Add medial tire decomposition paper

2026-06-08 15:34:53 -04:00
parent 6400fdfc5e
commit 20fe6c24ca
14 changed files with 2087 additions and 119 deletions
@@ -16,6 +16,7 @@ All papers are at `papers/<name>/paper.tex`. The current set:
 | `iterated_reduction_in_reduced_dual` | An Iterated Reduction in the Reduced Dual |
 | `level_resolutions_of_maximal_planar_graphs` | Level Resolutions of Maximal Planar Graphs |
 | `level_switching` | Level Switching |
 | `medial_tire_decompositions_of_plane_triangulations` | Medial Tire Decompositions of Plane Triangulations |
 | `nested_tire_decompositions_of_plane_triangulations` | Nested Tire Decompositions of Plane Triangulations |
 | `plane_depth` | Plane Depth |
 | `plane_depth_sequencing` | Plane Depth Sequencing |
@@ -0,0 +1,350 @@
 """Compare full and reduced medial tire graphs on generated tires.
 The new medial decomposition paper defines:
  * full medial tire graph: the subgraph of M(G) induced by medial
    vertices corresponding to edges incident to tread triangles;
  * reduced medial tire graph: delete same-boundary medial edges and
    chord-only medial edges.
 For a tire tread inside an ambient triangulation, the medial edges
 visible in the tread come from annular triangular faces.  This script
 checks whether any same-boundary medial edges are actually present in
 that model.  It also compares against the older standalone drawing
 model, which added artificial outer/inner boundary faces.
 """
 from __future__ import annotations
 import argparse
 import itertools
 import random
 from collections import Counter
 Edge = tuple[int, int]
 MedialEdge = tuple[Edge, Edge]
 def random_tire(m: int, k: int, n_chords: int = 0, seed: int | None = None) -> dict:
    """Generate the same labelled annular tires used in earlier experiments."""
    rng = random.Random(seed)
    outer = list(range(m))
    inner = list(range(m, m + k))
    edges: set[Edge] = set()
    for i in range(m):
        edges.add(edge_key(outer[i], outer[(i + 1) % m]))
    for j in range(k):
        edges.add(edge_key(inner[j], inner[(j + 1) % k]))
    inner_chords = set()
    candidates = []
    for a in range(k):
        for b in range(a + 2, k):
            if not (a == 0 and b == k - 1):
                candidates.append((a, b))
    rng.shuffle(candidates)
    for a, b in candidates:
        if len(inner_chords) >= n_chords:
            break
        if any((a < a2 < b < b2) or (a2 < a < b2 < b) for a2, b2 in inner_chords):
            continue
        inner_chords.add((a, b))
        edges.add(edge_key(inner[a], inner[b]))
    edges.add(edge_key(outer[0], inner[0]))
    moves = ["O"] * m + ["I"] * k
    rng.shuffle(moves)
    triangles = []
    i, j = 0, 0
    for move in moves:
        if move == "O":
            tri = (outer[i % m], inner[j % k], outer[(i + 1) % m])
            triangles.append(tri)
            edges.add(edge_key(inner[j % k], outer[(i + 1) % m]))
            i += 1
        else:
            tri = (outer[i % m], inner[j % k], inner[(j + 1) % k])
            triangles.append(tri)
            edges.add(edge_key(outer[i % m], inner[(j + 1) % k]))
            j += 1
    return {
        "m": m,
        "k": k,
        "n_chords": len(inner_chords),
        "outer": outer,
        "inner": inner,
        "edges": sorted(edges),
        "triangles": triangles,
        "inner_chords": sorted(inner_chords),
        "lattice_path": "".join(moves),
        "seed": seed,
    }
 def tire_from_path(m: int, k: int, chords: tuple[tuple[int, int], ...], path: str) -> dict:
    outer = list(range(m))
    inner = list(range(m, m + k))
    edges: set[Edge] = set()
    for i in range(m):
        edges.add(edge_key(outer[i], outer[(i + 1) % m]))
    for j in range(k):
        edges.add(edge_key(inner[j], inner[(j + 1) % k]))
    for a, b in chords:
        edges.add(edge_key(inner[a], inner[b]))
    edges.add(edge_key(outer[0], inner[0]))
    triangles = []
    i, j = 0, 0
    for move in path:
        if move == "O":
            tri = (outer[i % m], inner[j % k], outer[(i + 1) % m])
            triangles.append(tri)
            edges.add(edge_key(inner[j % k], outer[(i + 1) % m]))
            i += 1
        else:
            tri = (outer[i % m], inner[j % k], inner[(j + 1) % k])
            triangles.append(tri)
            edges.add(edge_key(outer[i % m], inner[(j + 1) % k]))
            j += 1
    return {
        "m": m,
        "k": k,
        "n_chords": len(chords),
        "outer": outer,
        "inner": inner,
        "edges": sorted(edges),
        "triangles": triangles,
        "inner_chords": sorted(chords),
        "lattice_path": path,
        "seed": None,
    }
 def chord_crosses(c1: tuple[int, int], c2: tuple[int, int]) -> bool:
    a, b = c1
    c, d = c2
    return (a < c < b < d) or (c < a < d < b)
 def chord_sets(k: int, max_chords: int) -> list[tuple[tuple[int, int], ...]]:
    candidates = []
    for a in range(k):
        for b in range(a + 2, k):
            if not (a == 0 and b == k - 1):
                candidates.append((a, b))
    out = [()]
    def rec(start: int, chosen: tuple[tuple[int, int], ...]) -> None:
        if len(chosen) >= max_chords:
            return
        for idx in range(start, len(candidates)):
            chord = candidates[idx]
            if any(chord_crosses(chord, old) for old in chosen):
                continue
            nxt = chosen + (chord,)
            out.append(nxt)
            rec(idx + 1, nxt)
    rec(0, ())
    return out
 def lattice_paths(m: int, k: int):
    for o_positions in itertools.combinations(range(m + k), m):
        o_set = set(o_positions)
        yield "".join("O" if idx in o_set else "I" for idx in range(m + k))
 def edge_key(u: int, v: int) -> Edge:
    return tuple(sorted((u, v)))
 def face_edges(face: tuple[int, ...]) -> list[Edge]:
    return [edge_key(face[i], face[(i + 1) % len(face)]) for i in range(len(face))]
 def is_cycle_edge(edge: Edge, cycle: list[int]) -> bool:
    cycle_set = set(cycle)
    if not set(edge) <= cycle_set:
        return False
    n = len(cycle)
    idx = {v: i for i, v in enumerate(cycle)}
    a, b = idx[edge[0]], idx[edge[1]]
    return (a - b) % n in (1, n - 1)
 def is_inner_chord(edge: Edge, m: int, k: int) -> bool:
    u, v = edge
    if not (m <= u < m + k and m <= v < m + k):
        return False
    a, b = u - m, v - m
    d = abs(a - b)
    return min(d, k - d) != 1
 def suppress_reason(e1: Edge, e2: Edge, tire: dict) -> str | None:
    outer = tire["outer"]
    inner = tire["inner"]
    if is_cycle_edge(e1, outer) and is_cycle_edge(e2, outer):
        return "outer_boundary"
    if is_cycle_edge(e1, inner) and is_cycle_edge(e2, inner):
        return "inner_boundary"
    m, k = tire["m"], tire["k"]
    if is_inner_chord(e1, m, k) or is_inner_chord(e2, m, k):
        return "inner_chord"
    return None
 def medial_from_faces(faces: list[tuple[int, ...]], retained: set[Edge]) -> set[MedialEdge]:
    medial_edges: set[MedialEdge] = set()
    for face in faces:
        boundary = [e for e in face_edges(face) if e in retained]
        if len(boundary) < 2:
            continue
        for i, e in enumerate(boundary):
            nxt = boundary[(i + 1) % len(boundary)]
            if e != nxt:
                medial_edges.add(tuple(sorted((e, nxt))))
    return medial_edges
 def compare_tire(tire: dict, *, standalone_boundary_faces: bool) -> dict:
    annular_faces = [tuple(tri) for tri in tire["triangles"]]
    faces = list(annular_faces)
    if standalone_boundary_faces:
        faces.append(tuple(tire["outer"]))
        faces.append(tuple(reversed(tire["inner"])))
    # Definition 3.1 includes edges incident to at least one tread triangle.
    retained = {e for face in annular_faces for e in face_edges(face)}
    full_edges = medial_from_faces(faces, retained)
    removed = {me for me in full_edges if suppress_reason(me[0], me[1], tire)}
    reduced_edges = full_edges - removed
    reasons = Counter(suppress_reason(me[0], me[1], tire) for me in removed)
    reasons.pop(None, None)
    return {
        "vertices": len(retained),
        "full_edges": len(full_edges),
        "reduced_edges": len(reduced_edges),
        "removed": len(removed),
        "reasons": reasons,
        "examples": sorted(removed)[:5],
    }
 def run_sweep(args: argparse.Namespace) -> None:
    ambient_cases = 0
    ambient_differ = []
    standalone_cases = 0
    standalone_differ = []
    ambient_reasons: Counter[str] = Counter()
    standalone_reasons: Counter[str] = Counter()
    max_chords = args.max_chords
    for m in range(args.min_cycle, args.max_cycle + 1):
        for k in range(args.min_cycle, args.max_cycle + 1):
            for chords in range(max_chords + 1):
                for seed in range(args.seeds):
                    tire = random_tire(m=m, k=k, n_chords=chords, seed=seed)
                    ambient = compare_tire(tire, standalone_boundary_faces=False)
                    ambient_cases += 1
                    ambient_reasons.update(ambient["reasons"])
                    if ambient["removed"]:
                        ambient_differ.append((m, k, chords, seed, tire, ambient))
                    standalone = compare_tire(tire, standalone_boundary_faces=True)
                    standalone_cases += 1
                    standalone_reasons.update(standalone["reasons"])
                    if standalone["removed"]:
                        standalone_differ.append((m, k, chords, seed, tire, standalone))
    print("ambient tread-face model")
    print(f"  cases checked: {ambient_cases}")
    print(f"  cases where full != reduced: {len(ambient_differ)}")
    print(f"  removed-edge reasons: {dict(sorted(ambient_reasons.items()))}")
    if ambient_differ:
        m, k, chords, seed, tire, result = ambient_differ[0]
        print("  first difference:")
        print(f"    m={m} k={k} requested_chords={chords} seed={seed}")
        print(f"    path={tire['lattice_path']} chords={tire['inner_chords']}")
        print(f"    removed examples={result['examples']}")
    print()
    print("standalone tire-with-boundary-faces model")
    print(f"  cases checked: {standalone_cases}")
    print(f"  cases where full != reduced: {len(standalone_differ)}")
    print(f"  removed-edge reasons: {dict(sorted(standalone_reasons.items()))}")
    if standalone_differ:
        m, k, chords, seed, tire, result = standalone_differ[0]
        print("  first difference:")
        print(f"    m={m} k={k} requested_chords={chords} seed={seed}")
        print(f"    path={tire['lattice_path']} chords={tire['inner_chords']}")
        print(f"    full_edges={result['full_edges']} reduced_edges={result['reduced_edges']}")
        print(f"    removed examples={result['examples']}")
 def run_exhaustive(args: argparse.Namespace) -> None:
    ambient_cases = 0
    ambient_differ = []
    standalone_cases = 0
    standalone_differ = []
    for m in range(args.min_cycle, args.max_cycle + 1):
        for k in range(args.min_cycle, args.max_cycle + 1):
            for chords in chord_sets(k, args.max_chords):
                for path in lattice_paths(m, k):
                    tire = tire_from_path(m, k, chords, path)
                    ambient = compare_tire(tire, standalone_boundary_faces=False)
                    ambient_cases += 1
                    if ambient["removed"]:
                        ambient_differ.append((m, k, chords, path, ambient))
                    standalone = compare_tire(tire, standalone_boundary_faces=True)
                    standalone_cases += 1
                    if standalone["removed"]:
                        standalone_differ.append((m, k, chords, path, standalone))
    print("exhaustive ambient tread-face model")
    print(f"  cases checked: {ambient_cases}")
    print(f"  cases where full != reduced: {len(ambient_differ)}")
    if ambient_differ:
        m, k, chords, path, result = ambient_differ[0]
        print("  first difference:")
        print(f"    m={m} k={k} chords={chords} path={path}")
        print(f"    removed examples={result['examples']}")
    print()
    print("exhaustive standalone tire-with-boundary-faces model")
    print(f"  cases checked: {standalone_cases}")
    print(f"  cases where full != reduced: {len(standalone_differ)}")
    if standalone_differ:
        m, k, chords, path, result = standalone_differ[0]
        print("  first difference:")
        print(f"    m={m} k={k} chords={chords} path={path}")
        print(f"    full_edges={result['full_edges']} reduced_edges={result['reduced_edges']}")
        print(f"    removed examples={result['examples']}")
 def main() -> None:
    parser = argparse.ArgumentParser()
    parser.add_argument("--min-cycle", type=int, default=3)
    parser.add_argument("--max-cycle", type=int, default=8)
    parser.add_argument("--max-chords", type=int, default=3)
    parser.add_argument("--seeds", type=int, default=50)
    parser.add_argument("--exhaustive", action="store_true")
    args = parser.parse_args()
    if args.exhaustive:
        run_exhaustive(args)
    else:
        run_sweep(args)
 if __name__ == "__main__":
    main()
@@ -0,0 +1,82 @@
 # Full vs Reduced Medial Tire Findings
 Question: do Definition 3.1 (full medial tire graph) and Definition 3.2
 (reduced medial tire graph) differ?
 ## Experiment
 Script:
 ```bash
 python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py
 ```
 The script compares two models.
 - Ambient tread-face model: medial edges are contributed by annular
  triangular faces of the tire tread inside the ambient triangulation.
 - Standalone tire-with-boundary-faces model: the outer and inner
  boundary walks are also treated as faces, as in the older drawing
  script.
 ## Random Sweep
 Command:
 ```bash
 python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py
 ```
 Result:
 ```text
 ambient tread-face model
  cases checked: 7200
  cases where full != reduced: 0
  removed-edge reasons: {}
 standalone tire-with-boundary-faces model
  cases checked: 7200
  cases where full != reduced: 7200
  removed-edge reasons: {'inner_boundary': 39600, 'outer_boundary': 39600}
  first difference:
    m=3 k=3 requested_chords=0 seed=0
    path=IOOOII chords=[]
    full_edges=24 reduced_edges=18
    removed examples=[((0, 1), (0, 2)), ((0, 1), (1, 2)), ((0, 2), (1, 2)), ((3, 4), (3, 5)), ((3, 4), (4, 5))]
 ```
 ## Exhaustive Small Sweep
 Command:
 ```bash
 python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py --exhaustive --max-cycle 5 --max-chords 2
 ```
 Result:
 ```text
 exhaustive ambient tread-face model
  cases checked: 5578
  cases where full != reduced: 0
 exhaustive standalone tire-with-boundary-faces model
  cases checked: 5578
  cases where full != reduced: 5578
  first difference:
    m=3 k=3 chords=() path=OOOIII
    full_edges=24 reduced_edges=18
    removed examples=[((0, 1), (0, 2)), ((0, 1), (1, 2)), ((0, 2), (1, 2)), ((3, 4), (3, 5)), ((3, 4), (4, 5))]
 ```
 ## Interpretation
 For the intended ambient-triangulation definition, the experiments
 support the suspicion that Definition 3.1 and Definition 3.2 coincide:
 same-boundary medial edges do not arise from annular triangular tread
 faces, and inner chords are not incident to tread triangles.
 They differ only in the standalone tire-with-boundary-faces model,
 where the artificial outer and inner boundary faces create medial edges
 between consecutive boundary edges.
@@ -0,0 +1,34 @@
 \relax 
 \citation{bauerfeld-nested-tire-decompositions}
 \citation{bauerfeld-nested-tire-decompositions}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Background}}{1}{}\protected@file@percent }
 \citation{bauerfeld-nested-tire-decompositions}
 \citation{bauerfeld-nested-tire-decompositions}
 \newlabel{def:medial-graph}{{2.1}{2}}
 \newlabel{prop:medial-dual-invariance}{{2.3}{2}}
 \newlabel{cor:tait-medial}{{2.4}{2}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Medial tire pieces}}{2}{}\protected@file@percent }
 \newlabel{def:full-medial-tire}{{3.1}{2}}
 \newlabel{thm:annular-medial-colour-bound}{{3.3}{3}}
 \newlabel{def:boundary-medial-vertices}{{3.4}{3}}
 \newlabel{def:medial-restriction-relation}{{3.5}{3}}
 \citation{bauerfeld-nested-tire-decompositions}
 \citation{bauerfeld-nested-tire-decompositions}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{4}{}\protected@file@percent }
 \newlabel{cor:medial-tire-decomposition}{{4.1}{4}}
 \newlabel{def:compatible-family}{{4.2}{4}}
 \newlabel{prop:gluing-criterion}{{4.3}{4}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A medial pigeonhole programme}}{4}{}\protected@file@percent }
 \bibcite{bauerfeld-nested-tire-decompositions}{1}
 \bibcite{tait-original}{2}
 \newlabel{tocindent-1}{0pt}
 \newlabel{tocindent0}{12.7778pt}
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
 \newlabel{def:medial-boundary-state}{{5.1}{5}}
 \newlabel{conj:medial-chain-pigeonhole}{{5.2}{5}}
 \newlabel{conj:medial-route-fct}{{5.3}{5}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent }
 \gdef \@abspage@last{5}
@@ -0,0 +1,138 @@
 # Fdb version 3
 ["pdflatex"] 1780947173 "paper.tex" "paper.pdf" "paper" 1780947174
  "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1246382020 916 f87d7c45f9c908e672703b83b72241a3 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm" 1246382020 924 9904cf1d39e9767e7a3622f2a125a565 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1246382020 928 2dc8d444221b7a635bb58038579b861a ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1246382020 908 2921f8a10601f252058503cc6570e581 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm" 1246382020 940 75ac932a52f80982a9f8ea75d03a34cf ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1246382020 940 228d6584342e91276bf566bcf9716b83 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm" 1136768653 1328 c834bbb027764024c09d3d2bf908b5f0 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx8.tfm" 1136768653 1332 1fde11373e221473104d6cc5993f046e ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm" 1136768653 1300 63a6111ee6274895728663cf4b4e7e81 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm" 1136768653 1512 f21f83efb36853c0b70002322c1ab3ad ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm" 1136768653 1520 eccf95517727cb11801f4f1aee3a21b4 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm" 1136768653 1300 b62933e007d01cfd073f79b963c01526 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm" 1136768653 1292 21c1c5bfeaebccffdb478fd231a0997d ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmss10.tfm" 1136768653 1316 b636689f1933f24d1294acdf6041daaa ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmss8.tfm" 1136768653 1296 d77f431d10d47c8ea2cc18cf45346274 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm" 1136768653 1116 933a60c408fc0a863a92debe84b2d294 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1136768653 1120 8b7d695260f3cff42e636090a8002094 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm" 1136768653 1504 1747189e0441d1c18f3ea56fafc1c480 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1248133631 34811 78b52f49e893bcba91bd7581cdc144c0 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx8.pfb" 1248133631 32166 b0c356b15f19587482a9217ce1d8fa67 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1248133631 32001 6aeea3afe875097b1eb0da29acd61e28 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb" 1248133631 37912 77d683123f92148345f3fc36a38d9ab1 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1248133631 36281 c355509802a035cadc5f15869451dcee ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb" 1248133631 35469 70d41d2b9ea31d5d813066df7c99281c ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1248133631 32762 224316ccc9ad3ca0423a14971cfa7fc1 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1248133631 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmss10.pfb" 1248133631 24457 5cbb7bdf209d5d1ce9892a9b80a307cc ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb" 1248133631 32915 7bf7720c61a5b3a7ff25b0964421c9b6 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb" 1248133631 32587 1788b0c1c5b39540c96f5e42ccd6dae8 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1248133631 32716 08e384dc442464e7285e891af9f45947 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1248133631 37944 359e864bd06cde3b1cf57bb20757fb06 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb" 1248133631 35660 fb24af7afbadb71801619f1415838111 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb" 1248133631 31764 459c573c03a4949a528c2cc7f557e217 ""
  "/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii" 1461363279 71627 94eb9990bed73c364d7f53f960cc8c5b ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex" 1601326656 992 855ff26741653ab54814101ca36e153c ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex" 1601326656 43820 1fef971b75380574ab35a0d37fd92608 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex" 1601326656 19324 f4e4c6403dd0f1605fd20ed22fa79dea ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex" 1601326656 6038 ccb406740cc3f03bbfb58ad504fe8c27 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex" 1601326656 6944 e12f8f7a7364ddf66f93ba30fb3a3742 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex" 1601326656 4883 42daaf41e27c3735286e23e48d2d7af9 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex" 1601326656 2544 8c06d2a7f0f469616ac9e13db6d2f842 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex" 1601326656 44195 5e390c414de027626ca5e2df888fa68d ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex" 1601326656 17311 2ef6b2e29e2fc6a2fc8d6d652176e257 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex" 1601326656 21302 788a79944eb22192a4929e46963a3067 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex" 1601326656 9690 01feb7cde25d4293ef36eef45123eb80 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex" 1601326656 33335 dd1fa4814d4e51f18be97d88bf0da60c ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex" 1601326656 2965 4c2b1f4e0826925746439038172e5d6f ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex" 1601326656 5196 2cc249e0ee7e03da5f5f6589257b1e5b ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex" 1601326656 20726 d4c8db1e2e53b72721d29916314a22ea ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex" 1601326656 35249 abd4adf948f960299a4b3d27c5dddf46 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex" 1601326656 21989 fdc867d05d228316de137a9fc5ec3bbe ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex" 1601326656 8893 e851de2175338fdf7c17f3e091d94618 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex" 1601326656 4572 4a19637ef65ce88ad2f2d5064b69541d ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex" 1608933718 11518 738408f795261b70ce8dd47459171309 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex" 1621110968 186007 6e7dfe0bd57520fd5f91641aa72dcac8 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex" 1601326656 32995 ac577023e12c0e4bd8aa420b2e852d1a ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex" 1557692582 3063 8c415c68a0f3394e45cfeca0b65f6ee6 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex" 1601326656 521 8e224a7af69b7fee4451d1bf76b46654 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex" 1601326656 13391 84d29568c13bdce4133ab4a214711112 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex" 1601326656 104935 184ed87524e76d4957860df4ce0cd1c3 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex" 1601326656 10165 cec5fa73d49da442e56efc2d605ef154 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex" 1601326656 28178 41c17713108e0795aac6fef3d275fbca ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex" 1601326656 9989 c55967bf45126ff9b061fa2ca0c4694f ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex" 1601326656 3865 ac538ab80c5cf82b345016e474786549 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex" 1557692582 3177 27d85c44fbfe09ff3b2cf2879e3ea434 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex" 1621110968 11024 0179538121bc2dba172013a3ef89519f ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex" 1608933718 7854 4176998eeefd8745ac6d2d4bd9c98451 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex" 1601326656 3379 781797a101f647bab82741a99944a229 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex" 1601326656 92405 f515f31275db273f97b9d8f52e1b0736 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex" 1601326656 37376 11cd75aac3da1c1b152b2848f30adc14 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex" 1601326656 8471 c2883569d03f69e8e1cabfef4999cfd7 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex" 1601326656 21201 08d231a2386e2b61d64641c50dc15abd ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex" 1601326656 16121 346f9013d34804439f7436ff6786cef7 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex" 1621110968 44784 cedaa399d15f95e68e22906e2cc09ef8 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex" 1621110968 465 d68603f8b820ea4a08cce534944db581 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg" 1601326656 926 2963ea0dcf6cc6c0a770b69ec46a477b ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def" 1601326656 5546 f3f24d7898386cb7daac70bdd2c4d6dc ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def" 1601326656 12601 4786e597516eddd82097506db7cfa098 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex" 1621110968 61163 9b2eefc24e021323e0fc140e9826d016 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex" 1601326656 1896 b8e0ca0ac371d74c0ca05583f6313c91 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex" 1601326656 7778 53c8b5623d80238f6a20aa1df1868e63 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex" 1606168878 23997 a4bed72405fa644418bea7eac2887006 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex" 1621110968 37060 797782f0eb50075c9bc952374d9a659a ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex" 1601326656 37431 9abe862035de1b29c7a677f3205e3d9f ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex" 1601326656 4494 af17fb7efeafe423710479858e42fa7e ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex" 1601326656 7251 fb18c67117e09c64de82267e12cd8aa4 ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex" 1621110968 29274 e15c5b7157d21523bd9c9f1dfa146b8e ""
  "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def" 1621110968 6825 a2b0ea5b539dda0625e99dd15785ab59 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls" 1591045760 61881 a7369c346c2922a758ae6283cc1ed014 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty" 1359763108 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty" 1359763108 13829 94730e64147574077f8ecfea9bb69af4 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd" 1359763108 961 6518c6525a34feb5e8250ffa91731cff ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd" 1359763108 961 d02606146ba5601b5645f987c92e6193 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty" 1622667781 2222 da905dc1db75412efd2d8f67739f0596 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty" 1622667781 4173 bc0410bcccdff806d6132d3c1ef35481 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty" 1636758526 87648 07fbb6e9169e00cb2a2f40b31b2dbf3c ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty" 1636758526 4128 8eea906621b6639f7ba476a472036bbe ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty" 1636758526 2444 926f379cc60fcf0c6e3fee2223b4370d ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty" 1579991033 13886 d1306dcf79a944f6988e688c1785f9ce ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg" 1459978653 1213 620bba36b25224fa9b7e1ccb4ecb76fd ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg" 1465944070 1224 978390e9c2234eab29404bc21b268d1e ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def" 1601931164 19103 48d29b6e2a64cb717117ef65f107b404 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty" 1622581934 18399 7e40f80366dffb22c0e7b70517db5cb4 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty" 1636758526 7996 a8fb260d598dcaf305a7ae7b9c3e3229 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty" 1622581934 2671 4de6781a30211fe0ea4c672e4a2a8166 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty" 1636758526 4009 187ea2dc3194cd5a76cd99a8d7a6c4d0 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def" 1644269979 29921 d0acc05a38bd4aa3af2017f0b7c137ce ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg" 1279039959 678 4792914a8f45be57bb98413425e4c7af ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty" 1601326656 1090 bae35ef70b3168089ef166db3e66f5b2 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty" 1601326656 410 615550c46f918fcbee37641b02a862d9 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty" 1601326656 21013 f4ff83d25bb56552493b030f27c075ae ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty" 1601326656 989 c49c8ae06d96f8b15869da7428047b1e ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty" 1601326656 339 c2e180022e3afdb99c7d0ea5ce469b7d ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty" 1601326656 306 c56a323ca5bf9242f54474ced10fca71 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty" 1601326656 443 8c872229db56122037e86bcda49e14f3 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty" 1601326656 348 ee405e64380c11319f0e249fed57e6c5 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty" 1601326656 274 5ae372b7df79135d240456a1c6f2cf9a ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty" 1601326656 325 f9f16d12354225b7dd52a3321f085955 ""
  "/usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty" 1635798903 56029 3f7889dab51d620aa43177c391b7b190 ""
  "/usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf" 1646502317 40171 cdab547de63d26590bebb3baff566530 ""
  "/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 ""
  "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc ""
  "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c ""
  "paper.aux" 1780947174 1788 71344323fa0ab784f122b28ddb4904e9 "pdflatex"
  "paper.tex" 1780947165 17102 9b6f3bfc4420a1cb800822639b666f73 ""
  (generated)
  "paper.aux"
  "paper.log"
  "paper.pdf"
@@ -0,0 +1,474 @@
 PWD /Users/didericis/Code/math-research/papers/medial_tire_decompositions_of_plane_triangulations
 INPUT /usr/local/texlive/2022/texmf.cnf
 INPUT /usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf
 INPUT /usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt
 INPUT paper.tex
 OUTPUT paper.log
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
 INPUT ./paper.aux
 INPUT paper.aux
 INPUT paper.aux
 OUTPUT paper.aux
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
 INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
 INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
 INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
 OUTPUT paper.pdf
 INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmss10.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmss8.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmss8.tfm
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx8.tfm
 INPUT paper.aux
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx8.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmss10.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb
 INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb
@@ -0,0 +1,532 @@
 This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5)  8 JUN 2026 15:32
 entering extended mode
 restricted \write18 enabled.
 %&-line parsing enabled.
 **paper.tex
 (./paper.tex
 LaTeX2e <2021-11-15> patch level 1
 L3 programming layer <2022-02-24>
 (/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
 Document Class: amsart 2020/05/29 v2.20.6
 \linespacing=\dimen138
 \normalparindent=\dimen139
 \normaltopskip=\skip47
 (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
 Package: amsmath 2021/10/15 v2.17l AMS math features
 \@mathmargin=\skip48
 For additional information on amsmath, use the `?' option.
 (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
 Package: amstext 2021/08/26 v2.01 AMS text
 (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
 File: amsgen.sty 1999/11/30 v2.0 generic functions
 \@emptytoks=\toks16
 \ex@=\dimen140
 ))
 (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
 Package: amsbsy 1999/11/29 v1.2d Bold Symbols
 \pmbraise@=\dimen141
 )
 (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
 Package: amsopn 2021/08/26 v2.02 operator names
 )
 \inf@bad=\count185
 LaTeX Info: Redefining \frac on input line 234.
 \uproot@=\count186
 \leftroot@=\count187
 LaTeX Info: Redefining \overline on input line 399.
 \classnum@=\count188
 \DOTSCASE@=\count189
 LaTeX Info: Redefining \ldots on input line 496.
 LaTeX Info: Redefining \dots on input line 499.
 LaTeX Info: Redefining \cdots on input line 620.
 \Mathstrutbox@=\box50
 \strutbox@=\box51
 \big@size=\dimen142
 LaTeX Font Info:    Redeclaring font encoding OML on input line 743.
 LaTeX Font Info:    Redeclaring font encoding OMS on input line 744.
 \macc@depth=\count190
 \c@MaxMatrixCols=\count191
 \dotsspace@=\muskip16
 \c@parentequation=\count192
 \dspbrk@lvl=\count193
 \tag@help=\toks17
 \row@=\count194
 \column@=\count195
 \maxfields@=\count196
 \andhelp@=\toks18
 \eqnshift@=\dimen143
 \alignsep@=\dimen144
 \tagshift@=\dimen145
 \tagwidth@=\dimen146
 \totwidth@=\dimen147
 \lineht@=\dimen148
 \@envbody=\toks19
 \multlinegap=\skip49
 \multlinetaggap=\skip50
 \mathdisplay@stack=\toks20
 LaTeX Info: Redefining \[ on input line 2938.
 LaTeX Info: Redefining \] on input line 2939.
 )
 LaTeX Font Info:    Trying to load font information for U+msa on input line 397
 .
 (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
 File: umsa.fd 2013/01/14 v3.01 AMS symbols A
 )
 (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support
 \symAMSa=\mathgroup4
 \symAMSb=\mathgroup5
 LaTeX Font Info:    Redeclaring math symbol \hbar on input line 98.
 LaTeX Font Info:    Overwriting math alphabet `\mathfrak' in version `bold'
 (Font)                  U/euf/m/n --> U/euf/b/n on input line 106.
 )
 \copyins=\insert199
 \abstractbox=\box52
 \listisep=\skip51
 \c@part=\count197
 \c@section=\count198
 \c@subsection=\count266
 \c@subsubsection=\count267
 \c@paragraph=\count268
 \c@subparagraph=\count269
 \c@figure=\count270
 \c@table=\count271
 \abovecaptionskip=\skip52
 \belowcaptionskip=\skip53
 \captionindent=\dimen149
 \thm@style=\toks21
 \thm@bodyfont=\toks22
 \thm@headfont=\toks23
 \thm@notefont=\toks24
 \thm@headpunct=\toks25
 \thm@preskip=\skip54
 \thm@postskip=\skip55
 \thm@headsep=\skip56
 \dth@everypar=\toks26
 )
 (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
 Package: amssymb 2013/01/14 v3.01 AMS font symbols
 )
 (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
 Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR)
 (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
 Package: keyval 2014/10/28 v1.15 key=value parser (DPC)
 \KV@toks@=\toks27
 )
 (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
 Package: graphics 2021/03/04 v1.4d Standard LaTeX Graphics (DPC,SPQR)
 (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
 Package: trig 2021/08/11 v1.11 sin cos tan (DPC)
 )
 (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
 File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration
 )
 Package graphics Info: Driver file: pdftex.def on input line 107.
 (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
 File: pdftex.def 2020/10/05 v1.2a Graphics/color driver for pdftex
 ))
 \Gin@req@height=\dimen150
 \Gin@req@width=\dimen151
 )
 (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
 (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
 (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.te
 x
 \pgfutil@everybye=\toks28
 \pgfutil@tempdima=\dimen152
 \pgfutil@tempdimb=\dimen153
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-li
 sts.tex))
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def
 \pgfutil@abb=\box53
 ) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex)
 Package: pgfrcs 2021/05/15 v3.1.9a (3.1.9a)
 ))
 Package: pgf 2021/05/15 v3.1.9a (3.1.9a)
 (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
 (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
 Package: pgfsys 2021/05/15 v3.1.9a (3.1.9a)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
 \pgfkeys@pathtoks=\toks29
 \pgfkeys@temptoks=\toks30
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.c
 ode.tex
 \pgfkeys@tmptoks=\toks31
 ))
 \pgf@x=\dimen154
 \pgf@y=\dimen155
 \pgf@xa=\dimen156
 \pgf@ya=\dimen157
 \pgf@xb=\dimen158
 \pgf@yb=\dimen159
 \pgf@xc=\dimen160
 \pgf@yc=\dimen161
 \pgf@xd=\dimen162
 \pgf@yd=\dimen163
 \w@pgf@writea=\write3
 \r@pgf@reada=\read2
 \c@pgf@counta=\count272
 \c@pgf@countb=\count273
 \c@pgf@countc=\count274
 \c@pgf@countd=\count275
 \t@pgf@toka=\toks32
 \t@pgf@tokb=\toks33
 \t@pgf@tokc=\toks34
 \pgf@sys@id@count=\count276
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg
 File: pgf.cfg 2021/05/15 v3.1.9a (3.1.9a)
 )
 Driver file for pgf: pgfsys-pdftex.def
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.d
 ef
 File: pgfsys-pdftex.def 2021/05/15 v3.1.9a (3.1.9a)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-p
 df.def
 File: pgfsys-common-pdf.def 2021/05/15 v3.1.9a (3.1.9a)
 )))
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.
 code.tex
 File: pgfsyssoftpath.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgfsyssoftpath@smallbuffer@items=\count277
 \pgfsyssoftpath@bigbuffer@items=\count278
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.
 code.tex
 File: pgfsysprotocol.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 )) (/usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
 Package: xcolor 2021/10/31 v2.13 LaTeX color extensions (UK)
 (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
 File: color.cfg 2016/01/02 v1.6 sample color configuration
 )
 Package xcolor Info: Driver file: pdftex.def on input line 227.
 Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1352.
 Package xcolor Info: Model `hsb' substituted by `rgb' on input line 1356.
 Package xcolor Info: Model `RGB' extended on input line 1368.
 Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1370.
 Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1371.
 Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1372.
 Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1373.
 Package xcolor Info: Model `Gray' substituted by `gray' on input line 1374.
 Package xcolor Info: Model `wave' substituted by `hsb' on input line 1375.
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
 Package: pgfcore 2021/05/15 v3.1.9a (3.1.9a)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex
 \pgfmath@dimen=\dimen164
 \pgfmath@count=\count279
 \pgfmath@box=\box54
 \pgfmath@toks=\toks35
 \pgfmath@stack@operand=\toks36
 \pgfmath@stack@operation=\toks37
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.
 tex
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic
 .code.tex)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigo
 nometric.code.tex)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.rando
 m.code.tex)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.compa
 rison.code.tex)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.
 code.tex)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round
 .code.tex)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.
 code.tex)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integ
 erarithmetics.code.tex)))
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex
 \c@pgfmathroundto@lastzeros=\count280
 )) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.co
 de.tex
 File: pgfcorepoints.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgf@picminx=\dimen165
 \pgf@picmaxx=\dimen166
 \pgf@picminy=\dimen167
 \pgf@picmaxy=\dimen168
 \pgf@pathminx=\dimen169
 \pgf@pathmaxx=\dimen170
 \pgf@pathminy=\dimen171
 \pgf@pathmaxy=\dimen172
 \pgf@xx=\dimen173
 \pgf@xy=\dimen174
 \pgf@yx=\dimen175
 \pgf@yy=\dimen176
 \pgf@zx=\dimen177
 \pgf@zy=\dimen178
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconst
 ruct.code.tex
 File: pgfcorepathconstruct.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgf@path@lastx=\dimen179
 \pgf@path@lasty=\dimen180
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage
 .code.tex
 File: pgfcorepathusage.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgf@shorten@end@additional=\dimen181
 \pgf@shorten@start@additional=\dimen182
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.co
 de.tex
 File: pgfcorescopes.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgfpic=\box55
 \pgf@hbox=\box56
 \pgf@layerbox@main=\box57
 \pgf@picture@serial@count=\count281
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicst
 ate.code.tex
 File: pgfcoregraphicstate.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgflinewidth=\dimen183
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransform
 ations.code.tex
 File: pgfcoretransformations.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgf@pt@x=\dimen184
 \pgf@pt@y=\dimen185
 \pgf@pt@temp=\dimen186
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.cod
 e.tex
 File: pgfcorequick.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.c
 ode.tex
 File: pgfcoreobjects.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathproce
 ssing.code.tex
 File: pgfcorepathprocessing.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.co
 de.tex
 File: pgfcorearrows.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgfarrowsep=\dimen187
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.cod
 e.tex
 File: pgfcoreshade.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgf@max=\dimen188
 \pgf@sys@shading@range@num=\count282
 \pgf@shadingcount=\count283
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.cod
 e.tex
 File: pgfcoreimage.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.
 code.tex
 File: pgfcoreexternal.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgfexternal@startupbox=\box58
 ))
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.co
 de.tex
 File: pgfcorelayers.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretranspare
 ncy.code.tex
 File: pgfcoretransparency.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.
 code.tex
 File: pgfcorepatterns.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.
 tex
 File: pgfcorerdf.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 )))
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.cod
 e.tex
 File: pgfmoduleshapes.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgfnodeparttextbox=\box59
 )
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.
 tex
 File: pgfmoduleplot.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 )
 (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version
 -0-65.sty
 Package: pgfcomp-version-0-65 2021/05/15 v3.1.9a (3.1.9a)
 \pgf@nodesepstart=\dimen189
 \pgf@nodesepend=\dimen190
 )
 (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version
 -1-18.sty
 Package: pgfcomp-version-1-18 2021/05/15 v3.1.9a (3.1.9a)
 ))
 (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
 (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex)
 ) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex))
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
 Package: pgffor 2021/05/15 v3.1.9a (3.1.9a)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex)
 \pgffor@iter=\dimen191
 \pgffor@skip=\dimen192
 \pgffor@stack=\toks38
 \pgffor@toks=\toks39
 ))
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.cod
 e.tex
 Package: tikz 2021/05/15 v3.1.9a (3.1.9a)
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothan
 dlers.code.tex
 File: pgflibraryplothandlers.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgf@plot@mark@count=\count284
 \pgfplotmarksize=\dimen193
 )
 \tikz@lastx=\dimen194
 \tikz@lasty=\dimen195
 \tikz@lastxsaved=\dimen196
 \tikz@lastysaved=\dimen197
 \tikz@lastmovetox=\dimen198
 \tikz@lastmovetoy=\dimen256
 \tikzleveldistance=\dimen257
 \tikzsiblingdistance=\dimen258
 \tikz@figbox=\box60
 \tikz@figbox@bg=\box61
 \tikz@tempbox=\box62
 \tikz@tempbox@bg=\box63
 \tikztreelevel=\count285
 \tikznumberofchildren=\count286
 \tikznumberofcurrentchild=\count287
 \tikz@fig@count=\count288
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.cod
 e.tex
 File: pgfmodulematrix.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgfmatrixcurrentrow=\count289
 \pgfmatrixcurrentcolumn=\count290
 \pgf@matrix@numberofcolumns=\count291
 )
 \tikz@expandcount=\count292
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/librarie
 s/tikzlibrarytopaths.code.tex
 File: tikzlibrarytopaths.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 )))
 (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/librarie
 s/tikzlibrarybackgrounds.code.tex
 File: tikzlibrarybackgrounds.code.tex 2021/05/15 v3.1.9a (3.1.9a)
 \pgf@layerbox@background=\box64
 \pgf@layerboxsaved@background=\box65
 )
 \c@theorem=\count293
 (/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
 File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX)
 \l__color_backend_stack_int=\count294
 \l__pdf_internal_box=\box66
 )
 (./paper.aux)
 \openout1 = `paper.aux'.
 LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 29.
 LaTeX Font Info:    ... okay on input line 29.
 LaTeX Font Info:    Checking defaults for OMS/cmsy/m/n on input line 29.
 LaTeX Font Info:    ... okay on input line 29.
 LaTeX Font Info:    Checking defaults for OT1/cmr/m/n on input line 29.
 LaTeX Font Info:    ... okay on input line 29.
 LaTeX Font Info:    Checking defaults for T1/cmr/m/n on input line 29.
 LaTeX Font Info:    ... okay on input line 29.
 LaTeX Font Info:    Checking defaults for TS1/cmr/m/n on input line 29.
 LaTeX Font Info:    ... okay on input line 29.
 LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 29.
 LaTeX Font Info:    ... okay on input line 29.
 LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 29.
 LaTeX Font Info:    ... okay on input line 29.
 LaTeX Font Info:    Trying to load font information for U+msa on input line 29.
 (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
 File: umsa.fd 2013/01/14 v3.01 AMS symbols A
 )
 LaTeX Font Info:    Trying to load font information for U+msb on input line 29.
 (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
 File: umsb.fd 2013/01/14 v3.01 AMS symbols B
 )
 (/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
 [Loading MPS to PDF converter (version 2006.09.02).]
 \scratchcounter=\count295
 \scratchdimen=\dimen259
 \scratchbox=\box67
 \nofMPsegments=\count296
 \nofMParguments=\count297
 \everyMPshowfont=\toks40
 \MPscratchCnt=\count298
 \MPscratchDim=\dimen260
 \MPnumerator=\count299
 \makeMPintoPDFobject=\count300
 \everyMPtoPDFconversion=\toks41
 ) (/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
 Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf
 Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4
 85.
 (/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
 File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
 e
 ))
 [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
 [2] [3] [4] [5] (./paper.aux) ) 
 Here is how much of TeX's memory you used:
 13541 strings out of 478268
 270377 string characters out of 5846347
 536683 words of memory out of 5000000
 31371 multiletter control sequences out of 15000+600000
 476880 words of font info for 57 fonts, out of 8000000 for 9000
 1302 hyphenation exceptions out of 8191
 84i,8n,89p,414b,279s stack positions out of 10000i,1000n,20000p,200000b,200000s
 </usr/local/texlive/2022/texmf-dist/fonts/type1/
 public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/p
 ublic/amsfonts/cm/cmbx8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub
 lic/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub
 lic/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publ
 ic/amsfonts/cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public
 /amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
 msfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
 fonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
 nts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts
 /cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm
 /cmss10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/
 cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/c
 msy5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cms
 y6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7
 .pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.
 pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pf
 b></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam1
 0.pfb>
 Output written on paper.pdf (5 pages, 232700 bytes).
 PDF statistics:
 113 PDF objects out of 1000 (max. 8388607)
 69 compressed objects within 1 object stream
 0 named destinations out of 1000 (max. 500000)
 13 words of extra memory for PDF output out of 10000 (max. 10000000)
@@ -0,0 +1,405 @@
 %% filename: amsart-template.tex
 %% American Mathematical Society
 %% AMS-LaTeX v.2 template for use with amsart
 %% ====================================================================
 \documentclass{amsart}
 \usepackage{amssymb}
 \usepackage{graphicx}
 \usepackage{tikz}
 \usetikzlibrary{backgrounds}
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem{lemma}[theorem]{Lemma}
 \newtheorem{corollary}[theorem]{Corollary}
 \newtheorem{proposition}[theorem]{Proposition}
 \newtheorem{conjecture}[theorem]{Conjecture}
 \theoremstyle{definition}
 \newtheorem{definition}[theorem]{Definition}
 \newtheorem{example}[theorem]{Example}
 \newtheorem{xca}[theorem]{Exercise}
 \theoremstyle{remark}
 \newtheorem{remark}[theorem]{Remark}
 \numberwithin{equation}{section}
 \begin{document}
 \title{Medial Tire Decompositions of Plane Triangulations}
 %    author one information
 \author{Eric Bauerfeld}
 \address{}
 \curraddr{}
 \email{}
 \thanks{}
 \subjclass[2010]{Primary }
 \keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Four Colour Theorem}
 \date{}
 \dedicatory{}
 \begin{abstract}
 We use the nested tire decomposition of a plane triangulation to induce
 a decomposition of its full medial graph into medial tire subgraphs.
 For a plane triangulation $G$, the medial graph $M(G)$ is naturally
 isomorphic to the medial graph of the planar dual $G^*$, and proper
 $3$-vertex-colourings of $M(G)$ are equivalent to proper
 $3$-edge-colourings of the cubic dual.  Thus Tait's reformulation of
 the Four Colour Theorem may be studied through proper vertex
 $3$-colourings of medial subgraphs.  We define medial tire pieces,
 their boundary-state restriction relations, and a chain-pigeonhole
 conjecture for compatible medial boundary states across the tire tree.
 \end{abstract}
 \maketitle
 \section{Introduction}
 A classical theorem of Tait recasts the Four Colour Theorem in dual,
 edge-colouring terms: a plane triangulation $G$ is properly
 $4$-vertex-colourable if and only if its dual cubic graph $G^*$ is
 properly $3$-edge-colourable.  The present paper records a medial
 version of this viewpoint.  The vertices of the medial graph $M(G)$
 correspond to edges of $G$, and adjacency in $M(G)$ records
 consecutiveness of edges around vertices and faces of $G$.  Since
 planar duality interchanges vertices and faces while preserving the
 edge set, $M(G)$ is naturally isomorphic to $M(G^*)$.
 Consequently a proper vertex $3$-colouring of $M(G)$ is the same
 object as a proper edge $3$-colouring of $G^*$.  This suggests another
 route toward the Four Colour Theorem: rather than colouring the dual
 cubic graph directly, decompose the full medial graph into local
 annular pieces and try to prove that their proper vertex
 $3$-colouring boundary restrictions always compose.
 The structural input is the nested tire decomposition of
 \cite{bauerfeld-nested-tire-decompositions}.  A level source in a plane
 triangulation determines a rooted tree of tire treads.  Each tread is
 an annular triangulated region with an outer boundary, an inner
 outerplanar graph, and annular triangular faces.  We show that this
 decomposition induces a decomposition of $M(G)$ into medial tire
 subgraphs.  The boundary data of a medial tire are proper
 $3$-colourings of the medial vertices corresponding to boundary edges
 in the associated dual tire graph.
 \section{Background}
 Throughout, $G$ is a simple plane maximal planar graph with fixed
 embedding, and $G^*$ denotes its full planar dual.  We use the level
 source, dual depth, tire graph, tire tread, and tire-tree terminology
 of~\cite{bauerfeld-nested-tire-decompositions}.  In particular, a level
 source $S$ determines a rooted tire tree $\mathcal{T}(G,S)$ whose
 vertices are tire treads and whose parent-child relation records
 nested containment across level-cycle interfaces.
 \begin{definition}[Medial graph]
 \label{def:medial-graph}
 Let $H$ be a plane graph.  The \emph{medial graph} $M(H)$ has one
 vertex $m_e$ for each edge $e \in E(H)$.  Two medial vertices
 $m_e,m_f$ are adjacent whenever $e$ and $f$ are consecutive in the
 cyclic order of edges around a vertex of $H$ or around a face of $H$.
 The embedding is the standard one obtained by placing $m_e$ at the
 midpoint of $e$ and drawing medial edges through the vertex- and
 face-corners of $H$.
 \end{definition}
 \begin{remark}
 If $H$ has bridges or vertices of degree $1$, the usual medial
 construction may create parallel edges or loops depending on the
 chosen convention.  In this paper the main application is to plane
 triangulations and their cubic planar duals, where the medial graph is
 a loopless $4$-regular plane graph.
 \end{remark}
 \begin{proposition}[Medial dual invariance]
 \label{prop:medial-dual-invariance}
 Let $H$ be a connected plane graph and let $H^*$ be its planar dual.
 Then there is a natural plane-graph isomorphism
 \[
    M(H) \cong M(H^*).
 \]
 \end{proposition}
 \begin{proof}
 Each edge $e \in E(H)$ corresponds to a unique dual edge $e^* \in
 E(H^*)$, giving a bijection $m_e \mapsto m_{e^*}$ between the vertices
 of $M(H)$ and $M(H^*)$.  In $M(H)$ two vertices $m_e,m_f$ are adjacent
 exactly when $e$ and $f$ are consecutive around either a vertex or a
 face of $H$.  Under duality, vertices and faces are interchanged, and
 the cyclic order of the corresponding dual edges around the dual face
 or dual vertex is the same up to reversal.  Thus the same pairs are
 medial-adjacent in $M(H^*)$, and the midpoint construction identifies
 the two embedded medial graphs.
 \end{proof}
 \begin{corollary}[Tait colourings as medial vertex colourings]
 \label{cor:tait-medial}
 Let $G$ be a simple plane triangulation.  Proper vertex
 $3$-colourings of $M(G)$ are in natural bijection with proper
 $3$-edge-colourings of the cubic planar dual $G^*$.
 \end{corollary}
 \begin{proof}
 By Proposition~\ref{prop:medial-dual-invariance}, $M(G) \cong
 M(G^*)$.  Vertices of $M(G^*)$ correspond to edges of $G^*$, and two
 such vertices are adjacent exactly when the corresponding dual edges
 are incident and consecutive around a vertex or face of $G^*$.  Since
 $G^*$ is cubic, proper vertex $3$-colouring of $M(G^*)$ is therefore
 equivalent to assigning three colours to the edges of $G^*$ so that the
 three edges incident to each dual vertex receive pairwise distinct
 colours.
 \end{proof}
 \section{Medial tire pieces}
 \begin{definition}[Full medial tire graph]
 \label{def:full-medial-tire}
 Let $T$ be a tire tread in the tire tree $\mathcal{T}(G,S)$ supplied
 by~\cite{bauerfeld-nested-tire-decompositions}.  The \emph{full medial
 tire graph} of $T$, denoted $\mathsf{M}(T)$, is the subgraph of
 $M(G)$ induced by the medial vertices $m_e$ with $e$ an edge of $G$
 incident to at least one triangular face in the tread $T$.  The medial
 vertices corresponding to annular edges of $T$ are called
 \emph{annular medial vertices}.
 \end{definition}
 \begin{remark}
 In the ambient-triangulation setting, the full medial tire graph
 $\mathsf{M}(T)$ coincides with the omitted-edge medial tire graph
 studied in~\cite{bauerfeld-nested-tire-decompositions}.  Indeed, the
 medial edges of $\mathsf{M}(T)$ are contributed by corners of annular
 triangular tread faces.  Such a face contains at most one outer-boundary
 edge and at most one inner-boundary edge, so it does not contribute a
 medial edge between two outer-boundary edges or between two
 inner-boundary edges.  Similarly, chords of the inner outerplanar graph
 lie outside the annular tread and are not incident to annular tread
 faces.  Thus the deletion rule used for the earlier reduced medial tire
 graph removes no edges from the ambient object $\mathsf{M}(T)$.
 The distinction only appears in the standalone drawing convention where
 the outer and inner boundary walks are added as artificial faces before
 forming a medial graph.  Those artificial faces create same-boundary
 medial edges, and the reduced construction deletes them.
 \end{remark}
 \begin{theorem}[Annular medial colour bound]
 \label{thm:annular-medial-colour-bound}
 Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire tread with
 non-degenerate boundaries and simple inner boundary $B_{\mathrm{in}}$.
 Let $A(T)$ be the subgraph of $\mathsf{M}(T)$ induced by the annular
 medial vertices.  For a graph $H$, write $\operatorname{Col}_3(H)$ for
 the set of proper $3$-vertex-colourings of $H$.  Then $A(T)$ is a cycle
 and
 \[
    |\operatorname{Col}_3(\mathsf{M}(T))|
    \;\leq\; |\operatorname{Col}_3(A(T))|.
 \]
 \end{theorem}
 \begin{proof}
 Since the tread is a triangulated annulus with no vertices in its
 interior, each annular face has exactly one boundary edge, lying either
 on $B_{\mathrm{out}}$ or on $B_{\mathrm{in}}$, and exactly two annular
 edges.  As the annular faces are traversed cyclically around the tread,
 consecutive faces share one annular edge.  Equivalently, the annular
 edges occur in a cyclic order in which each annular face contains two
 consecutive annular edges.  Hence the subgraph of $\mathsf{M}(T)$
 induced by the annular medial vertices is a cycle.
 Consider the restriction map from proper $3$-colourings of
 $\mathsf{M}(T)$ to colourings of this annular medial cycle $A(T)$.  We
 claim that this map is injective.  Let $x$ be a non-annular medial
 vertex.  Then $x$ corresponds to an edge of $B_{\mathrm{out}}$ or
 $B_{\mathrm{in}}$: chords of $O$ are not incident to annular tread
 faces, and hence do not contribute vertices of $\mathsf{M}(T)$.  This
 boundary edge is incident to a unique annular face of the tread, and
 the other two edges of that face are annular edges.  Therefore $x$ is
 adjacent in $\mathsf{M}(T)$ to the two annular medial vertices
 corresponding to those two annular edges.
 Those two annular medial vertices are adjacent to each other, because
 their annular edges are consecutive on the same triangular annular
 face.  In any proper $3$-colouring they therefore receive two distinct
 colours, and $x$ is forced to receive the remaining third colour.  Thus
 every non-annular medial vertex has its colour uniquely determined by
 the colouring of $A(T)$.  Two colourings of $\mathsf{M}(T)$ with the
 same restriction to $A(T)$ are identical, so the restriction map is
 injective.  The stated inequality follows.
 \end{proof}
 \begin{definition}[Boundary medial vertices]
 \label{def:boundary-medial-vertices}
 Let $T$ be a tire tread and let $\Gamma_T$ be the corresponding dual
 tire subgraph in $G^*$.  A vertex $m_e \in V(\mathsf{M}(T))$ is an
 \emph{outer boundary medial vertex} if the corresponding dual edge
 $e^* \in E(G^*)$ lies on the outer boundary of $\Gamma_T$.  It is an
 \emph{inner boundary medial vertex} if $e^*$ lies on the inner boundary
 of $\Gamma_T$.  We write
 \[
    \partial_{\mathrm{out}}\mathsf{M}(T)
    \quad\text{and}\quad
    \partial_{\mathrm{in}}\mathsf{M}(T)
 \]
 for the two boundary sets.
 \end{definition}
 \begin{definition}[Medial tire restriction relation]
 \label{def:medial-restriction-relation}
 Let $\mathrm{Col}_3(X)$ denote the set of proper vertex
 $3$-colourings of the induced subgraph on a vertex set $X$.  The
 \emph{medial tire restriction relation} of $T$ is
 \[
    R_T \subseteq
    \mathrm{Col}_3(\partial_{\mathrm{out}}\mathsf{M}(T))
    \times
    \mathrm{Col}_3(\partial_{\mathrm{in}}\mathsf{M}(T)),
 \]
 where $(\alpha,\beta) \in R_T$ exactly when $\alpha \cup \beta$
 extends to a proper vertex $3$-colouring of $\mathsf{M}(T)$.
 \end{definition}
 \begin{remark}
 The definition deliberately records boundary colourings on medial
 vertices corresponding to boundary edges in the dual tire graph.  Under
 Corollary~\ref{cor:tait-medial}, these are precisely edge-colouring
 states on the boundary edges through which a dual tire piece meets its
 parent and children.
 \end{remark}
 \section{Decomposition}
 \begin{corollary}[Medial tire decomposition]
 \label{cor:medial-tire-decomposition}
 Let $G$ be a plane triangulation with level source $S$.  The tire-tree
 decomposition $\mathcal{T}(G,S)$ of
 \cite{bauerfeld-nested-tire-decompositions} induces a rooted
 decomposition of the full medial graph $M(G)$ into full medial tire
 graphs $\{\mathsf{M}(T): T \in V(\mathcal{T}(G,S))\}$, glued along
 their boundary medial vertex sets.
 \end{corollary}
 \begin{proof}
 By the tire-tread partition theorem of
 \cite{bauerfeld-nested-tire-decompositions}, the bounded triangular
 faces of $G$ are partitioned into nested tire treads, with intersections
 between parent and child treads occurring only along their level-cycle
 interface data.  Every edge of $G$ that is incident to a bounded face
 therefore belongs to the closure of at least one tire tread, and an
 edge lying in two closures lies on the interface between adjacent
 treads in the tire tree.  Passing to $M(G)$ sends edges of $G$ to
 medial vertices.  Thus each tread determines the induced subgraph
 $\mathsf{M}(T)$ on its incident edge set, and overlaps between two such
 subgraphs are exactly the medial vertices corresponding to interface
 edges, namely the appropriate boundary medial vertex sets.
 \end{proof}
 \begin{definition}[Compatible family of medial tire colourings]
 \label{def:compatible-family}
 A \emph{compatible family of medial tire colourings} on
 $\mathcal{T}(G,S)$ is a choice, for each tread $T$, of a proper
 vertex $3$-colouring $\varphi_T$ of $\mathsf{M}(T)$ such that whenever
 $T'$ is a child tread of $T$, the two colourings agree on
 $
    V(\mathsf{M}(T)) \cap V(\mathsf{M}(T')).
 $
 \end{definition}
 \begin{proposition}[Gluing criterion]
 \label{prop:gluing-criterion}
 The full medial graph $M(G)$ has a proper vertex $3$-colouring if and
 only if the tire tree $\mathcal{T}(G,S)$ admits a compatible family of
 medial tire colourings.
 \end{proposition}
 \begin{proof}
 A proper vertex $3$-colouring of $M(G)$ restricts to a proper vertex
 $3$-colouring of every induced subgraph $\mathsf{M}(T)$, and these
 restrictions agree on overlaps.
 Conversely, suppose a compatible family is given.  Define a colour on
 each vertex $m_e$ of $M(G)$ by choosing any tread $T$ with
 $m_e \in V(\mathsf{M}(T))$ and setting
 $\varphi(m_e)=\varphi_T(m_e)$.  Compatibility makes this independent of
 the choice of $T$.  Every medial edge of $M(G)$ is drawn in a corner of
 some bounded triangular face of $G$ or along the outer boundary
 interface.  The relevant incident primal edges lie together in the
 closure of a single tire tread or in a shared boundary interface, where
 properness is already enforced by one of the local colourings.  Hence
 $\varphi$ is a proper vertex $3$-colouring of $M(G)$.
 \end{proof}
 \section{A medial pigeonhole programme}
 The restriction relation $R_T$ records exactly the local information
 needed to pass a medial $3$-colouring through a tire.  In a nested
 chain
 \[
    T_0 \supset T_1 \supset \cdots \supset T_k,
 \]
 the outer boundary state of $T_{i+1}$ must match an inner boundary
 state allowed by $R_{T_i}$.  Thus a proof of the Four Colour Theorem in
 this framework would follow from a structural reason that these
 restriction sets cannot remain mutually disjoint along every branch of
 the tire tree.
 \begin{definition}[Medial boundary state]
 \label{def:medial-boundary-state}
 A \emph{medial boundary state} on a boundary set
 $\partial\mathsf{M}(T)$ is a proper vertex $3$-colouring of the
 subgraph induced by that boundary set, considered up to permutation of
 the three colours and the dihedral symmetries of the boundary walk
 when that boundary is a cycle.
 \end{definition}
 \begin{conjecture}[Medial chain-pigeonhole principle]
 \label{conj:medial-chain-pigeonhole}
 There is a function $N(k)$ such that the following holds.  Let
 $T_0 \supset T_1 \supset \cdots \supset T_{N(k)}$ be a nested chain of
 tire treads whose relevant boundary medial walks have length at most
 $k$.  Then two adjacent restriction relations in the chain have
 compatible medial boundary states after colour permutation and boundary
 symmetry.  Equivalently, the chain contains a local gluing step that
 cannot be obstructed by disjoint proper vertex $3$-colouring
 restrictions.
 \end{conjecture}
 \begin{conjecture}[Medial tire route to the Four Colour Theorem]
 \label{conj:medial-route-fct}
 For every plane triangulation $G$ and every level source $S$, the
 restriction relations $\{R_T : T \in V(\mathcal{T}(G,S))\}$ admit a
 compatible selection of boundary states across the tire tree.  Hence
 $M(G)$ is properly vertex $3$-colourable, $G^*$ is properly
 $3$-edge-colourable, and $G$ is properly $4$-vertex-colourable.
 \end{conjecture}
 \begin{remark}
 Conjecture~\ref{conj:medial-route-fct} is equivalent in strength to
 the Four Colour Theorem when combined with Tait's correspondence.  The
 point of the formulation is not to weaken the target theorem, but to
 move the obstruction into finite boundary-state restrictions carried by
 annular medial tire pieces.
 \end{remark}
 \begin{thebibliography}{9}
 \bibitem{bauerfeld-nested-tire-decompositions}
 E.~Bauerfeld,
 \emph{Nested Tire Decompositions of Plane Triangulations},
 manuscript (math-research repository), 2026.
 \bibitem{tait-original}
 P.~G. Tait,
 \emph{Remarks on the colourings of maps},
 Proceedings of the Royal Society of Edinburgh \textbf{10} (1880),
 729--729.
 \end{thebibliography}
 \end{document}
@@ -8,50 +8,48 @@
 \citation{dvorak-lidicky-cones}
 \citation{heesch-untersuchungen}
 \citation{robertson-sanders-seymour-thomas}
 \citation{robertson-sanders-seymour-thomas}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Related work.}}{1}{}\protected@file@percent }
 \citation{robertson-sanders-seymour-thomas}
 \newlabel{def:dual}{{1.3}{2}}
 \newlabel{def:dual-depth}{{1.4}{2}}
 \newlabel{def:dual-component}{{1.5}{2}}
 \newlabel{def:tire-graph}{{1.6}{2}}
 \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname  \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{3}{}\protected@file@percent }
 \newlabel{fig:dual-depth}{{1}{3}}
 \newlabel{def:tire-graph}{{1.6}{3}}
 \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm  {out}}$ a $6$-cycle on vertices $0,\dots  ,5$ (blue), inner boundary $B_{\mathrm  {in}}$ a $4$-cycle on vertices $6,\dots  ,9$ (red), inner outerplanar graph $O = B_{\mathrm  {in}} \cup \{7\text  {--}9\}$ (with one chord, orange), and $E_{\mathrm  {ann}}$ (grey) tiling the annulus between $B_{\mathrm  {out}}$ and $B_{\mathrm  {in}}$ by ten triangular faces.}}{4}{}\protected@file@percent }
 \newlabel{fig:tire-example}{{2}{4}}
 \newlabel{def:medial-tire-graph}{{1.7}{4}}
-\newlabel{thm:annular-medial-colour-bound}{{1.8}{4}}
+\newlabel{rem:tire-counts}{{1.8}{4}}
 \newlabel{prop:no-level-d-pinch}{{1.9}{4}}
 \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The medial tire graph for the tire in Figure\nonbreakingspace 2\hbox {}. The chord of $O$ is drawn faintly and omitted before taking the medial graph; medial edges between consecutive outer-boundary edges or consecutive inner-boundary edges are also omitted. Each medial vertex is placed at the midpoint of its corresponding retained tire edge.}}{5}{}\protected@file@percent }
 \newlabel{fig:medial-tire-example}{{3}{5}}
 \newlabel{rem:tire-counts}{{1.9}{5}}
 \newlabel{prop:no-level-d-pinch}{{1.10}{6}}
 \newlabel{lem:tire-component}{{1.11}{6}}
 \citation{bauerfeld-depth}
 \newlabel{lem:tire-component}{{1.10}{6}}
 \citation{bauerfeld-depth}
-\newlabel{thm:tread-partition}{{1.12}{8}}
+\newlabel{thm:tread-partition}{{1.11}{7}}
-\newlabel{rem:tire-component-degenerate}{{1.13}{8}}
+\newlabel{rem:tire-component-degenerate}{{1.12}{8}}
-\newlabel{rem:tire-no-extra-hypotheses}{{1.14}{8}}
+\newlabel{rem:tire-no-extra-hypotheses}{{1.13}{8}}
-\newlabel{thm:inner-dual-outerplanar}{{1.15}{9}}
+\newlabel{thm:inner-dual-outerplanar}{{1.14}{8}}
-\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 1 ($R$ = disk, $k = 6$). The apex $v_0$ sits at the centre; the non-degenerate boundary $B_{\mathrm  {non-deg}}$ (red) is the hexagonal outer cycle; spokes (grey) triangulate the disk into a fan of $6$ triangles around $v_0$. Each triangle has two spoke edges (interior, contributing $\Gamma $-edges) and one boundary edge (contributing a leaf in $D(T)$, no $\Gamma $-edge). The inner dual $\Gamma $ (blue) is the cycle $C_6$ formed by the six annular face centroids, a manifestly outerplanar graph.}}{10}{}\protected@file@percent }
+\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 1 ($R$ = disk, $k = 6$). The apex $v_0$ sits at the centre; the non-degenerate boundary $B_{\mathrm  {non-deg}}$ (red) is the hexagonal outer cycle; spokes (grey) triangulate the disk into a fan of $6$ triangles around $v_0$. Each triangle has two spoke edges (interior, contributing $\Gamma $-edges) and one boundary edge (contributing a leaf in $D(T)$, no $\Gamma $-edge). The inner dual $\Gamma $ (blue) is the cycle $C_6$ formed by the six annular face centroids, a manifestly outerplanar graph.}}{9}{}\protected@file@percent }
-\newlabel{fig:inner-dual-disk-case}{{4}{10}}
+\newlabel{fig:inner-dual-disk-case}{{4}{9}}
 \citation{bauerfeld-nested-tire-duals}
 \citation{bauerfeld-nested-tire-duals}
 \newlabel{rem:hamilton-cycle-spoke-only}{{1.15}{10}}
 \newlabel{rem:bridge-case-theta}{{1.16}{10}}
 \newlabel{thm:tread-tree}{{1.17}{10}}
 \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Case 2 ($R$ = annulus) with $O$ a barbell. $B_{\mathrm  {out}}$ is the outer hexagon (red); $O$ has two triangles $\{a_1, a_2, a_3\}$ and $\{b_1, b_2, b_3\}$ joined by the bridge $a_3\text  {--}b_1$ (all light red). The annulus is triangulated by $14$ annular triangles: $6$ ``outer-cap'' triangles (one per outer edge), $6$ ``inner-cap'' triangles (one per non-bridge edge of $O$), and $2$ ``bridge-cap'' triangles $\{u_0, a_3, b_1\}$ and $\{u_3, a_3, b_1\}$ adjacent to the bridge. Each blue dot sits at the centroid of an annular triangle; blue edges connect dual vertices whose triangles share an interior annular edge (spoke or bridge). The two bridge-cap vertices have $\Gamma $-degree $3$ (their triangles have no boundary edge) and are joined by the dashed blue \emph  {chord} corresponding to the bridge; the remaining $13$ edges form the Hamilton cycle that wraps around the annulus. All $14$ vertices lie on the outer face of the cycle-with-chord embedding, so $\Gamma \cong  \Theta (1, 7, 7)$ is outerplanar.}}{11}{}\protected@file@percent }
 \newlabel{fig:inner-dual-annulus-case}{{5}{11}}
-\newlabel{rem:hamilton-cycle-spoke-only}{{1.16}{11}}
+\newlabel{rem:tree-multiple-children}{{1.18}{12}}
-\newlabel{rem:bridge-case-theta}{{1.17}{11}}
+\newlabel{thm:tire-tree-decomposition}{{1.19}{12}}
 \newlabel{thm:tread-tree}{{1.18}{12}}
 \newlabel{rem:tree-multiple-children}{{1.19}{13}}
 \newlabel{thm:tire-tree-decomposition}{{1.20}{13}}
 \@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.20\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal  {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal  {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{15}{}\protected@file@percent }
 \newlabel{fig:tire-tree-decomposition}{{6}{15}}
 \newlabel{rem:tree-coloring-factorisation}{{1.21}{15}}
 \bibcite{tait-original}{1}
 \bibcite{bauerfeld-depth}{2}
 \bibcite{bauerfeld-nested-tire-duals}{3}
 \bibcite{birkhoff-reducibility}{4}
 \bibcite{birkhoff-lewis-chromatic}{5}
 \bibcite{tutte-four-colour-conjecture}{6}
 \newlabel{rem:tree-coloring-factorisation}{{1.20}{14}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{14}{}\protected@file@percent }
 \bibcite{tutte-algebraic-colorings}{7}
 \bibcite{tutte-chromatic-sums-1973}{8}
 \bibcite{heesch-untersuchungen}{9}
@@ -62,5 +60,6 @@
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{16}{}\protected@file@percent }
+\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.19\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal  {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal  {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{15}{}\protected@file@percent }
-\gdef \@abspage@last{16}
+\newlabel{fig:tire-tree-decomposition}{{6}{15}}
 \gdef \@abspage@last{15}
@@ -1,5 +1,5 @@
 # Fdb version 3
-["pdflatex"] 1780945550 "paper.tex" "paper.pdf" "paper" 1780945552
+["pdflatex"] 1780947176 "paper.tex" "paper.pdf" "paper" 1780947178
  "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df ""
  "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 ""
@@ -143,8 +143,8 @@
  "fig_medial_tire_example.png" 1780943640 209003 4349824e4f016bde4b938be6b2cb5b2c ""
  "fig_tire_example.png" 1779857443 104494 8f9ce26b469b4236b8b67829f73a5faa ""
  "fig_tire_tree_decomposition.png" 1780290287 372371 1b44f5a3e9f637d78ae951b1f2e3a89d ""
-  "paper.aux" 1780945552 6952 b88a9a8b764572f2fc3be94aabc8b157 "pdflatex"
+  "paper.aux" 1780947178 6896 03aab2e2511678ab091cd0356763e615 "pdflatex"
-  "paper.tex" 1780945491 60644 1d96ad32143c562e73fccdfacbbc9be2 ""
+  "paper.tex" 1780947122 58249 66f46a4d7e93e153ac47781a196e8b3d ""
  (generated)
  "paper.aux"
  "paper.log"
@@ -1,4 +1,4 @@
-This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5)  8 JUN 2026 15:05
+This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5)  8 JUN 2026 15:32
 entering extended mode
 restricted \write18 enabled.
 %&-line parsing enabled.
@@ -498,7 +498,7 @@ e
 <fig_dual_depth.png, id=21, 642.8015pt x 606.265pt>
 File: fig_dual_depth.png Graphic file (type png)
 <use fig_dual_depth.png>
-Package pdftex.def Info: fig_dual_depth.png  used on input line 163.
+Package pdftex.def Info: fig_dual_depth.png  used on input line 162.
 (pdftex.def)             Requested size: 251.9989pt x 237.67276pt.
@@ -508,71 +508,71 @@ LaTeX Warning: `h' float specifier changed to `ht'.
 <fig_tire_example.png, id=35, 559.64081pt x 375.804pt>
 File: fig_tire_example.png Graphic file (type png)
 <use fig_tire_example.png>
-Package pdftex.def Info: fig_tire_example.png  used on input line 242.
+Package pdftex.def Info: fig_tire_example.png  used on input line 241.
 (pdftex.def)             Requested size: 280.79956pt x 188.56097pt.
 <fig_medial_tire_example.png, id=36, 658.8615pt x 444.4605pt>
 File: fig_medial_tire_example.png Graphic file (type png)
 <use fig_medial_tire_example.png>
-Package pdftex.def Info: fig_medial_tire_example.png  used on input line 276.
+Package pdftex.def Info: fig_medial_tire_example.png  used on input line 275.
 (pdftex.def)             Requested size: 280.79956pt x 189.42558pt.
 LaTeX Warning: `h' float specifier changed to `ht'.
 [4 <./fig_tire_example.png>] [5 <./fig_medial_tire_example.png>] [6] [7]
-[8]
+[8] [9]
 LaTeX Warning: `h' float specifier changed to `ht'.
-[9] [10] [11]
+[10] [11] [12] [13]
-Underfull \vbox (badness 1635) has occurred while \output is active []
+<fig_tire_tree_decomposition.png, id=77, 1101.3145pt x 633.9685pt>
 [12]
 [13] [14]
 <fig_tire_tree_decomposition.png, id=80, 1101.3145pt x 633.9685pt>
 File: fig_tire_tree_decomposition.png Graphic file (type png)
 <use fig_tire_tree_decomposition.png>
 Package pdftex.def Info: fig_tire_tree_decomposition.png  used on input line 11
-53.
+06.
 (pdftex.def)             Requested size: 341.9989pt x 196.86678pt.
- [15 <./fig_tire_tree_decomposition.png>] [16] (./paper.aux) ) 
+
 LaTeX Warning: `h' float specifier changed to `ht'.
 [14] [15 <./fig_tire_tree_decomposition.png>] (./paper.aux) ) 
 Here is how much of TeX's memory you used:
- 14076 strings out of 478268
+ 14075 strings out of 478268
- 280038 string characters out of 5846347
+ 280005 string characters out of 5846347
- 565445 words of memory out of 5000000
+ 567436 words of memory out of 5000000
- 31898 multiletter control sequences out of 15000+600000
+ 31897 multiletter control sequences out of 15000+600000
 478218 words of font info for 62 fonts, out of 8000000 for 9000
 1302 hyphenation exceptions out of 8191
 84i,12n,89p,1168b,801s stack positions out of 10000i,1000n,20000p,200000b,200000s
-</usr/local/texli
+</usr/local/texliv
-ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texliv
+e/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive
-e/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx8.pfb></usr/local/texlive/
+/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx8.pfb></usr/local/texlive/2
-2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/
+022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2
-2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive/2
+022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive/20
-022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/20
+22/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/202
-22/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr/local/texlive/2022
+2/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr/local/texlive/2022/
-/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/t
+texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/te
-exmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/tex
+xmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texm
-mf-dist/fonts/type1/public/amsfonts/cm/cmmi9.pfb></usr/local/texlive/2022/texmf
+f-dist/fonts/type1/public/amsfonts/cm/cmmi9.pfb></usr/local/texlive/2022/texmf-
-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-d
+dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-di
-ist/fonts/type1/public/amsfonts/cm/cmr5.pfb></usr/local/texlive/2022/texmf-dist
+st/fonts/type1/public/amsfonts/cm/cmr5.pfb></usr/local/texlive/2022/texmf-dist/
-/fonts/type1/public/amsfonts/cm/cmr6.pfb></usr/local/texlive/2022/texmf-dist/fo
+fonts/type1/public/amsfonts/cm/cmr6.pfb></usr/local/texlive/2022/texmf-dist/fon
-nts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts
+ts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/
-/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/ty
+type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
-pe1/public/amsfonts/cm/cmr9.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1
+e1/public/amsfonts/cm/cmr9.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/
-/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/
+public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/p
-public/amsfonts/cm/cmsy5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pu
+ublic/amsfonts/cm/cmsy5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub
-blic/amsfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publ
+lic/amsfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publi
-ic/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public
+c/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
-/amsfonts/cm/cmsy9.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
+amsfonts/cm/cmsy9.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
-msfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
+sfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
-sfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
+fonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
-onts/symbols/msam10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
+nts/symbols/msam10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
-amsfonts/symbols/msbm10.pfb>
+msfonts/symbols/msbm10.pfb>
-Output written on paper.pdf (16 pages, 1082268 bytes).
+Output written on paper.pdf (15 pages, 1079047 bytes).
 PDF statistics:
- 189 PDF objects out of 1000 (max. 8388607)
+ 186 PDF objects out of 1000 (max. 8388607)
- 113 compressed objects within 2 object streams
+ 111 compressed objects within 2 object streams
 0 named destinations out of 1000 (max. 500000)
 33 words of extra memory for PDF output out of 10000 (max. 10000000)
@@ -52,9 +52,8 @@ $G$ induces a BFS layering of $G$ and endows the inner planar dual
 $G'$ with a \emph{dual depth} grading.  The basic object of study is
 the \emph{tire graph} $T$ --- a plane graph whose outer and inner
 boundaries bound a closed planar region, the \emph{tire tread} $R$,
-triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$.  We define
+triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$.  Our main
-medial tire graphs and prove a basic colour-count bound for their
+structural results are the
 annular medial cycle.  Our main structural results are the
 \emph{tire-component lemma}, the \emph{tire-tread partition theorem},
 and the rooted \emph{tire-tree decomposition}, which together organize
 the bounded faces of $G$ into nested tire treads.
@@ -283,52 +282,6 @@ corresponding retained tire edge.}
 \label{fig:medial-tire-example}
 \end{figure}
 \begin{theorem}[Annular medial colour bound]
 \label{thm:annular-medial-colour-bound}
 Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph with
 non-degenerate boundaries and simple inner boundary $B_{\mathrm{in}}$.
 Let $A(T)$ be the subgraph of $M_{\mathrm{tire}}(T)$ induced by the
 annular medial vertices.  For a graph $H$, write
 $\operatorname{Col}_3(H)$ for the set of proper $3$-vertex-colourings
 of $H$.  Then $A(T)$ is a cycle and
 \[
    |\operatorname{Col}_3(M_{\mathrm{tire}}(T))|
    \;\leq\; |\operatorname{Col}_3(A(T))|.
 \]
 \end{theorem}
 \begin{proof}
 Since the tread is a triangulated annulus with no vertices in its
 interior, each annular face has exactly one boundary edge, lying either
 on $B_{\mathrm{out}}$ or on $B_{\mathrm{in}}$, and exactly two annular
 edges.  As the annular faces are traversed cyclically around the tread,
 consecutive faces share one annular edge.  Equivalently, the annular
 edges occur in a cyclic order in which each annular face contains two
 consecutive annular edges.  Hence the subgraph of
 $M_{\mathrm{tire}}(T)$ induced by the annular medial vertices is a
 cycle.
 Consider the restriction map from proper $3$-colourings of
 $M_{\mathrm{tire}}(T)$ to colourings of this annular medial cycle
 $A(T)$.  We claim that this map is injective.  Let $x$ be a
 non-annular medial vertex.  Then $x$ corresponds to an edge of
 $B_{\mathrm{out}}$ or $B_{\mathrm{in}}$, since the chords of $O$ were
 omitted before forming $M_{\mathrm{tire}}(T)$.  This boundary edge is
 incident to a unique annular face of $T^{\circ}$, and the other two
 edges of that face are annular edges.  Therefore $x$ is adjacent in
 $M_{\mathrm{tire}}(T)$ to the two annular medial vertices corresponding
 to those two annular edges.
 Those two annular medial vertices are adjacent to each other, because
 their annular edges are consecutive on the same triangular annular
 face.  In any proper $3$-colouring they therefore receive two distinct
 colours, and $x$ is forced to receive the remaining third colour.
 Thus every non-annular medial vertex has its colour uniquely determined
 by the colouring of $A(T)$.  Two colourings of $M_{\mathrm{tire}}(T)$
 with the same restriction to $A(T)$ are identical, so the restriction
 map is injective.  The stated inequality follows.
 \end{proof}
 \begin{remark}
 \label{rem:tire-counts}
 Let $\mu = |V(B_{\mathrm{out}})|$ and $\nu = |V(B_{\mathrm{in}})|$.  By