diff --git a/papers/coloring_nested_tire_graphs/experiments/cut_tire_test.py b/papers/coloring_nested_tire_graphs/experiments/cut_tire_test.py
new file mode 100644
index 0000000..29c1c86
--- /dev/null
+++ b/papers/coloring_nested_tire_graphs/experiments/cut_tire_test.py
@@ -0,0 +1,244 @@
+"""Empirical test of the rainbow conjecture and König-lift conjecture
+on cut tires from the new (pendant-redefined) definition.
+
+For each cut tire in our Holton-McKay #0 example:
+  - Build the cut tire as a graph (face boundary + labelled pendants).
+  - Enumerate proper edge 3-colorings via Sage chromatic_polynomial
+    on the line graph (or direct brute force for small tires).
+  - Compute the projection support onto (out spoke colors, in spoke
+    colors) jointly.
+
+Tests:
+  Rainbow conjecture analog:
+    For tires whose face boundary is a simple cycle (spoke-only),
+    by Prop 1.13 of paper.tex, # colorings = 2^n + 2(-1)^n.
+    Spoke colors at each face-boundary vertex are uniquely
+    determined by the cycle coloring (the "third color"). So
+    projection support has specific structure.
+
+  Stronger rainbow: face boundary closed walk that's not simple
+    (= theta-graph-like). Apply Prop's from rainbow_proof.tex.
+
+  König-lift conjecture analog:
+    For pairs of adjacent cut tires (T_d, T_{d+1}) sharing in-spokes
+    of T_d with face-boundary edges of T_{d+1}, do the projection
+    supports overlap on the shared edges?
+"""
+import os
+import sys
+from itertools import permutations, product
+from collections import defaultdict
+
+from sage.all import Graph
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+sys.path.insert(0, HERE)
+from cut_depth_label import (
+    parse_planar_code, HM_FILE, find_six_edge_cut,
+    apply_procedure, compute_nice_layout,
+)
+from cut_tire import cut_tire_at
+
+
+def build_cut_tire_graph(tire, d):
+    """Build a Sage Graph for the cut tire (face boundary + pendants),
+    returning (G, edge_labels) where edge_labels[e] ∈
+    {'face', 'out', 'in'} classifies the edge."""
+    G = Graph(multiedges=False, loops=False)
+    edge_labels = {}
+    for (u, v) in tire['face_edges']:
+        G.add_edge(u, v)
+        edge_labels[(min(u, v), max(u, v))] = 'face'
+    # Add pendants
+    next_id = max(set(tire['face_vertices_unique']) | {0}) + 1000
+    pendant_map = {}
+    for (v, pid) in tire['out_spokes']:
+        nv = next_id; next_id += 1
+        G.add_edge(v, nv)
+        edge_labels[(min(v, nv), max(v, nv))] = 'out'
+        pendant_map[pid] = (v, nv)
+    for (v, pid) in tire['in_spokes']:
+        nv = next_id; next_id += 1
+        G.add_edge(v, nv)
+        edge_labels[(min(v, nv), max(v, nv))] = 'in'
+        pendant_map[pid] = (v, nv)
+    return G, edge_labels, pendant_map
+
+
+def enumerate_proper_edge_3colorings(G):
+    """Enumerate proper edge 3-colorings by computing the line graph
+    and its proper 3-vertex-colorings."""
+    L = G.line_graph(labels=False)
+    edges = list(G.edges(labels=False))
+    edges_norm = [(min(u, v), max(u, v)) for (u, v) in edges]
+    n_edges = len(edges_norm)
+    # Map L's vertices to edge indices
+    L_verts = list(L.vertices())
+    vertex_to_edge_idx = {}
+    for i, e in enumerate(edges):
+        e_can = frozenset(e)
+        for v in L_verts:
+            if frozenset(v) == e_can:
+                vertex_to_edge_idx[v] = i
+                break
+    # Brute force over 3^n_edges, filtering for proper-coloring constraint
+    colorings = []
+    for assignment in product((1, 2, 3), repeat=n_edges):
+        # Check: at each vertex of G, all incident edges have distinct colors
+        good = True
+        for v in G.vertices():
+            incident = [edges_norm.index((min(v, w), max(v, w)))
+                        for w in G.neighbors(v)]
+            colors_at_v = [assignment[i] for i in incident]
+            if len(set(colors_at_v)) != len(colors_at_v):
+                good = False
+                break
+        if good:
+            colorings.append(assignment)
+    return edges_norm, colorings
+
+
+def cut_tire_analysis(tire, d):
+    """Analyze a single cut tire: enumerate proper 3-edge-colorings,
+    compute spoke projections, check for S_3-orbit closure."""
+    G, edge_labels, _ = build_cut_tire_graph(tire, d)
+    n = G.size()
+    if n > 18:  # too slow
+        return None
+    edges_norm, colorings = enumerate_proper_edge_3colorings(G)
+    n_colorings = len(colorings)
+    # Projection: spoke (out + in) tuple
+    out_edge_indices = [i for i, e in enumerate(edges_norm)
+                        if edge_labels[e] == 'out']
+    in_edge_indices = [i for i, e in enumerate(edges_norm)
+                       if edge_labels[e] == 'in']
+    proj = set()
+    for c in colorings:
+        sigma = (tuple(c[i] for i in out_edge_indices),
+                 tuple(c[i] for i in in_edge_indices))
+        proj.add(sigma)
+    # S_3-closure check
+    s3_closed = True
+    for sigma in proj:
+        for pi in permutations((1, 2, 3)):
+            mapped = (tuple(pi[c - 1] for c in sigma[0]),
+                      tuple(pi[c - 1] for c in sigma[1]))
+            if mapped not in proj:
+                s3_closed = False
+                break
+        if not s3_closed:
+            break
+    # Count S_3-orbits
+    seen = set()
+    orbits = []
+    for sigma in sorted(proj):
+        if sigma in seen:
+            continue
+        orbit = set()
+        for pi in permutations((1, 2, 3)):
+            mapped = (tuple(pi[c - 1] for c in sigma[0]),
+                      tuple(pi[c - 1] for c in sigma[1]))
+            orbit.add(mapped)
+        orbit_in_proj = orbit & proj
+        orbits.append(sorted(orbit_in_proj))
+        seen |= orbit_in_proj
+    return {
+        'n_edges': n,
+        'n_colorings': n_colorings,
+        'projection_size': len(proj),
+        'S3_closed': s3_closed,
+        'n_orbits': len(orbits),
+        'orbit_sizes': sorted(len(o) for o in orbits),
+        'projection': proj,
+        'out_count': len(out_edge_indices),
+        'in_count': len(in_edge_indices),
+    }
+
+
+def konig_lift_test(t1, t2, d1, d2, analysis1, analysis2):
+    """For adjacent tires (t1 at depth d1, t2 at depth d2 = d1+1),
+    test König-lift compatibility.
+
+    Shared structure: in spokes of t1 correspond to a subset of
+    face-boundary edges of t2 (those that are also incident to t1's
+    face boundary).
+    """
+    # For now: compute the intersection of projections restricted to
+    # the in-spoke side of t1.  (Full identification of shared edges
+    # requires more careful bookkeeping; this is a first approximation.)
+    if d2 != d1 + 1:
+        return None
+    in_projections_t1 = set(sigma[1] for sigma in analysis1['projection'])
+    out_projections_t2 = set(sigma[0] for sigma in analysis2['projection'])
+    # These don't have a natural bijection without more setup; instead
+    # compare their sizes and S_3-symmetry.
+    return {
+        't1_in_size': len(in_projections_t1),
+        't2_out_size': len(out_projections_t2),
+        # The "shared boundary" between t1 and t2 isn't trivially
+        # identifiable from this aggregated data; flag as needing
+        # explicit shared-edge identification.
+    }
+
+
+def main():
+    gs = parse_planar_code(HM_FILE)
+    G = gs[0]
+    S, cut = find_six_edge_cut(G)
+    base_pos = compute_nice_layout(G)
+    S1 = frozenset(G.vertices()) - frozenset(S)
+    H1, pos1, ed1, _, _, _ = apply_procedure(
+        G, S1, cut, base_pos, '1',
+        pendant_start_id=max(G.vertices()) + 1 + 6)
+
+    print(f"G'_1 depths range: 0 to {max(ed1.values())}\n")
+
+    all_tires = {}  # (d, face_idx) -> tire dict + analysis
+    print("=" * 80)
+    print("Cut tire analysis (per tire)")
+    print("=" * 80)
+    for d in range(1, max(ed1.values()) + 1):
+        tires = cut_tire_at(H1, ed1, d)
+        for f_idx, tire in enumerate(tires):
+            n_face = tire['face_length']
+            n_out = len(tire['out_spokes'])
+            n_in = len(tire['in_spokes'])
+            n_edges = n_face + n_out + n_in
+            print(f'\ndepth {d}, face {f_idx}: |face|={n_face}, '
+                  f'out={n_out}, in={n_in}, total edges={n_edges}')
+            if n_edges > 18:
+                print(f'  ... too many edges to enumerate ({n_edges} > 18); skipping')
+                continue
+            analysis = cut_tire_analysis(tire, d)
+            if analysis is None:
+                continue
+            print(f"  Proper edge 3-colorings: {analysis['n_colorings']}")
+            print(f"  Projection (out, in) size: {analysis['projection_size']}")
+            print(f"  S_3-closed: {analysis['S3_closed']}")
+            print(f"  # S_3-orbits: {analysis['n_orbits']}, "
+                  f"orbit sizes: {analysis['orbit_sizes']}")
+            all_tires[(d, f_idx)] = (tire, analysis)
+
+    # Rainbow analysis: for each tire, check if its projection is
+    # related to a specific structured set (perms-per-face etc.)
+    print("\n" + "=" * 80)
+    print("Rainbow conjecture-style check")
+    print("=" * 80)
+    for (d, f_idx), (tire, analysis) in all_tires.items():
+        n_out = analysis['out_count']
+        n_in = analysis['in_count']
+        if n_out == 0 or n_in == 0:
+            print(f'\ndepth {d}, face {f_idx}: trivial '
+                  f'(one side has 0 spokes), skipping rainbow check')
+            continue
+        # Expected: if both sides have 3 spokes each, check perms-per-face
+        if n_out == 3 and n_in == 3:
+            expected_size = 36  # perms × perms
+            actual = analysis['projection_size']
+            print(f'\ndepth {d}, face {f_idx}: out={n_out}, in={n_in}; '
+                  f'predicted perms-per-face size = {expected_size}, '
+                  f'actual = {actual}')
+
+
+if __name__ == '__main__':
+    main()
diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_conjecture_tests.aux b/papers/coloring_nested_tire_graphs/notes/cut_tire_conjecture_tests.aux
new file mode 100644
index 0000000..259ef3a
--- /dev/null
+++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_conjecture_tests.aux
@@ -0,0 +1,6 @@
+\relax 
+\@writefile{toc}{\contentsline {paragraph}{What would test the rainbow conjecture properly.}{2}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{Cut-tire K\"onig-lift analog (tentative).}{2}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{What this would require to test.}{2}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{What's actually closer to provable.}{3}{}\protected@file@percent }
+\gdef \@abspage@last{3}
diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_conjecture_tests.log b/papers/coloring_nested_tire_graphs/notes/cut_tire_conjecture_tests.log
new file mode 100644
index 0000000..dae797c
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+\documentclass[11pt]{article}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage{graphicx}
+\usepackage{geometry}
+\usepackage{booktabs}
+\geometry{margin=1in}
+
+\title{Testing the rainbow and K\"onig-lift conjectures on cut tires}
+\author{}
+\date{}
+
+\newtheorem*{obs}{Observation}
+
+\begin{document}
+\maketitle
+
+\section*{What was tested}
+
+For the cut tires arising on $G'_1$ of Holton-McKay graph \#0 (with
+the matching $6$-edge cut from \texttt{cut\_depth\_label.tex}), under
+the redefined cut tire definition (face boundary $+$ labelled
+pendants at degree-$2$ vertices), I:
+\begin{enumerate}
+  \item Built each cut tire as a graph (face boundary plus pendants).
+  \item Brute-force enumerated proper edge $3$-colorings.
+  \item Computed the projection support $\{(\sigma_{\mathrm{out}},
+        \sigma_{\mathrm{in}}) : \chi \text{ proper}\}$.
+  \item Checked $S_3$-closure and orbit decomposition.
+\end{enumerate}
+
+\section*{Results}
+
+\begin{center}
+\small
+\begin{tabular}{lr|rrrrr|r|cl}
+\toprule
+$d$ & face & $|f|$ & out & in & $|E|$ & $\#$col. & $|\pi|$
+    & $S_3$-cl.\ & orbit sizes\\
+\midrule
+$1$ & $0$ & $12$ & $5$ & $0$ & $17$ & $96$  & $93$  & Yes & $[3, 6^{15}]$\\
+$1$ & $1$ & $4$  & $1$ & $0$ & $5$  & $6$   & $3$   & Yes & $[3]$\\
+$2$ & $0$ & $7$  & $4$ & $3$ & $14$ & $126$ & $126$ & Yes & $[6^{21}]$\\
+$2$ & $1$ & $7$  & $4$ & $3$ & $14$ & $126$ & $126$ & Yes & $[6^{21}]$\\
+$3$ & $0$--$2$ & $2$ & $0$ & $0$ & $2$ & $3$ & $1$ & Yes & $[1]$\\
+$4$ & $0$ & $4$  & $1$ & $0$ & $5$  & $6$   & $3$   & Yes & $[3]$\\
+$4$ & $1$ & $8$  & $2$ & $1$ & $11$ & $24$  & $21$  & Yes & $[3, 6^3]$\\
+$5$ & $0$ & $14$ & $4$ & $2$ & $20$ & --- (too big) & --- & --- & ---\\
+$5$ & $1$ & $2$  & $0$ & $0$ & $2$  & $3$   & $1$   & Yes & $[1]$\\
+$6$ & $0$ & $12$ & $3$ & $2$ & $17$ & $96$  & $93$  & Yes & $[3, 6^{15}]$\\
+$7$ & $0$ & $2$  & $0$ & $0$ & $2$  & $3$   & $1$   & Yes & $[1]$\\
+\bottomrule
+\end{tabular}
+\end{center}
+
+\section*{Observations}
+
+\begin{obs}[Universal $S_3$-closure]
+Every cut tire's projection $\pi(\sigma_{\mathrm{out}}, \sigma_{\mathrm{in}})$
+is closed under the diagonal $S_3$ action on the $3$ colors.  This
+is structural and expected: proper edge $3$-coloring is color-
+symmetric.
+\end{obs}
+
+\begin{obs}[Orbit sizes are $3$ or $6$ only]
+Every $S_3$-orbit in every cut tire projection has size $3$ (the
+constant-color orbit, when present) or $6$ (the generic orbit using
+all $3$ colors).  No size-$2$ orbits, which would correspond to
+$\sigma$'s with non-trivial $S_3$ stabilizer; these don't occur.
+\end{obs}
+
+\begin{obs}[Non-trivial cut tires have substantial projection support]
+The two ``main'' cut tires at depth $2$ (face length $7$, $4$ out
+$+\ 3$ in spokes) each have $126$ proper edge $3$-colorings, all
+$126$ distinct in their joint $(\sigma_{\mathrm{out}}, \sigma_{\mathrm{in}})$
+projection.  This is $21$ full $S_3$-orbits of size $6$.
+
+By contrast the cut tire at depth $1$ (face length $12$, $5$ out
+$+\ 0$ in) has $96$ colorings with $93$ distinct projections
+($16$ orbits, including one size-$3$ orbit and fifteen size-$6$);
+the small drop $96 - 93 = 3$ corresponds to the constant orbit.
+\end{obs}
+
+\section*{Why the rainbow conjecture is not directly testable here}
+
+The rainbow conjecture from \texttt{rainbow\_proof.tex} states:
+for an antipodal-chord SP tire $T = (m_1, (0, m/2), \mathrm{SP})$
+with $m \in \{4, 6\}$ even and $m_1 \ge m - 1$, the inner-spoke
+projection $\pi_D(\mathcal{C}(T))$ equals the perms-per-face set
+$\mathcal{P}_m$ (size $36$).
+
+For the rainbow conjecture to apply to a cut tire, the cut tire's
+face boundary structure would need to match the antipodal-chord SP
+structure: a cycle of length $m$ with $r = 2$ ``O-face''-analogous
+pieces, each containing $m/2$ boundary edges.
+
+The cut tires in our example have face boundaries of lengths
+$2, 4, 7, 8, 12, 14$, none of which structurally matches the
+$\theta(1, p, q)$-shape with the antipodal-chord SP convention.
+So the rainbow conjecture cannot be directly checked on these cut
+tires; what we can confirm is the weaker structural property
+($S_3$-closure, size-$6$ orbit structure), which is universal.
+
+\paragraph{What would test the rainbow conjecture properly.}
+Find a cut tire whose face boundary is a closed walk visiting each
+vertex twice, structured as $\theta(1, p, q)$ in the partial-tire-
+dual sense.  Such cut tires can arise when $H_d$ has a
+``pinch'' vertex (cut vertex with two faces sharing that vertex).
+The example $H_1$ has $4$ revisited vertices in its length-$12$
+face boundary, suggesting bridge-like structure; explicit
+identification of whether this matches $\theta(1, p, q)$ would be
+the next step.
+
+\section*{Why the K\"onig-lift conjecture is not directly testable here}
+
+The K\"onig-lift conjecture from \texttt{worst\_case\_proof\_sketch.tex}
+applies to pairs of adjacent SP tires $(T_1, T_2)$ sharing a cycle
+$\gamma$ where \emph{both} sides give direct $\gamma$-face partitions
+(both have chord(s) on $\gamma$).
+
+For cut tires at depths $d$ and $d + 1$: the ``shared'' structure
+is the bijection $\{\text{in spokes of } T_d\}
+\leftrightarrow \{\text{specific face boundary edges of } T_{d+1}\}$.
+This is not the same as ``both tires give $\gamma$-face partitions''
+because $T_{d+1}$'s face boundary is not (in general) a $\gamma$-cycle
+that $T_d$ also borders --- the cut tires sit in $H_d$ and $H_{d+1}$
+respectively, with different vertex sets.
+
+So the K\"onig-lift conjecture would need restatement for the
+cut-tire chain.  A correct restatement might say:
+
+\paragraph{Cut-tire K\"onig-lift analog (tentative).}
+For each pair of adjacent cut tires $(T_d, T_{d+1})$ in a chain, the
+bijection $\beta : \{\text{in spokes of } T_d\} \to
+\{\text{face-boundary edges of } T_{d+1} \text{ adjacent to } V(f_d)\}$
+preserves a Latin structure: any Latin $\sigma$ on the shared
+positions in $T_{d+1}$'s face boundary can be lifted to a proper
+edge $3$-coloring of $T_d$ via $\beta$, and vice versa.
+
+\paragraph{What this would require to test.}
+\begin{enumerate}
+  \item Explicit identification of the bijection $\beta$ for each
+        pair $(T_d, T_{d+1})$.  This requires tracking the planar
+        embedding inheritance from $G'_i$ through to $H_d$ and
+        $H_{d+1}$.
+  \item Computing the Latin structure on each $T_{d+1}$'s face
+        boundary edges (analogous to the perms-per-face structure
+        on $\gamma$).
+  \item Comparing the projected supports under $\beta$.
+\end{enumerate}
+
+None of this is automated in the present code.
+
+\section*{What can be concluded from the empirical data}
+
+\begin{enumerate}
+  \item \textbf{$S_3$-closure holds universally.}  Every cut tire
+        projection is $S_3$-symmetric, with orbits of size $3$ or $6$.
+        This matches the partial-tire-dual data
+        (\texttt{orbit\_decomposition.tex}, Obs.\ ``S3-closed'').
+  \item \textbf{Non-trivial cut tires have rich projections.}  At
+        depth $2$, the projection is $126 = 21 \cdot 6$ ($21$ full
+        $S_3$-orbits).  At depth $6$, $93 = 1 \cdot 3 + 15 \cdot 6$
+        ($16$ orbits).  This is substantially more than the
+        rainbow's $36 = 6 \cdot 6$ specific orbit, so cut tires here
+        are \emph{looser} than the rainbow case --- chain pigeonhole
+        should be \emph{easier} when projections are large.
+  \item \textbf{Trivial cut tires (length-$2$ faces) contribute
+        nothing.}  Most depths have at least one length-$2$ face in
+        $H_d$ (degenerate), which gives a trivial cut tire with no
+        spokes.  These are not informative for chain pigeonhole.
+  \item \textbf{Neither conjecture is directly applicable.}  The
+        rainbow conjecture requires antipodal-chord SP structure,
+        which our cut tires don't naturally have.  The K\"onig-lift
+        conjecture requires both sides give $\gamma$-face
+        partitions, which the cut-tire chain doesn't naturally
+        produce.
+\end{enumerate}
+
+\paragraph{What's actually closer to provable.}
+The empirical data suggests the projections are LARGE (and
+$S_3$-symmetric) rather than SMALL.  A natural chain pigeonhole
+statement is:
+
+\begin{quote}
+\textbf{Conjecture (loose).}  For each cut tire $T$ arising in the
+chain, the projection $\pi(T)$ has size $\ge c \cdot 6$ for some
+absolute constant $c$ depending only on $|E(T)|$, with $\pi(T)$
+$S_3$-closed.  Chain composition through $T_1, T_2, \ldots$ yields
+$|\mathcal{R}_i| \ge c'$ uniform in chain length.
+\end{quote}
+
+The constant $c$ might be derivable from Prop 1.13 of \texttt{paper.tex}
+($2^n + 2(-1)^n$ colorings for spoke-only $D(T)$).  An honest
+chain pigeonhole then asks: do the cut-tire chain compositions on
+both sides of the cut produce sufficient overlap to force
+$\mathcal{R}_0 \cap \mathcal{R}_1 \neq \emptyset$?
+
+This requires the full chain machinery, not just per-tire analysis.
+
+\end{document}