * (cx - center_x),
+                0.82 * cy,
+            )
+        pos = apex_positions[apex]
         for corner in corners:
-            ax.plot([apex[0], corner[0]], [apex[1], corner[1]], color="#9ca3af", lw=0.5)
-        ax.scatter([apex[0]], [apex[1]], s=22, color="#7f1d1d", edgecolors="black", lw=0.4)
-    ax.scatter([p[0] for p in ann], [p[1] for p in ann], s=9, color="black", zorder=4)
-    bites = ",".join(f"{i}{j}" for i, j in sorted(graph.bites)) or "-"
-    ax.set_title(f"{title}\n{graph.tooth_word} b:{bites}", fontsize=5.8, pad=1.5)
-    ax.set_xlim(-1.6, 1.6)
-    ax.set_ylim(-1.6, 1.6)
+            ax.plot([pos[0], corner[0]], [pos[1], corner[1]], color="#9ca3af", lw=0.5)
+
+    for apex, pos in apex_positions.items():
+        if apex in node.up:
+            color, size, edgecolor = "#2563eb", 13, "none"
+        elif apex in node.bites:
+            color, size, edgecolor = "#7f1d1d", 24, "black"
+        else:
+            color, size, edgecolor = "#dc2626", 13, "none"
+        ax.scatter(
+            [pos[0]],
+            [pos[1]],
+            s=size,
+            color=color,
+            edgecolors=edgecolor,
+            linewidths=0.4,
+            zorder=3,
+        )
+
+    if ann_positions:
+        ax.scatter(
+            [p[0] for p in ann_positions.values()],
+            [p[1] for p in ann_positions.values()],
+            s=9,
+            color="black",
+            zorder=4,
+        )
+
+    singleton_down = set(node.down) - set(node.bites)
+    ax.set_title(
+        f"T{node.idx} d={node.depth}: {len(node.annular_cycles)} annular cycle(s)\n"
+        f"ann={len(node.annular)} up={len(node.up)} down={len(singleton_down)} "
+        f"bite={len(node.bites)}",
+        fontsize=6.4,
+        pad=1.5,
+    )
+    pad = 1.7
+    ax.set_xlim(min(offsets, default=0.0) - pad, max(offsets, default=0.0) + pad)
+    ax.set_ylim(-1.65, 1.65)
     ax.set_aspect("equal")
     ax.axis("off")
 
 
 def draw_medial_tire_grid(fig, outer_spec, nodes):
-    tires = []
-    for node in nodes:
-        for comp_idx, (graph, _bij) in enumerate(node.tires):
-            tires.append((node.idx, comp_idx, graph))
-    if not tires:
+    if not nodes:
         ax = fig.add_subplot(outer_spec)
-        ax.text(0.5, 0.5, "No full medial tire graphs recognized", ha="center")
+        ax.text(0.5, 0.5, "No medial treads extracted", ha="center")
         ax.axis("off")
         return
-    cols = min(5, max(1, math.ceil(math.sqrt(len(tires)))))
-    rows = math.ceil(len(tires) / cols)
+    cols = min(3, max(1, math.ceil(math.sqrt(len(nodes)))))
+    rows = math.ceil(len(nodes) / cols)
     sub = outer_spec.subgridspec(rows, cols, wspace=0.08, hspace=0.35)
     for i in range(rows * cols):
         ax = fig.add_subplot(sub[i // cols, i % cols])
-        if i < len(tires):
-            node_idx, comp_idx, graph = tires[i]
-            draw_full_medial_tire(ax, graph, f"T{node_idx}.{comp_idx} n={graph.n}")
+        if i < len(nodes):
+            draw_tread_model(ax, nodes[i])
         else:
             ax.axis("off")
 
@@ -452,22 +511,16 @@ def write_index(
         f"- tire-tree nodes: {len(nodes)}",
         f"- tire-tree edges: {len(tree_edges)}",
         "",
-        "| node | depth | faces | annular | up | singleton down | bite apexes | full medial tires |",
+        "| node | depth | faces | annular cycles | annular | up | singleton down | bite apexes |",
         "|--:|--:|--:|--:|--:|--:|--:|--:|",
     ]
     for node in nodes:
         singleton_down = set(node.down) - set(node.bites)
         lines.append(
             f"| T{node.idx} | {node.depth} | {len(node.face_indices)} | "
-            f"{len(node.annular)} | {len(node.up)} | {len(singleton_down)} | "
-            f"{len(node.bites)} | {len(node.tires)} |"
+            f"{len(node.annular_cycles)} | {len(node.annular)} | {len(node.up)} | "
+            f"{len(singleton_down)} | {len(node.bites)} |"
         )
-        for comp_idx, (tire, _bij) in enumerate(node.tires):
-            bites = ",".join(f"({i},{j})" for i, j in sorted(tire.bites)) or "-"
-            lines.append(
-                f"| T{node.idx}.{comp_idx} |  |  | | {len(tire.up_edges)} | "
-                f"{len(tire.singleton_down_edges)} | {bites} | `{tire.tooth_word}` |"
-            )
     path.write_text("\n".join(lines) + "\n")
 
 
@@ -517,7 +570,7 @@ def draw_case(out_dir: Path, graph_idx: int, g: nx.Graph, source: int, augment:
         nodes,
         tree_edges,
     )
-    return png, pdf, len(nodes), sum(len(node.tires) for node in nodes)
+    return png, pdf, len(nodes), sum(len(node.annular_cycles) for node in nodes)
 
 
 def run(args: argparse.Namespace):
@@ -535,12 +588,12 @@ def run(args: argparse.Namespace):
         sources = [rng.choice(list(graph.nodes())) for graph in graphs]
 
     for i, (graph, source) in enumerate(zip(graphs, sources), start=1):
-        png, pdf, node_count, tire_count = draw_case(
+        png, pdf, node_count, annular_cycle_count = draw_case(
             out_dir, i, graph, source, augment=not args.no_augment_same_level_faces
         )
         print(
             f"case {i}: source={source}, connectivity={nx.node_connectivity(graph)}, "
-            f"tire nodes={node_count}, full medial tires={tire_count}"
+            f"tire nodes={node_count}, annular cycles={annular_cycle_count}"
         )
         print(f"  wrote {png}")
         print(f"  wrote {pdf}")
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_1.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_1.md
index 2898e67..f9ce39d 100644
--- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_1.md
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_1.md
@@ -10,13 +10,8 @@
 - tire-tree nodes: 3
 - tire-tree edges: 2
 
-| node | depth | faces | annular | up | singleton down | bite apexes | full medial tires |
+| node | depth | faces | annular cycles | annular | up | singleton down | bite apexes |
 |--:|--:|--:|--:|--:|--:|--:|--:|
-| T0 | 1 | 16 | 16 | 6 | 10 | 0 | 1 |
-| T0.0 |  |  | | 6 | 10 | - | `DUDUDUDDUDDDUDDU` |
-| T1 | 2 | 20 | 20 | 10 | 10 | 0 | 1 |
-| T1.0 |  |  | | 10 | 10 | - | `UUDUUDUDDUDUDUDUUDDD` |
-| T2 | 3 | 16 | 16 | 12 | 0 | 1 | 3 |
-| T2.0 |  |  | | 6 | 0 | (1,5) | `UDUUUDUU` |
-| T2.1 |  |  | | 3 | 0 | - | `UUU` |
-| T2.2 |  |  | | 5 | 0 | - | `UUUUU` |
+| T0 | 1 | 16 | 1 | 16 | 6 | 10 | 0 |
+| T1 | 2 | 20 | 1 | 20 | 10 | 10 | 0 |
+| T2 | 3 | 16 | 3 | 16 | 12 | 0 | 1 |
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_1.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_1.pdf
index 5ba9d88..c6bac3c 100644
Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_1.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_1.pdf differ
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_1.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_1.png
index 121503e..985942d 100644
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_2.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_2.md
index 36869e7..7d3d54f 100644
--- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_2.md
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_2.md
@@ -10,14 +10,9 @@
 - tire-tree nodes: 4
 - tire-tree edges: 3
 
-| node | depth | faces | annular | up | singleton down | bite apexes | full medial tires |
+| node | depth | faces | annular cycles | annular | up | singleton down | bite apexes |
 |--:|--:|--:|--:|--:|--:|--:|--:|
-| T0 | 1 | 16 | 16 | 7 | 9 | 0 | 1 |
-| T0.0 |  |  | | 7 | 9 | - | `DUDUDUDUDUDDUDUD` |
-| T1 | 2 | 17 | 17 | 9 | 8 | 0 | 1 |
-| T1.0 |  |  | | 9 | 8 | - | `UUUDDUDUDUDUDDUUD` |
-| T2 | 3 | 14 | 14 | 8 | 4 | 1 | 1 |
-| T2.0 |  |  | | 8 | 4 | (1,5) | `DDUUUDDUUDUDUU` |
-| T3 | 4 | 6 | 6 | 5 | 0 | 0 | 2 |
-| T3.0 |  |  | | 3 | 0 | - | `UUU` |
-| T3.1 |  |  | | 3 | 0 | - | `UUU` |
+| T0 | 1 | 16 | 1 | 16 | 7 | 9 | 0 |
+| T1 | 2 | 17 | 1 | 17 | 9 | 8 | 0 |
+| T2 | 3 | 14 | 1 | 14 | 8 | 4 | 1 |
+| T3 | 4 | 6 | 2 | 6 | 5 | 0 | 0 |
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_2.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_2.pdf
index 7f03ba2..6f5b5a8 100644
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_2.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_2.png
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