diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences.py b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences.py
new file mode 100644
index 0000000..81abca2
--- /dev/null
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences.py
@@ -0,0 +1,384 @@
+"""Singleton-down-apex colour sequences of Kempe-balanced 3-colourings.
+
+Inner-face counterpart of ``kempe_up_tooth_sequences.py``.  There the role of
+the distinguished valid face was the unique *outer* face, which carries every
+up-tooth apex; here we play the same game on an *inner* non-tooth face of B(T)
+(the root face, or a bite's inner-gap face), which carries the singleton
+down-tooth apexes assigned to it.
+
+For a fixed annular size ``n`` and a fixed count ``m`` we:
+
+  1. take every full medial tire graph M(T) with |A(T)| = n (one representative
+     per dihedral symmetry class) that has an inner non-tooth face F holding
+     exactly ``m`` singleton down-tooth apexes -- by Remark 3.8 every such face
+     holds 0 or >= 3, so m >= 3.  (At n = 9 each M(T) has at most one inner face
+     bearing singletons, so a graph and its face coincide one-to-one.)
+  2. enumerate the Kempe-balanced (``valid``) proper 3-colourings of M(T) and
+     read off the colour sequence of F's singleton down-tooth apexes d_i in
+     increasing annular-edge order (their cyclic order along F's arc);
+  3. reduce each sequence modulo the six colour permutations -- NOT modulo the
+     dihedral symmetry of the cycle -- to a canonical sequence;
+  4. group the configurations by the *set* of canonical down-apex sequences
+     they realise, and report how many share each set.
+
+The Kempe-balanced rule is global, so the same valid colourings are used as in
+the up-tooth experiment; only the apex set we read changes.  Note that the rule
+forces, on every valid face, each colour pair to meet the counted apexes an even
+number of times -- so the down-apex sequences obey the same equal-parity law as
+the up-tooth sequences.
+
+Run:  python3 kempe_down_face_sequences.py --n 9 --m 3
+"""
+
+from __future__ import annotations
+
+import argparse
+import math
+import os
+from collections import defaultdict
+
+from full_medial_tire_generator import (
+    FullMedialTireGraph,
+    generate,
+    innermost_bite,
+)
+from kempe_valid_colorings import classify_colorings
+from kempe_up_tooth_sequences import (
+    PALETTE,
+    PALETTE_NAME,
+    _parity_partitions,
+    _positions,
+    canonical_sequence,
+    compact_coloring,
+    seq_str,
+)
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+Coloring = dict[str, int]
+
+Bite = tuple[int, int]
+FaceKey = Bite | None  # None = root face
+
+
+def face_name(face: FaceKey) -> str:
+    return "root" if face is None else f"bite({face[0]},{face[1]})"
+
+
+def inner_face_singletons(graph: FullMedialTireGraph) -> dict[FaceKey, list[int]]:
+    """Map each inner non-tooth face to its singleton down-tooth edges (sorted)."""
+    perface: dict[FaceKey, list[int]] = defaultdict(list)
+    for e in graph.singleton_down_edges:
+        perface[innermost_bite(e, graph.bites)].append(e)
+    return {face: sorted(edges) for face, edges in perface.items()}
+
+
+def config_id(idx: int) -> str:
+    return f"C{idx:02d}"
+
+
+def describe_config(graph: FullMedialTireGraph, face: FaceKey, edges) -> str:
+    bites = ",".join(f"({i},{j})" for i, j in sorted(graph.bites)) or "-"
+    apexes = ",".join(f"d{e}" for e in edges)
+    return (f"word={graph.tooth_word} bites={bites} "
+            f"face={face_name(face)} apexes=[{apexes}]")
+
+
+# ---------------------------------------------------------------------------
+# Data collection.
+# ---------------------------------------------------------------------------
+
+class Config:
+    """A single (M(T), inner face) specimen with m singleton down apexes."""
+
+    def __init__(self, graph: FullMedialTireGraph, face: FaceKey, edges: list[int]):
+        self.graph = graph
+        self.face = face
+        self.edges = edges                       # singleton down edges on the face
+        self.apexes = [f"d{e}" for e in edges]   # their apex vertex names
+
+    def sequence(self, coloring: Coloring) -> tuple[int, ...]:
+        return tuple(coloring[v] for v in self.apexes)
+
+
+class Experiment:
+    def __init__(self, n: int, m: int):
+        self.n = n
+        self.m = m
+        self.configs: list[Config] = []
+        for g in generate(n, min_up_teeth=3, dedup=True):
+            for face, edges in inner_face_singletons(g).items():
+                if len(edges) == m:
+                    self.configs.append(Config(g, face, edges))
+
+        # per config: list of (coloring, canonical down-apex sequence)
+        self.colorings: list[list[tuple[Coloring, tuple[int, ...]]]] = []
+        # per config: set of canonical sequences it realises
+        self.config_seq_sets: list[frozenset[tuple[int, ...]]] = []
+        # canonical sequence -> list of (config_idx, coloring)
+        self.by_sequence: dict[tuple[int, ...], list[tuple[int, Coloring]]] = defaultdict(list)
+
+        for cidx, cfg in enumerate(self.configs):
+            entries: list[tuple[Coloring, tuple[int, ...]]] = []
+            seqs: set[tuple[int, ...]] = set()
+            for coloring, verdict in classify_colorings(cfg.graph, dedup_colors=True):
+                if not verdict.valid:
+                    continue
+                cseq = canonical_sequence(cfg.sequence(coloring))
+                entries.append((coloring, cseq))
+                seqs.add(cseq)
+                self.by_sequence[cseq].append((cidx, coloring))
+            self.colorings.append(entries)
+            self.config_seq_sets.append(frozenset(seqs))
+
+    def groups(self):
+        groups: dict[frozenset[tuple[int, ...]], list[int]] = defaultdict(list)
+        for cidx, sset in enumerate(self.config_seq_sets):
+            groups[sset].append(cidx)
+        return sorted(groups.items(), key=lambda kv: (-len(kv[1]), len(kv[0])))
+
+    def sequences(self) -> list[tuple[int, ...]]:
+        return sorted(self.by_sequence)
+
+
+# ---------------------------------------------------------------------------
+# Drawing (rings the singleton down apexes of the chosen inner face).
+# ---------------------------------------------------------------------------
+
+def _draw(ax, cfg: Config, coloring, title):
+    graph = cfg.graph
+    pos = _positions(graph)
+    for u, v in graph.edges():
+        ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]],
+                color="#bbbbbb", lw=0.5, zorder=1)
+    for k in range(graph.n):
+        a, b = f"a{k}", f"a{(k + 1) % graph.n}"
+        ax.plot([pos[a][0], pos[b][0]], [pos[a][1], pos[b][1]],
+                color="#666666", lw=1.0, zorder=2)
+    for v, (x, y) in pos.items():
+        is_bite = v.startswith("p")
+        ax.scatter([x], [y], s=34 if is_bite else 24, color=PALETTE[coloring[v]],
+                   edgecolors="black", linewidths=0.5 if is_bite else 0.3, zorder=3)
+    # ring the singleton down apexes of this inner face
+    dx = [pos[v][0] for v in cfg.apexes]
+    dy = [pos[v][1] for v in cfg.apexes]
+    ax.scatter(dx, dy, s=120, facecolors="none", edgecolors="#222222",
+               linewidths=1.4, zorder=4)
+    ax.set_xlim(-1.65, 1.65)
+    ax.set_ylim(-1.85, 1.65)
+    ax.set_aspect("equal")
+    ax.axis("off")
+    ax.set_title(title, fontsize=6, pad=1.5)
+
+
+def draw_sequence(exp: Experiment, seq, out_png, out_pdf):
+    import matplotlib
+    matplotlib.use("Agg")
+    import matplotlib.pyplot as plt
+
+    entries = exp.by_sequence[seq]
+    cols = 10
+    rows = math.ceil(len(entries) / cols)
+    fig, axes = plt.subplots(rows, cols, figsize=(cols * 1.5, rows * 1.7), squeeze=False)
+    for idx in range(rows * cols):
+        ax = axes[idx // cols][idx % cols]
+        if idx < len(entries):
+            cidx, coloring = entries[idx]
+            cfg = exp.configs[cidx]
+            dseq = seq_str(cfg.sequence(coloring))
+            _draw(ax, cfg, coloring, f"{config_id(cidx)}  d={dseq}")
+        else:
+            ax.axis("off")
+    fig.suptitle(
+        f"Kempe-balanced colourings with inner-face singleton-down-apex "
+        f"sequence {seq_str(seq)} (mod colour permutation)\n"
+        f"n={exp.n}, m={exp.m} down apexes on the face — {len(entries)} "
+        f"colourings on {len({c for c, _ in entries})} configs; "
+        f"black rings mark the face's down apexes",
+        fontsize=11, y=0.998,
+    )
+    fig.tight_layout(rect=(0, 0, 1, 0.96))
+    fig.savefig(out_png, dpi=170)
+    fig.savefig(out_pdf)
+    plt.close(fig)
+    print(f"wrote {out_png}")
+
+
+# ---------------------------------------------------------------------------
+# Markdown notes.
+# ---------------------------------------------------------------------------
+
+def write_sequence_note(exp: Experiment, seq, path, fig_name):
+    s = seq_str(seq)
+    by_config: dict[int, list[Coloring]] = defaultdict(list)
+    for cidx, coloring in exp.by_sequence[seq]:
+        by_config[cidx].append(coloring)
+
+    cm: dict[int, int] = {}
+    for c in seq:
+        cm[c] = cm.get(c, 0) + 1
+    counts = ", ".join(f"{v}×colour{k}" for k, v in sorted(cm.items()))
+
+    lines = []
+    lines.append(f"# Inner-face down-apex sequence `{s}`")
+    lines.append("")
+    lines.append(
+        f"Canonical colour sequence of the singleton down-tooth apexes on a "
+        f"single inner non-tooth face (read in increasing annular-edge order, "
+        f"reduced modulo the six colour permutations) for Kempe-balanced "
+        f"3-colourings of M(T) with **n = {exp.n}**, **m = {exp.m} singleton "
+        f"down apexes on the face**."
+    )
+    lines.append("")
+    lines.append(f"- Colour multiset: {counts}.")
+    lines.append(f"- Realised by **{len(by_config)}** of {len(exp.configs)} "
+                 f"configs (M(T), inner face).")
+    lines.append(f"- **{len(exp.by_sequence[seq])}** Kempe-balanced colourings "
+                 f"(mod colour permutation) produce it.")
+    lines.append(f"- Figure: `{fig_name}` (black rings mark the face's down apexes).")
+    lines.append("")
+    lines.append("Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` "
+                 "up-tooth apexes; `D[...]` singleton down apexes `d` and bite "
+                 "apexes `p`. Colours 0/1/2 = "
+                 + ", ".join(f"{c}:{PALETTE_NAME[c]}" for c in (0, 1, 2)) + ".")
+    lines.append("")
+    for cidx in sorted(by_config):
+        cfg = exp.configs[cidx]
+        cols = by_config[cidx]
+        lines.append(f"## {config_id(cidx)} — {describe_config(cfg.graph, cfg.face, cfg.edges)}")
+        lines.append("")
+        lines.append(f"{len(cols)} colouring(s) with down-apex sequence `{s}`:")
+        lines.append("")
+        for coloring in cols:
+            raw = seq_str(cfg.sequence(coloring))
+            lines.append(f"- face apexes (raw labels) `{raw}` → canonical `{s}`  ·  "
+                         f"`{compact_coloring(cfg.graph, coloring)}`")
+        lines.append("")
+    with open(path, "w") as fh:
+        fh.write("\n".join(lines) + "\n")
+    print(f"wrote {path}")
+
+
+def write_summary(exp: Experiment, path):
+    lines = []
+    lines.append(f"# Inner-face singleton-down-apex sequences of Kempe-balanced "
+                 f"colourings (n={exp.n}, m={exp.m})")
+    lines.append("")
+    lines.append(
+        f"Every full medial tire graph M(T) with |A(T)| = {exp.n} (one "
+        f"representative per dihedral class) that has an inner non-tooth face "
+        f"holding exactly {exp.m} singleton down-tooth apexes: "
+        f"**{len(exp.configs)} configs (M(T), inner face)**. For each we "
+        f"enumerate the Kempe-balanced (valid) proper 3-colourings (modulo "
+        f"colour permutation), read the down-apex colour sequence in increasing "
+        f"annular-edge order, and reduce it modulo colour permutation (NOT "
+        f"dihedral symmetry)."
+    )
+    lines.append("")
+    total = sum(len(c) for c in exp.colorings)
+    lines.append(f"- Total Kempe-balanced colourings (mod colour permutation): "
+                 f"**{total}**.")
+    lines.append(f"- Distinct canonical down-apex sequences overall: "
+                 f"**{len(exp.by_sequence)}**.")
+    lines.append("")
+    lines.append("## Distinct canonical down-apex sequences")
+    lines.append("")
+    lines.append("| sequence | colour multiset | #configs realising | #colourings |")
+    lines.append("|---|---|---|---|")
+    for seq in exp.sequences():
+        cm: dict[int, int] = {}
+        for c in seq:
+            cm[c] = cm.get(c, 0) + 1
+        cms = "+".join(str(v) for v in sorted(cm.values(), reverse=True))
+        ncfg = len({c for c, _ in exp.by_sequence[seq]})
+        lines.append(f"| `{seq_str(seq)}` | {cms} | {ncfg} | "
+                     f"{len(exp.by_sequence[seq])} |")
+    lines.append("")
+    parity = "even" if exp.m % 2 == 0 else "odd"
+    allowed = sorted(
+        {"+".join(str(v) for v in sorted(p, reverse=True))
+         for p in _parity_partitions(exp.m)}
+    )
+    lines.append("Note: every realised sequence has its three colour-counts of "
+                 "**equal parity** — exactly the Kempe-parity constraint on the "
+                 "inner face (each colour pair meets its singleton down apexes an "
+                 f"even number of times). With m = {exp.m} apexes (m is "
+                 f"{'even' if exp.m % 2 == 0 else 'odd'}) every count must be "
+                 f"**{parity}**, so the only admissible colour multisets are "
+                 + ", ".join(allowed) + ".")
+    lines.append("")
+    lines.append("## Step 4 — grouping configs by their set of unique down-apex "
+                 "sequences")
+    lines.append("")
+    groups = exp.groups()
+    lines.append(f"The {len(exp.configs)} configs fall into **{len(groups)}** "
+                 f"groups by the set of canonical down-apex sequences they "
+                 f"realise:")
+    lines.append("")
+    lines.append("| #configs | set of down-apex sequences | config ids |")
+    lines.append("|---|---|---|")
+    for sset, cidxs in groups:
+        seqs = ", ".join(f"`{seq_str(s)}`" for s in sorted(sset))
+        ids = ", ".join(config_id(i) for i in cidxs)
+        lines.append(f"| {len(cidxs)} | {{ {seqs} }} | {ids} |")
+    lines.append("")
+    lines.append("## Config atlas (ids)")
+    lines.append("")
+    lines.append("| id | word / bites / face / apexes | #Kempe-balanced | "
+                 "down-apex sequence set |")
+    lines.append("|---|---|---|---|")
+    for cidx, cfg in enumerate(exp.configs):
+        sset = exp.config_seq_sets[cidx]
+        seqs = ", ".join(f"`{seq_str(s)}`" for s in sorted(sset))
+        lines.append(f"| {config_id(cidx)} | "
+                     f"{describe_config(cfg.graph, cfg.face, cfg.edges)} | "
+                     f"{len(exp.colorings[cidx])} | {{ {seqs} }} |")
+    lines.append("")
+    lines.append("## Per-sequence notes")
+    lines.append("")
+    for seq in exp.sequences():
+        lines.append(f"- [`{seq_str(seq)}`](seq_{seq_str(seq)}.md) — "
+                     f"figure `seq_{seq_str(seq)}.png`")
+    lines.append("")
+    with open(path, "w") as fh:
+        fh.write("\n".join(lines) + "\n")
+    print(f"wrote {path}")
+
+
+# ---------------------------------------------------------------------------
+# Driver.
+# ---------------------------------------------------------------------------
+
+def run(args):
+    exp = Experiment(args.n, args.m)
+    print(f"n={args.n}, m={args.m}: {len(exp.configs)} configs (M(T), inner face)")
+    print(f"distinct canonical down-apex sequences: {len(exp.by_sequence)}")
+    for seq in exp.sequences():
+        nc = len({c for c, _ in exp.by_sequence[seq]})
+        print(f"  {seq_str(seq)}: {nc} configs, {len(exp.by_sequence[seq])} colourings")
+    print(f"groups by sequence-set: {len(exp.groups())}")
+
+    notes_dir = os.path.join(HERE, f"kempe_down_face_sequences_n{args.n}_m{args.m}")
+    os.makedirs(notes_dir, exist_ok=True)
+
+    write_summary(exp, os.path.join(notes_dir, "summary.md"))
+    for seq in exp.sequences():
+        s = seq_str(seq)
+        fig_name = f"seq_{s}.png"
+        write_sequence_note(exp, seq, os.path.join(notes_dir, f"seq_{s}.md"), fig_name)
+        if not args.no_figures:
+            draw_sequence(exp, seq,
+                          os.path.join(notes_dir, fig_name),
+                          os.path.join(notes_dir, f"seq_{s}.pdf"))
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--n", type=int, default=9)
+    parser.add_argument("--m", type=int, default=3,
+                        help="singleton down apexes on the inner face (>=3)")
+    parser.add_argument("--no-figures", action="store_true")
+    run(parser.parse_args())
+
+
+if __name__ == "__main__":
+    main()
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.md
new file mode 100644
index 0000000..d019b39
--- /dev/null
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.md
@@ -0,0 +1,382 @@
+# Inner-face down-apex sequence `012`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 3 singleton down apexes on the face**.
+
+- Colour multiset: 1×colour0, 1×colour1, 1×colour2.
+- Realised by **26** of 26 configs (M(T), inner face).
+- **241** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_012.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUUUUDDD bites=- face=root apexes=[d6,d7,d8]
+
+22 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101012  U[u0:2 u1:2 u2:2 u3:2 u4:2 u5:2]  D[d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010101021  U[u0:2 u1:2 u2:2 u3:2 u4:2 u5:2]  D[d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010102012  U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1]  D[d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010102021  U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1]  D[d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010121012  U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2]  D[d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010121021  U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2]  D[d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010201012  U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2]  D[d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010201021  U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2]  D[d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010202012  U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1]  D[d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202021  U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1]  D[d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010212012  U[u0:2 u1:2 u2:1 u3:0 u4:0 u5:1]  D[d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010212021  U[u0:2 u1:2 u2:1 u3:0 u4:0 u5:1]  D[d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012012012  U[u0:2 u1:0 u2:1 u3:2 u4:0 u5:1]  D[d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012012021  U[u0:2 u1:0 u2:1 u3:2 u4:0 u5:1]  D[d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012021012  U[u0:2 u1:0 u2:1 u3:1 u4:0 u5:2]  D[d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012021021  U[u0:2 u1:0 u2:1 u3:1 u4:0 u5:2]  D[d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012101012  U[u0:2 u1:0 u2:0 u3:2 u4:2 u5:2]  D[d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012101021  U[u0:2 u1:0 u2:0 u3:2 u4:2 u5:2]  D[d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012102012  U[u0:2 u1:0 u2:0 u3:2 u4:1 u5:1]  D[d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012102021  U[u0:2 u1:0 u2:0 u3:2 u4:1 u5:1]  D[d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012121012  U[u0:2 u1:0 u2:0 u3:0 u4:0 u5:2]  D[d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012121021  U[u0:2 u1:0 u2:0 u3:0 u4:0 u5:2]  D[d6:1 d7:0 d8:2]`
+
+## C01 — word=UUUUUDUDD bites=- face=root apexes=[d5,d7,d8]
+
+11 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101012  U[u0:2 u1:2 u2:2 u3:2 u4:2 u6:2]  D[d5:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010102021  U[u0:2 u1:2 u2:2 u3:2 u4:1 u6:1]  D[d5:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010121012  U[u0:2 u1:2 u2:2 u3:0 u4:0 u6:2]  D[d5:2 d7:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010201012  U[u0:2 u1:2 u2:1 u3:1 u4:2 u6:2]  D[d5:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202021  U[u0:2 u1:2 u2:1 u3:1 u4:1 u6:1]  D[d5:1 d7:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010212021  U[u0:2 u1:2 u2:1 u3:0 u4:0 u6:1]  D[d5:1 d7:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012012021  U[u0:2 u1:0 u2:1 u3:2 u4:0 u6:1]  D[d5:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012021012  U[u0:2 u1:0 u2:1 u3:1 u4:0 u6:2]  D[d5:2 d7:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012101012  U[u0:2 u1:0 u2:0 u3:2 u4:2 u6:2]  D[d5:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012102021  U[u0:2 u1:0 u2:0 u3:2 u4:1 u6:1]  D[d5:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012121012  U[u0:2 u1:0 u2:0 u3:0 u4:0 u6:2]  D[d5:2 d7:0 d8:1]`
+
+## C02 — word=UUUUDUUDD bites=- face=root apexes=[d4,d7,d8]
+
+17 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101012  U[u0:2 u1:2 u2:2 u3:2 u5:2 u6:2]  D[d4:2 d7:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101212  U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0]  D[d4:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010102021  U[u0:2 u1:2 u2:2 u3:2 u5:1 u6:1]  D[d4:1 d7:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010102121  U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0]  D[d4:1 d7:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010120121  U[u0:2 u1:2 u2:2 u3:0 u5:2 u6:0]  D[d4:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010201012  U[u0:2 u1:2 u2:1 u3:1 u5:2 u6:2]  D[d4:2 d7:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010201212  U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0]  D[d4:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202021  U[u0:2 u1:2 u2:1 u3:1 u5:1 u6:1]  D[d4:1 d7:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202121  U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0]  D[d4:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010210212  U[u0:2 u1:2 u2:1 u3:0 u5:1 u6:0]  D[d4:2 d7:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012010212  U[u0:2 u1:0 u2:1 u3:2 u5:1 u6:0]  D[d4:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012020121  U[u0:2 u1:0 u2:1 u3:1 u5:2 u6:0]  D[d4:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012101012  U[u0:2 u1:0 u2:0 u3:2 u5:2 u6:2]  D[d4:2 d7:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012101212  U[u0:2 u1:0 u2:0 u3:2 u5:0 u6:0]  D[d4:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012102021  U[u0:2 u1:0 u2:0 u3:2 u5:1 u6:1]  D[d4:1 d7:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012102121  U[u0:2 u1:0 u2:0 u3:2 u5:0 u6:0]  D[d4:1 d7:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012120121  U[u0:2 u1:0 u2:0 u3:0 u5:2 u6:0]  D[d4:1 d7:0 d8:2]`
+
+## C03 — word=UUUUDUDUD bites=- face=root apexes=[d4,d6,d8]
+
+26 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101212  U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0]  D[d4:2 d6:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010102121  U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0]  D[d4:1 d6:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010120121  U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0]  D[d4:1 d6:0 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010121012  U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0]  D[d4:0 d6:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010121021  U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0]  D[d4:0 d6:1 d8:2]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010121201  U[u0:2 u1:2 u2:2 u3:0 u5:0 u7:2]  D[d4:0 d6:1 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010201212  U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0]  D[d4:2 d6:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202121  U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0]  D[d4:1 d6:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010210212  U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0]  D[d4:2 d6:0 d8:1]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010212012  U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0]  D[d4:0 d6:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010212021  U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0]  D[d4:0 d6:1 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010212102  U[u0:2 u1:2 u2:1 u3:0 u5:0 u7:1]  D[d4:0 d6:2 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012010212  U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0]  D[d4:2 d6:0 d8:1]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012012012  U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0]  D[d4:0 d6:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012012021  U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0]  D[d4:0 d6:1 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012012102  U[u0:2 u1:0 u2:1 u3:2 u5:0 u7:1]  D[d4:0 d6:2 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012020121  U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0]  D[d4:1 d6:0 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012021012  U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0]  D[d4:0 d6:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012021021  U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0]  D[d4:0 d6:1 d8:2]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012021201  U[u0:2 u1:0 u2:1 u3:1 u5:0 u7:2]  D[d4:0 d6:1 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012101212  U[u0:2 u1:0 u2:0 u3:2 u5:0 u7:0]  D[d4:2 d6:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012102121  U[u0:2 u1:0 u2:0 u3:2 u5:0 u7:0]  D[d4:1 d6:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012120121  U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0]  D[d4:1 d6:0 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012121012  U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0]  D[d4:0 d6:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012121021  U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0]  D[d4:0 d6:1 d8:2]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012121201  U[u0:2 u1:0 u2:0 u3:0 u5:0 u7:2]  D[d4:0 d6:1 d8:2]`
+
+## C04 — word=UUUUDDDDD bites=(4,8) face=bite(4,8) apexes=[d5,d6,d7]
+
+12 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=010101021  U[u0:2 u1:2 u2:2 u3:2]  D[d5:2 d6:1 d7:0 p4_8:2]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010101201  U[u0:2 u1:2 u2:2 u3:2]  D[d5:0 d6:1 d7:2 p4_8:2]`
+- face apexes (raw labels) `120` → canonical `012`  ·  `A=010102012  U[u0:2 u1:2 u2:2 u3:2]  D[d5:1 d6:2 d7:0 p4_8:1]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010102102  U[u0:2 u1:2 u2:2 u3:2]  D[d5:0 d6:2 d7:1 p4_8:1]`
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=010201021  U[u0:2 u1:2 u2:1 u3:1]  D[d5:2 d6:1 d7:0 p4_8:2]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010201201  U[u0:2 u1:2 u2:1 u3:1]  D[d5:0 d6:1 d7:2 p4_8:2]`
+- face apexes (raw labels) `120` → canonical `012`  ·  `A=010202012  U[u0:2 u1:2 u2:1 u3:1]  D[d5:1 d6:2 d7:0 p4_8:1]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010202102  U[u0:2 u1:2 u2:1 u3:1]  D[d5:0 d6:2 d7:1 p4_8:1]`
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=012101021  U[u0:2 u1:0 u2:0 u3:2]  D[d5:2 d6:1 d7:0 p4_8:2]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012101201  U[u0:2 u1:0 u2:0 u3:2]  D[d5:0 d6:1 d7:2 p4_8:2]`
+- face apexes (raw labels) `120` → canonical `012`  ·  `A=012102012  U[u0:2 u1:0 u2:0 u3:2]  D[d5:1 d6:2 d7:0 p4_8:1]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012102102  U[u0:2 u1:0 u2:0 u3:2]  D[d5:0 d6:2 d7:1 p4_8:1]`
+
+## C05 — word=UUUDUUUDD bites=- face=root apexes=[d3,d7,d8]
+
+13 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101012  U[u0:2 u1:2 u2:2 u4:2 u5:2 u6:2]  D[d3:2 d7:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101212  U[u0:2 u1:2 u2:2 u4:2 u5:0 u6:0]  D[d3:2 d7:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010102012  U[u0:2 u1:2 u2:2 u4:1 u5:1 u6:2]  D[d3:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010201021  U[u0:2 u1:2 u2:1 u4:2 u5:2 u6:1]  D[d3:1 d7:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202021  U[u0:2 u1:2 u2:1 u4:1 u5:1 u6:1]  D[d3:1 d7:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202121  U[u0:2 u1:2 u2:1 u4:1 u5:0 u6:0]  D[d3:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012010212  U[u0:2 u1:0 u2:1 u4:2 u5:1 u6:0]  D[d3:2 d7:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012012012  U[u0:2 u1:0 u2:1 u4:0 u5:1 u6:2]  D[d3:2 d7:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012020121  U[u0:2 u1:0 u2:1 u4:1 u5:2 u6:0]  D[d3:1 d7:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012021021  U[u0:2 u1:0 u2:1 u4:0 u5:2 u6:1]  D[d3:1 d7:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012101012  U[u0:2 u1:0 u2:0 u4:2 u5:2 u6:2]  D[d3:2 d7:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012101212  U[u0:2 u1:0 u2:0 u4:2 u5:0 u6:0]  D[d3:2 d7:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012102012  U[u0:2 u1:0 u2:0 u4:1 u5:1 u6:2]  D[d3:2 d7:0 d8:1]`
+
+## C06 — word=UUUDUUDUD bites=- face=root apexes=[d3,d6,d8]
+
+20 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101212  U[u0:2 u1:2 u2:2 u4:2 u5:0 u7:0]  D[d3:2 d6:0 d8:1]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010120102  U[u0:2 u1:2 u2:2 u4:1 u5:2 u7:1]  D[d3:0 d6:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010120201  U[u0:2 u1:2 u2:2 u4:1 u5:1 u7:2]  D[d3:0 d6:1 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010121012  U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0]  D[d3:0 d6:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010121021  U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0]  D[d3:0 d6:1 d8:2]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010121201  U[u0:2 u1:2 u2:2 u4:0 u5:0 u7:2]  D[d3:0 d6:1 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202121  U[u0:2 u1:2 u2:1 u4:1 u5:0 u7:0]  D[d3:1 d6:0 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010210102  U[u0:2 u1:2 u2:1 u4:2 u5:2 u7:1]  D[d3:0 d6:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010210201  U[u0:2 u1:2 u2:1 u4:2 u5:1 u7:2]  D[d3:0 d6:1 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010212012  U[u0:2 u1:2 u2:1 u4:0 u5:1 u7:0]  D[d3:0 d6:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010212021  U[u0:2 u1:2 u2:1 u4:0 u5:1 u7:0]  D[d3:0 d6:1 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010212102  U[u0:2 u1:2 u2:1 u4:0 u5:0 u7:1]  D[d3:0 d6:2 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012010212  U[u0:2 u1:0 u2:1 u4:2 u5:1 u7:0]  D[d3:2 d6:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012020121  U[u0:2 u1:0 u2:1 u4:1 u5:2 u7:0]  D[d3:1 d6:0 d8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012101212  U[u0:2 u1:0 u2:0 u4:2 u5:0 u7:0]  D[d3:2 d6:0 d8:1]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012120102  U[u0:2 u1:0 u2:0 u4:1 u5:2 u7:1]  D[d3:0 d6:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012120201  U[u0:2 u1:0 u2:0 u4:1 u5:1 u7:2]  D[d3:0 d6:1 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012121012  U[u0:2 u1:0 u2:0 u4:0 u5:2 u7:0]  D[d3:0 d6:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012121021  U[u0:2 u1:0 u2:0 u4:0 u5:2 u7:0]  D[d3:0 d6:1 d8:2]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012121201  U[u0:2 u1:0 u2:0 u4:0 u5:0 u7:2]  D[d3:0 d6:1 d8:2]`
+
+## C07 — word=UUUDUDDDD bites=(3,5) face=root apexes=[d6,d7,d8]
+
+6 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101012  U[u0:2 u1:2 u2:2 u4:2]  D[d6:2 d7:0 d8:1 p3_5:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010101021  U[u0:2 u1:2 u2:2 u4:2]  D[d6:1 d7:0 d8:2 p3_5:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010202012  U[u0:2 u1:2 u2:1 u4:1]  D[d6:2 d7:0 d8:1 p3_5:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202021  U[u0:2 u1:2 u2:1 u4:1]  D[d6:1 d7:0 d8:2 p3_5:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012101012  U[u0:2 u1:0 u2:0 u4:2]  D[d6:2 d7:0 d8:1 p3_5:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012101021  U[u0:2 u1:0 u2:0 u4:2]  D[d6:1 d7:0 d8:2 p3_5:2]`
+
+## C08 — word=UUUDUDDDD bites=(3,8) face=bite(3,8) apexes=[d5,d6,d7]
+
+6 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=010101021  U[u0:2 u1:2 u2:2 u4:2]  D[d5:2 d6:1 d7:0 p3_8:2]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010101201  U[u0:2 u1:2 u2:2 u4:2]  D[d5:0 d6:1 d7:2 p3_8:2]`
+- face apexes (raw labels) `120` → canonical `012`  ·  `A=010202012  U[u0:2 u1:2 u2:1 u4:1]  D[d5:1 d6:2 d7:0 p3_8:1]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010202102  U[u0:2 u1:2 u2:1 u4:1]  D[d5:0 d6:2 d7:1 p3_8:1]`
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=012101021  U[u0:2 u1:0 u2:0 u4:2]  D[d5:2 d6:1 d7:0 p3_8:2]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012101201  U[u0:2 u1:0 u2:0 u4:2]  D[d5:0 d6:1 d7:2 p3_8:2]`
+
+## C09 — word=UUUDDUDDD bites=(4,6) face=root apexes=[d3,d7,d8]
+
+3 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101012  U[u0:2 u1:2 u2:2 u5:2]  D[d3:2 d7:0 d8:1 p4_6:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202021  U[u0:2 u1:2 u2:1 u5:1]  D[d3:1 d7:0 d8:2 p4_6:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012101012  U[u0:2 u1:0 u2:0 u5:2]  D[d3:2 d7:0 d8:1 p4_6:2]`
+
+## C10 — word=UUUDDUDDD bites=(3,8) face=bite(3,8) apexes=[d4,d6,d7]
+
+3 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=010101021  U[u0:2 u1:2 u2:2 u5:2]  D[d4:2 d6:1 d7:0 p3_8:2]`
+- face apexes (raw labels) `120` → canonical `012`  ·  `A=010202012  U[u0:2 u1:2 u2:1 u5:1]  D[d4:1 d6:2 d7:0 p3_8:1]`
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=012101021  U[u0:2 u1:0 u2:0 u5:2]  D[d4:2 d6:1 d7:0 p3_8:2]`
+
+## C11 — word=UUDUUDUUD bites=- face=root apexes=[d2,d5,d8]
+
+18 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101202  U[u0:2 u1:2 u3:2 u4:2 u6:1 u7:1]  D[d2:2 d5:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101212  U[u0:2 u1:2 u3:2 u4:2 u6:0 u7:0]  D[d2:2 d5:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010102102  U[u0:2 u1:2 u3:2 u4:1 u6:2 u7:1]  D[d2:2 d5:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010121202  U[u0:2 u1:2 u3:0 u4:0 u6:1 u7:1]  D[d2:2 d5:0 d8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010121212  U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0]  D[d2:2 d5:0 d8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010201201  U[u0:2 u1:2 u3:1 u4:2 u6:1 u7:2]  D[d2:1 d5:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202101  U[u0:2 u1:2 u3:1 u4:1 u6:2 u7:2]  D[d2:1 d5:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202121  U[u0:2 u1:2 u3:1 u4:1 u6:0 u7:0]  D[d2:1 d5:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010212101  U[u0:2 u1:2 u3:0 u4:0 u6:2 u7:2]  D[d2:1 d5:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010212121  U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0]  D[d2:1 d5:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012012101  U[u0:2 u1:0 u3:2 u4:0 u6:2 u7:2]  D[d2:1 d5:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012012121  U[u0:2 u1:0 u3:2 u4:0 u6:0 u7:0]  D[d2:1 d5:0 d8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012021201  U[u0:2 u1:0 u3:1 u4:0 u6:1 u7:2]  D[d2:1 d5:0 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012101012  U[u0:2 u1:0 u3:2 u4:2 u6:2 u7:0]  D[d2:0 d5:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012102021  U[u0:2 u1:0 u3:2 u4:1 u6:1 u7:0]  D[d2:0 d5:1 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012120102  U[u0:2 u1:0 u3:0 u4:1 u6:2 u7:1]  D[d2:0 d5:2 d8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012120201  U[u0:2 u1:0 u3:0 u4:1 u6:1 u7:2]  D[d2:0 d5:1 d8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012121012  U[u0:2 u1:0 u3:0 u4:0 u6:2 u7:0]  D[d2:0 d5:2 d8:1]`
+
+## C12 — word=UUDUUDDDD bites=(2,5) face=root apexes=[d6,d7,d8]
+
+10 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101012  U[u0:2 u1:2 u3:2 u4:2]  D[d6:2 d7:0 d8:1 p2_5:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010101021  U[u0:2 u1:2 u3:2 u4:2]  D[d6:1 d7:0 d8:2 p2_5:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010121012  U[u0:2 u1:2 u3:0 u4:0]  D[d6:2 d7:0 d8:1 p2_5:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010121021  U[u0:2 u1:2 u3:0 u4:0]  D[d6:1 d7:0 d8:2 p2_5:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010202012  U[u0:2 u1:2 u3:1 u4:1]  D[d6:2 d7:0 d8:1 p2_5:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202021  U[u0:2 u1:2 u3:1 u4:1]  D[d6:1 d7:0 d8:2 p2_5:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010212012  U[u0:2 u1:2 u3:0 u4:0]  D[d6:2 d7:0 d8:1 p2_5:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010212021  U[u0:2 u1:2 u3:0 u4:0]  D[d6:1 d7:0 d8:2 p2_5:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012012012  U[u0:2 u1:0 u3:2 u4:0]  D[d6:2 d7:0 d8:1 p2_5:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012012021  U[u0:2 u1:0 u3:2 u4:0]  D[d6:1 d7:0 d8:2 p2_5:1]`
+
+## C13 — word=UUDUDUDDD bites=(4,6) face=root apexes=[d2,d7,d8]
+
+5 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101012  U[u0:2 u1:2 u3:2 u5:2]  D[d2:2 d7:0 d8:1 p4_6:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010121212  U[u0:2 u1:2 u3:0 u5:0]  D[d2:2 d7:0 d8:1 p4_6:0]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202021  U[u0:2 u1:2 u3:1 u5:1]  D[d2:1 d7:0 d8:2 p4_6:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010212121  U[u0:2 u1:2 u3:0 u5:0]  D[d2:1 d7:0 d8:2 p4_6:0]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012012121  U[u0:2 u1:0 u3:2 u5:0]  D[d2:1 d7:0 d8:2 p4_6:0]`
+
+## C14 — word=UUDUDUDDD bites=(2,4) face=root apexes=[d6,d7,d8]
+
+6 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101012  U[u0:2 u1:2 u3:2 u5:2]  D[d6:2 d7:0 d8:1 p2_4:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010101021  U[u0:2 u1:2 u3:2 u5:2]  D[d6:1 d7:0 d8:2 p2_4:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010202012  U[u0:2 u1:2 u3:1 u5:1]  D[d6:2 d7:0 d8:1 p2_4:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202021  U[u0:2 u1:2 u3:1 u5:1]  D[d6:1 d7:0 d8:2 p2_4:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012121012  U[u0:2 u1:0 u3:0 u5:2]  D[d6:2 d7:0 d8:1 p2_4:0]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012121021  U[u0:2 u1:0 u3:0 u5:2]  D[d6:1 d7:0 d8:2 p2_4:0]`
+
+## C15 — word=UUDUDUDDD bites=(2,8) face=bite(2,8) apexes=[d4,d6,d7]
+
+5 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=010101021  U[u0:2 u1:2 u3:2 u5:2]  D[d4:2 d6:1 d7:0 p2_8:2]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010121201  U[u0:2 u1:2 u3:0 u5:0]  D[d4:0 d6:1 d7:2 p2_8:2]`
+- face apexes (raw labels) `120` → canonical `012`  ·  `A=010202012  U[u0:2 u1:2 u3:1 u5:1]  D[d4:1 d6:2 d7:0 p2_8:1]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010212102  U[u0:2 u1:2 u3:0 u5:0]  D[d4:0 d6:2 d7:1 p2_8:1]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012012102  U[u0:2 u1:0 u3:2 u5:0]  D[d4:0 d6:2 d7:1 p2_8:1]`
+
+## C16 — word=UUDUDDUDD bites=(5,7) face=root apexes=[d2,d4,d8]
+
+6 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010121212  U[u0:2 u1:2 u3:0 u6:0]  D[d2:2 d4:0 d8:1 p5_7:0]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010212121  U[u0:2 u1:2 u3:0 u6:0]  D[d2:1 d4:0 d8:2 p5_7:0]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012012121  U[u0:2 u1:0 u3:2 u6:0]  D[d2:1 d4:0 d8:2 p5_7:0]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012101212  U[u0:2 u1:0 u3:2 u6:0]  D[d2:0 d4:2 d8:1 p5_7:0]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012102121  U[u0:2 u1:0 u3:2 u6:0]  D[d2:0 d4:1 d8:2 p5_7:0]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012120101  U[u0:2 u1:0 u3:0 u6:2]  D[d2:0 d4:1 d8:2 p5_7:2]`
+
+## C17 — word=UUDUDDUDD bites=(2,4) face=root apexes=[d5,d7,d8]
+
+3 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101012  U[u0:2 u1:2 u3:2 u6:2]  D[d5:2 d7:0 d8:1 p2_4:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202021  U[u0:2 u1:2 u3:1 u6:1]  D[d5:1 d7:0 d8:2 p2_4:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012121012  U[u0:2 u1:0 u3:0 u6:2]  D[d5:2 d7:0 d8:1 p2_4:0]`
+
+## C18 — word=UUDUDDUDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d7]
+
+5 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010102101  U[u0:2 u1:2 u3:2 u6:2]  D[d4:1 d5:0 d7:2 p2_8:2]`
+- face apexes (raw labels) `120` → canonical `012`  ·  `A=010120121  U[u0:2 u1:2 u3:0 u6:0]  D[d4:1 d5:2 d7:0 p2_8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010201202  U[u0:2 u1:2 u3:1 u6:1]  D[d4:2 d5:0 d7:1 p2_8:1]`
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=010210212  U[u0:2 u1:2 u3:0 u6:0]  D[d4:2 d5:1 d7:0 p2_8:1]`
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=012010212  U[u0:2 u1:0 u3:2 u6:0]  D[d4:2 d5:1 d7:0 p2_8:1]`
+
+## C19 — word=UUDUDDDUD bites=(6,8) face=root apexes=[d2,d4,d5]
+
+3 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=010102101  U[u0:2 u1:2 u3:2 u7:2]  D[d2:2 d4:1 d5:0 p6_8:2]`
+- face apexes (raw labels) `120` → canonical `012`  ·  `A=010201202  U[u0:2 u1:2 u3:1 u7:1]  D[d2:1 d4:2 d5:0 p6_8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012120101  U[u0:2 u1:0 u3:0 u7:2]  D[d2:0 d4:1 d5:2 p6_8:2]`
+
+## C20 — word=UUDUDDDUD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6]
+
+10 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101201  U[u0:2 u1:2 u3:2 u7:2]  D[d4:2 d5:0 d6:1 p2_8:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010102101  U[u0:2 u1:2 u3:2 u7:2]  D[d4:1 d5:0 d6:2 p2_8:2]`
+- face apexes (raw labels) `120` → canonical `012`  ·  `A=010120121  U[u0:2 u1:2 u3:0 u7:0]  D[d4:1 d5:2 d6:0 p2_8:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010121021  U[u0:2 u1:2 u3:0 u7:0]  D[d4:0 d5:2 d6:1 p2_8:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010201202  U[u0:2 u1:2 u3:1 u7:1]  D[d4:2 d5:0 d6:1 p2_8:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202102  U[u0:2 u1:2 u3:1 u7:1]  D[d4:1 d5:0 d6:2 p2_8:1]`
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=010210212  U[u0:2 u1:2 u3:0 u7:0]  D[d4:2 d5:1 d6:0 p2_8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010212012  U[u0:2 u1:2 u3:0 u7:0]  D[d4:0 d5:1 d6:2 p2_8:1]`
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=012010212  U[u0:2 u1:0 u3:2 u7:0]  D[d4:2 d5:1 d6:0 p2_8:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012012012  U[u0:2 u1:0 u3:2 u7:0]  D[d4:0 d5:1 d6:2 p2_8:1]`
+
+## C21 — word=UUDDUUDDD bites=(3,6) face=root apexes=[d2,d7,d8]
+
+9 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101012  U[u0:2 u1:2 u4:2 u5:2]  D[d2:2 d7:0 d8:1 p3_6:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010102012  U[u0:2 u1:2 u4:1 u5:1]  D[d2:2 d7:0 d8:1 p3_6:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010120212  U[u0:2 u1:2 u4:1 u5:1]  D[d2:2 d7:0 d8:1 p3_6:0]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010121212  U[u0:2 u1:2 u4:0 u5:0]  D[d2:2 d7:0 d8:1 p3_6:0]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010201021  U[u0:2 u1:2 u4:2 u5:2]  D[d2:1 d7:0 d8:2 p3_6:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010202021  U[u0:2 u1:2 u4:1 u5:1]  D[d2:1 d7:0 d8:2 p3_6:1]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010210121  U[u0:2 u1:2 u4:2 u5:2]  D[d2:1 d7:0 d8:2 p3_6:0]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010212121  U[u0:2 u1:2 u4:0 u5:0]  D[d2:1 d7:0 d8:2 p3_6:0]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=012021021  U[u0:2 u1:0 u4:0 u5:2]  D[d2:1 d7:0 d8:2 p3_6:1]`
+
+## C22 — word=UUDDUDUDD bites=(5,7) face=root apexes=[d2,d3,d8]
+
+5 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010120202  U[u0:2 u1:2 u4:1 u6:1]  D[d2:2 d3:0 d8:1 p5_7:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010121212  U[u0:2 u1:2 u4:0 u6:0]  D[d2:2 d3:0 d8:1 p5_7:0]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010210101  U[u0:2 u1:2 u4:2 u6:2]  D[d2:1 d3:0 d8:2 p5_7:2]`
+- face apexes (raw labels) `102` → canonical `012`  ·  `A=010212121  U[u0:2 u1:2 u4:0 u6:0]  D[d2:1 d3:0 d8:2 p5_7:0]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012101212  U[u0:2 u1:0 u4:2 u6:0]  D[d2:0 d3:2 d8:1 p5_7:0]`
+
+## C23 — word=UUDDUDUDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d7]
+
+8 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=010102021  U[u0:2 u1:2 u4:1 u6:1]  D[d3:2 d5:1 d7:0 p2_8:2]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=010120201  U[u0:2 u1:2 u4:1 u6:1]  D[d3:0 d5:1 d7:2 p2_8:2]`
+- face apexes (raw labels) `120` → canonical `012`  ·  `A=010201012  U[u0:2 u1:2 u4:2 u6:2]  D[d3:1 d5:2 d7:0 p2_8:1]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=010210102  U[u0:2 u1:2 u4:2 u6:2]  D[d3:0 d5:2 d7:1 p2_8:1]`
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=012010212  U[u0:2 u1:0 u4:2 u6:0]  D[d3:2 d5:1 d7:0 p2_8:1]`
+- face apexes (raw labels) `210` → canonical `012`  ·  `A=012012012  U[u0:2 u1:0 u4:0 u6:2]  D[d3:2 d5:1 d7:0 p2_8:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=012012102  U[u0:2 u1:0 u4:0 u6:2]  D[d3:2 d5:0 d7:1 p2_8:1]`
+- face apexes (raw labels) `120` → canonical `012`  ·  `A=012021012  U[u0:2 u1:0 u4:0 u6:2]  D[d3:1 d5:2 d7:0 p2_8:1]`
+
+## C24 — word=UDUDUDUDD bites=(5,7) face=root apexes=[d1,d3,d8]
+
+6 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010120202  U[u0:2 u2:2 u4:1 u6:1]  D[d1:2 d3:0 d8:1 p5_7:1]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010121212  U[u0:2 u2:2 u4:0 u6:0]  D[d1:2 d3:0 d8:1 p5_7:0]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010210202  U[u0:2 u2:1 u4:2 u6:1]  D[d1:2 d3:0 d8:1 p5_7:1]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012010202  U[u0:2 u2:1 u4:2 u6:1]  D[d1:0 d3:2 d8:1 p5_7:1]`
+- face apexes (raw labels) `012` → canonical `012`  ·  `A=012020101  U[u0:2 u2:1 u4:1 u6:2]  D[d1:0 d3:1 d8:2 p5_7:2]`
+- face apexes (raw labels) `021` → canonical `012`  ·  `A=012101212  U[u0:2 u2:0 u4:2 u6:0]  D[d1:0 d3:2 d8:1 p5_7:0]`
+
+## C25 — word=UDUDUDUDD bites=(3,5) face=root apexes=[d1,d7,d8]
+
+3 colouring(s) with down-apex sequence `012`:
+
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010101012  U[u0:2 u2:2 u4:2 u6:2]  D[d1:2 d7:0 d8:1 p3_5:2]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010121212  U[u0:2 u2:2 u4:0 u6:0]  D[d1:2 d7:0 d8:1 p3_5:0]`
+- face apexes (raw labels) `201` → canonical `012`  ·  `A=010202012  U[u0:2 u2:1 u4:1 u6:2]  D[d1:2 d7:0 d8:1 p3_5:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/summary.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/summary.md
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+# Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=3)
+
+Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 3 singleton down-tooth apexes: **26 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in increasing annular-edge order, and reduce it modulo colour permutation (NOT dihedral symmetry).
+
+- Total Kempe-balanced colourings (mod colour permutation): **241**.
+- Distinct canonical down-apex sequences overall: **1**.
+
+## Distinct canonical down-apex sequences
+
+| sequence | colour multiset | #configs realising | #colourings |
+|---|---|---|---|
+| `012` | 1+1+1 | 26 | 241 |
+
+Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 3 apexes (m is odd) every count must be **odd**, so the only admissible colour multisets are 1+1+1.
+
+## Step 4 — grouping configs by their set of unique down-apex sequences
+
+The 26 configs fall into **1** groups by the set of canonical down-apex sequences they realise:
+
+| #configs | set of down-apex sequences | config ids |
+|---|---|---|
+| 26 | { `012` } | C00, C01, C02, C03, C04, C05, C06, C07, C08, C09, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19, C20, C21, C22, C23, C24, C25 |
+
+## Config atlas (ids)
+
+| id | word / bites / face / apexes | #Kempe-balanced | down-apex sequence set |
+|---|---|---|---|
+| C00 | word=UUUUUUDDD bites=- face=root apexes=[d6,d7,d8] | 22 | { `012` } |
+| C01 | word=UUUUUDUDD bites=- face=root apexes=[d5,d7,d8] | 11 | { `012` } |
+| C02 | word=UUUUDUUDD bites=- face=root apexes=[d4,d7,d8] | 17 | { `012` } |
+| C03 | word=UUUUDUDUD bites=- face=root apexes=[d4,d6,d8] | 26 | { `012` } |
+| C04 | word=UUUUDDDDD bites=(4,8) face=bite(4,8) apexes=[d5,d6,d7] | 12 | { `012` } |
+| C05 | word=UUUDUUUDD bites=- face=root apexes=[d3,d7,d8] | 13 | { `012` } |
+| C06 | word=UUUDUUDUD bites=- face=root apexes=[d3,d6,d8] | 20 | { `012` } |
+| C07 | word=UUUDUDDDD bites=(3,5) face=root apexes=[d6,d7,d8] | 6 | { `012` } |
+| C08 | word=UUUDUDDDD bites=(3,8) face=bite(3,8) apexes=[d5,d6,d7] | 6 | { `012` } |
+| C09 | word=UUUDDUDDD bites=(4,6) face=root apexes=[d3,d7,d8] | 3 | { `012` } |
+| C10 | word=UUUDDUDDD bites=(3,8) face=bite(3,8) apexes=[d4,d6,d7] | 3 | { `012` } |
+| C11 | word=UUDUUDUUD bites=- face=root apexes=[d2,d5,d8] | 18 | { `012` } |
+| C12 | word=UUDUUDDDD bites=(2,5) face=root apexes=[d6,d7,d8] | 10 | { `012` } |
+| C13 | word=UUDUDUDDD bites=(4,6) face=root apexes=[d2,d7,d8] | 5 | { `012` } |
+| C14 | word=UUDUDUDDD bites=(2,4) face=root apexes=[d6,d7,d8] | 6 | { `012` } |
+| C15 | word=UUDUDUDDD bites=(2,8) face=bite(2,8) apexes=[d4,d6,d7] | 5 | { `012` } |
+| C16 | word=UUDUDDUDD bites=(5,7) face=root apexes=[d2,d4,d8] | 6 | { `012` } |
+| C17 | word=UUDUDDUDD bites=(2,4) face=root apexes=[d5,d7,d8] | 3 | { `012` } |
+| C18 | word=UUDUDDUDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d7] | 5 | { `012` } |
+| C19 | word=UUDUDDDUD bites=(6,8) face=root apexes=[d2,d4,d5] | 3 | { `012` } |
+| C20 | word=UUDUDDDUD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6] | 10 | { `012` } |
+| C21 | word=UUDDUUDDD bites=(3,6) face=root apexes=[d2,d7,d8] | 9 | { `012` } |
+| C22 | word=UUDDUDUDD bites=(5,7) face=root apexes=[d2,d3,d8] | 5 | { `012` } |
+| C23 | word=UUDDUDUDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d7] | 8 | { `012` } |
+| C24 | word=UDUDUDUDD bites=(5,7) face=root apexes=[d1,d3,d8] | 6 | { `012` } |
+| C25 | word=UDUDUDUDD bites=(3,5) face=root apexes=[d1,d7,d8] | 3 | { `012` } |
+
+## Per-sequence notes
+
+- [`012`](seq_012.md) — figure `seq_012.png`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0000.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0000.md
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index 0000000..99eee45
--- /dev/null
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0000.md
@@ -0,0 +1,180 @@
+# Inner-face down-apex sequence `0000`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**.
+
+- Colour multiset: 4×colour0.
+- Realised by **21** of 23 configs (M(T), inner face).
+- **64** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_0000.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8]
+
+10 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=010120101  U[u0:2 u1:2 u2:2 u3:0 u4:1]  D[d5:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=010120202  U[u0:2 u1:2 u2:2 u3:0 u4:1]  D[d5:1 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=010210101  U[u0:2 u1:2 u2:1 u3:0 u4:2]  D[d5:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=010210202  U[u0:2 u1:2 u2:1 u3:0 u4:2]  D[d5:1 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012010101  U[u0:2 u1:0 u2:1 u3:2 u4:2]  D[d5:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012010202  U[u0:2 u1:0 u2:1 u3:2 u4:2]  D[d5:1 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012020101  U[u0:2 u1:0 u2:1 u3:1 u4:1]  D[d5:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u2:1 u3:1 u4:1]  D[d5:1 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012120101  U[u0:2 u1:0 u2:0 u3:0 u4:1]  D[d5:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012120202  U[u0:2 u1:0 u2:0 u3:0 u4:1]  D[d5:1 d6:1 d7:1 d8:1]`
+
+## C01 — word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8]
+
+5 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=010120202  U[u0:2 u1:2 u2:2 u3:0 u5:1]  D[d4:1 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=010210101  U[u0:2 u1:2 u2:1 u3:0 u5:2]  D[d4:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012010101  U[u0:2 u1:0 u2:1 u3:2 u5:2]  D[d4:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u2:1 u3:1 u5:1]  D[d4:1 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012120202  U[u0:2 u1:0 u2:0 u3:0 u5:1]  D[d4:1 d6:1 d7:1 d8:1]`
+
+## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8]
+
+5 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=010120202  U[u0:2 u1:2 u2:2 u3:0 u6:1]  D[d4:1 d5:1 d7:1 d8:1]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=010210101  U[u0:2 u1:2 u2:1 u3:0 u6:2]  D[d4:2 d5:2 d7:2 d8:2]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012010101  U[u0:2 u1:0 u2:1 u3:2 u6:2]  D[d4:2 d5:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u2:1 u3:1 u6:1]  D[d4:1 d5:1 d7:1 d8:1]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012120202  U[u0:2 u1:0 u2:0 u3:0 u6:1]  D[d4:1 d5:1 d7:1 d8:1]`
+
+## C03 — word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8]
+
+7 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=010102101  U[u0:2 u1:2 u2:2 u4:1 u5:0]  D[d3:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=010201202  U[u0:2 u1:2 u2:1 u4:2 u5:0]  D[d3:1 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012010101  U[u0:2 u1:0 u2:1 u4:2 u5:2]  D[d3:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012012101  U[u0:2 u1:0 u2:1 u4:0 u5:0]  D[d3:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u2:1 u4:1 u5:1]  D[d3:1 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012021202  U[u0:2 u1:0 u2:1 u4:0 u5:0]  D[d3:1 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012102101  U[u0:2 u1:0 u2:0 u4:1 u5:0]  D[d3:2 d6:2 d7:2 d8:2]`
+
+## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8]
+
+2 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012010101  U[u0:2 u1:0 u2:1 u4:2 u6:2]  D[d3:2 d5:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u2:1 u4:1 u6:1]  D[d3:1 d5:1 d7:1 d8:1]`
+
+## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8]
+
+2 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012010101  U[u0:2 u1:0 u2:1 u4:2 u7:2]  D[d3:2 d5:2 d6:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u2:1 u4:1 u7:1]  D[d3:1 d5:1 d6:1 d8:1]`
+
+## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8]
+
+7 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=010101201  U[u0:2 u1:2 u2:2 u5:0 u6:1]  D[d3:2 d4:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=010202102  U[u0:2 u1:2 u2:1 u5:0 u6:2]  D[d3:1 d4:1 d7:1 d8:1]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012010101  U[u0:2 u1:0 u2:1 u5:2 u6:2]  D[d3:2 d4:2 d7:2 d8:2]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012010201  U[u0:2 u1:0 u2:1 u5:1 u6:1]  D[d3:2 d4:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020102  U[u0:2 u1:0 u2:1 u5:2 u6:2]  D[d3:1 d4:1 d7:1 d8:1]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u2:1 u5:1 u6:1]  D[d3:1 d4:1 d7:1 d8:1]`
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012101201  U[u0:2 u1:0 u2:0 u5:0 u6:1]  D[d3:2 d4:2 d7:2 d8:2]`
+
+## C07 — word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7]
+
+4 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012010101  U[u0:2 u1:0 u2:1]  D[d4:2 d5:2 d6:2 d7:2 p3_8:2]`
+- face apexes (raw labels) `0000` → canonical `0000`  ·  `A=012012121  U[u0:2 u1:0 u2:1]  D[d4:0 d5:0 d6:0 d7:0 p3_8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u2:1]  D[d4:1 d5:1 d6:1 d7:1 p3_8:1]`
+- face apexes (raw labels) `0000` → canonical `0000`  ·  `A=012021212  U[u0:2 u1:0 u2:1]  D[d4:0 d5:0 d6:0 d7:0 p3_8:1]`
+
+## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8]
+
+4 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=010120101  U[u0:2 u1:2 u3:0 u4:1 u6:2]  D[d2:2 d5:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=010210202  U[u0:2 u1:2 u3:0 u4:2 u6:1]  D[d2:1 d5:1 d7:1 d8:1]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012010202  U[u0:2 u1:0 u3:2 u4:2 u6:1]  D[d2:1 d5:1 d7:1 d8:1]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u3:1 u4:1 u6:1]  D[d2:1 d5:1 d7:1 d8:1]`
+
+## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8]
+
+4 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=010101201  U[u0:2 u1:2 u3:2 u5:0 u6:1]  D[d2:2 d4:2 d7:2 d8:2]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=010202102  U[u0:2 u1:2 u3:1 u5:0 u6:2]  D[d2:1 d4:1 d7:1 d8:1]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020102  U[u0:2 u1:0 u3:1 u5:2 u6:2]  D[d2:1 d4:1 d7:1 d8:1]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u3:1 u5:1 u6:1]  D[d2:1 d4:1 d7:1 d8:1]`
+
+## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8]
+
+1 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u3:1 u5:1 u7:1]  D[d2:1 d4:1 d6:1 d8:1]`
+
+## C11 — word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012020101  U[u0:2 u1:0 u3:1]  D[d5:2 d6:2 d7:2 d8:2 p2_4:1]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u3:1]  D[d5:1 d6:1 d7:1 d8:1 p2_4:1]`
+
+## C12 — word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7]
+
+2 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u3:1]  D[d4:1 d5:1 d6:1 d7:1 p2_8:1]`
+- face apexes (raw labels) `0000` → canonical `0000`  ·  `A=012021212  U[u0:2 u1:0 u3:1]  D[d4:0 d5:0 d6:0 d7:0 p2_8:1]`
+
+## C13 — word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u4:1]  D[d2:1 d6:1 d7:1 d8:1 p3_5:1]`
+
+## C14 — word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7]
+
+1 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u4:1]  D[d3:1 d5:1 d6:1 d7:1 p2_8:1]`
+
+## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8]
+
+1 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u5:1]  D[d2:1 d3:1 d7:1 d8:1 p4_6:1]`
+
+## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7]
+
+1 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012020202  U[u0:2 u1:0 u5:1]  D[d3:1 d4:1 d6:1 d7:1 p2_8:1]`
+
+## C17 — word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=010212101  U[u0:2 u2:1 u4:0]  D[d1:2 d6:2 d7:2 d8:2 p3_5:0]`
+
+## C18 — word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `2222` → canonical `0000`  ·  `A=012120101  U[u0:2 u2:0 u4:1]  D[d5:2 d6:2 d7:2 d8:2 p1_3:0]`
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012120202  U[u0:2 u2:0 u4:1]  D[d5:1 d6:1 d7:1 d8:1 p1_3:0]`
+
+## C20 — word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `1111` → canonical `0000`  ·  `A=012120202  U[u0:2 u2:0 u5:1]  D[d4:1 d6:1 d7:1 d8:1 p1_3:0]`
+
+## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7]
+
+1 colouring(s) with down-apex sequence `0000`:
+
+- face apexes (raw labels) `0000` → canonical `0000`  ·  `A=010212121  U[u0:2 u2:1 u5:0]  D[d3:0 d4:0 d6:0 d7:0 p1_8:2]`
+
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.md
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+# Inner-face down-apex sequence `0011`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1.
+- Realised by **23** of 23 configs (M(T), inner face).
+- **108** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_0011.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8]
+
+10 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=010120102  U[u0:2 u1:2 u2:2 u3:0 u4:1]  D[d5:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=010120201  U[u0:2 u1:2 u2:2 u3:0 u4:1]  D[d5:1 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=010210102  U[u0:2 u1:2 u2:1 u3:0 u4:2]  D[d5:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=010210201  U[u0:2 u1:2 u2:1 u3:0 u4:2]  D[d5:1 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012010102  U[u0:2 u1:0 u2:1 u3:2 u4:2]  D[d5:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012010201  U[u0:2 u1:0 u2:1 u3:2 u4:2]  D[d5:1 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012020102  U[u0:2 u1:0 u2:1 u3:1 u4:1]  D[d5:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020201  U[u0:2 u1:0 u2:1 u3:1 u4:1]  D[d5:1 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012120102  U[u0:2 u1:0 u2:0 u3:0 u4:1]  D[d5:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012120201  U[u0:2 u1:0 u2:0 u3:0 u4:1]  D[d5:1 d6:1 d7:2 d8:2]`
+
+## C01 — word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8]
+
+5 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=010120201  U[u0:2 u1:2 u2:2 u3:0 u5:1]  D[d4:1 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=010210102  U[u0:2 u1:2 u2:1 u3:0 u5:2]  D[d4:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012010102  U[u0:2 u1:0 u2:1 u3:2 u5:2]  D[d4:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020201  U[u0:2 u1:0 u2:1 u3:1 u5:1]  D[d4:1 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012120201  U[u0:2 u1:0 u2:0 u3:0 u5:1]  D[d4:1 d6:1 d7:2 d8:2]`
+
+## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8]
+
+15 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=010120201  U[u0:2 u1:2 u2:2 u3:0 u6:1]  D[d4:1 d5:1 d7:2 d8:2]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=010121201  U[u0:2 u1:2 u2:2 u3:0 u6:1]  D[d4:0 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=010121202  U[u0:2 u1:2 u2:2 u3:0 u6:1]  D[d4:0 d5:0 d7:1 d8:1]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=010210102  U[u0:2 u1:2 u2:1 u3:0 u6:2]  D[d4:2 d5:2 d7:1 d8:1]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=010212101  U[u0:2 u1:2 u2:1 u3:0 u6:2]  D[d4:0 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=010212102  U[u0:2 u1:2 u2:1 u3:0 u6:2]  D[d4:0 d5:0 d7:1 d8:1]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012010102  U[u0:2 u1:0 u2:1 u3:2 u6:2]  D[d4:2 d5:2 d7:1 d8:1]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=012012101  U[u0:2 u1:0 u2:1 u3:2 u6:2]  D[d4:0 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012012102  U[u0:2 u1:0 u2:1 u3:2 u6:2]  D[d4:0 d5:0 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020201  U[u0:2 u1:0 u2:1 u3:1 u6:1]  D[d4:1 d5:1 d7:2 d8:2]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=012021201  U[u0:2 u1:0 u2:1 u3:1 u6:1]  D[d4:0 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012021202  U[u0:2 u1:0 u2:1 u3:1 u6:1]  D[d4:0 d5:0 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012120201  U[u0:2 u1:0 u2:0 u3:0 u6:1]  D[d4:1 d5:1 d7:2 d8:2]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=012121201  U[u0:2 u1:0 u2:0 u3:0 u6:1]  D[d4:0 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012121202  U[u0:2 u1:0 u2:0 u3:0 u6:1]  D[d4:0 d5:0 d7:1 d8:1]`
+
+## C03 — word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8]
+
+7 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=010102102  U[u0:2 u1:2 u2:2 u4:1 u5:0]  D[d3:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=010201201  U[u0:2 u1:2 u2:1 u4:2 u5:0]  D[d3:1 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012010102  U[u0:2 u1:0 u2:1 u4:2 u5:2]  D[d3:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012012102  U[u0:2 u1:0 u2:1 u4:0 u5:0]  D[d3:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020201  U[u0:2 u1:0 u2:1 u4:1 u5:1]  D[d3:1 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012021201  U[u0:2 u1:0 u2:1 u4:0 u5:0]  D[d3:1 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012102102  U[u0:2 u1:0 u2:0 u4:1 u5:0]  D[d3:2 d6:2 d7:1 d8:1]`
+
+## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8]
+
+8 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=010121201  U[u0:2 u1:2 u2:2 u4:0 u6:1]  D[d3:0 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=010121202  U[u0:2 u1:2 u2:2 u4:0 u6:1]  D[d3:0 d5:0 d7:1 d8:1]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=010212101  U[u0:2 u1:2 u2:1 u4:0 u6:2]  D[d3:0 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=010212102  U[u0:2 u1:2 u2:1 u4:0 u6:2]  D[d3:0 d5:0 d7:1 d8:1]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012010102  U[u0:2 u1:0 u2:1 u4:2 u6:2]  D[d3:2 d5:2 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020201  U[u0:2 u1:0 u2:1 u4:1 u6:1]  D[d3:1 d5:1 d7:2 d8:2]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=012121201  U[u0:2 u1:0 u2:0 u4:0 u6:1]  D[d3:0 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012121202  U[u0:2 u1:0 u2:0 u4:0 u6:1]  D[d3:0 d5:0 d7:1 d8:1]`
+
+## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8]
+
+3 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=010121202  U[u0:2 u1:2 u2:2 u4:0 u7:1]  D[d3:0 d5:0 d6:1 d8:1]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=010212101  U[u0:2 u1:2 u2:1 u4:0 u7:2]  D[d3:0 d5:0 d6:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012121202  U[u0:2 u1:0 u2:0 u4:0 u7:1]  D[d3:0 d5:0 d6:1 d8:1]`
+
+## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8]
+
+13 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=010101202  U[u0:2 u1:2 u2:2 u5:0 u6:1]  D[d3:2 d4:2 d7:1 d8:1]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=010121201  U[u0:2 u1:2 u2:2 u5:0 u6:1]  D[d3:0 d4:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=010121202  U[u0:2 u1:2 u2:2 u5:0 u6:1]  D[d3:0 d4:0 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=010202101  U[u0:2 u1:2 u2:1 u5:0 u6:2]  D[d3:1 d4:1 d7:2 d8:2]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=010212101  U[u0:2 u1:2 u2:1 u5:0 u6:2]  D[d3:0 d4:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=010212102  U[u0:2 u1:2 u2:1 u5:0 u6:2]  D[d3:0 d4:0 d7:1 d8:1]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012010102  U[u0:2 u1:0 u2:1 u5:2 u6:2]  D[d3:2 d4:2 d7:1 d8:1]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012010202  U[u0:2 u1:0 u2:1 u5:1 u6:1]  D[d3:2 d4:2 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020101  U[u0:2 u1:0 u2:1 u5:2 u6:2]  D[d3:1 d4:1 d7:2 d8:2]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020201  U[u0:2 u1:0 u2:1 u5:1 u6:1]  D[d3:1 d4:1 d7:2 d8:2]`
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012101202  U[u0:2 u1:0 u2:0 u5:0 u6:1]  D[d3:2 d4:2 d7:1 d8:1]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=012121201  U[u0:2 u1:0 u2:0 u5:0 u6:1]  D[d3:0 d4:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012121202  U[u0:2 u1:0 u2:0 u5:0 u6:1]  D[d3:0 d4:0 d7:1 d8:1]`
+
+## C07 — word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7]
+
+4 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `2200` → canonical `0011`  ·  `A=012010121  U[u0:2 u1:0 u2:1]  D[d4:2 d5:2 d6:0 d7:0 p3_8:2]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=012012101  U[u0:2 u1:0 u2:1]  D[d4:0 d5:0 d6:2 d7:2 p3_8:2]`
+- face apexes (raw labels) `1100` → canonical `0011`  ·  `A=012020212  U[u0:2 u1:0 u2:1]  D[d4:1 d5:1 d6:0 d7:0 p3_8:1]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012021202  U[u0:2 u1:0 u2:1]  D[d4:0 d5:0 d6:1 d7:1 p3_8:1]`
+
+## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8]
+
+10 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=010120102  U[u0:2 u1:2 u3:0 u4:1 u6:2]  D[d2:2 d5:2 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=010210201  U[u0:2 u1:2 u3:0 u4:2 u6:1]  D[d2:1 d5:1 d7:2 d8:2]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012010201  U[u0:2 u1:0 u3:2 u4:2 u6:1]  D[d2:1 d5:1 d7:2 d8:2]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020201  U[u0:2 u1:0 u3:1 u4:1 u6:1]  D[d2:1 d5:1 d7:2 d8:2]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=012101201  U[u0:2 u1:0 u3:2 u4:2 u6:1]  D[d2:0 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012101202  U[u0:2 u1:0 u3:2 u4:2 u6:1]  D[d2:0 d5:0 d7:1 d8:1]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=012102101  U[u0:2 u1:0 u3:2 u4:1 u6:2]  D[d2:0 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012102102  U[u0:2 u1:0 u3:2 u4:1 u6:2]  D[d2:0 d5:0 d7:1 d8:1]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=012121201  U[u0:2 u1:0 u3:0 u4:0 u6:1]  D[d2:0 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012121202  U[u0:2 u1:0 u3:0 u4:0 u6:1]  D[d2:0 d5:0 d7:1 d8:1]`
+
+## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8]
+
+6 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=010101202  U[u0:2 u1:2 u3:2 u5:0 u6:1]  D[d2:2 d4:2 d7:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=010202101  U[u0:2 u1:2 u3:1 u5:0 u6:2]  D[d2:1 d4:1 d7:2 d8:2]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020101  U[u0:2 u1:0 u3:1 u5:2 u6:2]  D[d2:1 d4:1 d7:2 d8:2]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020201  U[u0:2 u1:0 u3:1 u5:1 u6:1]  D[d2:1 d4:1 d7:2 d8:2]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=012121201  U[u0:2 u1:0 u3:0 u5:0 u6:1]  D[d2:0 d4:0 d7:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012121202  U[u0:2 u1:0 u3:0 u5:0 u6:1]  D[d2:0 d4:0 d7:1 d8:1]`
+
+## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8]
+
+4 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=010101202  U[u0:2 u1:2 u3:2 u5:0 u7:1]  D[d2:2 d4:2 d6:1 d8:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=010202101  U[u0:2 u1:2 u3:1 u5:0 u7:2]  D[d2:1 d4:1 d6:2 d8:2]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020101  U[u0:2 u1:0 u3:1 u5:2 u7:2]  D[d2:1 d4:1 d6:2 d8:2]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012121202  U[u0:2 u1:0 u3:0 u5:0 u7:1]  D[d2:0 d4:0 d6:1 d8:1]`
+
+## C11 — word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012020102  U[u0:2 u1:0 u3:1]  D[d5:2 d6:2 d7:1 d8:1 p2_4:1]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020201  U[u0:2 u1:0 u3:1]  D[d5:1 d6:1 d7:2 d8:2 p2_4:1]`
+
+## C12 — word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7]
+
+2 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `1100` → canonical `0011`  ·  `A=012020212  U[u0:2 u1:0 u3:1]  D[d4:1 d5:1 d6:0 d7:0 p2_8:1]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012021202  U[u0:2 u1:0 u3:1]  D[d4:0 d5:0 d6:1 d7:1 p2_8:1]`
+
+## C13 — word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020201  U[u0:2 u1:0 u4:1]  D[d2:1 d6:1 d7:2 d8:2 p3_5:1]`
+
+## C14 — word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7]
+
+1 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `1100` → canonical `0011`  ·  `A=012020212  U[u0:2 u1:0 u4:1]  D[d3:1 d5:1 d6:0 d7:0 p2_8:1]`
+
+## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8]
+
+3 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012020201  U[u0:2 u1:0 u5:1]  D[d2:1 d3:1 d7:2 d8:2 p4_6:1]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=012120201  U[u0:2 u1:0 u5:1]  D[d2:0 d3:0 d7:2 d8:2 p4_6:1]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012120202  U[u0:2 u1:0 u5:1]  D[d2:0 d3:0 d7:1 d8:1 p4_6:1]`
+
+## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7]
+
+3 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012010202  U[u0:2 u1:0 u5:1]  D[d3:2 d4:2 d6:1 d7:1 p2_8:1]`
+- face apexes (raw labels) `2200` → canonical `0011`  ·  `A=012010212  U[u0:2 u1:0 u5:1]  D[d3:2 d4:2 d6:0 d7:0 p2_8:1]`
+- face apexes (raw labels) `1100` → canonical `0011`  ·  `A=012020212  U[u0:2 u1:0 u5:1]  D[d3:1 d4:1 d6:0 d7:0 p2_8:1]`
+
+## C17 — word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=010212102  U[u0:2 u2:1 u4:0]  D[d1:2 d6:2 d7:1 d8:1 p3_5:0]`
+
+## C18 — word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=012120102  U[u0:2 u2:0 u4:1]  D[d5:2 d6:2 d7:1 d8:1 p1_3:0]`
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012120201  U[u0:2 u2:0 u4:1]  D[d5:1 d6:1 d7:2 d8:2 p1_3:0]`
+
+## C19 — word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8]
+
+2 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=012120201  U[u0:2 u2:0 u5:1]  D[d1:0 d3:0 d7:2 d8:2 p4_6:1]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012120202  U[u0:2 u2:0 u5:1]  D[d1:0 d3:0 d7:1 d8:1 p4_6:1]`
+
+## C20 — word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=012120201  U[u0:2 u2:0 u5:1]  D[d4:1 d6:1 d7:2 d8:2 p1_3:0]`
+
+## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7]
+
+3 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `1122` → canonical `0011`  ·  `A=010202101  U[u0:2 u2:1 u5:0]  D[d3:1 d4:1 d6:2 d7:2 p1_8:2]`
+- face apexes (raw labels) `1100` → canonical `0011`  ·  `A=010202121  U[u0:2 u2:1 u5:0]  D[d3:1 d4:1 d6:0 d7:0 p1_8:2]`
+- face apexes (raw labels) `0022` → canonical `0011`  ·  `A=010212101  U[u0:2 u2:1 u5:0]  D[d3:0 d4:0 d6:2 d7:2 p1_8:2]`
+
+## C22 — word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8]
+
+2 colouring(s) with down-apex sequence `0011`:
+
+- face apexes (raw labels) `2211` → canonical `0011`  ·  `A=010120202  U[u0:2 u3:0 u6:1]  D[d1:2 d2:2 d4:1 d8:1 p5_7:1]`
+- face apexes (raw labels) `0011` → canonical `0011`  ·  `A=012120202  U[u0:2 u3:0 u6:1]  D[d1:0 d2:0 d4:1 d8:1 p5_7:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0101.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0101.md
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@@ -0,0 +1,124 @@
+# Inner-face down-apex sequence `0101`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1.
+- Realised by **12** of 23 configs (M(T), inner face).
+- **53** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_0101.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8]
+
+5 colouring(s) with down-apex sequence `0101`:
+
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=010121021  U[u0:2 u1:2 u2:2 u3:0 u6:1]  D[d4:0 d5:2 d7:0 d8:2]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=010212012  U[u0:2 u1:2 u2:1 u3:0 u6:2]  D[d4:0 d5:1 d7:0 d8:1]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=012012012  U[u0:2 u1:0 u2:1 u3:2 u6:2]  D[d4:0 d5:1 d7:0 d8:1]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=012021021  U[u0:2 u1:0 u2:1 u3:1 u6:1]  D[d4:0 d5:2 d7:0 d8:2]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=012121021  U[u0:2 u1:0 u2:0 u3:0 u6:1]  D[d4:0 d5:2 d7:0 d8:2]`
+
+## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8]
+
+9 colouring(s) with down-apex sequence `0101`:
+
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=010120121  U[u0:2 u1:2 u2:2 u4:1 u6:0]  D[d3:0 d5:2 d7:0 d8:2]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=010120212  U[u0:2 u1:2 u2:2 u4:1 u6:0]  D[d3:0 d5:1 d7:0 d8:1]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=010121021  U[u0:2 u1:2 u2:2 u4:0 u6:1]  D[d3:0 d5:2 d7:0 d8:2]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=010210121  U[u0:2 u1:2 u2:1 u4:2 u6:0]  D[d3:0 d5:2 d7:0 d8:2]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=010210212  U[u0:2 u1:2 u2:1 u4:2 u6:0]  D[d3:0 d5:1 d7:0 d8:1]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=010212012  U[u0:2 u1:2 u2:1 u4:0 u6:2]  D[d3:0 d5:1 d7:0 d8:1]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=012120121  U[u0:2 u1:0 u2:0 u4:1 u6:0]  D[d3:0 d5:2 d7:0 d8:2]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=012120212  U[u0:2 u1:0 u2:0 u4:1 u6:0]  D[d3:0 d5:1 d7:0 d8:1]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=012121021  U[u0:2 u1:0 u2:0 u4:0 u6:1]  D[d3:0 d5:2 d7:0 d8:2]`
+
+## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8]
+
+11 colouring(s) with down-apex sequence `0101`:
+
+- face apexes (raw labels) `2121` → canonical `0101`  ·  `A=010102012  U[u0:2 u1:2 u2:2 u4:1 u7:0]  D[d3:2 d5:1 d6:2 d8:1]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=010120121  U[u0:2 u1:2 u2:2 u4:1 u7:0]  D[d3:0 d5:2 d6:0 d8:2]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=010120212  U[u0:2 u1:2 u2:2 u4:1 u7:0]  D[d3:0 d5:1 d6:0 d8:1]`
+- face apexes (raw labels) `1212` → canonical `0101`  ·  `A=010201021  U[u0:2 u1:2 u2:1 u4:2 u7:0]  D[d3:1 d5:2 d6:1 d8:2]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=010210121  U[u0:2 u1:2 u2:1 u4:2 u7:0]  D[d3:0 d5:2 d6:0 d8:2]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=010210212  U[u0:2 u1:2 u2:1 u4:2 u7:0]  D[d3:0 d5:1 d6:0 d8:1]`
+- face apexes (raw labels) `2121` → canonical `0101`  ·  `A=012012012  U[u0:2 u1:0 u2:1 u4:0 u7:0]  D[d3:2 d5:1 d6:2 d8:1]`
+- face apexes (raw labels) `1212` → canonical `0101`  ·  `A=012021021  U[u0:2 u1:0 u2:1 u4:0 u7:0]  D[d3:1 d5:2 d6:1 d8:2]`
+- face apexes (raw labels) `2121` → canonical `0101`  ·  `A=012102012  U[u0:2 u1:0 u2:0 u4:1 u7:0]  D[d3:2 d5:1 d6:2 d8:1]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=012120121  U[u0:2 u1:0 u2:0 u4:1 u7:0]  D[d3:0 d5:2 d6:0 d8:2]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=012120212  U[u0:2 u1:0 u2:0 u4:1 u7:0]  D[d3:0 d5:1 d6:0 d8:1]`
+
+## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8]
+
+3 colouring(s) with down-apex sequence `0101`:
+
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=010120212  U[u0:2 u1:2 u2:2 u5:1 u6:0]  D[d3:0 d4:1 d7:0 d8:1]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=010210121  U[u0:2 u1:2 u2:1 u5:2 u6:0]  D[d3:0 d4:2 d7:0 d8:2]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=012120212  U[u0:2 u1:0 u2:0 u5:1 u6:0]  D[d3:0 d4:1 d7:0 d8:1]`
+
+## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8]
+
+5 colouring(s) with down-apex sequence `0101`:
+
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=012101021  U[u0:2 u1:0 u3:2 u4:2 u6:1]  D[d2:0 d5:2 d7:0 d8:2]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=012102012  U[u0:2 u1:0 u3:2 u4:1 u6:2]  D[d2:0 d5:1 d7:0 d8:1]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=012120121  U[u0:2 u1:0 u3:0 u4:1 u6:0]  D[d2:0 d5:2 d7:0 d8:2]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=012120212  U[u0:2 u1:0 u3:0 u4:1 u6:0]  D[d2:0 d5:1 d7:0 d8:1]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=012121021  U[u0:2 u1:0 u3:0 u4:0 u6:1]  D[d2:0 d5:2 d7:0 d8:2]`
+
+## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8]
+
+3 colouring(s) with down-apex sequence `0101`:
+
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=012101021  U[u0:2 u1:0 u3:2 u5:2 u6:1]  D[d2:0 d4:2 d7:0 d8:2]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=012102012  U[u0:2 u1:0 u3:2 u5:1 u6:2]  D[d2:0 d4:1 d7:0 d8:1]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=012120212  U[u0:2 u1:0 u3:0 u5:1 u6:0]  D[d2:0 d4:1 d7:0 d8:1]`
+
+## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8]
+
+8 colouring(s) with down-apex sequence `0101`:
+
+- face apexes (raw labels) `2121` → canonical `0101`  ·  `A=010102012  U[u0:2 u1:2 u3:2 u5:1 u7:0]  D[d2:2 d4:1 d6:2 d8:1]`
+- face apexes (raw labels) `2121` → canonical `0101`  ·  `A=010102102  U[u0:2 u1:2 u3:2 u5:0 u7:1]  D[d2:2 d4:1 d6:2 d8:1]`
+- face apexes (raw labels) `2121` → canonical `0101`  ·  `A=010120102  U[u0:2 u1:2 u3:0 u5:2 u7:1]  D[d2:2 d4:1 d6:2 d8:1]`
+- face apexes (raw labels) `1212` → canonical `0101`  ·  `A=010201021  U[u0:2 u1:2 u3:1 u5:2 u7:0]  D[d2:1 d4:2 d6:1 d8:2]`
+- face apexes (raw labels) `1212` → canonical `0101`  ·  `A=010201201  U[u0:2 u1:2 u3:1 u5:0 u7:2]  D[d2:1 d4:2 d6:1 d8:2]`
+- face apexes (raw labels) `1212` → canonical `0101`  ·  `A=010210201  U[u0:2 u1:2 u3:0 u5:1 u7:2]  D[d2:1 d4:2 d6:1 d8:2]`
+- face apexes (raw labels) `1212` → canonical `0101`  ·  `A=012010201  U[u0:2 u1:0 u3:2 u5:1 u7:2]  D[d2:1 d4:2 d6:1 d8:2]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=012120212  U[u0:2 u1:0 u3:0 u5:1 u7:0]  D[d2:0 d4:1 d6:0 d8:1]`
+
+## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8]
+
+1 colouring(s) with down-apex sequence `0101`:
+
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=012102021  U[u0:2 u1:0 u5:1]  D[d2:0 d3:2 d7:0 d8:2 p4_6:1]`
+
+## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7]
+
+1 colouring(s) with down-apex sequence `0101`:
+
+- face apexes (raw labels) `2020` → canonical `0101`  ·  `A=012012012  U[u0:2 u1:0 u5:1]  D[d3:2 d4:0 d6:2 d7:0 p2_8:1]`
+
+## C19 — word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8]
+
+3 colouring(s) with down-apex sequence `0101`:
+
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=012012121  U[u0:2 u2:1 u5:0]  D[d1:0 d3:2 d7:0 d8:2 p4_6:0]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=012021212  U[u0:2 u2:1 u5:0]  D[d1:0 d3:1 d7:0 d8:1 p4_6:0]`
+- face apexes (raw labels) `0202` → canonical `0101`  ·  `A=012102021  U[u0:2 u2:0 u5:1]  D[d1:0 d3:2 d7:0 d8:2 p4_6:1]`
+
+## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7]
+
+1 colouring(s) with down-apex sequence `0101`:
+
+- face apexes (raw labels) `1212` → canonical `0101`  ·  `A=010201201  U[u0:2 u2:1 u5:0]  D[d3:1 d4:2 d6:1 d7:2 p1_8:2]`
+
+## C22 — word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8]
+
+3 colouring(s) with down-apex sequence `0101`:
+
+- face apexes (raw labels) `2121` → canonical `0101`  ·  `A=010201212  U[u0:2 u3:1 u6:0]  D[d1:2 d2:1 d4:2 d8:1 p5_7:0]`
+- face apexes (raw labels) `2121` → canonical `0101`  ·  `A=010210202  U[u0:2 u3:0 u6:1]  D[d1:2 d2:1 d4:2 d8:1 p5_7:1]`
+- face apexes (raw labels) `0101` → canonical `0101`  ·  `A=012021212  U[u0:2 u3:1 u6:0]  D[d1:0 d2:1 d4:0 d8:1 p5_7:0]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0101.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0101.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.md
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+# Inner-face down-apex sequence `0110`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1.
+- Realised by **19** of 23 configs (M(T), inner face).
+- **73** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_0110.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8]
+
+10 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=010120121  U[u0:2 u1:2 u2:2 u3:0 u4:1]  D[d5:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=010120212  U[u0:2 u1:2 u2:2 u3:0 u4:1]  D[d5:1 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=010210121  U[u0:2 u1:2 u2:1 u3:0 u4:2]  D[d5:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=010210212  U[u0:2 u1:2 u2:1 u3:0 u4:2]  D[d5:1 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012010121  U[u0:2 u1:0 u2:1 u3:2 u4:2]  D[d5:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012010212  U[u0:2 u1:0 u2:1 u3:2 u4:2]  D[d5:1 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012020121  U[u0:2 u1:0 u2:1 u3:1 u4:1]  D[d5:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012020212  U[u0:2 u1:0 u2:1 u3:1 u4:1]  D[d5:1 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012120121  U[u0:2 u1:0 u2:0 u3:0 u4:1]  D[d5:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012120212  U[u0:2 u1:0 u2:0 u3:0 u4:1]  D[d5:1 d6:0 d7:0 d8:1]`
+
+## C01 — word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8]
+
+5 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=010120212  U[u0:2 u1:2 u2:2 u3:0 u5:1]  D[d4:1 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=010210121  U[u0:2 u1:2 u2:1 u3:0 u5:2]  D[d4:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012010121  U[u0:2 u1:0 u2:1 u3:2 u5:2]  D[d4:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012020212  U[u0:2 u1:0 u2:1 u3:1 u5:1]  D[d4:1 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012120212  U[u0:2 u1:0 u2:0 u3:0 u5:1]  D[d4:1 d6:0 d7:0 d8:1]`
+
+## C03 — word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8]
+
+7 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=010102121  U[u0:2 u1:2 u2:2 u4:1 u5:0]  D[d3:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=010201212  U[u0:2 u1:2 u2:1 u4:2 u5:0]  D[d3:1 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012010121  U[u0:2 u1:0 u2:1 u4:2 u5:2]  D[d3:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012012121  U[u0:2 u1:0 u2:1 u4:0 u5:0]  D[d3:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012020212  U[u0:2 u1:0 u2:1 u4:1 u5:1]  D[d3:1 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012021212  U[u0:2 u1:0 u2:1 u4:0 u5:0]  D[d3:1 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012102121  U[u0:2 u1:0 u2:0 u4:1 u5:0]  D[d3:2 d6:0 d7:0 d8:2]`
+
+## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8]
+
+5 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=010102121  U[u0:2 u1:2 u2:2 u4:1 u6:0]  D[d3:2 d5:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=010201212  U[u0:2 u1:2 u2:1 u4:2 u6:0]  D[d3:1 d5:0 d7:0 d8:1]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012012121  U[u0:2 u1:0 u2:1 u4:0 u6:0]  D[d3:2 d5:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012021212  U[u0:2 u1:0 u2:1 u4:0 u6:0]  D[d3:1 d5:0 d7:0 d8:1]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012102121  U[u0:2 u1:0 u2:0 u4:1 u6:0]  D[d3:2 d5:0 d7:0 d8:2]`
+
+## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8]
+
+12 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2112` → canonical `0110`  ·  `A=010102021  U[u0:2 u1:2 u2:2 u4:1 u7:0]  D[d3:2 d5:1 d6:1 d8:2]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=010102121  U[u0:2 u1:2 u2:2 u4:1 u7:0]  D[d3:2 d5:0 d6:0 d8:2]`
+- face apexes (raw labels) `1221` → canonical `0110`  ·  `A=010201012  U[u0:2 u1:2 u2:1 u4:2 u7:0]  D[d3:1 d5:2 d6:2 d8:1]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=010201212  U[u0:2 u1:2 u2:1 u4:2 u7:0]  D[d3:1 d5:0 d6:0 d8:1]`
+- face apexes (raw labels) `2112` → canonical `0110`  ·  `A=012010201  U[u0:2 u1:0 u2:1 u4:2 u7:2]  D[d3:2 d5:1 d6:1 d8:2]`
+- face apexes (raw labels) `2112` → canonical `0110`  ·  `A=012012021  U[u0:2 u1:0 u2:1 u4:0 u7:0]  D[d3:2 d5:1 d6:1 d8:2]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012012121  U[u0:2 u1:0 u2:1 u4:0 u7:0]  D[d3:2 d5:0 d6:0 d8:2]`
+- face apexes (raw labels) `1221` → canonical `0110`  ·  `A=012020102  U[u0:2 u1:0 u2:1 u4:1 u7:1]  D[d3:1 d5:2 d6:2 d8:1]`
+- face apexes (raw labels) `1221` → canonical `0110`  ·  `A=012021012  U[u0:2 u1:0 u2:1 u4:0 u7:0]  D[d3:1 d5:2 d6:2 d8:1]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012021212  U[u0:2 u1:0 u2:1 u4:0 u7:0]  D[d3:1 d5:0 d6:0 d8:1]`
+- face apexes (raw labels) `2112` → canonical `0110`  ·  `A=012102021  U[u0:2 u1:0 u2:0 u4:1 u7:0]  D[d3:2 d5:1 d6:1 d8:2]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012102121  U[u0:2 u1:0 u2:0 u4:1 u7:0]  D[d3:2 d5:0 d6:0 d8:2]`
+
+## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8]
+
+4 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012012021  U[u0:2 u1:0 u2:1 u5:1 u6:1]  D[d3:2 d4:0 d7:0 d8:2]`
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012012121  U[u0:2 u1:0 u2:1 u5:0 u6:0]  D[d3:2 d4:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012021012  U[u0:2 u1:0 u2:1 u5:2 u6:2]  D[d3:1 d4:0 d7:0 d8:1]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012021212  U[u0:2 u1:0 u2:1 u5:0 u6:0]  D[d3:1 d4:0 d7:0 d8:1]`
+
+## C07 — word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7]
+
+4 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2112` → canonical `0110`  ·  `A=012010201  U[u0:2 u1:0 u2:1]  D[d4:2 d5:1 d6:1 d7:2 p3_8:2]`
+- face apexes (raw labels) `0110` → canonical `0110`  ·  `A=012012021  U[u0:2 u1:0 u2:1]  D[d4:0 d5:1 d6:1 d7:0 p3_8:2]`
+- face apexes (raw labels) `1221` → canonical `0110`  ·  `A=012020102  U[u0:2 u1:0 u2:1]  D[d4:1 d5:2 d6:2 d7:1 p3_8:1]`
+- face apexes (raw labels) `0220` → canonical `0110`  ·  `A=012021012  U[u0:2 u1:0 u2:1]  D[d4:0 d5:2 d6:2 d7:0 p3_8:1]`
+
+## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8]
+
+3 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=010102121  U[u0:2 u1:2 u3:2 u4:1 u6:0]  D[d2:2 d5:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=010201212  U[u0:2 u1:2 u3:1 u4:2 u6:0]  D[d2:1 d5:0 d7:0 d8:1]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012021212  U[u0:2 u1:0 u3:1 u4:0 u6:0]  D[d2:1 d5:0 d7:0 d8:1]`
+
+## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8]
+
+5 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=010121021  U[u0:2 u1:2 u3:0 u5:2 u6:1]  D[d2:2 d4:0 d7:0 d8:2]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=010212012  U[u0:2 u1:2 u3:0 u5:1 u6:2]  D[d2:1 d4:0 d7:0 d8:1]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012012012  U[u0:2 u1:0 u3:2 u5:1 u6:2]  D[d2:1 d4:0 d7:0 d8:1]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012021012  U[u0:2 u1:0 u3:1 u5:2 u6:2]  D[d2:1 d4:0 d7:0 d8:1]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012021212  U[u0:2 u1:0 u3:1 u5:0 u6:0]  D[d2:1 d4:0 d7:0 d8:1]`
+
+## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8]
+
+6 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2112` → canonical `0110`  ·  `A=010102021  U[u0:2 u1:2 u3:2 u5:1 u7:0]  D[d2:2 d4:1 d6:1 d8:2]`
+- face apexes (raw labels) `2112` → canonical `0110`  ·  `A=010120201  U[u0:2 u1:2 u3:0 u5:1 u7:2]  D[d2:2 d4:1 d6:1 d8:2]`
+- face apexes (raw labels) `1221` → canonical `0110`  ·  `A=010201012  U[u0:2 u1:2 u3:1 u5:2 u7:0]  D[d2:1 d4:2 d6:2 d8:1]`
+- face apexes (raw labels) `1221` → canonical `0110`  ·  `A=010210102  U[u0:2 u1:2 u3:0 u5:2 u7:1]  D[d2:1 d4:2 d6:2 d8:1]`
+- face apexes (raw labels) `1221` → canonical `0110`  ·  `A=012010102  U[u0:2 u1:0 u3:2 u5:2 u7:1]  D[d2:1 d4:2 d6:2 d8:1]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012021212  U[u0:2 u1:0 u3:1 u5:0 u7:0]  D[d2:1 d4:0 d6:0 d8:1]`
+
+## C11 — word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012020121  U[u0:2 u1:0 u3:1]  D[d5:2 d6:0 d7:0 d8:2 p2_4:1]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012020212  U[u0:2 u1:0 u3:1]  D[d5:1 d6:0 d7:0 d8:1 p2_4:1]`
+
+## C12 — word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7]
+
+2 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `1221` → canonical `0110`  ·  `A=012020102  U[u0:2 u1:0 u3:1]  D[d4:1 d5:2 d6:2 d7:1 p2_8:1]`
+- face apexes (raw labels) `0220` → canonical `0110`  ·  `A=012021012  U[u0:2 u1:0 u3:1]  D[d4:0 d5:2 d6:2 d7:0 p2_8:1]`
+
+## C13 — word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012020212  U[u0:2 u1:0 u4:1]  D[d2:1 d6:0 d7:0 d8:1 p3_5:1]`
+
+## C14 — word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7]
+
+1 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `1221` → canonical `0110`  ·  `A=012020102  U[u0:2 u1:0 u4:1]  D[d3:1 d5:2 d6:2 d7:1 p2_8:1]`
+
+## C17 — word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=010212121  U[u0:2 u2:1 u4:0]  D[d1:2 d6:0 d7:0 d8:2 p3_5:0]`
+
+## C18 — word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=012120121  U[u0:2 u2:0 u4:1]  D[d5:2 d6:0 d7:0 d8:2 p1_3:0]`
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012120212  U[u0:2 u2:0 u4:1]  D[d5:1 d6:0 d7:0 d8:1 p1_3:0]`
+
+## C19 — word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8]
+
+1 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2002` → canonical `0110`  ·  `A=010212121  U[u0:2 u2:1 u5:0]  D[d1:2 d3:0 d7:0 d8:2 p4_6:0]`
+
+## C20 — word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `1001` → canonical `0110`  ·  `A=012120212  U[u0:2 u2:0 u5:1]  D[d4:1 d6:0 d7:0 d8:1 p1_3:0]`
+
+## C22 — word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8]
+
+1 colouring(s) with down-apex sequence `0110`:
+
+- face apexes (raw labels) `2112` → canonical `0110`  ·  `A=010202121  U[u0:2 u3:1 u6:0]  D[d1:2 d2:1 d4:1 d8:2 p5_7:0]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/summary.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/summary.md
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+# Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=4)
+
+Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 4 singleton down-tooth apexes: **23 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in increasing annular-edge order, and reduce it modulo colour permutation (NOT dihedral symmetry).
+
+- Total Kempe-balanced colourings (mod colour permutation): **298**.
+- Distinct canonical down-apex sequences overall: **4**.
+
+## Distinct canonical down-apex sequences
+
+| sequence | colour multiset | #configs realising | #colourings |
+|---|---|---|---|
+| `0000` | 4 | 21 | 64 |
+| `0011` | 2+2 | 23 | 108 |
+| `0101` | 2+2 | 12 | 53 |
+| `0110` | 2+2 | 19 | 73 |
+
+Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 4 apexes (m is even) every count must be **even**, so the only admissible colour multisets are 2+2, 4.
+
+## Step 4 — grouping configs by their set of unique down-apex sequences
+
+The 23 configs fall into **4** groups by the set of canonical down-apex sequences they realise:
+
+| #configs | set of down-apex sequences | config ids |
+|---|---|---|
+| 11 | { `0000`, `0011`, `0110` } | C00, C01, C03, C07, C11, C12, C13, C14, C17, C18, C20 |
+| 6 | { `0000`, `0011`, `0101`, `0110` } | C04, C05, C06, C08, C09, C10 |
+| 4 | { `0000`, `0011`, `0101` } | C02, C15, C16, C21 |
+| 2 | { `0011`, `0101`, `0110` } | C19, C22 |
+
+## Config atlas (ids)
+
+| id | word / bites / face / apexes | #Kempe-balanced | down-apex sequence set |
+|---|---|---|---|
+| C00 | word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8] | 30 | { `0000`, `0011`, `0110` } |
+| C01 | word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8] | 15 | { `0000`, `0011`, `0110` } |
+| C02 | word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8] | 25 | { `0000`, `0011`, `0101` } |
+| C03 | word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8] | 21 | { `0000`, `0011`, `0110` } |
+| C04 | word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8] | 24 | { `0000`, `0011`, `0101`, `0110` } |
+| C05 | word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8] | 28 | { `0000`, `0011`, `0101`, `0110` } |
+| C06 | word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8] | 27 | { `0000`, `0011`, `0101`, `0110` } |
+| C07 | word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7] | 12 | { `0000`, `0011`, `0110` } |
+| C08 | word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8] | 22 | { `0000`, `0011`, `0101`, `0110` } |
+| C09 | word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8] | 18 | { `0000`, `0011`, `0101`, `0110` } |
+| C10 | word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8] | 19 | { `0000`, `0011`, `0101`, `0110` } |
+| C11 | word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8] | 6 | { `0000`, `0011`, `0110` } |
+| C12 | word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7] | 6 | { `0000`, `0011`, `0110` } |
+| C13 | word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8] | 3 | { `0000`, `0011`, `0110` } |
+| C14 | word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7] | 3 | { `0000`, `0011`, `0110` } |
+| C15 | word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8] | 5 | { `0000`, `0011`, `0101` } |
+| C16 | word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7] | 5 | { `0000`, `0011`, `0101` } |
+| C17 | word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8] | 3 | { `0000`, `0011`, `0110` } |
+| C18 | word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8] | 6 | { `0000`, `0011`, `0110` } |
+| C19 | word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8] | 6 | { `0011`, `0101`, `0110` } |
+| C20 | word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8] | 3 | { `0000`, `0011`, `0110` } |
+| C21 | word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7] | 5 | { `0000`, `0011`, `0101` } |
+| C22 | word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8] | 6 | { `0011`, `0101`, `0110` } |
+
+## Per-sequence notes
+
+- [`0000`](seq_0000.md) — figure `seq_0000.png`
+- [`0011`](seq_0011.md) — figure `seq_0011.png`
+- [`0101`](seq_0101.md) — figure `seq_0101.png`
+- [`0110`](seq_0110.md) — figure `seq_0110.png`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.md
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+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.md
@@ -0,0 +1,91 @@
+# Inner-face down-apex sequence `00012`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
+
+- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
+- Realised by **10** of 10 configs (M(T), inner face).
+- **30** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_00012.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
+
+6 colouring(s) with down-apex sequence `00012`:
+
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=010101012  U[u0:2 u1:2 u2:2 u3:2]  D[d4:2 d5:2 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=010102021  U[u0:2 u1:2 u2:2 u3:2]  D[d4:1 d5:1 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=010201012  U[u0:2 u1:2 u2:1 u3:1]  D[d4:2 d5:2 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=010202021  U[u0:2 u1:2 u2:1 u3:1]  D[d4:1 d5:1 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=012101012  U[u0:2 u1:0 u2:0 u3:2]  D[d4:2 d5:2 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=012102021  U[u0:2 u1:0 u2:0 u3:2]  D[d4:1 d5:1 d6:1 d7:0 d8:2]`
+
+## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `00012`:
+
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=010101012  U[u0:2 u1:2 u2:2 u4:2]  D[d3:2 d5:2 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=010202021  U[u0:2 u1:2 u2:1 u4:1]  D[d3:1 d5:1 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=012101012  U[u0:2 u1:0 u2:0 u4:2]  D[d3:2 d5:2 d6:2 d7:0 d8:1]`
+
+## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `00012`:
+
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=010101012  U[u0:2 u1:2 u2:2 u5:2]  D[d3:2 d4:2 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=010202021  U[u0:2 u1:2 u2:1 u5:1]  D[d3:1 d4:1 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=012101012  U[u0:2 u1:0 u2:0 u5:2]  D[d3:2 d4:2 d6:2 d7:0 d8:1]`
+
+## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
+
+5 colouring(s) with down-apex sequence `00012`:
+
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=010101012  U[u0:2 u1:2 u3:2 u4:2]  D[d2:2 d5:2 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=010121012  U[u0:2 u1:2 u3:0 u4:0]  D[d2:2 d5:2 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=010202021  U[u0:2 u1:2 u3:1 u4:1]  D[d2:1 d5:1 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=010212021  U[u0:2 u1:2 u3:0 u4:0]  D[d2:1 d5:1 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=012012021  U[u0:2 u1:0 u3:2 u4:0]  D[d2:1 d5:1 d6:1 d7:0 d8:2]`
+
+## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `00012`:
+
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=010101012  U[u0:2 u1:2 u3:2 u5:2]  D[d2:2 d4:2 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=010202021  U[u0:2 u1:2 u3:1 u5:1]  D[d2:1 d4:1 d6:1 d7:0 d8:2]`
+
+## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
+
+2 colouring(s) with down-apex sequence `00012`:
+
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=010101012  U[u0:2 u1:2 u3:2 u6:2]  D[d2:2 d4:2 d5:2 d7:0 d8:1]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=010202021  U[u0:2 u1:2 u3:1 u6:1]  D[d2:1 d4:1 d5:1 d7:0 d8:2]`
+
+## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
+
+1 colouring(s) with down-apex sequence `00012`:
+
+- face apexes (raw labels) `00012` → canonical `00012`  ·  `A=012121201  U[u0:2 u1:0 u3:0 u7:2]  D[d2:0 d4:0 d5:0 d6:1 d8:2]`
+
+## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
+
+5 colouring(s) with down-apex sequence `00012`:
+
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=010101012  U[u0:2 u1:2 u4:2 u5:2]  D[d2:2 d3:2 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=010102012  U[u0:2 u1:2 u4:1 u5:1]  D[d2:2 d3:2 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=010201021  U[u0:2 u1:2 u4:2 u5:2]  D[d2:1 d3:1 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=010202021  U[u0:2 u1:2 u4:1 u5:1]  D[d2:1 d3:1 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=012021021  U[u0:2 u1:0 u4:0 u5:2]  D[d2:1 d3:1 d6:1 d7:0 d8:2]`
+
+## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
+
+2 colouring(s) with down-apex sequence `00012`:
+
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=010101012  U[u0:2 u1:2 u4:2 u6:2]  D[d2:2 d3:2 d5:2 d7:0 d8:1]`
+- face apexes (raw labels) `11102` → canonical `00012`  ·  `A=010202021  U[u0:2 u1:2 u4:1 u6:1]  D[d2:1 d3:1 d5:1 d7:0 d8:2]`
+
+## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `00012`:
+
+- face apexes (raw labels) `22201` → canonical `00012`  ·  `A=010101012  U[u0:2 u2:2 u4:2 u6:2]  D[d1:2 d3:2 d5:2 d7:0 d8:1]`
+
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.md
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+# Inner-face down-apex sequence `00102`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
+
+- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
+- Realised by **6** of 10 configs (M(T), inner face).
+- **13** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_00102.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
+
+6 colouring(s) with down-apex sequence `00102`:
+
+- face apexes (raw labels) `00201` → canonical `00102`  ·  `A=010121012  U[u0:2 u1:2 u2:2 u5:2]  D[d3:0 d4:0 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `00102` → canonical `00102`  ·  `A=010121021  U[u0:2 u1:2 u2:2 u5:2]  D[d3:0 d4:0 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `00201` → canonical `00102`  ·  `A=010212012  U[u0:2 u1:2 u2:1 u5:1]  D[d3:0 d4:0 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `00102` → canonical `00102`  ·  `A=010212021  U[u0:2 u1:2 u2:1 u5:1]  D[d3:0 d4:0 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `00201` → canonical `00102`  ·  `A=012121012  U[u0:2 u1:0 u2:0 u5:2]  D[d3:0 d4:0 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `00102` → canonical `00102`  ·  `A=012121021  U[u0:2 u1:0 u2:0 u5:2]  D[d3:0 d4:0 d6:1 d7:0 d8:2]`
+
+## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `00102`:
+
+- face apexes (raw labels) `00201` → canonical `00102`  ·  `A=012121012  U[u0:2 u1:0 u3:0 u5:2]  D[d2:0 d4:0 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `00102` → canonical `00102`  ·  `A=012121021  U[u0:2 u1:0 u3:0 u5:2]  D[d2:0 d4:0 d6:1 d7:0 d8:2]`
+
+## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `00102`:
+
+- face apexes (raw labels) `00201` → canonical `00102`  ·  `A=012121012  U[u0:2 u1:0 u3:0 u6:2]  D[d2:0 d4:0 d5:2 d7:0 d8:1]`
+
+## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `00102`:
+
+- face apexes (raw labels) `00201` → canonical `00102`  ·  `A=012121012  U[u0:2 u1:0 u4:0 u5:2]  D[d2:0 d3:0 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `00102` → canonical `00102`  ·  `A=012121021  U[u0:2 u1:0 u4:0 u5:2]  D[d2:0 d3:0 d6:1 d7:0 d8:2]`
+
+## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `00102`:
+
+- face apexes (raw labels) `00201` → canonical `00102`  ·  `A=012121012  U[u0:2 u1:0 u4:0 u6:2]  D[d2:0 d3:0 d5:2 d7:0 d8:1]`
+
+## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `00102`:
+
+- face apexes (raw labels) `00201` → canonical `00102`  ·  `A=012121012  U[u0:2 u2:0 u4:0 u6:2]  D[d1:0 d3:0 d5:2 d7:0 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.png
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.md
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@@ -0,0 +1,86 @@
+# Inner-face down-apex sequence `00120`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
+
+- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
+- Realised by **9** of 10 configs (M(T), inner face).
+- **30** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_00120.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
+
+6 colouring(s) with down-apex sequence `00120`:
+
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=010101021  U[u0:2 u1:2 u2:2 u3:2]  D[d4:2 d5:2 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=010102012  U[u0:2 u1:2 u2:2 u3:2]  D[d4:1 d5:1 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=010201021  U[u0:2 u1:2 u2:1 u3:1]  D[d4:2 d5:2 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=010202012  U[u0:2 u1:2 u2:1 u3:1]  D[d4:1 d5:1 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=012101021  U[u0:2 u1:0 u2:0 u3:2]  D[d4:2 d5:2 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=012102012  U[u0:2 u1:0 u2:0 u3:2]  D[d4:1 d5:1 d6:2 d7:0 d8:1]`
+
+## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `00120`:
+
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=010101021  U[u0:2 u1:2 u2:2 u4:2]  D[d3:2 d5:2 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=010202012  U[u0:2 u1:2 u2:1 u4:1]  D[d3:1 d5:1 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=012101021  U[u0:2 u1:0 u2:0 u4:2]  D[d3:2 d5:2 d6:1 d7:0 d8:2]`
+
+## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `00120`:
+
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=010101021  U[u0:2 u1:2 u2:2 u5:2]  D[d3:2 d4:2 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=010202012  U[u0:2 u1:2 u2:1 u5:1]  D[d3:1 d4:1 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=012101021  U[u0:2 u1:0 u2:0 u5:2]  D[d3:2 d4:2 d6:1 d7:0 d8:2]`
+
+## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
+
+5 colouring(s) with down-apex sequence `00120`:
+
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=010101021  U[u0:2 u1:2 u3:2 u4:2]  D[d2:2 d5:2 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=010121021  U[u0:2 u1:2 u3:0 u4:0]  D[d2:2 d5:2 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=010202012  U[u0:2 u1:2 u3:1 u4:1]  D[d2:1 d5:1 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=010212012  U[u0:2 u1:2 u3:0 u4:0]  D[d2:1 d5:1 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=012012012  U[u0:2 u1:0 u3:2 u4:0]  D[d2:1 d5:1 d6:2 d7:0 d8:1]`
+
+## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `00120`:
+
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=010101021  U[u0:2 u1:2 u3:2 u5:2]  D[d2:2 d4:2 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=010202012  U[u0:2 u1:2 u3:1 u5:1]  D[d2:1 d4:1 d6:2 d7:0 d8:1]`
+
+## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
+
+2 colouring(s) with down-apex sequence `00120`:
+
+- face apexes (raw labels) `22012` → canonical `00120`  ·  `A=010101201  U[u0:2 u1:2 u3:2 u7:2]  D[d2:2 d4:2 d5:0 d6:1 d8:2]`
+- face apexes (raw labels) `11021` → canonical `00120`  ·  `A=010202102  U[u0:2 u1:2 u3:1 u7:1]  D[d2:1 d4:1 d5:0 d6:2 d8:1]`
+
+## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
+
+5 colouring(s) with down-apex sequence `00120`:
+
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=010101021  U[u0:2 u1:2 u4:2 u5:2]  D[d2:2 d3:2 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=010102021  U[u0:2 u1:2 u4:1 u5:1]  D[d2:2 d3:2 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=010201012  U[u0:2 u1:2 u4:2 u5:2]  D[d2:1 d3:1 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=010202012  U[u0:2 u1:2 u4:1 u5:1]  D[d2:1 d3:1 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=012021012  U[u0:2 u1:0 u4:0 u5:2]  D[d2:1 d3:1 d6:2 d7:0 d8:1]`
+
+## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
+
+3 colouring(s) with down-apex sequence `00120`:
+
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=010102021  U[u0:2 u1:2 u4:1 u6:1]  D[d2:2 d3:2 d5:1 d7:0 d8:2]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=010201012  U[u0:2 u1:2 u4:2 u6:2]  D[d2:1 d3:1 d5:2 d7:0 d8:1]`
+- face apexes (raw labels) `11201` → canonical `00120`  ·  `A=012021012  U[u0:2 u1:0 u4:0 u6:2]  D[d2:1 d3:1 d5:2 d7:0 d8:1]`
+
+## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `00120`:
+
+- face apexes (raw labels) `22102` → canonical `00120`  ·  `A=010102021  U[u0:2 u2:2 u4:1 u6:1]  D[d1:2 d3:2 d5:1 d7:0 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.md
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@@ -0,0 +1,62 @@
+# Inner-face down-apex sequence `01002`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
+
+- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
+- Realised by **7** of 10 configs (M(T), inner face).
+- **16** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_01002.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `01002`:
+
+- face apexes (raw labels) `01002` → canonical `01002`  ·  `A=010120121  U[u0:2 u1:2 u2:2 u5:2]  D[d3:0 d4:1 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `02001` → canonical `01002`  ·  `A=010210212  U[u0:2 u1:2 u2:1 u5:1]  D[d3:0 d4:2 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `01002` → canonical `01002`  ·  `A=012120121  U[u0:2 u1:0 u2:0 u5:2]  D[d3:0 d4:1 d6:0 d7:0 d8:2]`
+
+## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `01002`:
+
+- face apexes (raw labels) `02001` → canonical `01002`  ·  `A=012101212  U[u0:2 u1:0 u3:2 u5:0]  D[d2:0 d4:2 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `01002` → canonical `01002`  ·  `A=012102121  U[u0:2 u1:0 u3:2 u5:0]  D[d2:0 d4:1 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `01002` → canonical `01002`  ·  `A=012120121  U[u0:2 u1:0 u3:0 u5:2]  D[d2:0 d4:1 d6:0 d7:0 d8:2]`
+
+## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
+
+2 colouring(s) with down-apex sequence `01002`:
+
+- face apexes (raw labels) `02001` → canonical `01002`  ·  `A=012101212  U[u0:2 u1:0 u3:2 u6:0]  D[d2:0 d4:2 d5:0 d7:0 d8:1]`
+- face apexes (raw labels) `01002` → canonical `01002`  ·  `A=012102121  U[u0:2 u1:0 u3:2 u6:0]  D[d2:0 d4:1 d5:0 d7:0 d8:2]`
+
+## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
+
+5 colouring(s) with down-apex sequence `01002`:
+
+- face apexes (raw labels) `20221` → canonical `01002`  ·  `A=010121012  U[u0:2 u1:2 u3:0 u7:0]  D[d2:2 d4:0 d5:2 d6:2 d8:1]`
+- face apexes (raw labels) `10112` → canonical `01002`  ·  `A=010212021  U[u0:2 u1:2 u3:0 u7:0]  D[d2:1 d4:0 d5:1 d6:1 d8:2]`
+- face apexes (raw labels) `10112` → canonical `01002`  ·  `A=012012021  U[u0:2 u1:0 u3:2 u7:0]  D[d2:1 d4:0 d5:1 d6:1 d8:2]`
+- face apexes (raw labels) `02001` → canonical `01002`  ·  `A=012101212  U[u0:2 u1:0 u3:2 u7:0]  D[d2:0 d4:2 d5:0 d6:0 d8:1]`
+- face apexes (raw labels) `01002` → canonical `01002`  ·  `A=012102121  U[u0:2 u1:0 u3:2 u7:0]  D[d2:0 d4:1 d5:0 d6:0 d8:2]`
+
+## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `01002`:
+
+- face apexes (raw labels) `02001` → canonical `01002`  ·  `A=012101212  U[u0:2 u1:0 u4:2 u5:0]  D[d2:0 d3:2 d6:0 d7:0 d8:1]`
+
+## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `01002`:
+
+- face apexes (raw labels) `02001` → canonical `01002`  ·  `A=012101212  U[u0:2 u1:0 u4:2 u6:0]  D[d2:0 d3:2 d5:0 d7:0 d8:1]`
+
+## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `01002`:
+
+- face apexes (raw labels) `02001` → canonical `01002`  ·  `A=012101212  U[u0:2 u2:0 u4:2 u6:0]  D[d1:0 d3:2 d5:0 d7:0 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.md
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@@ -0,0 +1,43 @@
+# Inner-face down-apex sequence `01020`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
+
+- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
+- Realised by **4** of 10 configs (M(T), inner face).
+- **12** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_01020.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
+
+3 colouring(s) with down-apex sequence `01020`:
+
+- face apexes (raw labels) `21202` → canonical `01020`  ·  `A=010120121  U[u0:2 u1:2 u3:0 u6:0]  D[d2:2 d4:1 d5:2 d7:0 d8:2]`
+- face apexes (raw labels) `12101` → canonical `01020`  ·  `A=010210212  U[u0:2 u1:2 u3:0 u6:0]  D[d2:1 d4:2 d5:1 d7:0 d8:1]`
+- face apexes (raw labels) `12101` → canonical `01020`  ·  `A=012010212  U[u0:2 u1:0 u3:2 u6:0]  D[d2:1 d4:2 d5:1 d7:0 d8:1]`
+
+## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
+
+6 colouring(s) with down-apex sequence `01020`:
+
+- face apexes (raw labels) `21202` → canonical `01020`  ·  `A=010120121  U[u0:2 u1:2 u3:0 u7:0]  D[d2:2 d4:1 d5:2 d6:0 d8:2]`
+- face apexes (raw labels) `20212` → canonical `01020`  ·  `A=010121021  U[u0:2 u1:2 u3:0 u7:0]  D[d2:2 d4:0 d5:2 d6:1 d8:2]`
+- face apexes (raw labels) `12101` → canonical `01020`  ·  `A=010210212  U[u0:2 u1:2 u3:0 u7:0]  D[d2:1 d4:2 d5:1 d6:0 d8:1]`
+- face apexes (raw labels) `10121` → canonical `01020`  ·  `A=010212012  U[u0:2 u1:2 u3:0 u7:0]  D[d2:1 d4:0 d5:1 d6:2 d8:1]`
+- face apexes (raw labels) `12101` → canonical `01020`  ·  `A=012010212  U[u0:2 u1:0 u3:2 u7:0]  D[d2:1 d4:2 d5:1 d6:0 d8:1]`
+- face apexes (raw labels) `10121` → canonical `01020`  ·  `A=012012012  U[u0:2 u1:0 u3:2 u7:0]  D[d2:1 d4:0 d5:1 d6:2 d8:1]`
+
+## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
+
+2 colouring(s) with down-apex sequence `01020`:
+
+- face apexes (raw labels) `12101` → canonical `01020`  ·  `A=012010212  U[u0:2 u1:0 u4:2 u6:0]  D[d2:1 d3:2 d5:1 d7:0 d8:1]`
+- face apexes (raw labels) `12101` → canonical `01020`  ·  `A=012012012  U[u0:2 u1:0 u4:0 u6:2]  D[d2:1 d3:2 d5:1 d7:0 d8:1]`
+
+## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `01020`:
+
+- face apexes (raw labels) `21202` → canonical `01020`  ·  `A=010201021  U[u0:2 u2:1 u4:2 u6:1]  D[d1:2 d3:1 d5:2 d7:0 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.png
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.md
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@@ -0,0 +1,89 @@
+# Inner-face down-apex sequence `01112`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
+
+- Colour multiset: 1×colour0, 3×colour1, 1×colour2.
+- Realised by **9** of 10 configs (M(T), inner face).
+- **33** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_01112.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
+
+6 colouring(s) with down-apex sequence `01112`:
+
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=010101212  U[u0:2 u1:2 u2:2 u3:2]  D[d4:2 d5:0 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=010102121  U[u0:2 u1:2 u2:2 u3:2]  D[d4:1 d5:0 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=010201212  U[u0:2 u1:2 u2:1 u3:1]  D[d4:2 d5:0 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=010202121  U[u0:2 u1:2 u2:1 u3:1]  D[d4:1 d5:0 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=012101212  U[u0:2 u1:0 u2:0 u3:2]  D[d4:2 d5:0 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=012102121  U[u0:2 u1:0 u2:0 u3:2]  D[d4:1 d5:0 d6:0 d7:0 d8:2]`
+
+## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `01112`:
+
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=010101212  U[u0:2 u1:2 u2:2 u4:2]  D[d3:2 d5:0 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=010202121  U[u0:2 u1:2 u2:1 u4:1]  D[d3:1 d5:0 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=012101212  U[u0:2 u1:0 u2:0 u4:2]  D[d3:2 d5:0 d6:0 d7:0 d8:1]`
+
+## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
+
+5 colouring(s) with down-apex sequence `01112`:
+
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=010101212  U[u0:2 u1:2 u3:2 u4:2]  D[d2:2 d5:0 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=010121212  U[u0:2 u1:2 u3:0 u4:0]  D[d2:2 d5:0 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=010202121  U[u0:2 u1:2 u3:1 u4:1]  D[d2:1 d5:0 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=010212121  U[u0:2 u1:2 u3:0 u4:0]  D[d2:1 d5:0 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=012012121  U[u0:2 u1:0 u3:2 u4:0]  D[d2:1 d5:0 d6:0 d7:0 d8:2]`
+
+## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `01112`:
+
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=010121212  U[u0:2 u1:2 u3:0 u5:0]  D[d2:2 d4:0 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=010212121  U[u0:2 u1:2 u3:0 u5:0]  D[d2:1 d4:0 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=012012121  U[u0:2 u1:0 u3:2 u5:0]  D[d2:1 d4:0 d6:0 d7:0 d8:2]`
+
+## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
+
+3 colouring(s) with down-apex sequence `01112`:
+
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=010121212  U[u0:2 u1:2 u3:0 u6:0]  D[d2:2 d4:0 d5:0 d7:0 d8:1]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=010212121  U[u0:2 u1:2 u3:0 u6:0]  D[d2:1 d4:0 d5:0 d7:0 d8:2]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=012012121  U[u0:2 u1:0 u3:2 u6:0]  D[d2:1 d4:0 d5:0 d7:0 d8:2]`
+
+## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
+
+6 colouring(s) with down-apex sequence `01112`:
+
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=010121212  U[u0:2 u1:2 u3:0 u7:0]  D[d2:2 d4:0 d5:0 d6:0 d8:1]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=010212121  U[u0:2 u1:2 u3:0 u7:0]  D[d2:1 d4:0 d5:0 d6:0 d8:2]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=012012121  U[u0:2 u1:0 u3:2 u7:0]  D[d2:1 d4:0 d5:0 d6:0 d8:2]`
+- face apexes (raw labels) `02221` → canonical `01112`  ·  `A=012101012  U[u0:2 u1:0 u3:2 u7:0]  D[d2:0 d4:2 d5:2 d6:2 d8:1]`
+- face apexes (raw labels) `01112` → canonical `01112`  ·  `A=012102021  U[u0:2 u1:0 u3:2 u7:0]  D[d2:0 d4:1 d5:1 d6:1 d8:2]`
+- face apexes (raw labels) `01112` → canonical `01112`  ·  `A=012120201  U[u0:2 u1:0 u3:0 u7:2]  D[d2:0 d4:1 d5:1 d6:1 d8:2]`
+
+## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
+
+4 colouring(s) with down-apex sequence `01112`:
+
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=010120212  U[u0:2 u1:2 u4:1 u5:1]  D[d2:2 d3:0 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=010121212  U[u0:2 u1:2 u4:0 u5:0]  D[d2:2 d3:0 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=010210121  U[u0:2 u1:2 u4:2 u5:2]  D[d2:1 d3:0 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=010212121  U[u0:2 u1:2 u4:0 u5:0]  D[d2:1 d3:0 d6:0 d7:0 d8:2]`
+
+## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
+
+2 colouring(s) with down-apex sequence `01112`:
+
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=010121212  U[u0:2 u1:2 u4:0 u6:0]  D[d2:2 d3:0 d5:0 d7:0 d8:1]`
+- face apexes (raw labels) `10002` → canonical `01112`  ·  `A=010212121  U[u0:2 u1:2 u4:0 u6:0]  D[d2:1 d3:0 d5:0 d7:0 d8:2]`
+
+## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `01112`:
+
+- face apexes (raw labels) `20001` → canonical `01112`  ·  `A=010121212  U[u0:2 u2:2 u4:0 u6:0]  D[d1:2 d3:0 d5:0 d7:0 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.md
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+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.md
@@ -0,0 +1,43 @@
+# Inner-face down-apex sequence `01121`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
+
+- Colour multiset: 1×colour0, 3×colour1, 1×colour2.
+- Realised by **4** of 10 configs (M(T), inner face).
+- **12** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_01121.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
+
+5 colouring(s) with down-apex sequence `01121`:
+
+- face apexes (raw labels) `21101` → canonical `01121`  ·  `A=010102012  U[u0:2 u1:2 u3:2 u6:2]  D[d2:2 d4:1 d5:1 d7:0 d8:1]`
+- face apexes (raw labels) `21101` → canonical `01121`  ·  `A=010120212  U[u0:2 u1:2 u3:0 u6:0]  D[d2:2 d4:1 d5:1 d7:0 d8:1]`
+- face apexes (raw labels) `12202` → canonical `01121`  ·  `A=010201021  U[u0:2 u1:2 u3:1 u6:1]  D[d2:1 d4:2 d5:2 d7:0 d8:2]`
+- face apexes (raw labels) `12202` → canonical `01121`  ·  `A=010210121  U[u0:2 u1:2 u3:0 u6:0]  D[d2:1 d4:2 d5:2 d7:0 d8:2]`
+- face apexes (raw labels) `12202` → canonical `01121`  ·  `A=012010121  U[u0:2 u1:0 u3:2 u6:0]  D[d2:1 d4:2 d5:2 d7:0 d8:2]`
+
+## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
+
+5 colouring(s) with down-apex sequence `01121`:
+
+- face apexes (raw labels) `21101` → canonical `01121`  ·  `A=010120212  U[u0:2 u1:2 u3:0 u7:0]  D[d2:2 d4:1 d5:1 d6:0 d8:1]`
+- face apexes (raw labels) `12202` → canonical `01121`  ·  `A=010210121  U[u0:2 u1:2 u3:0 u7:0]  D[d2:1 d4:2 d5:2 d6:0 d8:2]`
+- face apexes (raw labels) `12202` → canonical `01121`  ·  `A=012010121  U[u0:2 u1:0 u3:2 u7:0]  D[d2:1 d4:2 d5:2 d6:0 d8:2]`
+- face apexes (raw labels) `02212` → canonical `01121`  ·  `A=012101021  U[u0:2 u1:0 u3:2 u7:0]  D[d2:0 d4:2 d5:2 d6:1 d8:2]`
+- face apexes (raw labels) `01121` → canonical `01121`  ·  `A=012102012  U[u0:2 u1:0 u3:2 u7:0]  D[d2:0 d4:1 d5:1 d6:2 d8:1]`
+
+## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `01121`:
+
+- face apexes (raw labels) `12202` → canonical `01121`  ·  `A=012010121  U[u0:2 u1:0 u4:2 u6:0]  D[d2:1 d3:2 d5:2 d7:0 d8:2]`
+
+## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `01121`:
+
+- face apexes (raw labels) `21101` → canonical `01121`  ·  `A=010202012  U[u0:2 u2:1 u4:1 u6:2]  D[d1:2 d3:1 d5:1 d7:0 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.md
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+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.md
@@ -0,0 +1,88 @@
+# Inner-face down-apex sequence `01200`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
+
+- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
+- Realised by **9** of 10 configs (M(T), inner face).
+- **32** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_01200.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
+
+6 colouring(s) with down-apex sequence `01200`:
+
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=010101201  U[u0:2 u1:2 u2:2 u3:2]  D[d4:2 d5:0 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `10211` → canonical `01200`  ·  `A=010102102  U[u0:2 u1:2 u2:2 u3:2]  D[d4:1 d5:0 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=010201201  U[u0:2 u1:2 u2:1 u3:1]  D[d4:2 d5:0 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `10211` → canonical `01200`  ·  `A=010202102  U[u0:2 u1:2 u2:1 u3:1]  D[d4:1 d5:0 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=012101201  U[u0:2 u1:0 u2:0 u3:2]  D[d4:2 d5:0 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `10211` → canonical `01200`  ·  `A=012102102  U[u0:2 u1:0 u2:0 u3:2]  D[d4:1 d5:0 d6:2 d7:1 d8:1]`
+
+## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `01200`:
+
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=010101201  U[u0:2 u1:2 u2:2 u4:2]  D[d3:2 d5:0 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `10211` → canonical `01200`  ·  `A=010202102  U[u0:2 u1:2 u2:1 u4:1]  D[d3:1 d5:0 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=012101201  U[u0:2 u1:0 u2:0 u4:2]  D[d3:2 d5:0 d6:1 d7:2 d8:2]`
+
+## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
+
+5 colouring(s) with down-apex sequence `01200`:
+
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=010101201  U[u0:2 u1:2 u3:2 u4:2]  D[d2:2 d5:0 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=010121201  U[u0:2 u1:2 u3:0 u4:0]  D[d2:2 d5:0 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `10211` → canonical `01200`  ·  `A=010202102  U[u0:2 u1:2 u3:1 u4:1]  D[d2:1 d5:0 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `10211` → canonical `01200`  ·  `A=010212102  U[u0:2 u1:2 u3:0 u4:0]  D[d2:1 d5:0 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `10211` → canonical `01200`  ·  `A=012012102  U[u0:2 u1:0 u3:2 u4:0]  D[d2:1 d5:0 d6:2 d7:1 d8:1]`
+
+## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `01200`:
+
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=010121201  U[u0:2 u1:2 u3:0 u5:0]  D[d2:2 d4:0 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `10211` → canonical `01200`  ·  `A=010212102  U[u0:2 u1:2 u3:0 u5:0]  D[d2:1 d4:0 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `10211` → canonical `01200`  ·  `A=012012102  U[u0:2 u1:0 u3:2 u5:0]  D[d2:1 d4:0 d6:2 d7:1 d8:1]`
+
+## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
+
+2 colouring(s) with down-apex sequence `01200`:
+
+- face apexes (raw labels) `21022` → canonical `01200`  ·  `A=010102101  U[u0:2 u1:2 u3:2 u6:2]  D[d2:2 d4:1 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `12011` → canonical `01200`  ·  `A=010201202  U[u0:2 u1:2 u3:1 u6:1]  D[d2:1 d4:2 d5:0 d7:1 d8:1]`
+
+## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
+
+2 colouring(s) with down-apex sequence `01200`:
+
+- face apexes (raw labels) `21022` → canonical `01200`  ·  `A=010102101  U[u0:2 u1:2 u3:2 u7:2]  D[d2:2 d4:1 d5:0 d6:2 d8:2]`
+- face apexes (raw labels) `12011` → canonical `01200`  ·  `A=010201202  U[u0:2 u1:2 u3:1 u7:1]  D[d2:1 d4:2 d5:0 d6:1 d8:1]`
+
+## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
+
+4 colouring(s) with down-apex sequence `01200`:
+
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=010120201  U[u0:2 u1:2 u4:1 u5:1]  D[d2:2 d3:0 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=010121201  U[u0:2 u1:2 u4:0 u5:0]  D[d2:2 d3:0 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `10211` → canonical `01200`  ·  `A=010210102  U[u0:2 u1:2 u4:2 u5:2]  D[d2:1 d3:0 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `10211` → canonical `01200`  ·  `A=010212102  U[u0:2 u1:2 u4:0 u5:0]  D[d2:1 d3:0 d6:2 d7:1 d8:1]`
+
+## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
+
+3 colouring(s) with down-apex sequence `01200`:
+
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=010120201  U[u0:2 u1:2 u4:1 u6:1]  D[d2:2 d3:0 d5:1 d7:2 d8:2]`
+- face apexes (raw labels) `10211` → canonical `01200`  ·  `A=010210102  U[u0:2 u1:2 u4:2 u6:2]  D[d2:1 d3:0 d5:2 d7:1 d8:1]`
+- face apexes (raw labels) `12011` → canonical `01200`  ·  `A=012012102  U[u0:2 u1:0 u4:0 u6:2]  D[d2:1 d3:2 d5:0 d7:1 d8:1]`
+
+## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
+
+4 colouring(s) with down-apex sequence `01200`:
+
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=010120201  U[u0:2 u2:2 u4:1 u6:1]  D[d1:2 d3:0 d5:1 d7:2 d8:2]`
+- face apexes (raw labels) `21022` → canonical `01200`  ·  `A=010201201  U[u0:2 u2:1 u4:2 u6:1]  D[d1:2 d3:1 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `21022` → canonical `01200`  ·  `A=010202101  U[u0:2 u2:1 u4:1 u6:2]  D[d1:2 d3:1 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `20122` → canonical `01200`  ·  `A=010210201  U[u0:2 u2:1 u4:2 u6:1]  D[d1:2 d3:0 d5:1 d7:2 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.png
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.md
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@@ -0,0 +1,56 @@
+# Inner-face down-apex sequence `01211`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
+
+- Colour multiset: 1×colour0, 3×colour1, 1×colour2.
+- Realised by **6** of 10 configs (M(T), inner face).
+- **15** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_01211.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `01211`:
+
+- face apexes (raw labels) `01211` → canonical `01211`  ·  `A=010120102  U[u0:2 u1:2 u2:2 u5:2]  D[d3:0 d4:1 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `02122` → canonical `01211`  ·  `A=010210201  U[u0:2 u1:2 u2:1 u5:1]  D[d3:0 d4:2 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `01211` → canonical `01211`  ·  `A=012120102  U[u0:2 u1:0 u2:0 u5:2]  D[d3:0 d4:1 d6:2 d7:1 d8:1]`
+
+## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `01211`:
+
+- face apexes (raw labels) `02122` → canonical `01211`  ·  `A=012101201  U[u0:2 u1:0 u3:2 u5:0]  D[d2:0 d4:2 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `01211` → canonical `01211`  ·  `A=012102102  U[u0:2 u1:0 u3:2 u5:0]  D[d2:0 d4:1 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `01211` → canonical `01211`  ·  `A=012120102  U[u0:2 u1:0 u3:0 u5:2]  D[d2:0 d4:1 d6:2 d7:1 d8:1]`
+
+## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
+
+3 colouring(s) with down-apex sequence `01211`:
+
+- face apexes (raw labels) `21011` → canonical `01211`  ·  `A=010102102  U[u0:2 u1:2 u3:2 u6:2]  D[d2:2 d4:1 d5:0 d7:1 d8:1]`
+- face apexes (raw labels) `12022` → canonical `01211`  ·  `A=010201201  U[u0:2 u1:2 u3:1 u6:1]  D[d2:1 d4:2 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `01211` → canonical `01211`  ·  `A=012120102  U[u0:2 u1:0 u3:0 u6:2]  D[d2:0 d4:1 d5:2 d7:1 d8:1]`
+
+## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `01211`:
+
+- face apexes (raw labels) `02122` → canonical `01211`  ·  `A=012101201  U[u0:2 u1:0 u4:2 u5:0]  D[d2:0 d3:2 d6:1 d7:2 d8:2]`
+
+## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `01211`:
+
+- face apexes (raw labels) `12022` → canonical `01211`  ·  `A=012012101  U[u0:2 u1:0 u4:0 u6:2]  D[d2:1 d3:2 d5:0 d7:2 d8:2]`
+
+## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
+
+4 colouring(s) with down-apex sequence `01211`:
+
+- face apexes (raw labels) `21011` → canonical `01211`  ·  `A=010201202  U[u0:2 u2:1 u4:2 u6:1]  D[d1:2 d3:1 d5:0 d7:1 d8:1]`
+- face apexes (raw labels) `21011` → canonical `01211`  ·  `A=010202102  U[u0:2 u2:1 u4:1 u6:2]  D[d1:2 d3:1 d5:0 d7:1 d8:1]`
+- face apexes (raw labels) `02122` → canonical `01211`  ·  `A=012010201  U[u0:2 u2:1 u4:2 u6:1]  D[d1:0 d3:2 d5:1 d7:2 d8:2]`
+- face apexes (raw labels) `01211` → canonical `01211`  ·  `A=012020102  U[u0:2 u2:1 u4:1 u6:2]  D[d1:0 d3:1 d5:2 d7:1 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.png
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.md
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+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.md
@@ -0,0 +1,97 @@
+# Inner-face down-apex sequence `01222`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
+
+- Colour multiset: 1×colour0, 1×colour1, 3×colour2.
+- Realised by **10** of 10 configs (M(T), inner face).
+- **36** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_01222.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
+
+6 colouring(s) with down-apex sequence `01222`:
+
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=010101202  U[u0:2 u1:2 u2:2 u3:2]  D[d4:2 d5:0 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `10222` → canonical `01222`  ·  `A=010102101  U[u0:2 u1:2 u2:2 u3:2]  D[d4:1 d5:0 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=010201202  U[u0:2 u1:2 u2:1 u3:1]  D[d4:2 d5:0 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `10222` → canonical `01222`  ·  `A=010202101  U[u0:2 u1:2 u2:1 u3:1]  D[d4:1 d5:0 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=012101202  U[u0:2 u1:0 u2:0 u3:2]  D[d4:2 d5:0 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `10222` → canonical `01222`  ·  `A=012102101  U[u0:2 u1:0 u2:0 u3:2]  D[d4:1 d5:0 d6:2 d7:2 d8:2]`
+
+## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `01222`:
+
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=010101202  U[u0:2 u1:2 u2:2 u4:2]  D[d3:2 d5:0 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `10222` → canonical `01222`  ·  `A=010202101  U[u0:2 u1:2 u2:1 u4:1]  D[d3:1 d5:0 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=012101202  U[u0:2 u1:0 u2:0 u4:2]  D[d3:2 d5:0 d6:1 d7:1 d8:1]`
+
+## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `01222`:
+
+- face apexes (raw labels) `01222` → canonical `01222`  ·  `A=010120101  U[u0:2 u1:2 u2:2 u5:2]  D[d3:0 d4:1 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `02111` → canonical `01222`  ·  `A=010210202  U[u0:2 u1:2 u2:1 u5:1]  D[d3:0 d4:2 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `01222` → canonical `01222`  ·  `A=012120101  U[u0:2 u1:0 u2:0 u5:2]  D[d3:0 d4:1 d6:2 d7:2 d8:2]`
+
+## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
+
+5 colouring(s) with down-apex sequence `01222`:
+
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=010101202  U[u0:2 u1:2 u3:2 u4:2]  D[d2:2 d5:0 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=010121202  U[u0:2 u1:2 u3:0 u4:0]  D[d2:2 d5:0 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `10222` → canonical `01222`  ·  `A=010202101  U[u0:2 u1:2 u3:1 u4:1]  D[d2:1 d5:0 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `10222` → canonical `01222`  ·  `A=010212101  U[u0:2 u1:2 u3:0 u4:0]  D[d2:1 d5:0 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `10222` → canonical `01222`  ·  `A=012012101  U[u0:2 u1:0 u3:2 u4:0]  D[d2:1 d5:0 d6:2 d7:2 d8:2]`
+
+## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
+
+6 colouring(s) with down-apex sequence `01222`:
+
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=010121202  U[u0:2 u1:2 u3:0 u5:0]  D[d2:2 d4:0 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `10222` → canonical `01222`  ·  `A=010212101  U[u0:2 u1:2 u3:0 u5:0]  D[d2:1 d4:0 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `10222` → canonical `01222`  ·  `A=012012101  U[u0:2 u1:0 u3:2 u5:0]  D[d2:1 d4:0 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `02111` → canonical `01222`  ·  `A=012101202  U[u0:2 u1:0 u3:2 u5:0]  D[d2:0 d4:2 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `01222` → canonical `01222`  ·  `A=012102101  U[u0:2 u1:0 u3:2 u5:0]  D[d2:0 d4:1 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `01222` → canonical `01222`  ·  `A=012120101  U[u0:2 u1:0 u3:0 u5:2]  D[d2:0 d4:1 d6:2 d7:2 d8:2]`
+
+## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `01222`:
+
+- face apexes (raw labels) `01222` → canonical `01222`  ·  `A=012120101  U[u0:2 u1:0 u3:0 u6:2]  D[d2:0 d4:1 d5:2 d7:2 d8:2]`
+
+## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
+
+1 colouring(s) with down-apex sequence `01222`:
+
+- face apexes (raw labels) `01222` → canonical `01222`  ·  `A=012120101  U[u0:2 u1:0 u3:0 u7:2]  D[d2:0 d4:1 d5:2 d6:2 d8:2]`
+
+## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
+
+5 colouring(s) with down-apex sequence `01222`:
+
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=010120202  U[u0:2 u1:2 u4:1 u5:1]  D[d2:2 d3:0 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=010121202  U[u0:2 u1:2 u4:0 u5:0]  D[d2:2 d3:0 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `10222` → canonical `01222`  ·  `A=010210101  U[u0:2 u1:2 u4:2 u5:2]  D[d2:1 d3:0 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `10222` → canonical `01222`  ·  `A=010212101  U[u0:2 u1:2 u4:0 u5:0]  D[d2:1 d3:0 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `02111` → canonical `01222`  ·  `A=012101202  U[u0:2 u1:0 u4:2 u5:0]  D[d2:0 d3:2 d6:1 d7:1 d8:1]`
+
+## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
+
+2 colouring(s) with down-apex sequence `01222`:
+
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=010120202  U[u0:2 u1:2 u4:1 u6:1]  D[d2:2 d3:0 d5:1 d7:1 d8:1]`
+- face apexes (raw labels) `10222` → canonical `01222`  ·  `A=010210101  U[u0:2 u1:2 u4:2 u6:2]  D[d2:1 d3:0 d5:2 d7:2 d8:2]`
+
+## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
+
+4 colouring(s) with down-apex sequence `01222`:
+
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=010120202  U[u0:2 u2:2 u4:1 u6:1]  D[d1:2 d3:0 d5:1 d7:1 d8:1]`
+- face apexes (raw labels) `20111` → canonical `01222`  ·  `A=010210202  U[u0:2 u2:1 u4:2 u6:1]  D[d1:2 d3:0 d5:1 d7:1 d8:1]`
+- face apexes (raw labels) `02111` → canonical `01222`  ·  `A=012010202  U[u0:2 u2:1 u4:2 u6:1]  D[d1:0 d3:2 d5:1 d7:1 d8:1]`
+- face apexes (raw labels) `01222` → canonical `01222`  ·  `A=012020101  U[u0:2 u2:1 u4:1 u6:2]  D[d1:0 d3:1 d5:2 d7:2 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/summary.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/summary.md
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+# Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=5)
+
+Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 5 singleton down-tooth apexes: **10 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in increasing annular-edge order, and reduce it modulo colour permutation (NOT dihedral symmetry).
+
+- Total Kempe-balanced colourings (mod colour permutation): **229**.
+- Distinct canonical down-apex sequences overall: **10**.
+
+## Distinct canonical down-apex sequences
+
+| sequence | colour multiset | #configs realising | #colourings |
+|---|---|---|---|
+| `00012` | 3+1+1 | 10 | 30 |
+| `00102` | 3+1+1 | 6 | 13 |
+| `00120` | 3+1+1 | 9 | 30 |
+| `01002` | 3+1+1 | 7 | 16 |
+| `01020` | 3+1+1 | 4 | 12 |
+| `01112` | 3+1+1 | 9 | 33 |
+| `01121` | 3+1+1 | 4 | 12 |
+| `01200` | 3+1+1 | 9 | 32 |
+| `01211` | 3+1+1 | 6 | 15 |
+| `01222` | 3+1+1 | 10 | 36 |
+
+Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 5 apexes (m is odd) every count must be **odd**, so the only admissible colour multisets are 3+1+1.
+
+## Step 4 — grouping configs by their set of unique down-apex sequences
+
+The 10 configs fall into **6** groups by the set of canonical down-apex sequences they realise:
+
+| #configs | set of down-apex sequences | config ids |
+|---|---|---|
+| 3 | { `00012`, `00120`, `01112`, `01200`, `01222` } | C00, C01, C03 |
+| 2 | { `00012`, `00102`, `00120`, `01002`, `01112`, `01200`, `01211`, `01222` } | C04, C07 |
+| 2 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | C08, C09 |
+| 1 | { `00012`, `00102`, `00120`, `01002`, `01211`, `01222` } | C02 |
+| 1 | { `00012`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01222` } | C06 |
+| 1 | { `00012`, `00102`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | C05 |
+
+## Config atlas (ids)
+
+| id | word / bites / face / apexes | #Kempe-balanced | down-apex sequence set |
+|---|---|---|---|
+| C00 | word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8] | 30 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
+| C01 | word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8] | 15 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
+| C02 | word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] | 21 | { `00012`, `00102`, `00120`, `01002`, `01211`, `01222` } |
+| C03 | word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8] | 25 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
+| C04 | word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] | 24 | { `00012`, `00102`, `00120`, `01002`, `01112`, `01200`, `01211`, `01222` } |
+| C05 | word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] | 22 | { `00012`, `00102`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
+| C06 | word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] | 28 | { `00012`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01222` } |
+| C07 | word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] | 27 | { `00012`, `00102`, `00120`, `01002`, `01112`, `01200`, `01211`, `01222` } |
+| C08 | word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] | 18 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
+| C09 | word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] | 19 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
+
+## Per-sequence notes
+
+- [`00012`](seq_00012.md) — figure `seq_00012.png`
+- [`00102`](seq_00102.md) — figure `seq_00102.png`
+- [`00120`](seq_00120.md) — figure `seq_00120.png`
+- [`01002`](seq_01002.md) — figure `seq_01002.png`
+- [`01020`](seq_01020.md) — figure `seq_01020.png`
+- [`01112`](seq_01112.md) — figure `seq_01112.png`
+- [`01121`](seq_01121.md) — figure `seq_01121.png`
+- [`01200`](seq_01200.md) — figure `seq_01200.png`
+- [`01211`](seq_01211.md) — figure `seq_01211.png`
+- [`01222`](seq_01222.md) — figure `seq_01222.png`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000000.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000000.md
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+# Inner-face down-apex sequence `000000`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 6×colour0.
+- Realised by **4** of 7 configs (M(T), inner face).
+- **5** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_000000.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `000000`:
+
+- face apexes (raw labels) `222222` → canonical `000000`  ·  `A=012010101  U[u0:2 u1:0 u2:1]  D[d3:2 d4:2 d5:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `111111` → canonical `000000`  ·  `A=012020202  U[u0:2 u1:0 u2:1]  D[d3:1 d4:1 d5:1 d6:1 d7:1 d8:1]`
+
+## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `000000`:
+
+- face apexes (raw labels) `111111` → canonical `000000`  ·  `A=012020202  U[u0:2 u1:0 u3:1]  D[d2:1 d4:1 d5:1 d6:1 d7:1 d8:1]`
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `000000`:
+
+- face apexes (raw labels) `111111` → canonical `000000`  ·  `A=012020202  U[u0:2 u1:0 u4:1]  D[d2:1 d3:1 d5:1 d6:1 d7:1 d8:1]`
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `000000`:
+
+- face apexes (raw labels) `111111` → canonical `000000`  ·  `A=012020202  U[u0:2 u1:0 u5:1]  D[d2:1 d3:1 d4:1 d6:1 d7:1 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000000.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000000.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.md
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+# Inner-face down-apex sequence `000011`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 4×colour0, 2×colour1.
+- Realised by **5** of 7 configs (M(T), inner face).
+- **7** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_000011.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `000011`:
+
+- face apexes (raw labels) `222211` → canonical `000011`  ·  `A=012010102  U[u0:2 u1:0 u2:1]  D[d3:2 d4:2 d5:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `111122` → canonical `000011`  ·  `A=012020201  U[u0:2 u1:0 u2:1]  D[d3:1 d4:1 d5:1 d6:1 d7:2 d8:2]`
+
+## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `000011`:
+
+- face apexes (raw labels) `111122` → canonical `000011`  ·  `A=012020201  U[u0:2 u1:0 u3:1]  D[d2:1 d4:1 d5:1 d6:1 d7:2 d8:2]`
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `000011`:
+
+- face apexes (raw labels) `111122` → canonical `000011`  ·  `A=012020201  U[u0:2 u1:0 u4:1]  D[d2:1 d3:1 d5:1 d6:1 d7:2 d8:2]`
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `000011`:
+
+- face apexes (raw labels) `111122` → canonical `000011`  ·  `A=012020201  U[u0:2 u1:0 u5:1]  D[d2:1 d3:1 d4:1 d6:1 d7:2 d8:2]`
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+2 colouring(s) with down-apex sequence `000011`:
+
+- face apexes (raw labels) `000022` → canonical `000011`  ·  `A=012121201  U[u0:2 u3:0 u6:1]  D[d1:0 d2:0 d4:0 d5:0 d7:2 d8:2]`
+- face apexes (raw labels) `000011` → canonical `000011`  ·  `A=012121202  U[u0:2 u3:0 u6:1]  D[d1:0 d2:0 d4:0 d5:0 d7:1 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.md
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+# Inner-face down-apex sequence `000101`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 4×colour0, 2×colour1.
+- Realised by **1** of 7 configs (M(T), inner face).
+- **1** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_000101.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `000101`:
+
+- face apexes (raw labels) `000202` → canonical `000101`  ·  `A=012121021  U[u0:2 u3:0 u6:1]  D[d1:0 d2:0 d4:0 d5:2 d7:0 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.pdf
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+# Inner-face down-apex sequence `000110`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 4×colour0, 2×colour1.
+- Realised by **4** of 7 configs (M(T), inner face).
+- **5** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_000110.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `000110`:
+
+- face apexes (raw labels) `222002` → canonical `000110`  ·  `A=012010121  U[u0:2 u1:0 u2:1]  D[d3:2 d4:2 d5:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `111001` → canonical `000110`  ·  `A=012020212  U[u0:2 u1:0 u2:1]  D[d3:1 d4:1 d5:1 d6:0 d7:0 d8:1]`
+
+## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `000110`:
+
+- face apexes (raw labels) `111001` → canonical `000110`  ·  `A=012020212  U[u0:2 u1:0 u3:1]  D[d2:1 d4:1 d5:1 d6:0 d7:0 d8:1]`
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `000110`:
+
+- face apexes (raw labels) `111001` → canonical `000110`  ·  `A=012020212  U[u0:2 u1:0 u4:1]  D[d2:1 d3:1 d5:1 d6:0 d7:0 d8:1]`
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `000110`:
+
+- face apexes (raw labels) `111001` → canonical `000110`  ·  `A=012020212  U[u0:2 u1:0 u5:1]  D[d2:1 d3:1 d4:1 d6:0 d7:0 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.md
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+# Inner-face down-apex sequence `001001`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 4×colour0, 2×colour1.
+- Realised by **4** of 7 configs (M(T), inner face).
+- **6** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_001001.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `001001`:
+
+- face apexes (raw labels) `002002` → canonical `001001`  ·  `A=012120121  U[u0:2 u1:0 u4:1]  D[d2:0 d3:0 d5:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `001001` → canonical `001001`  ·  `A=012120212  U[u0:2 u1:0 u4:1]  D[d2:0 d3:0 d5:1 d6:0 d7:0 d8:1]`
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `001001`:
+
+- face apexes (raw labels) `001001` → canonical `001001`  ·  `A=012120212  U[u0:2 u1:0 u5:1]  D[d2:0 d3:0 d4:1 d6:0 d7:0 d8:1]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `001001`:
+
+- face apexes (raw labels) `002002` → canonical `001001`  ·  `A=012120121  U[u0:2 u2:0 u4:1]  D[d1:0 d3:0 d5:2 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `001001` → canonical `001001`  ·  `A=012120212  U[u0:2 u2:0 u4:1]  D[d1:0 d3:0 d5:1 d6:0 d7:0 d8:1]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `001001`:
+
+- face apexes (raw labels) `001001` → canonical `001001`  ·  `A=012120212  U[u0:2 u2:0 u5:1]  D[d1:0 d3:0 d4:1 d6:0 d7:0 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.md
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+# Inner-face down-apex sequence `001010`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 4×colour0, 2×colour1.
+- Realised by **1** of 7 configs (M(T), inner face).
+- **1** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_001010.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `001010`:
+
+- face apexes (raw labels) `220202` → canonical `001010`  ·  `A=010121021  U[u0:2 u3:0 u6:1]  D[d1:2 d2:2 d4:0 d5:2 d7:0 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.md
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@@ -0,0 +1,37 @@
+# Inner-face down-apex sequence `001100`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 4×colour0, 2×colour1.
+- Realised by **4** of 7 configs (M(T), inner face).
+- **6** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_001100.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `001100`:
+
+- face apexes (raw labels) `221122` → canonical `001100`  ·  `A=012010201  U[u0:2 u1:0 u2:1]  D[d3:2 d4:2 d5:1 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `112211` → canonical `001100`  ·  `A=012020102  U[u0:2 u1:0 u2:1]  D[d3:1 d4:1 d5:2 d6:2 d7:1 d8:1]`
+
+## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `001100`:
+
+- face apexes (raw labels) `112211` → canonical `001100`  ·  `A=012020102  U[u0:2 u1:0 u3:1]  D[d2:1 d4:1 d5:2 d6:2 d7:1 d8:1]`
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `001100`:
+
+- face apexes (raw labels) `112211` → canonical `001100`  ·  `A=012020102  U[u0:2 u1:0 u4:1]  D[d2:1 d3:1 d5:2 d6:2 d7:1 d8:1]`
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+2 colouring(s) with down-apex sequence `001100`:
+
+- face apexes (raw labels) `221122` → canonical `001100`  ·  `A=010120201  U[u0:2 u3:0 u6:1]  D[d1:2 d2:2 d4:1 d5:1 d7:2 d8:2]`
+- face apexes (raw labels) `220022` → canonical `001100`  ·  `A=010121201  U[u0:2 u3:0 u6:1]  D[d1:2 d2:2 d4:0 d5:0 d7:2 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.md
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+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.md
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+# Inner-face down-apex sequence `001111`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 4×colour1.
+- Realised by **7** of 7 configs (M(T), inner face).
+- **12** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_001111.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `001111`:
+
+- face apexes (raw labels) `221111` → canonical `001111`  ·  `A=012010202  U[u0:2 u1:0 u2:1]  D[d3:2 d4:2 d5:1 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `112222` → canonical `001111`  ·  `A=012020101  U[u0:2 u1:0 u2:1]  D[d3:1 d4:1 d5:2 d6:2 d7:2 d8:2]`
+
+## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `001111`:
+
+- face apexes (raw labels) `112222` → canonical `001111`  ·  `A=012020101  U[u0:2 u1:0 u3:1]  D[d2:1 d4:1 d5:2 d6:2 d7:2 d8:2]`
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `001111`:
+
+- face apexes (raw labels) `112222` → canonical `001111`  ·  `A=012020101  U[u0:2 u1:0 u4:1]  D[d2:1 d3:1 d5:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `002222` → canonical `001111`  ·  `A=012120101  U[u0:2 u1:0 u4:1]  D[d2:0 d3:0 d5:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `001111` → canonical `001111`  ·  `A=012120202  U[u0:2 u1:0 u4:1]  D[d2:0 d3:0 d5:1 d6:1 d7:1 d8:1]`
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `001111`:
+
+- face apexes (raw labels) `001111` → canonical `001111`  ·  `A=012120202  U[u0:2 u1:0 u5:1]  D[d2:0 d3:0 d4:1 d6:1 d7:1 d8:1]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `001111`:
+
+- face apexes (raw labels) `002222` → canonical `001111`  ·  `A=012120101  U[u0:2 u2:0 u4:1]  D[d1:0 d3:0 d5:2 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `001111` → canonical `001111`  ·  `A=012120202  U[u0:2 u2:0 u4:1]  D[d1:0 d3:0 d5:1 d6:1 d7:1 d8:1]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `001111`:
+
+- face apexes (raw labels) `001111` → canonical `001111`  ·  `A=012120202  U[u0:2 u2:0 u5:1]  D[d1:0 d3:0 d4:1 d6:1 d7:1 d8:1]`
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+2 colouring(s) with down-apex sequence `001111`:
+
+- face apexes (raw labels) `221111` → canonical `001111`  ·  `A=010120202  U[u0:2 u3:0 u6:1]  D[d1:2 d2:2 d4:1 d5:1 d7:1 d8:1]`
+- face apexes (raw labels) `001111` → canonical `001111`  ·  `A=012120202  U[u0:2 u3:0 u6:1]  D[d1:0 d2:0 d4:1 d5:1 d7:1 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.md
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@@ -0,0 +1,44 @@
+# Inner-face down-apex sequence `001122`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **5** of 7 configs (M(T), inner face).
+- **8** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_001122.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `001122`:
+
+- face apexes (raw labels) `002211` → canonical `001122`  ·  `A=012120102  U[u0:2 u1:0 u4:1]  D[d2:0 d3:0 d5:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `001122` → canonical `001122`  ·  `A=012120201  U[u0:2 u1:0 u4:1]  D[d2:0 d3:0 d5:1 d6:1 d7:2 d8:2]`
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `001122`:
+
+- face apexes (raw labels) `001122` → canonical `001122`  ·  `A=012120201  U[u0:2 u1:0 u5:1]  D[d2:0 d3:0 d4:1 d6:1 d7:2 d8:2]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `001122`:
+
+- face apexes (raw labels) `002211` → canonical `001122`  ·  `A=012120102  U[u0:2 u2:0 u4:1]  D[d1:0 d3:0 d5:2 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `001122` → canonical `001122`  ·  `A=012120201  U[u0:2 u2:0 u4:1]  D[d1:0 d3:0 d5:1 d6:1 d7:2 d8:2]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `001122`:
+
+- face apexes (raw labels) `001122` → canonical `001122`  ·  `A=012120201  U[u0:2 u2:0 u5:1]  D[d1:0 d3:0 d4:1 d6:1 d7:2 d8:2]`
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+2 colouring(s) with down-apex sequence `001122`:
+
+- face apexes (raw labels) `220011` → canonical `001122`  ·  `A=010121202  U[u0:2 u3:0 u6:1]  D[d1:2 d2:2 d4:0 d5:0 d7:1 d8:1]`
+- face apexes (raw labels) `001122` → canonical `001122`  ·  `A=012120201  U[u0:2 u3:0 u6:1]  D[d1:0 d2:0 d4:1 d5:1 d7:2 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.md
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@@ -0,0 +1,30 @@
+# Inner-face down-apex sequence `001221`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **3** of 7 configs (M(T), inner face).
+- **4** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_001221.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `001221`:
+
+- face apexes (raw labels) `221001` → canonical `001221`  ·  `A=012010212  U[u0:2 u1:0 u2:1]  D[d3:2 d4:2 d5:1 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `112002` → canonical `001221`  ·  `A=012020121  U[u0:2 u1:0 u2:1]  D[d3:1 d4:1 d5:2 d6:0 d7:0 d8:2]`
+
+## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `001221`:
+
+- face apexes (raw labels) `112002` → canonical `001221`  ·  `A=012020121  U[u0:2 u1:0 u3:1]  D[d2:1 d4:1 d5:2 d6:0 d7:0 d8:2]`
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `001221`:
+
+- face apexes (raw labels) `112002` → canonical `001221`  ·  `A=012020121  U[u0:2 u1:0 u4:1]  D[d2:1 d3:1 d5:2 d6:0 d7:0 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.png
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.md
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--- /dev/null
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.md
@@ -0,0 +1,38 @@
+# Inner-face down-apex sequence `010001`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 4×colour0, 2×colour1.
+- Realised by **4** of 7 configs (M(T), inner face).
+- **7** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_010001.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `010001`:
+
+- face apexes (raw labels) `020002` → canonical `010001`  ·  `A=012102121  U[u0:2 u1:0 u4:1]  D[d2:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `010001`:
+
+- face apexes (raw labels) `020002` → canonical `010001`  ·  `A=012012121  U[u0:2 u2:1 u4:0]  D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `010001` → canonical `010001`  ·  `A=012021212  U[u0:2 u2:1 u4:0]  D[d1:0 d3:1 d5:0 d6:0 d7:0 d8:1]`
+- face apexes (raw labels) `020002` → canonical `010001`  ·  `A=012102121  U[u0:2 u2:0 u4:1]  D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `010001`:
+
+- face apexes (raw labels) `020002` → canonical `010001`  ·  `A=012012121  U[u0:2 u2:1 u5:0]  D[d1:0 d3:2 d4:0 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `010001` → canonical `010001`  ·  `A=012021212  U[u0:2 u2:1 u5:0]  D[d1:0 d3:1 d4:0 d6:0 d7:0 d8:1]`
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `010001`:
+
+- face apexes (raw labels) `010001` → canonical `010001`  ·  `A=012021212  U[u0:2 u3:1 u6:0]  D[d1:0 d2:1 d4:0 d5:0 d7:0 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.md
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+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.md
@@ -0,0 +1,23 @@
+# Inner-face down-apex sequence `010100`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 4×colour0, 2×colour1.
+- Realised by **2** of 7 configs (M(T), inner face).
+- **2** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_010100.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `010100`:
+
+- face apexes (raw labels) `212122` → canonical `010100`  ·  `A=010201201  U[u0:2 u2:1 u5:0]  D[d1:2 d3:1 d4:2 d6:1 d7:2 d8:2]`
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `010100`:
+
+- face apexes (raw labels) `212122` → canonical `010100`  ·  `A=010210201  U[u0:2 u3:0 u6:1]  D[d1:2 d2:1 d4:2 d5:1 d7:2 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.png
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.md
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+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.md
@@ -0,0 +1,39 @@
+# Inner-face down-apex sequence `010111`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 4×colour1.
+- Realised by **4** of 7 configs (M(T), inner face).
+- **8** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_010111.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `010111`:
+
+- face apexes (raw labels) `020222` → canonical `010111`  ·  `A=012102101  U[u0:2 u1:0 u4:1]  D[d2:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `010111`:
+
+- face apexes (raw labels) `020222` → canonical `010111`  ·  `A=012012101  U[u0:2 u2:1 u4:0]  D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `010111` → canonical `010111`  ·  `A=012021202  U[u0:2 u2:1 u4:0]  D[d1:0 d3:1 d5:0 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `020222` → canonical `010111`  ·  `A=012102101  U[u0:2 u2:0 u4:1]  D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `010111`:
+
+- face apexes (raw labels) `212111` → canonical `010111`  ·  `A=010201202  U[u0:2 u2:1 u5:0]  D[d1:2 d3:1 d4:2 d6:1 d7:1 d8:1]`
+- face apexes (raw labels) `020222` → canonical `010111`  ·  `A=012012101  U[u0:2 u2:1 u5:0]  D[d1:0 d3:2 d4:0 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `010111` → canonical `010111`  ·  `A=012021202  U[u0:2 u2:1 u5:0]  D[d1:0 d3:1 d4:0 d6:1 d7:1 d8:1]`
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `010111`:
+
+- face apexes (raw labels) `212111` → canonical `010111`  ·  `A=010210202  U[u0:2 u3:0 u6:1]  D[d1:2 d2:1 d4:2 d5:1 d7:1 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.md
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+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.md
@@ -0,0 +1,32 @@
+# Inner-face down-apex sequence `010122`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **3** of 7 configs (M(T), inner face).
+- **6** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_010122.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `010122`:
+
+- face apexes (raw labels) `020211` → canonical `010122`  ·  `A=012102102  U[u0:2 u1:0 u4:1]  D[d2:0 d3:2 d5:0 d6:2 d7:1 d8:1]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `010122`:
+
+- face apexes (raw labels) `020211` → canonical `010122`  ·  `A=012012102  U[u0:2 u2:1 u4:0]  D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `010122` → canonical `010122`  ·  `A=012021201  U[u0:2 u2:1 u4:0]  D[d1:0 d3:1 d5:0 d6:1 d7:2 d8:2]`
+- face apexes (raw labels) `020211` → canonical `010122`  ·  `A=012102102  U[u0:2 u2:0 u4:1]  D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `010122`:
+
+- face apexes (raw labels) `020211` → canonical `010122`  ·  `A=012012102  U[u0:2 u2:1 u5:0]  D[d1:0 d3:2 d4:0 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `010122` → canonical `010122`  ·  `A=012021201  U[u0:2 u2:1 u5:0]  D[d1:0 d3:1 d4:0 d6:1 d7:2 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.md
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+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.md
@@ -0,0 +1,23 @@
+# Inner-face down-apex sequence `010221`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **2** of 7 configs (M(T), inner face).
+- **2** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_010221.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `010221`:
+
+- face apexes (raw labels) `212001` → canonical `010221`  ·  `A=010201212  U[u0:2 u2:1 u5:0]  D[d1:2 d3:1 d4:2 d6:0 d7:0 d8:1]`
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `010221`:
+
+- face apexes (raw labels) `212001` → canonical `010221`  ·  `A=010201212  U[u0:2 u3:1 u6:0]  D[d1:2 d2:1 d4:2 d5:0 d7:0 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.md
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@@ -0,0 +1,43 @@
+# Inner-face down-apex sequence `011000`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 4×colour0, 2×colour1.
+- Realised by **5** of 7 configs (M(T), inner face).
+- **7** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_011000.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `011000`:
+
+- face apexes (raw labels) `200222` → canonical `011000`  ·  `A=012012101  U[u0:2 u1:0 u2:1]  D[d3:2 d4:0 d5:0 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `100111` → canonical `011000`  ·  `A=012021202  U[u0:2 u1:0 u2:1]  D[d3:1 d4:0 d5:0 d6:1 d7:1 d8:1]`
+
+## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011000`:
+
+- face apexes (raw labels) `100111` → canonical `011000`  ·  `A=012021202  U[u0:2 u1:0 u3:1]  D[d2:1 d4:0 d5:0 d6:1 d7:1 d8:1]`
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011000`:
+
+- face apexes (raw labels) `122111` → canonical `011000`  ·  `A=012010202  U[u0:2 u1:0 u5:1]  D[d2:1 d3:2 d4:2 d6:1 d7:1 d8:1]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011000`:
+
+- face apexes (raw labels) `200222` → canonical `011000`  ·  `A=010212101  U[u0:2 u2:1 u4:0]  D[d1:2 d3:0 d5:0 d6:2 d7:2 d8:2]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `011000`:
+
+- face apexes (raw labels) `211222` → canonical `011000`  ·  `A=010202101  U[u0:2 u2:1 u5:0]  D[d1:2 d3:1 d4:1 d6:2 d7:2 d8:2]`
+- face apexes (raw labels) `200222` → canonical `011000`  ·  `A=010212101  U[u0:2 u2:1 u5:0]  D[d1:2 d3:0 d4:0 d6:2 d7:2 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.md
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@@ -0,0 +1,23 @@
+# Inner-face down-apex sequence `011011`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 4×colour1.
+- Realised by **2** of 7 configs (M(T), inner face).
+- **2** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_011011.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011011`:
+
+- face apexes (raw labels) `122122` → canonical `011011`  ·  `A=012010201  U[u0:2 u1:0 u5:1]  D[d2:1 d3:2 d4:2 d6:1 d7:2 d8:2]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011011`:
+
+- face apexes (raw labels) `211211` → canonical `011011`  ·  `A=010202102  U[u0:2 u2:1 u5:0]  D[d1:2 d3:1 d4:1 d6:2 d7:1 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.md
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@@ -0,0 +1,36 @@
+# Inner-face down-apex sequence `011022`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **4** of 7 configs (M(T), inner face).
+- **5** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_011022.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `011022`:
+
+- face apexes (raw labels) `200211` → canonical `011022`  ·  `A=012012102  U[u0:2 u1:0 u2:1]  D[d3:2 d4:0 d5:0 d6:2 d7:1 d8:1]`
+- face apexes (raw labels) `100122` → canonical `011022`  ·  `A=012021201  U[u0:2 u1:0 u2:1]  D[d3:1 d4:0 d5:0 d6:1 d7:2 d8:2]`
+
+## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011022`:
+
+- face apexes (raw labels) `100122` → canonical `011022`  ·  `A=012021201  U[u0:2 u1:0 u3:1]  D[d2:1 d4:0 d5:0 d6:1 d7:2 d8:2]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011022`:
+
+- face apexes (raw labels) `200211` → canonical `011022`  ·  `A=010212102  U[u0:2 u2:1 u4:0]  D[d1:2 d3:0 d5:0 d6:2 d7:1 d8:1]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011022`:
+
+- face apexes (raw labels) `200211` → canonical `011022`  ·  `A=010212102  U[u0:2 u2:1 u5:0]  D[d1:2 d3:0 d4:0 d6:2 d7:1 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.md
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+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.md
@@ -0,0 +1,17 @@
+# Inner-face down-apex sequence `011101`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 4×colour1.
+- Realised by **1** of 7 configs (M(T), inner face).
+- **1** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_011101.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `011101`:
+
+- face apexes (raw labels) `011101` → canonical `011101`  ·  `A=012020212  U[u0:2 u3:1 u6:0]  D[d1:0 d2:1 d4:1 d5:1 d7:0 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.md
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@@ -0,0 +1,36 @@
+# Inner-face down-apex sequence `011110`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 4×colour1.
+- Realised by **4** of 7 configs (M(T), inner face).
+- **5** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_011110.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `011110`:
+
+- face apexes (raw labels) `200002` → canonical `011110`  ·  `A=012012121  U[u0:2 u1:0 u2:1]  D[d3:2 d4:0 d5:0 d6:0 d7:0 d8:2]`
+- face apexes (raw labels) `100001` → canonical `011110`  ·  `A=012021212  U[u0:2 u1:0 u2:1]  D[d3:1 d4:0 d5:0 d6:0 d7:0 d8:1]`
+
+## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011110`:
+
+- face apexes (raw labels) `100001` → canonical `011110`  ·  `A=012021212  U[u0:2 u1:0 u3:1]  D[d2:1 d4:0 d5:0 d6:0 d7:0 d8:1]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011110`:
+
+- face apexes (raw labels) `200002` → canonical `011110`  ·  `A=010212121  U[u0:2 u2:1 u4:0]  D[d1:2 d3:0 d5:0 d6:0 d7:0 d8:2]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011110`:
+
+- face apexes (raw labels) `200002` → canonical `011110`  ·  `A=010212121  U[u0:2 u2:1 u5:0]  D[d1:2 d3:0 d4:0 d6:0 d7:0 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.md
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@@ -0,0 +1,17 @@
+# Inner-face down-apex sequence `011202`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **1** of 7 configs (M(T), inner face).
+- **1** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_011202.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `011202`:
+
+- face apexes (raw labels) `011202` → canonical `011202`  ·  `A=012020121  U[u0:2 u3:1 u6:0]  D[d1:0 d2:1 d4:1 d5:2 d7:0 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.md
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@@ -0,0 +1,29 @@
+# Inner-face down-apex sequence `011220`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **3** of 7 configs (M(T), inner face).
+- **3** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_011220.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011220`:
+
+- face apexes (raw labels) `122001` → canonical `011220`  ·  `A=012010212  U[u0:2 u1:0 u5:1]  D[d2:1 d3:2 d4:2 d6:0 d7:0 d8:1]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `011220`:
+
+- face apexes (raw labels) `211002` → canonical `011220`  ·  `A=010202121  U[u0:2 u2:1 u5:0]  D[d1:2 d3:1 d4:1 d6:0 d7:0 d8:2]`
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `011220`:
+
+- face apexes (raw labels) `211002` → canonical `011220`  ·  `A=010202121  U[u0:2 u3:1 u6:0]  D[d1:2 d2:1 d4:1 d5:0 d7:0 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.png
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index 0000000..7411a47
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.md
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+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.md
@@ -0,0 +1,30 @@
+# Inner-face down-apex sequence `012012`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **3** of 7 configs (M(T), inner face).
+- **4** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_012012.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `012012`:
+
+- face apexes (raw labels) `201201` → canonical `012012`  ·  `A=012012012  U[u0:2 u1:0 u2:1]  D[d3:2 d4:0 d5:1 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `102102` → canonical `012012`  ·  `A=012021021  U[u0:2 u1:0 u2:1]  D[d3:1 d4:0 d5:2 d6:1 d7:0 d8:2]`
+
+## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `012012`:
+
+- face apexes (raw labels) `102102` → canonical `012012`  ·  `A=012021021  U[u0:2 u1:0 u3:1]  D[d2:1 d4:0 d5:2 d6:1 d7:0 d8:2]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `012012`:
+
+- face apexes (raw labels) `201201` → canonical `012012`  ·  `A=010212012  U[u0:2 u2:1 u4:0]  D[d1:2 d3:0 d5:1 d6:2 d7:0 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.png
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.md
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index 0000000..90fbf18
--- /dev/null
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.md
@@ -0,0 +1,17 @@
+# Inner-face down-apex sequence `012021`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **1** of 7 configs (M(T), inner face).
+- **1** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_012021.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `012021`:
+
+- face apexes (raw labels) `120102` → canonical `012021`  ·  `A=012012021  U[u0:2 u1:0 u5:1]  D[d2:1 d3:2 d4:0 d6:1 d7:0 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.png
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index 0000000..242eb30
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.md
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index 0000000..e8d53df
--- /dev/null
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.md
@@ -0,0 +1,37 @@
+# Inner-face down-apex sequence `012102`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **4** of 7 configs (M(T), inner face).
+- **6** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_012102.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `012102`:
+
+- face apexes (raw labels) `021201` → canonical `012102`  ·  `A=012102012  U[u0:2 u1:0 u4:1]  D[d2:0 d3:2 d5:1 d6:2 d7:0 d8:1]`
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `012102`:
+
+- face apexes (raw labels) `021201` → canonical `012102`  ·  `A=012102012  U[u0:2 u1:0 u5:1]  D[d2:0 d3:2 d4:1 d6:2 d7:0 d8:1]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `012102`:
+
+- face apexes (raw labels) `021201` → canonical `012102`  ·  `A=012012012  U[u0:2 u2:1 u4:0]  D[d1:0 d3:2 d5:1 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `012102` → canonical `012102`  ·  `A=012021021  U[u0:2 u2:1 u4:0]  D[d1:0 d3:1 d5:2 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `021201` → canonical `012102`  ·  `A=012102012  U[u0:2 u2:0 u4:1]  D[d1:0 d3:2 d5:1 d6:2 d7:0 d8:1]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `012102`:
+
+- face apexes (raw labels) `021201` → canonical `012102`  ·  `A=012102012  U[u0:2 u2:0 u5:1]  D[d1:0 d3:2 d4:1 d6:2 d7:0 d8:1]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.png
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.md
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+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.md
@@ -0,0 +1,23 @@
+# Inner-face down-apex sequence `012120`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **2** of 7 configs (M(T), inner face).
+- **2** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_012120.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `012120`:
+
+- face apexes (raw labels) `120201` → canonical `012120`  ·  `A=012012012  U[u0:2 u1:0 u5:1]  D[d2:1 d3:2 d4:0 d6:2 d7:0 d8:1]`
+
+## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
+
+1 colouring(s) with down-apex sequence `012120`:
+
+- face apexes (raw labels) `210102` → canonical `012120`  ·  `A=010212021  U[u0:2 u3:0 u6:1]  D[d1:2 d2:1 d4:0 d5:1 d7:0 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.png
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.md
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--- /dev/null
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.md
@@ -0,0 +1,37 @@
+# Inner-face down-apex sequence `012201`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **4** of 7 configs (M(T), inner face).
+- **6** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_012201.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `012201`:
+
+- face apexes (raw labels) `021102` → canonical `012201`  ·  `A=012102021  U[u0:2 u1:0 u4:1]  D[d2:0 d3:2 d5:1 d6:1 d7:0 d8:2]`
+
+## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `012201`:
+
+- face apexes (raw labels) `021102` → canonical `012201`  ·  `A=012102021  U[u0:2 u1:0 u5:1]  D[d2:0 d3:2 d4:1 d6:1 d7:0 d8:2]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+3 colouring(s) with down-apex sequence `012201`:
+
+- face apexes (raw labels) `021102` → canonical `012201`  ·  `A=012012021  U[u0:2 u2:1 u4:0]  D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `012201` → canonical `012201`  ·  `A=012021012  U[u0:2 u2:1 u4:0]  D[d1:0 d3:1 d5:2 d6:2 d7:0 d8:1]`
+- face apexes (raw labels) `021102` → canonical `012201`  ·  `A=012102021  U[u0:2 u2:0 u4:1]  D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]`
+
+## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `012201`:
+
+- face apexes (raw labels) `021102` → canonical `012201`  ·  `A=012102021  U[u0:2 u2:0 u5:1]  D[d1:0 d3:2 d4:1 d6:1 d7:0 d8:2]`
+
diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.pdf
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.png
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diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.md
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index 0000000..87efeb3
--- /dev/null
+++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.md
@@ -0,0 +1,30 @@
+# Inner-face down-apex sequence `012210`
+
+Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
+
+- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
+- Realised by **3** of 7 configs (M(T), inner face).
+- **4** Kempe-balanced colourings (mod colour permutation) produce it.
+- Figure: `seq_012210.png` (black rings mark the face's down apexes).
+
+Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
+
+## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
+
+2 colouring(s) with down-apex sequence `012210`:
+
+- face apexes (raw labels) `201102` → canonical `012210`  ·  `A=012012021  U[u0:2 u1:0 u2:1]  D[d3:2 d4:0 d5:1 d6:1 d7:0 d8:2]`
+- face apexes (raw labels) `102201` → canonical `012210`  ·  `A=012021012  U[u0:2 u1:0 u2:1]  D[d3:1 d4:0 d5:2 d6:2 d7:0 d8:1]`
+
+## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `012210`:
+
+- face apexes (raw labels) `102201` → canonical `012210`  ·  `A=012021012  U[u0:2 u1:0 u3:1]  D[d2:1 d4:0 d5:2 d6:2 d7:0 d8:1]`
+
+## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
+
+1 colouring(s) with down-apex sequence `012210`:
+
+- face apexes (raw labels) `201102` → canonical `012210`  ·  `A=010212021  U[u0:2 u2:1 u4:0]  D[d1:2 d3:0 d5:1 d6:1 d7:0 d8:2]`
+
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+# Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=6)
+
+Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 6 singleton down-tooth apexes: **7 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in increasing annular-edge order, and reduce it modulo colour permutation (NOT dihedral symmetry).
+
+- Total Kempe-balanced colourings (mod colour permutation): **127**.
+- Distinct canonical down-apex sequences overall: **28**.
+
+## Distinct canonical down-apex sequences
+
+| sequence | colour multiset | #configs realising | #colourings |
+|---|---|---|---|
+| `000000` | 6 | 4 | 5 |
+| `000011` | 4+2 | 5 | 7 |
+| `000101` | 4+2 | 1 | 1 |
+| `000110` | 4+2 | 4 | 5 |
+| `001001` | 4+2 | 4 | 6 |
+| `001010` | 4+2 | 1 | 1 |
+| `001100` | 4+2 | 4 | 6 |
+| `001111` | 4+2 | 7 | 12 |
+| `001122` | 2+2+2 | 5 | 8 |
+| `001221` | 2+2+2 | 3 | 4 |
+| `010001` | 4+2 | 4 | 7 |
+| `010100` | 4+2 | 2 | 2 |
+| `010111` | 4+2 | 4 | 8 |
+| `010122` | 2+2+2 | 3 | 6 |
+| `010221` | 2+2+2 | 2 | 2 |
+| `011000` | 4+2 | 5 | 7 |
+| `011011` | 4+2 | 2 | 2 |
+| `011022` | 2+2+2 | 4 | 5 |
+| `011101` | 4+2 | 1 | 1 |
+| `011110` | 4+2 | 4 | 5 |
+| `011202` | 2+2+2 | 1 | 1 |
+| `011220` | 2+2+2 | 3 | 3 |
+| `012012` | 2+2+2 | 3 | 4 |
+| `012021` | 2+2+2 | 1 | 1 |
+| `012102` | 2+2+2 | 4 | 6 |
+| `012120` | 2+2+2 | 2 | 2 |
+| `012201` | 2+2+2 | 4 | 6 |
+| `012210` | 2+2+2 | 3 | 4 |
+
+Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 6 apexes (m is even) every count must be **even**, so the only admissible colour multisets are 2+2+2, 4+2, 6.
+
+## Step 4 — grouping configs by their set of unique down-apex sequences
+
+The 7 configs fall into **6** groups by the set of canonical down-apex sequences they realise:
+
+| #configs | set of down-apex sequences | config ids |
+|---|---|---|
+| 2 | { `000000`, `000011`, `000110`, `001100`, `001111`, `001221`, `011000`, `011022`, `011110`, `012012`, `012210` } | C00, C01 |
+| 1 | { `000000`, `000011`, `000110`, `001001`, `001100`, `001111`, `001122`, `001221`, `010001`, `010111`, `010122`, `012102`, `012201` } | C02 |
+| 1 | { `000000`, `000011`, `000110`, `001001`, `001111`, `001122`, `011000`, `011011`, `011220`, `012021`, `012102`, `012120`, `012201` } | C03 |
+| 1 | { `001001`, `001111`, `001122`, `010001`, `010111`, `010122`, `011000`, `011022`, `011110`, `012012`, `012102`, `012201`, `012210` } | C04 |
+| 1 | { `000011`, `000101`, `001010`, `001100`, `001111`, `001122`, `010001`, `010100`, `010111`, `010221`, `011101`, `011202`, `011220`, `012120` } | C06 |
+| 1 | { `001001`, `001111`, `001122`, `010001`, `010100`, `010111`, `010122`, `010221`, `011000`, `011011`, `011022`, `011110`, `011220`, `012102`, `012201` } | C05 |
+
+## Config atlas (ids)
+
+| id | word / bites / face / apexes | #Kempe-balanced | down-apex sequence set |
+|---|---|---|---|
+| C00 | word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8] | 22 | { `000000`, `000011`, `000110`, `001100`, `001111`, `001221`, `011000`, `011022`, `011110`, `012012`, `012210` } |
+| C01 | word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] | 11 | { `000000`, `000011`, `000110`, `001100`, `001111`, `001221`, `011000`, `011022`, `011110`, `012012`, `012210` } |
+| C02 | word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] | 17 | { `000000`, `000011`, `000110`, `001001`, `001100`, `001111`, `001122`, `001221`, `010001`, `010111`, `010122`, `012102`, `012201` } |
+| C03 | word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] | 13 | { `000000`, `000011`, `000110`, `001001`, `001111`, `001122`, `011000`, `011011`, `011220`, `012021`, `012102`, `012120`, `012201` } |
+| C04 | word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] | 26 | { `001001`, `001111`, `001122`, `010001`, `010111`, `010122`, `011000`, `011022`, `011110`, `012012`, `012102`, `012201`, `012210` } |
+| C05 | word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] | 20 | { `001001`, `001111`, `001122`, `010001`, `010100`, `010111`, `010122`, `010221`, `011000`, `011011`, `011022`, `011110`, `011220`, `012102`, `012201` } |
+| C06 | word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] | 18 | { `000011`, `000101`, `001010`, `001100`, `001111`, `001122`, `010001`, `010100`, `010111`, `010221`, `011101`, `011202`, `011220`, `012120` } |
+
+## Per-sequence notes
+
+- [`000000`](seq_000000.md) — figure `seq_000000.png`
+- [`000011`](seq_000011.md) — figure `seq_000011.png`
+- [`000101`](seq_000101.md) — figure `seq_000101.png`
+- [`000110`](seq_000110.md) — figure `seq_000110.png`
+- [`001001`](seq_001001.md) — figure `seq_001001.png`
+- [`001010`](seq_001010.md) — figure `seq_001010.png`
+- [`001100`](seq_001100.md) — figure `seq_001100.png`
+- [`001111`](seq_001111.md) — figure `seq_001111.png`
+- [`001122`](seq_001122.md) — figure `seq_001122.png`
+- [`001221`](seq_001221.md) — figure `seq_001221.png`
+- [`010001`](seq_010001.md) — figure `seq_010001.png`
+- [`010100`](seq_010100.md) — figure `seq_010100.png`
+- [`010111`](seq_010111.md) — figure `seq_010111.png`
+- [`010122`](seq_010122.md) — figure `seq_010122.png`
+- [`010221`](seq_010221.md) — figure `seq_010221.png`
+- [`011000`](seq_011000.md) — figure `seq_011000.png`
+- [`011011`](seq_011011.md) — figure `seq_011011.png`
+- [`011022`](seq_011022.md) — figure `seq_011022.png`
+- [`011101`](seq_011101.md) — figure `seq_011101.png`
+- [`011110`](seq_011110.md) — figure `seq_011110.png`
+- [`011202`](seq_011202.md) — figure `seq_011202.png`
+- [`011220`](seq_011220.md) — figure `seq_011220.png`
+- [`012012`](seq_012012.md) — figure `seq_012012.png`
+- [`012021`](seq_012021.md) — figure `seq_012021.png`
+- [`012102`](seq_012102.md) — figure `seq_012102.png`
+- [`012120`](seq_012120.md) — figure `seq_012120.png`
+- [`012201`](seq_012201.md) — figure `seq_012201.png`
+- [`012210`](seq_012210.md) — figure `seq_012210.png`
+