diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_medial_annular_cycle_condition.py b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_medial_annular_cycle_condition.py
new file mode 100644
index 0000000..8c6bbd5
--- /dev/null
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_medial_annular_cycle_condition.py
@@ -0,0 +1,358 @@
+"""Check the medial annular-cycle almost-two-colour condition.
+
+For each generated plane triangulation G and each requested level source:
+
+  1. Build the full medial graph M(G).
+  2. Find depth-component tire annular medial subgraphs.
+  3. Enumerate simple cycles in those annular subgraphs.
+  4. Search for a proper vertex 3-colouring of M(G) such that every
+     such cycle uses two colours except at at most one vertex.
+
+Run with Sage, for example:
+
+  sage -python papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_medial_annular_cycle_condition.py --n-min 4 --n-max 8
+"""
+
+from __future__ import annotations
+
+import argparse
+from collections import defaultdict, deque
+from itertools import combinations
+from typing import Any, Iterable, Iterator, Sequence, cast
+
+from sage.all import Graph, graphs  # type: ignore[attr-defined]  # pylint: disable=no-name-in-module
+from sage.graphs.graph_coloring import all_graph_colorings  # type: ignore[attr-defined]  # pylint: disable=no-name-in-module
+
+
+Edge = tuple[Any, Any]
+Coloring = dict[Edge, int]
+Source = tuple[Any, ...]
+
+
+def vertex_key(v: Any) -> str:
+    return repr(v)
+
+
+def edge_key(u: Any, v: Any) -> Edge:
+    return (u, v) if vertex_key(u) <= vertex_key(v) else (v, u)
+
+
+def is_induced_cycle(g: Graph, vertices: Sequence[Any]) -> bool:
+    if len(vertices) < 3:
+        return False
+    h = cast(Graph, g.subgraph(list(vertices)))
+    return h.is_connected() and h.num_edges() == len(vertices) and all(
+        h.degree(v) == 2 for v in h.vertices()
+    )
+
+
+def induced_cycle_sources(g: Graph, max_size: int | None = None) -> Iterator[Source]:
+    vertices = sorted(g.vertices(), key=vertex_key)
+    upper = len(vertices) if max_size is None else min(max_size, len(vertices))
+    for k in range(3, upper + 1):
+        for subset in combinations(vertices, k):
+            if is_induced_cycle(g, subset):
+                yield tuple(subset)
+
+
+def level_sources(g: Graph, mode: str, max_cycle_source_size: int | None) -> Iterator[Source]:
+    if mode in ("vertex", "all"):
+        for v in sorted(g.vertices(), key=vertex_key):
+            yield (v,)
+    if mode in ("cycle", "all"):
+        yield from induced_cycle_sources(g, max_cycle_source_size)
+
+
+def distances_from_source(g: Graph, source: Source) -> dict[Any, int]:
+    if len(source) == 1:
+        return dict(g.shortest_path_lengths(source[0]))
+    distances = {v: 0 for v in source}
+    queue: deque[Any] = deque(source)
+    while queue:
+        v = queue.popleft()
+        for w in g.neighbor_iterator(v):
+            if w in distances:
+                continue
+            distances[w] = distances[v] + 1
+            queue.append(w)
+    return distances
+
+
+def embedded_copy(g: Graph) -> Graph:
+    emb = cast(Graph, g.copy())
+    if not emb.is_planar(set_embedding=True):
+        raise ValueError("graph is not planar")
+    return emb
+
+
+def medial_graph(g: Graph) -> Graph:
+    """Build the full medial graph from the embedding rotation at vertices."""
+    emb = embedded_copy(g)
+    rotation = emb.get_embedding()
+    m = Graph()
+    medial_vertices = [edge_key(u, v) for u, v, _ in emb.edge_iterator()]
+    m.add_vertices(medial_vertices)
+    for v, neighbors in rotation.items():
+        if len(neighbors) < 2:
+            continue
+        n = len(neighbors)
+        for i in range(n):
+            e1 = edge_key(v, neighbors[i])
+            e2 = edge_key(v, neighbors[(i + 1) % n])
+            if e1 != e2:
+                m.add_edge(e1, e2)
+    return m
+
+
+def face_vertices(face: Sequence[tuple[Any, Any]]) -> set[Any]:
+    out: set[Any] = set()
+    for u, v in face:
+        out.add(u)
+        out.add(v)
+    return out
+
+
+def face_edges(face: Sequence[tuple[Any, Any]]) -> set[Edge]:
+    return {edge_key(u, v) for u, v in face}
+
+
+def dual_components_by_depth(
+    g: Graph, source: Source
+) -> list[tuple[int, list[int], set[Edge]]]:
+    """Return (depth, face-indices, annular-edge-set) for each depth component."""
+    emb = embedded_copy(g)
+    distances = distances_from_source(emb, source)
+    faces = emb.faces()
+    f_vertices = [face_vertices(face) for face in faces]
+    f_edges = [face_edges(face) for face in faces]
+    depths = [min(distances[v] for v in verts) for verts in f_vertices]
+
+    edge_faces: dict[Edge, list[int]] = defaultdict(list)
+    for idx, edges in enumerate(f_edges):
+        for edge in edges:
+            edge_faces[edge].append(idx)
+
+    dual_adj: dict[int, set[int]] = defaultdict(set)
+    for incident in edge_faces.values():
+        for a in range(len(incident)):
+            for b in range(a + 1, len(incident)):
+                dual_adj[incident[a]].add(incident[b])
+                dual_adj[incident[b]].add(incident[a])
+
+    components = []
+    seen = [False] * len(faces)
+    for start in range(len(faces)):
+        if seen[start]:
+            continue
+        depth = depths[start]
+        comp = [start]
+        seen[start] = True
+        stack = [start]
+        while stack:
+            f = stack.pop()
+            for h in dual_adj[f]:
+                if not seen[h] and depths[h] == depth:
+                    seen[h] = True
+                    comp.append(h)
+                    stack.append(h)
+
+        annular_edges: set[Edge] = set()
+        for f in comp:
+            for u, v in f_edges[f]:
+                if {distances[u], distances[v]} == {depth, depth + 1}:
+                    annular_edges.add(edge_key(u, v))
+        if len(annular_edges) >= 3:
+            components.append((depth, comp, annular_edges))
+    return components
+
+
+def simple_cycle_vertex_sets(g: Graph) -> set[frozenset[Any]]:
+    vertices = sorted(g.vertices(), key=repr)
+    index = {v: i for i, v in enumerate(vertices)}
+    cycles: set[frozenset[Any]] = set()
+
+    def dfs(start: Any, current: Any, path: list[Any], seen: set[Any]) -> None:
+        for nxt in g.neighbor_iterator(current):
+            if nxt == start:
+                if len(path) >= 3:
+                    cycles.add(frozenset(path))
+                continue
+            if nxt in seen or index[nxt] <= index[start]:
+                continue
+            seen.add(nxt)
+            path.append(nxt)
+            dfs(start, nxt, path, seen)
+            path.pop()
+            seen.remove(nxt)
+
+    for start in vertices:
+        dfs(start, start, [start], {start})
+    return cycles
+
+
+def annular_medial_cycles(g: Graph, source: Source) -> list[frozenset[Edge]]:
+    m = medial_graph(g)
+    cycles: list[frozenset[Edge]] = []
+    seen: set[frozenset[Edge]] = set()
+    for _depth, _faces, annular_edges in dual_components_by_depth(g, source):
+        sub = cast(Graph, m.subgraph(list(annular_edges)))
+        for cycle in simple_cycle_vertex_sets(sub):
+            typed = frozenset(cast(Iterable[Edge], cycle))
+            if typed not in seen:
+                seen.add(typed)
+                cycles.append(typed)
+    return cycles
+
+
+def almost_two_coloured(cycle: frozenset[Edge], coloring: Coloring) -> bool:
+    counts = defaultdict(int)
+    for vertex in cycle:
+        counts[coloring[vertex]] += 1
+    return min(counts.get(c, 0) for c in range(3)) <= 1
+
+
+def first_cycle_violation(
+    cycles: Sequence[frozenset[Edge]], coloring: Coloring
+) -> frozenset[Edge] | None:
+    for cycle in cycles:
+        if not almost_two_coloured(cycle, coloring):
+            return cycle
+    return None
+
+
+def color_counts(cycle: frozenset[Edge], coloring: Coloring) -> dict[int, int]:
+    counts = {0: 0, 1: 0, 2: 0}
+    for vertex in cycle:
+        counts[coloring[vertex]] += 1
+    return counts
+
+
+def coloring_witness(
+    m: Graph,
+    cycles: Sequence[frozenset[Edge]],
+    max_colorings: int | None,
+) -> tuple[Coloring | None, int, bool, frozenset[Edge] | None, Coloring | None]:
+    checked = 0
+    last_violation = None
+    for raw in all_graph_colorings(m, 3, vertex_color_dict=True):
+        coloring = cast(Coloring, raw)
+        checked += 1
+        violation = first_cycle_violation(cycles, coloring)
+        if violation is None:
+            return coloring, checked, True, None, None
+        last_violation = violation
+        if max_colorings is not None and checked >= max_colorings:
+            return None, checked, False, last_violation, coloring
+    return None, checked, True, last_violation, coloring
+
+
+def source_label(source: Source) -> str:
+    if len(source) == 1:
+        return f"vertex:{source[0]}"
+    return "cycle:{" + ",".join(map(str, source)) + "}"
+
+
+def graphs_to_check(n: int, max_graphs: int | None):
+    for idx, g in enumerate(graphs.triangulations(n)):
+        if max_graphs is not None and idx >= max_graphs:
+            break
+        yield idx, cast(Graph, g)
+
+
+def run(args: argparse.Namespace) -> None:
+    total_cases = 0
+    skipped_no_cycles = 0
+    witnesses = 0
+    failures = []
+    inconclusive = []
+
+    for n in range(args.n_min, args.n_max + 1):
+        print(f"n={n}")
+        for graph_idx, g in graphs_to_check(n, args.max_graphs_per_n):
+            m = medial_graph(g)
+            sources = list(level_sources(g, args.sources, args.max_cycle_source_size))
+            if args.max_sources_per_graph is not None:
+                sources = sources[: args.max_sources_per_graph]
+            for source in sources:
+                cycles = annular_medial_cycles(g, source)
+                if not cycles:
+                    skipped_no_cycles += 1
+                    continue
+                total_cases += 1
+                witness, checked, exhausted, violation, last_coloring = coloring_witness(
+                    m, cycles, args.max_colorings
+                )
+                if witness is not None:
+                    witnesses += 1
+                    if args.verbose:
+                        print(
+                            f"  graph={graph_idx} source={source_label(source)} "
+                            f"cycles={len(cycles)} witness_after={checked}"
+                        )
+                    continue
+                record = {
+                    "n": n,
+                    "graph_idx": graph_idx,
+                    "graph_edges": sorted(edge_key(u, v) for u, v, _ in g.edge_iterator()),
+                    "source": source_label(source),
+                    "cycles": len(cycles),
+                    "checked": checked,
+                    "exhausted": exhausted,
+                    "violation_size": len(violation) if violation else None,
+                }
+                if args.failure_details and violation is not None and last_coloring is not None:
+                    record["violation_cycle"] = sorted(violation)
+                    record["violation_counts"] = color_counts(violation, last_coloring)
+                    record["violation_coloring"] = {
+                        edge: last_coloring[edge] for edge in sorted(violation)
+                    }
+                if exhausted:
+                    failures.append(record)
+                    print("  FAILURE", record)
+                    if args.stop_on_failure:
+                        print_summary(total_cases, skipped_no_cycles, witnesses, failures, inconclusive)
+                        return
+                else:
+                    inconclusive.append(record)
+                    print("  INCONCLUSIVE", record)
+
+    print_summary(total_cases, skipped_no_cycles, witnesses, failures, inconclusive)
+
+
+def print_summary(
+    total_cases: int,
+    skipped_no_cycles: int,
+    witnesses: int,
+    failures: Sequence[dict],
+    inconclusive: Sequence[dict],
+) -> None:
+    print()
+    print("summary")
+    print(f"  checked source decompositions with annular cycles: {total_cases}")
+    print(f"  skipped source decompositions with no annular cycles: {skipped_no_cycles}")
+    print(f"  witnesses found: {witnesses}")
+    print(f"  failures: {len(failures)}")
+    print(f"  inconclusive: {len(inconclusive)}")
+    if failures:
+        print(f"  first failure: {failures[0]}")
+    if inconclusive:
+        print(f"  first inconclusive: {inconclusive[0]}")
+
+
+def main() -> None:
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--n-min", type=int, default=4)
+    parser.add_argument("--n-max", type=int, default=8)
+    parser.add_argument("--sources", choices=("vertex", "cycle", "all"), default="vertex")
+    parser.add_argument("--max-cycle-source-size", type=int, default=6)
+    parser.add_argument("--max-graphs-per-n", type=int)
+    parser.add_argument("--max-sources-per-graph", type=int)
+    parser.add_argument("--max-colorings", type=int)
+    parser.add_argument("--stop-on-failure", action="store_true")
+    parser.add_argument("--failure-details", action="store_true")
+    parser.add_argument("--verbose", action="store_true")
+    run(parser.parse_args())
+
+
+if __name__ == "__main__":
+    main()
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/medial_annular_cycle_condition_findings.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/medial_annular_cycle_condition_findings.md
new file mode 100644
index 0000000..2136fdc
--- /dev/null
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/medial_annular_cycle_condition_findings.md
@@ -0,0 +1,83 @@
+# Medial Annular Cycle Condition Findings
+
+Question: for an actual plane triangulation and a tire decomposition,
+does there exist a proper vertex 3-coloring of the full medial graph
+such that every medial annular simple cycle uses two colors except at
+at most one vertex?
+
+The experiment is graph-level, not local-pattern-level:
+
+1. Build the full medial graph `M(G)` from the embedding rotation.
+2. Enumerate proper vertex 3-colorings of `M(G)`.
+3. For each level-source tire decomposition, build the annular medial
+   subgraph of each depth-component tire.
+4. Enumerate simple cycles in those annular medial subgraphs.
+5. Search for a proper medial coloring satisfying the almost-two-color
+   condition on every such cycle.
+
+## First Counterexample Found
+
+Command:
+
+```bash
+sage -python papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_medial_annular_cycle_condition.py --n-min 7 --n-max 7 --stop-on-failure --failure-details
+```
+
+Result:
+
+```text
+n=7
+  FAILURE {
+    'n': 7,
+    'graph_idx': 0,
+    'graph_edges': [
+      (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7),
+      (2, 3), (2, 6), (2, 7), (3, 4), (3, 5), (3, 6),
+      (4, 5), (5, 6), (6, 7)
+    ],
+    'source': 'vertex:1',
+    'cycles': 1,
+    'checked': 6,
+    'exhausted': True,
+    'violation_size': 6,
+    'violation_cycle': [
+      (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7)
+    ],
+    'violation_counts': {0: 2, 1: 2, 2: 2},
+    'violation_coloring': {
+      (1, 2): 2, (1, 3): 1, (1, 4): 0,
+      (1, 5): 2, (1, 6): 0, (1, 7): 1
+    }
+  }
+```
+
+Interpretation: for the vertex-source decomposition at source `1`, the
+only annular medial simple cycle is the six medial vertices
+corresponding to the six edges incident to `1`.  Sage enumerated all six
+proper vertex 3-colorings of the full medial graph.  Every one violates
+the almost-two-color condition on that cycle; the displayed coloring has
+color counts `{0: 2, 1: 2, 2: 2}` on the annular cycle.
+
+Thus the conjecture, in the form "for every possible tire decomposition
+there exists a proper medial 3-coloring satisfying the condition on all
+medial annular simple cycles", is false.
+
+## Default Sweep
+
+Command:
+
+```bash
+sage -python papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_medial_annular_cycle_condition.py --n-min 4 --n-max 8
+```
+
+Summary:
+
+```text
+checked source decompositions with annular cycles: 168
+skipped source decompositions with no annular cycles: 0
+witnesses found: 111
+failures: 57
+inconclusive: 0
+```
+
+Failures begin at `n=7` for vertex-source decompositions.
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/medial_tire_restriction_findings.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/medial_tire_restriction_findings.md
new file mode 100644
index 0000000..f1668d8
--- /dev/null
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/medial_tire_restriction_findings.md
@@ -0,0 +1,178 @@
+# Medial Tire Restriction Findings
+
+Question: among possible full medial tire graphs, which examples most
+restrict the inner boundary coloring relative to the outer boundary
+coloring?
+
+## Model
+
+The experiment uses the ambient full-medial tread model.
+
+- Annular medial vertices form a cycle `A_0, ..., A_{n-1}`.
+- Each annular face contributes one boundary-medial vertex `B_i`.
+- `B_i` is adjacent to `A_i` and `A_{i+1}`.
+- `B_i` is tagged `O` or `I` according to whether the corresponding
+  primal boundary edge lies on the outer or inner boundary.
+
+Boundary states are quotient by color permutations only.  Boundary
+order is fixed: rotations and reversals are not identified.
+
+Inner chords of the inner outerplanar graph do not affect this ambient
+full medial tire graph, because they are not incident to annular tread
+faces.
+
+## Metrics
+
+For the quotient transfer relation
+
+```text
+R_T subset outer_state_orbits x inner_state_orbits
+```
+
+the script reports:
+
+- `min_outer_extensions`: minimum number of compatible inner state
+  orbits over realized outer state orbits.
+- `avg_outer_extensions`: average number of compatible inner state
+  orbits over realized outer state orbits.
+- `relation_density_all`: `|R_T|` divided by all possible outer/inner
+  state orbit pairs.
+- `blocked_outer_orbits`: number of outer boundary state orbits that
+  are not compatible with any inner boundary state orbit.
+- `blocked_inner_orbits`: number of inner boundary state orbits that
+  are not compatible with any outer boundary state orbit.
+
+## Exhaustive Sweep Through 10 Faces
+
+Command:
+
+```bash
+python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/medial_tire_restriction_search.py --exhaustive --min-faces 6 --max-faces 10 --limit 3
+```
+
+Result:
+
+```text
+graphs analyzed: 1974
+boundary states quotient by color permutations only; no rotations or reversals
+
+most restrictive by min_outer_extensions
+  types=IIIOOO
+    faces=6 outer=3 inner=3 outer_orbits=5 inner_orbits=5
+    realized_outer=4 realized_inner=4 relation=10
+    blocked_outer=1 (0.200) blocked_inner=1 (0.200)
+    min_outer_extensions=1 avg_outer_extensions=2.500
+    density_all=0.400000 density_realized_outer=0.500000
+    worst_outer_states=((0, 1, 2),)
+  types=IIOIOO
+    faces=6 outer=3 inner=3 outer_orbits=5 inner_orbits=5
+    realized_outer=5 realized_inner=5 relation=9
+    blocked_outer=0 (0.000) blocked_inner=0 (0.000)
+    min_outer_extensions=1 avg_outer_extensions=1.800
+    density_all=0.360000 density_realized_outer=0.360000
+    worst_outer_states=((0, 0, 1), (0, 1, 2))
+  types=IIOOIO
+    faces=6 outer=3 inner=3 outer_orbits=5 inner_orbits=5
+    realized_outer=5 realized_inner=5 relation=9
+    blocked_outer=0 (0.000) blocked_inner=0 (0.000)
+    min_outer_extensions=1 avg_outer_extensions=1.800
+    density_all=0.360000 density_realized_outer=0.360000
+    worst_outer_states=((0, 1, 1), (0, 1, 2))
+
+most restrictive by avg_outer_extensions
+  types=IIOOOO
+    faces=6 outer=4 inner=2 outer_orbits=14 inner_orbits=2
+    realized_outer=8 realized_inner=2 relation=8
+    blocked_outer=6 (0.429) blocked_inner=0 (0.000)
+    min_outer_extensions=1 avg_outer_extensions=1.000
+    density_all=0.285714 density_realized_outer=0.500000
+  types=IOIOOO
+    faces=6 outer=4 inner=2 outer_orbits=14 inner_orbits=2
+    realized_outer=9 realized_inner=2 relation=9
+    blocked_outer=5 (0.357) blocked_inner=0 (0.000)
+    min_outer_extensions=1 avg_outer_extensions=1.000
+    density_all=0.321429 density_realized_outer=0.500000
+  types=IOOIOO
+    faces=6 outer=4 inner=2 outer_orbits=14 inner_orbits=2
+    realized_outer=9 realized_inner=2 relation=9
+    blocked_outer=5 (0.357) blocked_inner=0 (0.000)
+    min_outer_extensions=1 avg_outer_extensions=1.000
+    density_all=0.321429 density_realized_outer=0.500000
+
+most restrictive by relation_density_all
+  types=IIIIIIIIIO
+    faces=10 outer=1 inner=9 outer_orbits=1 inner_orbits=3281
+    realized_outer=1 realized_inner=171 relation=171
+    blocked_outer=0 (0.000) blocked_inner=3110 (0.948)
+    min_outer_extensions=171 avg_outer_extensions=171.000
+    density_all=0.052118 density_realized_outer=0.052118
+```
+
+## Random Sweep Through 14 Faces
+
+Command:
+
+```bash
+python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/medial_tire_restriction_search.py --samples 5000 --max-faces 14 --limit 3
+```
+
+Result:
+
+```text
+graphs analyzed: 3227
+
+most restrictive by min_outer_extensions
+  types=IIIOOO
+    min_outer_extensions=1 avg_outer_extensions=2.500
+    density_all=0.400000 density_realized_outer=0.500000
+  types=IIOIOO
+    min_outer_extensions=1 avg_outer_extensions=1.800
+    density_all=0.360000 density_realized_outer=0.360000
+  types=IIOOIO
+    min_outer_extensions=1 avg_outer_extensions=1.800
+    density_all=0.360000 density_realized_outer=0.360000
+
+most restrictive by avg_outer_extensions
+  types=IIOOOO
+    min_outer_extensions=1 avg_outer_extensions=1.000
+    density_all=0.285714 density_realized_outer=0.500000
+  types=IOIOOO
+    min_outer_extensions=1 avg_outer_extensions=1.000
+    density_all=0.321429 density_realized_outer=0.500000
+  types=IOOIOO
+    min_outer_extensions=1 avg_outer_extensions=1.000
+    density_all=0.321429 density_realized_outer=0.500000
+
+most restrictive by relation_density_all
+  types=IIIOIIIIIIIIII
+    faces=14 outer=1 inner=13 relation=2731
+    min_outer_extensions=2731 avg_outer_extensions=2731.000
+    density_all=0.010278 density_realized_outer=0.010278
+  types=OOOOOOOIOOOOOO
+    faces=14 outer=13 inner=1 relation=2731
+    min_outer_extensions=1 avg_outer_extensions=1.000
+    density_all=0.010278 density_realized_outer=1.000000
+  types=IIIOOIIIIIIIII
+    faces=14 outer=2 inner=12 relation=2048
+    min_outer_extensions=683 avg_outer_extensions=1024.000
+    density_all=0.011561 density_realized_outer=0.011561
+```
+
+## Interpretation
+
+The most restrictive small examples for extension counts appear at six
+annular faces.  Alternating or clustered `O/I` boundary-face patterns
+with a small boundary on one side can force each realized outer state
+to a single inner orbit.
+
+The raw density metric favors highly unbalanced boundaries, such as one
+outer boundary state position and many inner positions.  This may be a
+less useful notion of "worst case" unless boundary sizes are fixed or
+the density is normalized within a fixed `(outer, inner)` size class.
+
+The blocked-state metrics make the asymmetry explicit.  For instance,
+`IIIOOO` blocks one of the five possible outer state orbits, while
+`IIOIOO` and `IIOOIO` block none but still contain outer states with a
+unique compatible inner state orbit.  Highly unbalanced cases can block
+most state orbits on the larger boundary simply because the annular
+cycle has far fewer colorings than the unconstrained ordered boundary.
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/medial_tire_restriction_search.py b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/medial_tire_restriction_search.py
new file mode 100644
index 0000000..c339b51
--- /dev/null
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/medial_tire_restriction_search.py
@@ -0,0 +1,294 @@
+"""Search boundary-state restrictions for possible full medial tire graphs.
+
+Model.
+  A full medial tire graph has an annular medial cycle A_0,...,A_{n-1}.
+  Each annular face contributes one boundary-medial vertex B_i adjacent
+  to A_i and A_{i+1}; B_i is tagged as outer or inner.  This is the
+  ambient tread-face model from the medial tire decomposition paper.
+
+Boundary order is the cyclic order inherited from the annular face
+sequence.  Boundary states are quotient by S_3 color relabeling only:
+no rotations or reversals are identified.
+"""
+
+from __future__ import annotations
+
+import argparse
+import itertools
+import random
+from collections import defaultdict
+from dataclasses import dataclass
+from functools import lru_cache
+
+
+Coloring = tuple[int, ...]
+
+
+def canonical_color_orbit(coloring: Coloring) -> Coloring:
+    """Canonical representative modulo color renaming, preserving order."""
+    mapping: dict[int, int] = {}
+    out = []
+    for color in coloring:
+        if color not in mapping:
+            mapping[color] = len(mapping)
+        out.append(mapping[color])
+    return tuple(out)
+
+
+@lru_cache(maxsize=None)
+def all_state_orbits(length: int) -> frozenset[Coloring]:
+    return frozenset({
+        canonical_color_orbit(tuple(colors))
+        for colors in itertools.product(range(3), repeat=length)
+    })
+
+
+@lru_cache(maxsize=None)
+def cycle_3_colorings(n: int) -> list[Coloring]:
+    colors = [-1] * n
+    out: list[Coloring] = []
+
+    def rec(idx: int) -> None:
+        if idx == n:
+            if colors[-1] != colors[0]:
+                out.append(tuple(colors))
+            return
+        for color in range(3):
+            if idx > 0 and colors[idx - 1] == color:
+                continue
+            if idx == n - 1 and colors[0] == color:
+                continue
+            colors[idx] = color
+            rec(idx + 1)
+            colors[idx] = -1
+
+    rec(0)
+    return out
+
+
+def boundary_coloring_from_annular(
+    annular_coloring: Coloring,
+    face_types: str,
+    boundary_type: str,
+) -> Coloring:
+    values = []
+    n = len(face_types)
+    for idx, face_type in enumerate(face_types):
+        if face_type != boundary_type:
+            continue
+        left = annular_coloring[idx]
+        right = annular_coloring[(idx + 1) % n]
+        values.append(({0, 1, 2} - {left, right}).pop())
+    return tuple(values)
+
+
+@dataclass(frozen=True)
+class RestrictionMetrics:
+    face_types: str
+    n_faces: int
+    n_outer: int
+    n_inner: int
+    outer_orbits: int
+    inner_orbits: int
+    realized_outer_orbits: int
+    realized_inner_orbits: int
+    blocked_outer_orbits: int
+    blocked_inner_orbits: int
+    blocked_outer_fraction: float
+    blocked_inner_fraction: float
+    relation_size: int
+    min_outer_extensions: int
+    avg_outer_extensions: float
+    relation_density_all: float
+    relation_density_realized_outer: float
+    worst_outer_states: tuple[Coloring, ...]
+
+
+@lru_cache(maxsize=None)
+def analyze_face_types(face_types: str) -> RestrictionMetrics:
+    n = len(face_types)
+    n_outer = face_types.count("O")
+    n_inner = face_types.count("I")
+    possible_outer = all_state_orbits(n_outer)
+    possible_inner = all_state_orbits(n_inner)
+
+    relation: set[tuple[Coloring, Coloring]] = set()
+    by_outer: dict[Coloring, set[Coloring]] = defaultdict(set)
+
+    for annular in cycle_3_colorings(n):
+        outer = canonical_color_orbit(
+            boundary_coloring_from_annular(annular, face_types, "O")
+        )
+        inner = canonical_color_orbit(
+            boundary_coloring_from_annular(annular, face_types, "I")
+        )
+        relation.add((outer, inner))
+        by_outer[outer].add(inner)
+
+    realized_outer = set(by_outer)
+    realized_inner = {inner for _, inner in relation}
+    extension_counts = {outer: len(inners) for outer, inners in by_outer.items()}
+    min_extensions = min(extension_counts.values()) if extension_counts else 0
+    worst_outer = tuple(
+        sorted(outer for outer, count in extension_counts.items() if count == min_extensions)
+    )
+    avg_extensions = (
+        sum(extension_counts.values()) / len(extension_counts)
+        if extension_counts
+        else 0.0
+    )
+    all_denominator = len(possible_outer) * len(possible_inner)
+    realized_denominator = len(realized_outer) * len(possible_inner)
+
+    return RestrictionMetrics(
+        face_types=face_types,
+        n_faces=n,
+        n_outer=n_outer,
+        n_inner=n_inner,
+        outer_orbits=len(possible_outer),
+        inner_orbits=len(possible_inner),
+        realized_outer_orbits=len(realized_outer),
+        realized_inner_orbits=len(realized_inner),
+        blocked_outer_orbits=len(possible_outer) - len(realized_outer),
+        blocked_inner_orbits=len(possible_inner) - len(realized_inner),
+        blocked_outer_fraction=(
+            (len(possible_outer) - len(realized_outer)) / len(possible_outer)
+            if possible_outer
+            else 0.0
+        ),
+        blocked_inner_fraction=(
+            (len(possible_inner) - len(realized_inner)) / len(possible_inner)
+            if possible_inner
+            else 0.0
+        ),
+        relation_size=len(relation),
+        min_outer_extensions=min_extensions,
+        avg_outer_extensions=avg_extensions,
+        relation_density_all=len(relation) / all_denominator if all_denominator else 0.0,
+        relation_density_realized_outer=(
+            len(relation) / realized_denominator if realized_denominator else 0.0
+        ),
+        worst_outer_states=worst_outer[:5],
+    )
+
+
+def random_face_types(rng: random.Random, n: int, min_outer: int, min_inner: int) -> str:
+    if min_outer + min_inner > n:
+        raise ValueError("minimum outer/inner counts exceed n")
+    values = ["O"] * min_outer + ["I"] * min_inner
+    values.extend(rng.choice("OI") for _ in range(n - len(values)))
+    rng.shuffle(values)
+    return "".join(values)
+
+
+def all_face_types(n: int, min_outer: int, min_inner: int):
+    for values in itertools.product("OI", repeat=n):
+        face_types = "".join(values)
+        if face_types.count("O") >= min_outer and face_types.count("I") >= min_inner:
+            yield face_types
+
+
+def best_by(metrics: list[RestrictionMetrics], key_name: str, limit: int):
+    return sorted(metrics, key=lambda m: (getattr(m, key_name), m.n_faces, m.face_types))[
+        :limit
+    ]
+
+
+def print_metric(metric: RestrictionMetrics) -> None:
+    print(f"  types={metric.face_types}")
+    print(
+        f"    faces={metric.n_faces} outer={metric.n_outer} inner={metric.n_inner} "
+        f"outer_orbits={metric.outer_orbits} inner_orbits={metric.inner_orbits}"
+    )
+    print(
+        f"    realized_outer={metric.realized_outer_orbits} "
+        f"realized_inner={metric.realized_inner_orbits} relation={metric.relation_size}"
+    )
+    print(
+        f"    blocked_outer={metric.blocked_outer_orbits} "
+        f"({metric.blocked_outer_fraction:.3f}) "
+        f"blocked_inner={metric.blocked_inner_orbits} "
+        f"({metric.blocked_inner_fraction:.3f})"
+    )
+    print(
+        f"    min_outer_extensions={metric.min_outer_extensions} "
+        f"avg_outer_extensions={metric.avg_outer_extensions:.3f}"
+    )
+    print(
+        f"    density_all={metric.relation_density_all:.6f} "
+        f"density_realized_outer={metric.relation_density_realized_outer:.6f}"
+    )
+    print(f"    worst_outer_states={metric.worst_outer_states}")
+
+
+def run(args: argparse.Namespace) -> None:
+    rng = random.Random(args.seed)
+    seen: set[str] = set()
+    metrics: list[RestrictionMetrics] = []
+
+    if args.exhaustive:
+        for n in range(args.min_faces, args.max_faces + 1):
+            for face_types in all_face_types(n, args.min_outer, args.min_inner):
+                metrics.append(analyze_face_types(face_types))
+    else:
+        for _ in range(args.samples):
+            n = rng.randint(args.min_faces, args.max_faces)
+            face_types = random_face_types(rng, n, args.min_outer, args.min_inner)
+            if face_types in seen:
+                continue
+            seen.add(face_types)
+            metrics.append(analyze_face_types(face_types))
+
+    print(f"graphs analyzed: {len(metrics)}")
+    print(
+        "boundary states quotient by color permutations only; "
+        "no rotations or reversals"
+    )
+    print()
+
+    print("most restrictive by min_outer_extensions")
+    for metric in best_by(metrics, "min_outer_extensions", args.limit):
+        print_metric(metric)
+    print()
+
+    print("most restrictive by avg_outer_extensions")
+    for metric in best_by(metrics, "avg_outer_extensions", args.limit):
+        print_metric(metric)
+    print()
+
+    print("most restrictive by relation_density_all")
+    for metric in best_by(metrics, "relation_density_all", args.limit):
+        print_metric(metric)
+    print()
+
+    print("most restrictive by blocked_outer_orbits")
+    for metric in sorted(
+        metrics,
+        key=lambda m: (-m.blocked_outer_orbits, m.n_faces, m.face_types),
+    )[: args.limit]:
+        print_metric(metric)
+    print()
+
+    print("most restrictive by blocked_outer_fraction")
+    for metric in sorted(
+        metrics,
+        key=lambda m: (-m.blocked_outer_fraction, m.n_faces, m.face_types),
+    )[: args.limit]:
+        print_metric(metric)
+
+
+def main() -> None:
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--min-faces", type=int, default=6)
+    parser.add_argument("--max-faces", type=int, default=10)
+    parser.add_argument("--samples", type=int, default=1000)
+    parser.add_argument("--seed", type=int, default=0)
+    parser.add_argument("--min-outer", type=int, default=1)
+    parser.add_argument("--min-inner", type=int, default=1)
+    parser.add_argument("--limit", type=int, default=5)
+    parser.add_argument("--exhaustive", action="store_true")
+    run(parser.parse_args())
+
+
+if __name__ == "__main__":
+    main()
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/paper.aux b/papers/medial_tire_decompositions_of_plane_triangulations/paper.aux
index 3dc986c..511ea74 100644
--- a/papers/medial_tire_decompositions_of_plane_triangulations/paper.aux
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/paper.aux
@@ -15,20 +15,25 @@
 \newlabel{def:medial-restriction-relation}{{3.5}{3}}
 \citation{bauerfeld-nested-tire-decompositions}
 \citation{bauerfeld-nested-tire-decompositions}
+\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Three six-face full medial tire graphs found by the boundary-state restriction search. Black vertices are annular medial vertices; blue vertices are outer boundary medial vertices and red vertices are inner boundary medial vertices. The word below each diagram records the outer/inner type of the six annular faces in cyclic order. Boundary states are identified only up to colour permutation, not by rotation or reflection of the boundary order.}}{4}{}\protected@file@percent }
+\newlabel{fig:medial-restriction-worst-cases}{{1}{4}}
+\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A proper vertex $3$-colouring of the full medial graph of the first seven-vertex counterexample found by the experiment. The medial vertex labelled $ij$ corresponds to the edge $(i,j)$ of the triangulation. For the vertex-source decomposition at source $1$, the highlighted annular medial cycle has colour counts $(2,2,2)$, so it is not coloured with two colours except at at most one vertex.}}{4}{}\protected@file@percent }
+\newlabel{fig:medial-annular-cycle-counterexample}{{2}{4}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{4}{}\protected@file@percent }
 \newlabel{cor:medial-tire-decomposition}{{4.1}{4}}
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-\newlabel{prop:gluing-criterion}{{4.3}{4}}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A medial pigeonhole programme}}{4}{}\protected@file@percent }
+\newlabel{def:compatible-family}{{4.2}{5}}
+\newlabel{prop:gluing-criterion}{{4.3}{5}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A medial pigeonhole programme}}{5}{}\protected@file@percent }
+\newlabel{def:medial-boundary-state}{{5.1}{5}}
+\newlabel{conj:medial-chain-pigeonhole}{{5.2}{5}}
+\newlabel{conj:medial-route-fct}{{5.3}{6}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Kempe-cycle conservation across medial tires}}{6}{}\protected@file@percent }
 \bibcite{bauerfeld-nested-tire-decompositions}{1}
 \bibcite{tait-original}{2}
 \newlabel{tocindent-1}{0pt}
 \newlabel{tocindent0}{12.7778pt}
 \newlabel{tocindent1}{17.77782pt}
-\newlabel{tocindent2}{0pt}
+\newlabel{tocindent2}{29.38873pt}
 \newlabel{tocindent3}{0pt}
-\newlabel{def:medial-boundary-state}{{5.1}{5}}
-\newlabel{conj:medial-chain-pigeonhole}{{5.2}{5}}
-\newlabel{conj:medial-route-fct}{{5.3}{5}}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent }
-\gdef \@abspage@last{5}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent }
+\gdef \@abspage@last{8}
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/paper.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/paper.pdf
index 88f68f0..7528a59 100644
Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/paper.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/paper.pdf differ
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex b/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex
index a89b557..2aff88d 100644
--- a/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex
@@ -234,6 +234,188 @@ same restriction to $A(T)$ are identical, so the restriction map is
 injective.  The stated inequality follows.
 \end{proof}
 
+\begin{figure}[h]
+\centering
+\begin{tikzpicture}[scale=0.82,
+    ann/.style={circle, fill=black, inner sep=1.3pt},
+    outv/.style={circle, draw=blue!70!black, fill=blue!12, inner sep=1.7pt},
+    inv/.style={circle, draw=red!70!black, fill=red!12, inner sep=1.7pt},
+    mededge/.style={black!62, line width=0.45pt},
+    attach/.style={black!45, line width=0.38pt}]
+
+\newcommand{\hexannulus}[1]{
+  \coordinate (#1a0) at (90:1);
+  \coordinate (#1a1) at (30:1);
+  \coordinate (#1a2) at (-30:1);
+  \coordinate (#1a3) at (-90:1);
+  \coordinate (#1a4) at (-150:1);
+  \coordinate (#1a5) at (150:1);
+  \draw[mededge] (#1a0)--(#1a1)--(#1a2)--(#1a3)--(#1a4)--(#1a5)--cycle;
+  \foreach \i in {0,...,5} \node[ann] at (#1a\i) {};
+}
+
+\begin{scope}[shift={(-4.8,0)}]
+  \hexannulus{x}
+  \coordinate (xb0) at (60:0.48);
+  \coordinate (xb1) at (0:0.48);
+  \coordinate (xb2) at (-60:0.48);
+  \coordinate (xb3) at (-120:1.46);
+  \coordinate (xb4) at (180:1.46);
+  \coordinate (xb5) at (120:1.46);
+  \foreach \i/\j in {0/1,1/2,2/3,3/4,4/5} {
+    \draw[attach] (xb\i)--(xa\i);
+    \draw[attach] (xb\i)--(xa\j);
+  }
+  \draw[attach] (xb5)--(xa5);
+  \draw[attach] (xb5)--(xa0);
+  \foreach \i in {0,1,2} \node[inv] at (xb\i) {\scriptsize I};
+  \foreach \i in {3,4,5} \node[outv] at (xb\i) {\scriptsize O};
+  \node at (0,-1.86) {\scriptsize \texttt{IIIOOO}};
+  \node at (0,-2.15) {\scriptsize $\min |R_T(\alpha)|=1$};
+\end{scope}
+
+\begin{scope}
+  \hexannulus{y}
+  \coordinate (yb0) at (60:0.48);
+  \coordinate (yb1) at (0:0.48);
+  \coordinate (yb2) at (-60:1.46);
+  \coordinate (yb3) at (-120:0.48);
+  \coordinate (yb4) at (180:1.46);
+  \coordinate (yb5) at (120:1.46);
+  \foreach \i/\j in {0/1,1/2,2/3,3/4,4/5} {
+    \draw[attach] (yb\i)--(ya\i);
+    \draw[attach] (yb\i)--(ya\j);
+  }
+  \draw[attach] (yb5)--(ya5);
+  \draw[attach] (yb5)--(ya0);
+  \foreach \i in {0,1,3} \node[inv] at (yb\i) {\scriptsize I};
+  \foreach \i in {2,4,5} \node[outv] at (yb\i) {\scriptsize O};
+  \node at (0,-1.86) {\scriptsize \texttt{IIOIOO}};
+  \node at (0,-2.15) {\scriptsize $\min |R_T(\alpha)|=1$};
+\end{scope}
+
+\begin{scope}[shift={(4.8,0)}]
+  \hexannulus{z}
+  \coordinate (zb0) at (60:0.48);
+  \coordinate (zb1) at (0:0.48);
+  \coordinate (zb2) at (-60:1.46);
+  \coordinate (zb3) at (-120:1.46);
+  \coordinate (zb4) at (180:0.48);
+  \coordinate (zb5) at (120:1.46);
+  \foreach \i/\j in {0/1,1/2,2/3,3/4,4/5} {
+    \draw[attach] (zb\i)--(za\i);
+    \draw[attach] (zb\i)--(za\j);
+  }
+  \draw[attach] (zb5)--(za5);
+  \draw[attach] (zb5)--(za0);
+  \foreach \i in {0,1,4} \node[inv] at (zb\i) {\scriptsize I};
+  \foreach \i in {2,3,5} \node[outv] at (zb\i) {\scriptsize O};
+  \node at (0,-1.86) {\scriptsize \texttt{IIOOIO}};
+  \node at (0,-2.15) {\scriptsize $\min |R_T(\alpha)|=1$};
+\end{scope}
+\end{tikzpicture}
+\caption{Three six-face full medial tire graphs found by the boundary-state
+restriction search.  Black vertices are annular medial vertices; blue
+vertices are outer boundary medial vertices and red vertices are inner
+boundary medial vertices.  The word below each diagram records the
+outer/inner type of the six annular faces in cyclic order.  Boundary
+states are identified only up to colour permutation, not by rotation or
+reflection of the boundary order.}
+\label{fig:medial-restriction-worst-cases}
+\end{figure}
+
+\begin{figure}[h]
+\centering
+\begin{tikzpicture}[scale=0.82,
+    mededge/.style={black!38, line width=0.42pt},
+    cycleedge/.style={black, line width=1.25pt},
+    czero/.style={circle, draw=blue!65!black, fill=blue!18, inner sep=1.6pt},
+    cone/.style={circle, draw=red!65!black, fill=red!18, inner sep=1.6pt},
+    ctwo/.style={circle, draw=green!45!black, fill=green!20, inner sep=1.6pt}]
+
+\coordinate (n12) at (90:1.45);
+\coordinate (n13) at (30:1.45);
+\coordinate (n14) at (-30:1.45);
+\coordinate (n15) at (-90:1.45);
+\coordinate (n16) at (-150:1.45);
+\coordinate (n17) at (150:1.45);
+\coordinate (n23) at (1.85,1.55);
+\coordinate (n34) at (2.45,0.00);
+\coordinate (n45) at (1.85,-1.55);
+\coordinate (n56) at (-0.15,-2.45);
+\coordinate (n67) at (-1.85,-1.55);
+\coordinate (n27) at (-1.85,1.55);
+\coordinate (n26) at (-0.30,2.55);
+\coordinate (n36) at (0.30,0.55);
+\coordinate (n35) at (0.45,-0.85);
+
+\draw[mededge] (n12)--(n13);
+\draw[mededge] (n12)--(n17);
+\draw[mededge] (n12)--(n23);
+\draw[mededge] (n12)--(n27);
+\draw[mededge] (n13)--(n14);
+\draw[mededge] (n13)--(n23);
+\draw[mededge] (n13)--(n34);
+\draw[mededge] (n14)--(n15);
+\draw[mededge] (n14)--(n34);
+\draw[mededge] (n14)--(n45);
+\draw[mededge] (n15)--(n16);
+\draw[mededge] (n15)--(n45);
+\draw[mededge] (n15)--(n56);
+\draw[mededge] (n16)--(n17);
+\draw[mededge] (n16)--(n56);
+\draw[mededge] (n16)--(n67);
+\draw[mededge] (n17)--(n27);
+\draw[mededge] (n17)--(n67);
+\draw[mededge] (n23)--(n26);
+\draw[mededge] (n23)--(n36);
+\draw[mededge] (n26)--(n27);
+\draw[mededge] (n26)--(n36);
+\draw[mededge] (n26)--(n67);
+\draw[mededge] (n27)--(n67);
+\draw[mededge] (n34)--(n35);
+\draw[mededge] (n34)--(n45);
+\draw[mededge] (n35)--(n36);
+\draw[mededge] (n35)--(n45);
+\draw[mededge] (n35)--(n56);
+\draw[mededge] (n36)--(n56);
+
+\draw[cycleedge] (n12)--(n13)--(n14)--(n15)--(n16)--(n17)--cycle;
+
+\node[czero] (v12) at (n12) {\scriptsize $12$};
+\node[cone]  (v13) at (n13) {\scriptsize $13$};
+\node[ctwo]  (v14) at (n14) {\scriptsize $14$};
+\node[czero] (v15) at (n15) {\scriptsize $15$};
+\node[ctwo]  (v16) at (n16) {\scriptsize $16$};
+\node[cone]  (v17) at (n17) {\scriptsize $17$};
+\node[ctwo]  (v23) at (n23) {\scriptsize $23$};
+\node[cone]  (v26) at (n26) {\scriptsize $26$};
+\node[ctwo]  (v27) at (n27) {\scriptsize $27$};
+\node[czero] (v34) at (n34) {\scriptsize $34$};
+\node[ctwo]  (v35) at (n35) {\scriptsize $35$};
+\node[czero] (v36) at (n36) {\scriptsize $36$};
+\node[cone]  (v45) at (n45) {\scriptsize $45$};
+\node[cone]  (v56) at (n56) {\scriptsize $56$};
+\node[czero] (v67) at (n67) {\scriptsize $67$};
+
+\node[anchor=west] at (3.0,1.45) {\scriptsize colour $0$};
+\node[czero] at (2.82,1.45) {};
+\node[anchor=west] at (3.0,1.05) {\scriptsize colour $1$};
+\node[cone] at (2.82,1.05) {};
+\node[anchor=west] at (3.0,0.65) {\scriptsize colour $2$};
+\node[ctwo] at (2.82,0.65) {};
+\node[anchor=west, text width=2.35cm] at (2.82,-0.15)
+  {\scriptsize thick cycle: annular medial cycle for source $1$};
+\end{tikzpicture}
+\caption{A proper vertex $3$-colouring of the full medial graph of the
+first seven-vertex counterexample found by the experiment.  The medial
+vertex labelled $ij$ corresponds to the edge $(i,j)$ of the
+triangulation.  For the vertex-source decomposition at source $1$, the
+highlighted annular medial cycle has colour counts $(2,2,2)$, so it is
+not coloured with two colours except at at most one vertex.}
+\label{fig:medial-annular-cycle-counterexample}
+\end{figure}
+
 \begin{definition}[Boundary medial vertices]
 \label{def:boundary-medial-vertices}
 Let $T$ be a tire tread and let $\Gamma_T$ be the corresponding dual
@@ -387,6 +569,221 @@ move the obstruction into finite boundary-state restrictions carried by
 annular medial tire pieces.
 \end{remark}
 
+\subsection{Kempe-cycle conservation across medial tires}
+
+We now record an additional structure carried by proper
+$3$-colourings of medial graphs.  This structure will be useful for
+describing how colourings glue across level cycles.
+
+Let $G$ be a plane triangulation and let $M=M(G)$ be its medial graph.
+Let
+\[
+  \varphi:V(M)\to\{1,2,3\}
+\]
+be a proper $3$-colouring of $M$.  For a two-element colour set
+$P=\{a,b\}\subseteq\{1,2,3\}$, let $M_P$ denote the subgraph of $M$
+induced by the vertices of colours $a$ and $b$.
+
+Since $M$ is $4$-regular and $\varphi$ is proper, every vertex of
+$M_P$ has degree $2$ in $M_P$.  Hence every component of $M_P$ is a
+cycle.  We call these components the $P$-Kempe cycles of $\varphi$.
+
+\begin{lemma}[Kempe cycles are cycles]
+Let $G$ be a plane triangulation, let $M=M(G)$, and let
+$\varphi$ be a proper $3$-colouring of $M$.  For each
+$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$, every component of $M_P$ is a cycle.
+\end{lemma}
+
+\begin{proof}
+Let $v\in V(M_P)$.  In the medial graph $M$, the vertex $v$ has degree
+$4$.  Since $\varphi$ is a proper $3$-colouring, none of the neighbours
+of $v$ has colour $\varphi(v)$.  Thus all four neighbours of $v$ have
+one of the two colours different from $\varphi(v)$.
+
+In the medial graph of a plane triangulation, the neighbours of a
+medial vertex occur in two opposite pairs corresponding to the two
+faces incident with the corresponding edge of $G$.  Around each such
+triangular face, the three medial vertices receive all three colours.
+Consequently, at $v$ there are exactly two neighbours of each colour
+different from $\varphi(v)$.  It follows that, in the subgraph induced
+by any two colours $P$, every vertex has degree $2$.  Hence each
+component of $M_P$ is a cycle.
+\end{proof}
+
+Let $T$ be a medial tire region.  We regard $T$ as an annular transition
+region whose boundary consists of one outer level cycle and finitely
+many inner level cycles:
+\[
+  \partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
+\]
+Here $C_0$ is the outer level cycle of $T$, and the cycles
+$C_1,\ldots,C_m$ are the inner level cycles.  Each inner level cycle
+$C_i$ is also the outer level cycle of the corresponding child region
+in the tire tree.
+
+The following lemma is the basic conservation principle.
+
+\begin{lemma}[Kempe-cycle conservation across level cycles]
+Let $C$ be a level cycle of $M$ separating a parent side from a child
+side.  Let $K$ be a $P$-Kempe cycle for some
+$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$.  Then $K$ cannot enter the child side
+of $C$ without also leaving it.
+
+Equivalently, the incidences of $K$ with $C$ are paired by the
+components of $K$ lying on the child side of $C$, and also paired by the
+components of $K$ lying on the parent side of $C$.
+\end{lemma}
+
+\begin{proof}
+By the preceding lemma, $K$ is a cycle.  The level cycle $C$ separates
+the sphere into two closed regions, which we call the parent side and
+the child side.  Consider the intersection of $K$ with one of these
+regions.  Since $K$ is a cycle, no component of this intersection can
+have exactly one boundary endpoint on $C$.  Each component is either
+closed within the region, or is a path with two boundary endpoints on
+$C$.  Thus every entrance through $C$ is paired with an exit through
+$C$.
+\end{proof}
+
+More generally, let $T$ be a medial tire region with boundary
+\[
+  \partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
+\]
+For a $P$-Kempe cycle $K$, every component of $K\cap T$ is either a
+cycle contained in $T$, or a path with two endpoints on
+$\partial T$.  Thus the $P$-Kempe arcs inside $T$ define a pairing of
+the $P$-coloured boundary incidences of
+\[
+  C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
+\]
+This motivates the following refinement of boundary states.
+
+\begin{definition}[Kempe-enhanced boundary state]
+Let $T$ be a medial tire region with outer level cycle $C_0$ and inner
+level cycles $C_1,\ldots,C_m$.  Let
+\[
+  \mathcal C(T)=C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
+\]
+A \emph{Kempe-enhanced boundary state} on $T$ consists of the following
+data:
+\begin{enumerate}
+  \item a boundary colouring
+  \[
+    \alpha:V(\mathcal C(T))\to\{1,2,3\};
+  \]
+  \item for each colour pair
+  \[
+    P\in\{\{1,2\},\{2,3\},\{3,1\}\},
+  \]
+  a pairing $\pi_P$ of the $P$-coloured boundary incidences of
+  $\mathcal C(T)$ induced by the $P$-Kempe arcs lying inside $T$.
+\end{enumerate}
+We write such a state as
+\[
+  \kappa=(\alpha,\pi_{12},\pi_{23},\pi_{31}).
+\]
+\end{definition}
+
+Given a proper $3$-colouring $\varphi$ of the medial tire graph
+$M(T)$, the restriction of $\varphi$ to the boundary level cycles gives
+the boundary colouring $\alpha$, while the two-colour Kempe arcs inside
+$T$ give the pairings $\pi_{12},\pi_{23},\pi_{31}$.  Thus $\varphi$
+determines a Kempe-enhanced boundary state, denoted
+\[
+  \kappa_T(\varphi).
+\]
+
+\begin{definition}[Kempe-enhanced restriction relation]
+The \emph{Kempe-enhanced restriction relation} of $T$ is
+\[
+  \mathcal K_T
+  =
+  \left\{
+    \kappa_T(\varphi):
+    \varphi \text{ is a proper }3\text{-colouring of } M(T)
+  \right\}.
+\]
+This refines the ordinary boundary-colouring relation by recording not
+only which boundary colourings extend across $T$, but also how the
+two-colour Kempe cycles are routed through the annular tire region.
+\end{definition}
+
+The annular structure of a tire is useful in two distinct ways.  First,
+it gives a bounded transition region between level cycles: the colouring
+of the annular medial cycle controls, and in many cases determines, the
+colouring of the remaining medial tire vertices.  Thus the number of
+possible transition states is bounded in terms of the annular structure,
+rather than the total size of the subtree below the tire.  Second, it
+describes how the outer level cycle and the inner level cycles are
+related by Kempe arcs.  The level cycles are the gluing interfaces, while
+the annular tire is the transition operator between them.
+
+\begin{definition}[Kempe-compatible gluing]
+Let $T$ be a medial tire region and let $U$ be a child region glued to
+$T$ along a common level cycle $C$.  Thus $C$ is an inner level cycle of
+$T$ and the outer level cycle of $U$.
+
+Let
+\[
+  \kappa_T=(\alpha_T,\pi^T_{12},\pi^T_{23},\pi^T_{31})
+  \in \mathcal K_T
+\]
+and
+\[
+  \kappa_U=(\alpha_U,\pi^U_{12},\pi^U_{23},\pi^U_{31})
+  \in \mathcal K_U.
+\]
+We say that $\kappa_T$ and $\kappa_U$ are \emph{Kempe-compatible along
+$C$} if:
+\begin{enumerate}
+  \item the boundary colourings agree on $C$:
+  \[
+    \alpha_T|_{V(C)}=\alpha_U|_{V(C)};
+  \]
+  \item for each colour pair
+  \[
+    P\in\{\{1,2\},\{2,3\},\{3,1\}\},
+  \]
+  the pairings $\pi^T_P$ and $\pi^U_P$ compose along the
+  $P$-coloured incidences of $C$ without producing an unpaired endpoint.
+\end{enumerate}
+When these conditions hold, the composed pairings determine a
+Kempe-enhanced boundary state on the exposed boundary of
+$T\cup_C U$.
+\end{definition}
+
+In these terms, gluing local colourings is not merely a matter of
+matching boundary colours.  The colourings must also route their
+two-colour Kempe arcs compatibly across every shared level cycle.  The
+ordinary restriction relation records whether a boundary colouring can
+be extended locally; the Kempe-enhanced relation additionally records
+the conservation of Kempe-cycle flow through the annular transition
+region.
+
+For a tire with one outer level cycle and several inner level cycles,
+\[
+  \partial T=C_0\sqcup C_1\sqcup\cdots\sqcup C_m,
+\]
+the parent tire may correlate the boundary states on the different
+inner cycles.  The Kempe-enhanced relation records this correlation as
+a system of pairings among the $P$-coloured incidences of all boundary
+level cycles simultaneously.  Thus one should view a medial tire as a
+multi-output transition operator
+\[
+  \mathcal K_T:
+  C_0 \leadsto (C_1,\ldots,C_m),
+\]
+rather than as an independent collection of binary transitions.
+
+The guiding principle is therefore:
+
+\begin{quote}
+Level cycles are the interfaces used for gluing, while annular tire
+regions are the bounded transition regions that route Kempe cycles
+between those interfaces.
+\end{quote}
+
+
 \begin{thebibliography}{9}
 
 \bibitem{bauerfeld-nested-tire-decompositions}