From 5d81c1ed44859d2324d7709a71ef756f48b23c1c Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 02:11:33 -0400 Subject: [PATCH] coloring_nested_tire_graphs: investigate chord case in O for tire face connectors MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Companion to tire_fiber_data.tex. The chord case forces a modelling choice for the surrounding G: Steiner-rich (each B_in edge gets its own sub-triangle, chords are invisible to T'_{f'}) vs Steiner-poor (each O-face is one G-face, chord set determines feasibility). Under SP, a tire with k > 3 is infeasible unless O already contains enough chords to keep every O-face at most 3 B_in edges. With chords: support is positive but much smaller than SR — π_D never reaches 3^k, and π_D's shrinkage is m-independent. More chords = smaller faces = MORE support (each chord splits a hard constraint into easier ones). Side-by-side data table for (m, k) ∈ {3,4,5,6,...}² with various chord sets, plus 3-page note documenting the modelling choice and implications for chain pigeonhole. Co-Authored-By: Claude Opus 4.7 --- .../experiments/tire_fiber_chords.py | 191 +++++++++++ .../experiments/tire_fiber_chords_data.txt | 22 ++ .../notes/tire_fiber_chords.aux | 7 + .../notes/tire_fiber_chords.log | 320 ++++++++++++++++++ .../notes/tire_fiber_chords.pdf | Bin 0 -> 212866 bytes .../notes/tire_fiber_chords.tex | 230 +++++++++++++ 6 files changed, 770 insertions(+) create mode 100644 papers/coloring_nested_tire_graphs/experiments/tire_fiber_chords.py create mode 100644 papers/coloring_nested_tire_graphs/experiments/tire_fiber_chords_data.txt create mode 100644 papers/coloring_nested_tire_graphs/notes/tire_fiber_chords.aux create mode 100644 papers/coloring_nested_tire_graphs/notes/tire_fiber_chords.log create mode 100644 papers/coloring_nested_tire_graphs/notes/tire_fiber_chords.pdf create mode 100644 papers/coloring_nested_tire_graphs/notes/tire_fiber_chords.tex diff --git a/papers/coloring_nested_tire_graphs/experiments/tire_fiber_chords.py b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_chords.py new file mode 100644 index 0000000..b1e2de0 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_chords.py @@ -0,0 +1,191 @@ +"""Investigate fiber distributions when the inner outerplanar graph O +has non-trivial chords. + +A chord in O does NOT change the dual annular cycle T'_ann (which is +still C_{m+k}). What it changes is the inside-O face structure: with +j non-crossing chords, O has j+1 inside-O faces. Each face f dualizes +to a single vertex u_f in G', whose degree in T'_{f'} equals the +number of B_in edges on the boundary of f. + +Concretely: if a face f contains B_in edges e_{a_1}, ..., e_{a_d}, +then u_f is incident in T'_{f'} to the d D-triangle positions +p_{a_1}, ..., p_{a_d} on the dual cycle (one per B_in edge). Proper +edge 3-coloring at u_f requires those d spoke colors to be pairwise +distinct. + +Feasibility constraint: at k=3 colors, a face with d > 3 incident +B_in edges has no proper edge 3-coloring of u_f, so the tire admits +no global coloring. All examples below satisfy d <= 3 per face. +""" +from collections import Counter +from itertools import product + +COLORS = (1, 2, 3) + + +def d_positions_for(m: int, k: int) -> list[int]: + """Return cycle positions of D-triangles on the dual cycle C_{m+k} + for a balanced (Bresenham-style) alternating triangulation, in + the cyclic order matching B_in edges e_0, e_1, ..., e_{k-1}.""" + n = m + k + return [round(i * n / k) % n for i in range(k)] + + +def compute_faces_from_chords(k: int, chords: list[tuple[int, int]]) -> list[list[int]]: + """Compute face partition of B_in = C_k cut by non-crossing chords. + + Each face is returned as a list of B_in edge indices belonging to + it. For non-crossing chords: edge e_a belongs to the smallest + enclosing chord-arc, where chord (i, j) with i < j encloses e_a + iff i <= a < j. Edges not enclosed by any chord belong to the + 'outermost' face.""" + chords_norm = [tuple(sorted([a, b])) for (a, b) in chords] + + def enclosing_chord(edge: int): + candidates = [(j - i, i, j) for (i, j) in chords_norm if i <= edge < j] + if not candidates: + return None + candidates.sort() + _, i, j = candidates[0] + return (i, j) + + chord_to_face = {} + next_face = 0 + edge_to_face = {} + for a in range(k): + ch = enclosing_chord(a) + if ch not in chord_to_face: + chord_to_face[ch] = next_face + next_face += 1 + edge_to_face[a] = chord_to_face[ch] + + faces = [[] for _ in range(next_face)] + for a in range(k): + faces[edge_to_face[a]].append(a) + return faces + + +def fiber_distribution(m: int, k: int, chords: list[tuple[int, int]]): + """Compute fiber distribution N(T'_{f'}; σ) over σ ∈ {1,2,3}^n for + the tire with given m, k, and chord set in O. + + Returns (fibers, d_positions, faces_as_cycle_positions). + """ + n = m + k + d_positions = d_positions_for(m, k) + o_faces = compute_faces_from_chords(k, chords) + d_positions_by_face = [ + [d_positions[a] for a in face_edges] + for face_edges in o_faces + ] + + fibers: Counter[tuple[int, ...]] = Counter() + for ce in product(COLORS, repeat=n): + if not all(ce[i] != ce[(i + 1) % n] for i in range(n)): + continue + sigma = tuple( + ({1, 2, 3} - {ce[(i - 1) % n], ce[i]}).pop() + for i in range(n) + ) + ok = True + for positions in d_positions_by_face: + colors = [sigma[p] for p in positions] + if len(set(colors)) != len(colors): + ok = False + break + if ok: + fibers[sigma] += 1 + return fibers, d_positions, d_positions_by_face + + +def projection_support(fibers, positions: list[int]) -> set[tuple[int, ...]]: + return {tuple(sigma[p] for p in positions) for sigma in fibers} + + +CASES = [ + # (m, k, chords, label) + (4, 4, [], "(4,4) no chord [baseline]"), + (4, 4, [(0, 2)], "(4,4) chord (0,2) faces 2+2"), + (5, 5, [], "(5,5) no chord [baseline]"), + (5, 5, [(0, 2)], "(5,5) chord (0,2) faces 2+3"), + (5, 5, [(0, 3)], "(5,5) chord (0,3) faces 3+2"), + (6, 6, [], "(6,6) no chord [baseline]"), + (6, 6, [(0, 3)], "(6,6) chord (0,3) antipodal faces 3+3"), + (6, 6, [(0, 2), (3, 5)], "(6,6) chords (0,2)(3,5) faces 2+2+2"), + # k=4 with smaller outer cycle: + (3, 4, [], "(3,4) no chord"), + (3, 4, [(0, 2)], "(3,4) chord (0,2)"), + # larger n for k=4: + (6, 4, [], "(6,4) no chord"), + (6, 4, [(0, 2)], "(6,4) chord (0,2)"), +] + + +def u_positions_for(m: int, k: int) -> list[int]: + """U-positions (outer-spoke cycle positions) = complement of D-positions.""" + n = m + k + d = set(d_positions_for(m, k)) + return [p for p in range(n) if p not in d] + + +def spoke_only_fiber_distribution(n: int): + """Step-1 baseline: G_n = C_n + n independent pendants (Steiner-rich, + each B_in edge gets its own inside-O sub-triangle). No chord + constraints; every σ that comes from a proper cycle 3-coloring is + realisable.""" + fibers: Counter[tuple[int, ...]] = Counter() + for ce in product(COLORS, repeat=n): + if not all(ce[i] != ce[(i + 1) % n] for i in range(n)): + continue + sigma = tuple( + ({1, 2, 3} - {ce[(i - 1) % n], ce[i]}).pop() + for i in range(n) + ) + fibers[sigma] += 1 + return dict(fibers) + + +def main(): + print("Two models compared per (m, k, chord-set):") + print(" STEINER-RICH: G triangulates each O-face using internal Steiner vertices,") + print(" so each B_in edge dualizes to its own (degree-1) inside-O triangle.") + print(" Chord set has NO effect on T'_{f'}; equivalent to step-1 baseline.") + print(" STEINER-POOR: G does NOT further sub-triangulate O-faces beyond O itself.") + print(" Each O-face becomes one G-face with degree = (B_in-edge count).") + print(" Edge-3-colorable only if every O-face has ≤ 3 B_in edges.") + print() + print(f"{'case':<42s} {'n':>3s} " + f"{'P_e SR':>7s} {'D SR':>7s} {'U SR':>7s} | " + f"{'P_e SP':>7s} {'D SP':>7s} {'U SP':>7s} | " + f"faces (cycle-position groups)") + print("-" * 145) + for m, k, chords, label in CASES: + n = m + k + # Steiner-poor (with chord-induced constraints): + fibers_sp, d_pos, faces_cp = fiber_distribution(m, k, chords) + u_pos = u_positions_for(m, k) + pe_sp = sum(fibers_sp.values()) + d_proj_sp = projection_support(fibers_sp, d_pos) + u_proj_sp = projection_support(fibers_sp, u_pos) + + # Steiner-rich baseline (chord-set ignored): + fibers_sr = spoke_only_fiber_distribution(n) + pe_sr = sum(fibers_sr.values()) + d_proj_sr = projection_support(fibers_sr, d_pos) + u_proj_sr = projection_support(fibers_sr, u_pos) + + d_uni = 3 ** k + u_uni = 3 ** m + faces_str = " | ".join( + "{" + ",".join(map(str, sorted(f))) + "}" for f in faces_cp + ) + print( + f"{label:<42s} {n:>3d} " + f"{pe_sr:>7d} {len(d_proj_sr):>3d}/{d_uni:<3d} {len(u_proj_sr):>3d}/{u_uni:<3d} | " + f"{pe_sp:>7d} {len(d_proj_sp):>3d}/{d_uni:<3d} {len(u_proj_sp):>3d}/{u_uni:<3d} | " + f"{faces_str}" + ) + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/experiments/tire_fiber_chords_data.txt b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_chords_data.txt new file mode 100644 index 0000000..c35d402 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_chords_data.txt @@ -0,0 +1,22 @@ +Two models compared per (m, k, chord-set): + STEINER-RICH: G triangulates each O-face using internal Steiner vertices, + so each B_in edge dualizes to its own (degree-1) inside-O triangle. + Chord set has NO effect on T'_{f'}; equivalent to step-1 baseline. + STEINER-POOR: G does NOT further sub-triangulate O-faces beyond O itself. + Each O-face becomes one G-face with degree = (B_in-edge count). + Edge-3-colorable only if every O-face has ≤ 3 B_in edges. + +case n P_e SR D SR U SR | P_e SP D SP U SP | faces (cycle-position groups) +------------------------------------------------------------------------------------------------------------------------------------------------- +(4,4) no chord [baseline] 8 258 81/81 81/81 | 0 0/81 0/81 | {0,2,4,6} +(4,4) chord (0,2) faces 2+2 8 258 81/81 81/81 | 144 36/81 54/81 | {0,2} | {4,6} +(5,5) no chord [baseline] 10 1026 243/243 243/243 | 0 0/243 0/243 | {0,2,4,6,8} +(5,5) chord (0,2) faces 2+3 10 1026 243/243 243/243 | 240 36/243 90/243 | {0,2} | {4,6,8} +(5,5) chord (0,3) faces 3+2 10 1026 243/243 243/243 | 240 36/243 90/243 | {0,2,4} | {6,8} +(6,6) no chord [baseline] 12 4098 729/729 729/729 | 0 0/729 0/729 | {0,2,4,6,8,10} +(6,6) chord (0,3) antipodal faces 3+3 12 4098 729/729 729/729 | 408 36/729 90/729 | {0,2,4} | {6,8,10} +(6,6) chords (0,2)(3,5) faces 2+2+2 12 4098 729/729 729/729 | 1536 216/729 456/729 | {0,2} | {4,10} | {6,8} +(3,4) no chord 7 126 78/81 27/27 | 0 0/81 0/27 | {0,2,4,5} +(3,4) chord (0,2) 7 126 78/81 27/27 | 48 36/81 24/27 | {0,2} | {4,5} +(6,4) no chord 10 1026 81/81 549/729 | 0 0/81 0/729 | {0,2,5,8} +(6,4) chord (0,2) 10 1026 81/81 549/729 | 480 36/81 342/729 | {0,2} | {5,8} diff --git a/papers/coloring_nested_tire_graphs/notes/tire_fiber_chords.aux b/papers/coloring_nested_tire_graphs/notes/tire_fiber_chords.aux new file mode 100644 index 0000000..516ad95 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/tire_fiber_chords.aux @@ -0,0 +1,7 @@ +\relax +\newlabel{obs:sp-empty}{{}{2}} +\newlabel{obs:more-chords-more-support}{{}{2}} +\newlabel{obs:pi-D-m-independent}{{}{2}} +\newlabel{obs:u-shrinks}{{}{2}} +\newlabel{obs:sp-chord-rule}{{}{3}} +\gdef \@abspage@last{3} diff --git a/papers/coloring_nested_tire_graphs/notes/tire_fiber_chords.log b/papers/coloring_nested_tire_graphs/notes/tire_fiber_chords.log new file mode 100644 index 0000000..7441c64 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/tire_fiber_chords.log @@ -0,0 +1,320 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) 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+\date{} + +\newtheorem*{obs}{Observation} + +\begin{document} +\maketitle + +\section*{What this is} + +A companion to \texttt{tire\_fiber\_data.tex}: that note enumerated +fiber distributions in the spoke-only setting (no chords in the inner +outerplanar graph $O$). This note investigates what happens when $O$ +has chords --- specifically when the surrounding $G$ does \emph{not} +introduce Steiner vertices inside $B_{\mathrm{in}}$, so that each face +of $O$ in the tire's embedding remains a single face of $G$. +Script: \texttt{experiments/tire\_fiber\_chords.py}. Output: +\texttt{experiments/tire\_fiber\_chords\_data.txt}. + +\section*{The modelling choice that chords force on us} + +In the spoke-only enumeration (\texttt{tire\_fiber\_data.tex}) the +inner-spoke part of $T'_{f'}$ was simply $k$ pendant edges, one per +$B_{\mathrm{in}}$ edge. That picture corresponds to a surrounding +maximal planar graph $G$ that \emph{aggressively triangulates} the +disk inside $B_{\mathrm{in}}$ using interior (Steiner) vertices not in +$V(T)$, so each $B_{\mathrm{in}}$ edge gets its own dedicated inside-$O$ +sub-triangle. + +When we add chords to $O$, there is a second axis of choice: does $G$ +sub-triangulate the resulting $O$-faces, or not? Two extreme models: + +\begin{itemize} + \item \textbf{Steiner-rich (SR).} $G$ adds whatever vertices and + edges it needs inside each $O$-face so that every + $B_{\mathrm{in}}$ edge has its own sub-triangle. The + inner-spoke graph is $k$ pendants \emph{regardless of the + chord set} --- chords are invisible to $T'_{f'}$. This is the + step-1 baseline. + \item \textbf{Steiner-poor (SP).} $G$ does not further + sub-triangulate $O$-faces beyond what $O$ itself provides. + Each face of $O$ becomes a single face of $G$, whose dual + vertex has degree (in $G'$, and hence in $T'_{f'}$) equal to + the number of $B_{\mathrm{in}}$ edges on the face boundary. + Edge $3$-colourable only if \emph{every $O$-face has at most + $3$ $B_{\mathrm{in}}$ edges}. +\end{itemize} + +Real surrounding triangulations $G$ live somewhere between these two, +and the choice is not part of the tire structure itself --- it is a +property of how $G$ embeds the tire. But the two extremes already +say something useful about chord effects. + +\section*{Side-by-side data} + +\begin{center} +\scriptsize +\begin{tabular}{l r | r r r | r r r} +\toprule +& & \multicolumn{3}{c|}{\textbf{Steiner-rich}} + & \multicolumn{3}{c}{\textbf{Steiner-poor}}\\ +case & $n$ & $P_e$ & $|\pi_D(\mathcal{C})|/3^k$ & $|\pi_U(\mathcal{C})|/3^m$ + & $P_e$ & $|\pi_D(\mathcal{C})|/3^k$ & $|\pi_U(\mathcal{C})|/3^m$ \\ +\midrule +(4,4) no chord & 8 & 258 & 81/81 & 81/81 & 0 & 0/81 & 0/81 \\ +(4,4) chord (0,2), faces 2+2 & 8 & 258 & 81/81 & 81/81 & 144 & 36/81 & 54/81 \\ +(5,5) no chord & 10 & 1026 & 243/243 & 243/243 & 0 & 0/243 & 0/243 \\ +(5,5) chord (0,2), faces 2+3 & 10 & 1026 & 243/243 & 243/243 & 240 & 36/243 & 90/243 \\ +(5,5) chord (0,3), faces 3+2 & 10 & 1026 & 243/243 & 243/243 & 240 & 36/243 & 90/243 \\ +(6,6) no chord & 12 & 4098 & 729/729 & 729/729 & 0 & 0/729 & 0/729 \\ +(6,6) chord (0,3) antipodal, 3+3 & 12 & 4098 & 729/729 & 729/729 & 408 & 36/729 & 90/729 \\ +(6,6) chords (0,2)(3,5), 2+2+2 & 12 & 4098 & 729/729 & 729/729 & 1536 & 216/729 & 456/729 \\ +(3,4) no chord & 7 & 126 & 78/81 & 27/27 & 0 & 0/81 & 0/27 \\ +(3,4) chord (0,2) & 7 & 126 & 78/81 & 27/27 & 48 & 36/81 & 24/27 \\ +(6,4) no chord & 10 & 1026 & 81/81 & 549/729 & 0 & 0/81 & 0/729 \\ +(6,4) chord (0,2) & 10 & 1026 & 81/81 & 549/729 & 480 & 36/81 & 342/729 \\ +\bottomrule +\end{tabular} +\end{center} + +\noindent +($\pi_D$ projects $\sigma$ onto the $k$ inner-direction spoke positions +on the dual cycle; $\pi_U$ onto the $m$ outer-direction positions. +``Chord (a,b)'' means $O = B_{\mathrm{in}} \cup \{v_a v_b\}$.) + +\section*{Observations} + +\begin{obs}[SP is empty without chords for $k > 3$] +\label{obs:sp-empty} +If $O$ has no chord and $k \geq 4$, then $O$ has a single inner face +with $k$ incident $B_{\mathrm{in}}$ edges. Under SP this is a single +$G'$-vertex of degree $k$, and proper edge $3$-colouring needs $k$ +distinct colours at that vertex, which is impossible for $k \geq 4$. +\textbf{So the SP model forces chords once $k \geq 4$}: tires whose +inner boundary is longer than $3$ are only feasible if $O$ already +contains enough chords to break every face down to $\leq 3$ +$B_{\mathrm{in}}$ edges. +\end{obs} + +\noindent +This is the structural reason chord-aware modelling matters. Under +SR, you can pretend chord sets do not exist; under SP, you cannot. + +\begin{obs}[Adding chords \emph{enlarges} the SP support] +\label{obs:more-chords-more-support} +Comparing the two $(6,6)$ chord rows: +\begin{itemize} + \item Antipodal $(0,3)$ chord: face sizes $3{+}3$, gives + $P_e = 408$, $\pi_D$ support $36/729 \approx 4.9\%$. + \item Two chords $(0,2),(3,5)$: face sizes $2{+}2{+}2$, gives + $P_e = 1536$, $\pi_D$ support $216/729 \approx 29.6\%$. +\end{itemize} +Smaller faces $=$ weaker constraints $=$ more realisable +configurations. This is the opposite of what one might expect at +first glance (``more chords $=$ more edges $=$ more constraints'') --- +each chord splits a hard constraint into easier ones. +\end{obs} + +\begin{obs}[$\pi_D$ support depends only on chord structure, not on $m$] +\label{obs:pi-D-m-independent} +For chord set $\{(0,2)\}$ in $k = 4$, the $\pi_D$ support is +$36/81 = 4/9$ for every tested $m \in \{3, 4, 6\}$. This is structural: +the chord constraint pins down which spoke-quadruples are realisable +without involving the outer cycle at all. The $\pi_U$ support, by +contrast, varies with $m$. +\end{obs} + +\begin{obs}[SP support shrinks the $\pi_U$ support too] +\label{obs:u-shrinks} +Even though chord constraints live on the inner side, they propagate +to the outer side via the cycle. For $(m, k) = (4, 4)$: +\begin{itemize} + \item SR: $\pi_U$ saturates $3^4$. + \item SP with chord $(0,2)$: $\pi_U = 54/81 = 2/3$. +\end{itemize} +The chord-induced constraints on D-position spokes propagate around +the cycle through the proper-edge-colouring constraint, reducing the +set of cycle colourings, hence reducing $\pi_U$ as well. +\end{obs} + +\section*{Implications for the chain-pigeonhole step} + +In the spoke-only / SR step-1 analysis we observed that whenever +$|B_{\mathrm{out}}| \geq |B_{\mathrm{in}}|$ the inner-side projection +saturates the full ring-coloring universe $3^k$, so chain +compatibility on a shared cycle was trivially nonempty whenever at +least one of the two adjacent tires had the longer non-shared +boundary. + +Under SP this clean saturation breaks down on the chord side: + +\begin{itemize} + \item For a chorded tire, $\pi_D$ never reaches $3^k$ --- the + chord constraints permanently rule out some inner-spoke + configurations. The realisable set is small but non-empty + (e.g., $36/81$ at $k = 4$). + \item The shrinkage is the same regardless of $m$ on the inner + (D) side, so making the outer boundary very long does + \emph{not} help recover saturation on the chord side. + \item Two adjacent SP tires sharing a cycle $\gamma$ both have + shrunk projections; whether they intersect is now a real + question. E.g., projecting $\pi_D$ down to a $4/9$-fraction + of $\{1,2,3\}^k$ from each side: do the two $4/9$-sets always + meet? +\end{itemize} + +\noindent +That last question is the natural follow-up experiment (and is +step~2 of the action items). + +\subsection*{The structural fact worth stating} + +\begin{obs}[Steiner-poor chord-rule] +\label{obs:sp-chord-rule} +Under the SP model, a tire $T$ is edge-$3$-colourable on its face +connector $T'_{f'}$ iff its inner outerplanar graph $O$ \emph{already +triangulates the inside of $B_{\mathrm{in}}$ up to faces of $B_{\mathrm{in}}$- +size $\leq 3$}. Equivalently: $O$ must contain enough chords that +every $O$-face has at most $3$ $B_{\mathrm{in}}$ edges on its boundary. +\end{obs} + +\noindent +For a level-source BFS decomposition with $O$ chosen minimally (just +$B_{\mathrm{in}}$ itself, no chords), every tire with $k \geq 4$ fails +the SP edge-$3$-colouring test \emph{locally}. Globally this is no +contradiction --- the surrounding maximal planar $G$ does add Steiner +vertices and edges inside $B_{\mathrm{in}}$, making us closer to SR +than to SP. But the SP model is a worst-case envelope on what +purely local data implies, and shows that any structural argument that +ignores the inside-$B_{\mathrm{in}}$ triangulation is incomplete. + +\section*{Caveats / what we did not compute} + +\begin{enumerate} + \item \textbf{Only the two extremes.} Real $G$ triangulations lie + between SR (every $B_{\mathrm{in}}$ edge gets its own + Steiner-sub-triangle) and SP (no further sub-triangulation + at all). Intermediate cases (e.g., fan-triangulating one + $O$-face from a $V(B_{\mathrm{in}})$ vertex without adding + Steiner points) are not enumerated here, though the data + suggests they sit between. + \item \textbf{Outer side always SR.} We have always treated the + outer side ($B_{\mathrm{out}}$) as having no chord-induced + structure --- $B_{\mathrm{out}}$ is just a cycle in the tire + definition, so this is correct, but the analogous question + ``what if the region outside $B_{\mathrm{out}}$ is poorly + triangulated'' is not addressed. + \item \textbf{$\Delta \leq 3$ on $O$ ignored.} The chord + configurations tested above include cases where chord + endpoints have higher degree in $T$; the menagerie's + $\Delta \leq 3$ constraint is not enforced. This is fine + for the SP model in isolation but matters when composing + with the menagerie counting formulas. + \item \textbf{Adjacency intersection (step 2).} We have only + computed single-tire fiber distributions. The chain- + pigeonhole step itself --- intersecting $\pi_D$ of one tire + with $\pi_U$ of the adjacent tire on a shared cycle --- is + deferred. +\end{enumerate} + +\end{document}