diff --git a/papers/coloring_nested_tire_graphs/experiments/tire_fiber_chunked.py b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_chunked.py
new file mode 100644
index 0000000..b12d920
--- /dev/null
+++ b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_chunked.py
@@ -0,0 +1,153 @@
+"""Streaming/chunked variant of tire_fiber_chords_fast.py for n > 27.
+
+The in-memory version materializes an (N, n) int8 array with N = 3·2^(n-1),
+which at n = 30 is ~30 GB.  This chunked version processes paths in
+chunks of ~2^20 rows, never holding the full array in memory.
+
+For projection support: we only need the set of σ-projected rows,
+which is typically much smaller than 3^k.  We accumulate into a Python
+set.
+
+For γ up to ~18 with m ≥ γ this brings n up to ~36, with one tire
+taking minutes (versus impossible).
+"""
+from __future__ import annotations
+
+import numpy as np
+
+from tire_fiber_chords import (
+    d_positions_for,
+    u_positions_for,
+    compute_faces_from_chords,
+)
+from tire_fiber_chords_fast import OTHER_NP, induced_sigma_vec
+
+
+def proper_cycle_colorings_chunked(n: int, chunk_size: int = 1 << 20):
+    """Yield (rows, n) int8 chunks of proper edge 3-colorings of C_n.
+
+    Each chunk is a numpy array; chunks together cover every proper
+    edge 3-coloring exactly once.  After filtering for the cycle closure
+    constraint, a chunk may be smaller than chunk_size."""
+    if n < 2:
+        return
+
+    block_size = 1 << (n - 1)
+    eff_chunk = min(chunk_size, block_size)
+
+    for c0 in (1, 2, 3):
+        for start in range(0, block_size, eff_chunk):
+            end = min(start + eff_chunk, block_size)
+            n_rows = end - start
+            arr = np.empty((n_rows, n), dtype=np.int8)
+            arr[:, 0] = c0
+
+            row_idx = np.arange(start, end, dtype=np.int64)
+            for i in range(1, n):
+                bits = ((row_idx >> (i - 1)) & 1).astype(np.int8)
+                prev = arr[:, i - 1]
+                arr[:, i] = OTHER_NP[prev, bits]
+
+            mask = arr[:, -1] != arr[:, 0]
+            if mask.any():
+                yield arr[mask]
+
+
+def projection_support_streaming(
+    m: int,
+    k: int,
+    chords,
+    positions: list[int],
+    chunk_size: int = 1 << 20,
+) -> set:
+    """Compute the projected support {σ[positions] : σ realisable}
+    without materializing the full σ array.
+
+    Returns: set of tuples in {1,2,3}^len(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
+    ]
+
+    support: set[tuple[int, ...]] = set()
+    for c_chunk in proper_cycle_colorings_chunked(n, chunk_size):
+        sigma = induced_sigma_vec(c_chunk)
+
+        mask = np.ones(sigma.shape[0], dtype=bool)
+        for face in d_positions_by_face:
+            if len(face) <= 1:
+                continue
+            for i in range(len(face)):
+                for j in range(i + 1, len(face)):
+                    mask &= (sigma[:, face[i]] != sigma[:, face[j]])
+
+        if not mask.any():
+            continue
+        sigma_ok = sigma[mask]
+        projected = sigma_ok[:, positions]
+        support.update(map(tuple, projected.tolist()))
+
+    return support
+
+
+def spoke_only_projection_support_streaming(
+    n: int, positions: list[int], chunk_size: int = 1 << 20
+) -> set:
+    """Steiner-rich baseline (no chord constraints)."""
+    support: set[tuple[int, ...]] = set()
+    for c_chunk in proper_cycle_colorings_chunked(n, chunk_size):
+        sigma = induced_sigma_vec(c_chunk)
+        projected = sigma[:, positions]
+        support.update(map(tuple, projected.tolist()))
+    return support
+
+
+if __name__ == '__main__':
+    import time
+    from tire_fiber_chords_fast import fiber_distribution_fast, projection_support_fast
+
+    print("Validation: chunked vs in-memory")
+    print("-" * 60)
+    cases = [
+        (4, 4, [(0, 2)]),
+        (6, 6, [(0, 3)]),
+        (6, 9, [(0, 2), (3, 5), (6, 8)]),
+        (9, 9, [(0, 3), (4, 7)]),
+        (12, 12, [(0, 3), (4, 7), (8, 11)]),
+    ]
+    for m, k, ch in cases:
+        n = m + k
+        d_pos = d_positions_for(m, k)
+
+        t0 = time.time()
+        fibers_mem, _, _ = fiber_distribution_fast(m, k, ch)
+        S_mem = projection_support_fast(fibers_mem, d_pos)
+        dt_mem = time.time() - t0
+
+        t0 = time.time()
+        S_chunked = projection_support_streaming(m, k, ch, d_pos)
+        dt_chunked = time.time() - t0
+
+        ok = S_mem == S_chunked
+        status = "OK" if ok else "MISMATCH"
+        print(f"(m={m}, k={k}, n={n}): mem {dt_mem:.2f}s, chunked {dt_chunked:.2f}s, "
+              f"|S|={len(S_chunked)}, {status}")
+
+    print()
+    print("Stress test (n > 24):")
+    print("-" * 60)
+    cases_big = [
+        (12, 13, [(0, 3), (4, 7), (8, 11), (12, 1)]),    # n=25
+        (15, 15, [(0, 3), (4, 7), (8, 11), (12, 14), (12, 0)]),  # n=30
+    ]
+    for m, k, ch in cases_big:
+        n = m + k
+        d_pos = d_positions_for(m, k)
+        t0 = time.time()
+        S = projection_support_streaming(m, k, ch, d_pos)
+        dt = time.time() - t0
+        print(f"(m={m}, k={k}, n={n}): {dt:.1f}s, |S|={len(S)}")
diff --git a/papers/coloring_nested_tire_graphs/experiments/tire_fiber_threshold_search.py b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_threshold_search.py
new file mode 100644
index 0000000..09dce77
--- /dev/null
+++ b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_threshold_search.py
@@ -0,0 +1,236 @@
+"""Counterexample search at γ ∈ {13, 14, 15}, threshold m ≥ γ.
+
+The earlier search (tire_fiber_counterexample_search.py) capped n ≤ 27
+to fit fiber arrays in memory, which forced m_1 < γ at γ ≥ 13 — so it
+never tested the threshold-satisfying regime above γ = 12.  This
+script uses tire_fiber_chunked.projection_support_streaming to push
+n up to 30+ (at γ = 15, m = 15, the bottleneck is ~7 min per tire).
+
+Configs include all-3-faces chord matchings (where they exist —
+γ ∈ {15} via the multi-chord 'ring' (0,3),(3,6),...,(γ-3,γ),(0,γ-3))
+plus mixed-face chord matchings (γ ∈ {13, 14}).
+
+Shares the same counterexample_search.log with the earlier sweep;
+deduplication via pair-keys lets this be resumed mid-flight.
+"""
+from __future__ import annotations
+
+import json
+import time
+from pathlib import Path
+
+from tire_fiber_chords import (
+    compute_faces_from_chords,
+    d_positions_for,
+    u_positions_for,
+)
+from tire_fiber_chunked import projection_support_streaming
+
+
+HERE = Path(__file__).resolve().parent
+LOG_PATH = HERE / "counterexample_search.log"
+
+
+def log_event(event: dict) -> None:
+    event.setdefault('wall_time', time.time())
+    with open(LOG_PATH, 'a') as f:
+        f.write(json.dumps(event, default=str) + "\n")
+        f.flush()
+
+
+def has_all_three_faces(k: int, chords) -> bool:
+    if not chords:
+        return k == 3
+    faces = compute_faces_from_chords(k, chords)
+    return all(len(f) == 3 for f in faces)
+
+
+def make_key(gamma, m_1, ch1, model1, k_2, ch2, model2) -> str:
+    ch1_str = json.dumps(sorted([sorted(c) for c in ch1])) if ch1 else "[]"
+    ch2_str = json.dumps(sorted([sorted(c) for c in ch2])) if ch2 else "[]"
+    return f"gamma={gamma}|t1=({m_1},{ch1_str},{model1})|t2=({k_2},{ch2_str},{model2})"
+
+
+def read_existing_keys() -> set[str]:
+    keys = set()
+    if not LOG_PATH.exists():
+        return keys
+    with open(LOG_PATH) as f:
+        for line in f:
+            try:
+                ev = json.loads(line)
+            except json.JSONDecodeError:
+                continue
+            if ev.get('type') == 'pair' and 'key' in ev:
+                keys.add(ev['key'])
+    return keys
+
+
+# T1 chord configurations at the higher γ.
+T1_CONFIGS = {
+    13: [
+        # Mixed-face (no all-3 possible at γ=13, since 13 not divisible by 3)
+        [(0, 3), (4, 7), (8, 11), (12, 1)],
+        [(0, 3), (3, 6), (6, 9), (9, 12), (12, 0)],
+    ],
+    14: [
+        # Mixed-face
+        [(0, 3), (3, 6), (6, 9), (9, 12), (12, 0)],
+        [(0, 3), (4, 7), (8, 11), (12, 0)],
+    ],
+    15: [
+        # all-3 "ring" config: 5 chords, all faces size 3
+        [(0, 3), (3, 6), (6, 9), (9, 12), (0, 12)],
+        # mixed-face alternative
+        [(0, 3), (4, 7), (8, 11), (12, 14), (12, 0)],
+    ],
+}
+
+# T2 chord configurations (only k_2 ≥ γ to satisfy threshold).
+T2_CONFIGS = {
+    13: [
+        [(0, 3), (4, 7), (8, 11), (12, 1)],
+    ],
+    14: [
+        [(0, 3), (3, 6), (6, 9), (9, 12), (12, 0)],
+    ],
+    15: [
+        [(0, 3), (3, 6), (6, 9), (9, 12), (0, 12)],
+        [(0, 3), (4, 7), (8, 11), (12, 14), (12, 0)],
+    ],
+}
+
+
+# In-memory caches so that a fixed (m_1, gamma, ch1) doesn't get
+# re-streamed across multiple T2 variants.
+_pi_d_cache: dict = {}
+_pi_u_cache: dict = {}
+
+
+def _cache_key(args):
+    return tuple(args[:-1]) + (tuple(tuple(sorted(c)) for c in args[-1]),)
+
+
+def pi_D_cached(m_1: int, gamma: int, ch1: list) -> set:
+    key = (m_1, gamma, tuple(tuple(sorted(c)) for c in ch1))
+    if key not in _pi_d_cache:
+        d_pos = d_positions_for(m_1, gamma)
+        _pi_d_cache[key] = projection_support_streaming(m_1, gamma, ch1, d_pos)
+    return _pi_d_cache[key]
+
+
+def pi_U_cached(gamma: int, k_2: int, ch2: list) -> set:
+    key = (gamma, k_2, tuple(tuple(sorted(c)) for c in ch2))
+    if key not in _pi_u_cache:
+        u_pos = u_positions_for(gamma, k_2)
+        _pi_u_cache[key] = projection_support_streaming(gamma, k_2, ch2, u_pos)
+    return _pi_u_cache[key]
+
+
+def test_pair(gamma: int, m_1: int, ch1, k_2: int, ch2) -> dict:
+    t0 = time.time()
+    S1 = pi_D_cached(m_1, gamma, ch1)
+    S2 = pi_U_cached(gamma, k_2, ch2)
+    fwd = S1 & S2
+    S2_rev = {s[::-1] for s in S2}
+    rev = S1 & S2_rev
+    dt = time.time() - t0
+    return {
+        'type': 'pair',
+        'key': make_key(gamma, m_1, ch1, 'SP', k_2, ch2, 'SP'),
+        'gamma': gamma,
+        't1': {'m_1': m_1, 'chords': ch1, 'model': 'SP'},
+        't2': {'k_2': k_2, 'chords': ch2, 'model': 'SP'},
+        's1_size': len(S1),
+        's2_size': len(S2),
+        'fwd_size': len(fwd),
+        'rev_size': len(rev),
+        'compatible': bool(fwd or rev),
+        'time_seconds': round(dt, 3),
+        't1_all_three': has_all_three_faces(gamma, ch1),
+        't2_all_three': has_all_three_faces(k_2, ch2),
+    }
+
+
+def main():
+    already = read_existing_keys()
+    log_event({'type': 'session_start', 'session_kind': 'threshold-search-streaming'})
+    print(f"[start] {len(already)} pairs already tested", flush=True)
+
+    n_tested = 0
+    n_ce = 0
+    n_ce_all3 = 0
+    t_start = time.time()
+
+    for gamma in [13, 14, 15]:
+        cs1_list = T1_CONFIGS.get(gamma, [])
+        if not cs1_list:
+            continue
+        log_event({'type': 'k_start', 'gamma': gamma,
+                   'session_kind': 'threshold-search-streaming',
+                   'n_t1_configs': len(cs1_list)})
+
+        # We only test the threshold m_1 = γ here (n_1 = 2γ).  Larger m_1
+        # would saturate even more easily.
+        m_1 = gamma
+
+        for ch1 in cs1_list:
+            for k_2 in sorted(T2_CONFIGS.keys()):
+                if k_2 < gamma:
+                    continue  # threshold: k_2 ≥ γ
+                for ch2 in T2_CONFIGS[k_2]:
+                    key = make_key(gamma, m_1, ch1, 'SP', k_2, ch2, 'SP')
+                    if key in already:
+                        continue
+
+                    t0 = time.time()
+                    print(f"[{(time.time()-t_start)/60:.1f}m] testing γ={gamma} "
+                          f"T1=(m={m_1}, ch={ch1}), T2=(k={k_2}, ch={ch2})",
+                          flush=True)
+                    try:
+                        result = test_pair(gamma, m_1, ch1, k_2, ch2)
+                    except Exception as e:
+                        log_event({'type': 'exception', 'key': key,
+                                   'error': type(e).__name__ + ": " + str(e)})
+                        print(f"  exception: {e}", flush=True)
+                        continue
+                    log_event(result)
+                    already.add(key)
+                    n_tested += 1
+                    if not result['compatible']:
+                        n_ce += 1
+                        if result.get('t1_all_three') and result.get('t2_all_three'):
+                            n_ce_all3 += 1
+                            log_event({'type': 'COUNTEREXAMPLE',
+                                       'strict_latin': True,
+                                       'threshold': True,
+                                       'pair': result})
+                            print(f"  *** THRESHOLD STRICT-LATIN COUNTEREXAMPLE ***",
+                                  flush=True)
+                        else:
+                            log_event({'type': 'COUNTEREXAMPLE',
+                                       'pair': result})
+                            print(f"  *** counterexample (not strict-Latin) ***",
+                                  flush=True)
+                    print(f"  -> |S1|={result['s1_size']}, |S2|={result['s2_size']}, "
+                          f"fwd={result['fwd_size']}, rev={result['rev_size']}, "
+                          f"compat={result['compatible']} ({result['time_seconds']:.1f}s)",
+                          flush=True)
+        # Free the γ-specific caches before moving to the next γ
+        _pi_d_cache.clear()
+        _pi_u_cache.clear()
+
+    elapsed_h = (time.time() - t_start) / 3600
+    log_event({'type': 'session_end',
+               'session_kind': 'threshold-search-streaming',
+               'n_tested': n_tested,
+               'n_ce': n_ce,
+               'n_ce_strict_latin_threshold': n_ce_all3,
+               'elapsed_hours': elapsed_h})
+    print(f"\n[done] tested {n_tested}, CE {n_ce}, "
+          f"strict-Latin threshold CE {n_ce_all3}, "
+          f"elapsed {elapsed_h:.2f} hours", flush=True)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/papers/coloring_nested_tire_graphs/notes/worst_case_proof_sketch.aux b/papers/coloring_nested_tire_graphs/notes/worst_case_proof_sketch.aux
new file mode 100644
index 0000000..95d9805
--- /dev/null
+++ b/papers/coloring_nested_tire_graphs/notes/worst_case_proof_sketch.aux
@@ -0,0 +1,5 @@
+\relax 
+\newlabel{prop:konig-overlap}{{}{1}}
+\newlabel{prop:lower-bound-tight}{{}{1}}
+\newlabel{conj:t2-induces-partition}{{}{2}}
+\gdef \@abspage@last{3}
diff --git a/papers/coloring_nested_tire_graphs/notes/worst_case_proof_sketch.log b/papers/coloring_nested_tire_graphs/notes/worst_case_proof_sketch.log
new file mode 100644
index 0000000..d906b95
--- /dev/null
+++ b/papers/coloring_nested_tire_graphs/notes/worst_case_proof_sketch.log
@@ -0,0 +1,280 @@
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+\documentclass[11pt]{article}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage{graphicx}
+\usepackage{geometry}
+\geometry{margin=1in}
+
+\title{A proof attempt at the worst-case overlap}
+\author{}
+\date{}
+
+\newtheorem*{thm}{Theorem}
+\newtheorem*{prop}{Proposition}
+\newtheorem*{lem}{Lemma}
+\newtheorem*{conj}{Conjecture}
+
+\begin{document}
+\maketitle
+
+\section*{Setup}
+
+Let $\gamma$ be a cycle of length $k$ (divisible by $3$ for the
+``all-3'' case), and let $T_1, T_2$ be two adjacent SP tires sharing
+$\gamma$ with the threshold condition
+$|B_{\mathrm{out}}^{(1)}| \geq k$ and $|B_{\mathrm{in}}^{(2)}| \geq k$.
+Assume both tires' inner outerplanar graphs are configured so that
+every $O$-face has exactly $3$ $B_{\mathrm{in}}$ edges (strict Latin
+case).
+
+The empirical worst-case overlap is exactly $|S_3| = 6$ elements,
+forming a single $S_3$-orbit (see
+\texttt{tire\_fiber\_step2\_large.tex}, Obs. on the $S_3$-orbit
+structure).  This note sketches a proof.
+
+\section*{What I can prove cleanly: both sides give $\gamma$-face partitions}
+
+\begin{prop}[König $\Rightarrow$ overlap $\geq 6$]
+\label{prop:konig-overlap}
+Suppose both $T_1$ and $T_2$ induce \emph{direct} all-$3$ face
+partitions $\mathcal{F}_1, \mathcal{F}_2$ of $E(\gamma)$.  Then
+\[
+   |\mathcal{L}(\gamma, \mathcal{F}_1) \cap \mathcal{L}(\gamma,
+      \mathcal{F}_2)| \;\geq\; 6.
+\]
+\end{prop}
+
+\begin{proof}[Proof sketch]
+Define a bipartite graph $G$:
+\begin{itemize}
+  \item Left vertices: $\mathcal{F}_1$-faces (there are $k/3$ of
+        them).
+  \item Right vertices: $\mathcal{F}_2$-faces (also $k/3$).
+  \item One edge from $F \in \mathcal{F}_1$ to $F' \in \mathcal{F}_2$
+        for each $\gamma$-edge $e \in F \cap F'$.
+\end{itemize}
+Since each $\gamma$-edge belongs to exactly one face of each
+partition, $\gamma$-edges biject with edges of $G$, and $|E(G)| = k$.
+Each face has $3$ $\gamma$-edges, so every vertex of $G$ has degree
+exactly $3$.  Thus $G$ is $3$-regular bipartite.
+
+By König's edge-coloring theorem, every bipartite graph admits a
+proper $\Delta$-edge-coloring; here $\Delta = 3$.  So $G$ has a
+proper $3$-edge-coloring $\chi : E(G) \to \{1,2,3\}$.  Transport
+$\chi$ back to $\gamma$-edges via the bijection: define
+$\sigma(e) := \chi(e)$.  Then for any face $F \in \mathcal{F}_1$,
+the three $\gamma$-edges in $F$ are the three $G$-edges incident
+to the $F$-vertex (degree $3$), and $\chi$ being proper forces these
+to use all three colors --- so $\sigma|_F$ is a permutation of
+$\{1,2,3\}$.  Same for $\mathcal{F}_2$.
+
+Hence $\sigma \in \mathcal{L}(\gamma, \mathcal{F}_1) \cap
+\mathcal{L}(\gamma, \mathcal{F}_2)$.  The $S_3$ action on colours
+sends this $\sigma$ to $|S_3| = 6$ distinct elements of the
+intersection (they're distinct because $\sigma$ uses all three
+colours).
+\end{proof}
+
+\begin{prop}[Lower bound is achieved]
+\label{prop:lower-bound-tight}
+The lower bound $|{\cdots}| \geq 6$ in
+Prop.~\ref{prop:konig-overlap} is achieved exactly when $G$ has a
+\emph{unique} proper $3$-edge-coloring up to $S_3$-relabelling ---
+equivalently, when $G$ is a single closed $C_{2k/3}$-cycle (with
+two alternating perfect matchings, plus a third perfect matching
+forced by the remaining edges).  In particular, the worst case is
+$G \cong K_{3,3}$ (when $k = 9$) or $G$ a single cycle, both of
+which arise empirically.
+\end{prop}
+
+(Sketch: a $3$-regular bipartite graph has $\geq 6$ proper $3$-edge-
+colourings, with equality iff the colouring is essentially unique.
+The unique-up-to-$S_3$ case is exactly when $G$ is highly symmetric.
+This matches the empirical observation that worst-case intersections
+are single $S_3$-orbits.)
+
+\section*{What remains: the gap in the actual setting}
+
+In the real chain-pigeonhole setup, $T_1$'s chord is on $B_{\mathrm{in}}^{(1)}
+= \gamma$ (so $T_1$ \emph{does} give a direct $\gamma$-face partition
+$\mathcal{F}_1$), but $T_2$'s chord is on $B_{\mathrm{in}}^{(2)}$,
+\emph{not} on $\gamma$.  So $T_2$ does not directly give a
+$\gamma$-face partition.
+
+\subsection*{Conjectural extension to the real case}
+
+\begin{conj}[Induced $\gamma$-partition from $T_2$]
+\label{conj:t2-induces-partition}
+For any SP tire $T_2$ with $B_{\mathrm{in}}^{(2)}$-side chord
+structure $O^{(2)}$ such that every $O^{(2)}$-face has exactly $3$
+$B_{\mathrm{in}}^{(2)}$-edges, the outer-spoke support
+$\pi_U(\mathcal{C}(T_2)) \subseteq \{1,2,3\}^\gamma$ contains a
+Latin subset $\mathcal{L}(\gamma, \widetilde{\mathcal{F}_2})$ for
+some induced face partition $\widetilde{\mathcal{F}_2}$ of $\gamma$
+into triples, where $\widetilde{\mathcal{F}_2}$ is determined by how
+$T_2$'s annular triangulation distributes $B_{\mathrm{in}}^{(2)}$ faces
+across $\gamma$-edges.
+\end{conj}
+
+If Conjecture~\ref{conj:t2-induces-partition} holds, then
+Prop.~\ref{prop:konig-overlap} immediately gives the worst-case
+overlap $\geq 6$ for general adjacent tire pairs.
+
+\subsection*{Why this is plausible}
+
+At $\gamma = 6$, $T_1 = (m_1, (0,3), \mathrm{SP})$ vs $T_2 = (m_2,
+B_{\mathrm{in}}^{(2)} = C_3, \mathrm{SP})$, the empirical $6$-element
+intersection consists of patterns $(a, b, c, b, c, a)$ which:
+\begin{itemize}
+  \item $T_1$'s $\gamma$-face partition $\{0,1,2\} \mid \{3,4,5\}$:
+        $\sigma|_{0,1,2} = (a, b, c)$ is a permutation,
+        $\sigma|_{3,4,5} = (b, c, a)$ is a permutation.  $\checkmark$
+  \item The pattern uses each colour exactly twice (positions $\{0,5\},
+        \{1,3\}, \{2,4\}$).  These pair into $3$ groups of $2$ --- but
+        not in a way that's directly a Latin $\gamma$-partition of
+        size $3$.
+\end{itemize}
+The $T_2$-side constraint forces a structural pattern (equal pairs at
+specific positions); the empirical fact is that $T_2$'s induced
+constraint on $\gamma$ \emph{is} ``something like a Latin partition,''
+but I haven't yet found the precise statement.
+
+\subsection*{One concrete attempt at the induced partition}
+
+For $T_2$ with $B_{\mathrm{in}}^{(2)} = C_{k_2}$ and balanced
+annular triangulation, the dual cycle of $T_2$ alternates between
+D-triangles (one $B_{\mathrm{in}}^{(2)}$-edge each) and U-triangles
+(one $\gamma$-edge each).  Each $B_{\mathrm{in}}^{(2)}$-edge has, in
+some sense, two ``adjacent'' $\gamma$-edges via the two annular edges
+of its D-triangle.
+
+\emph{Candidate induced partition.}  Group the $\gamma$-edges by
+``which $B_{\mathrm{in}}^{(2)}$-face's pair of $\gamma$-neighbours
+they share an annular edge with.''  When $k_2 = k/3$ (so there are
+$k/3$ D-triangles, each accounting for $3$ $\gamma$-edges via its $2$
+annular edges), this gives a partition of $E(\gamma)$ into $k/3$
+triples --- the right shape.
+
+Verifying that this candidate partition gives the correct Latin
+subset of $\pi_U$ is an open computation.
+
+\section*{Where the lower bound of $|S_3| = 6$ comes from independently}
+
+Setting aside the construction, the lower bound of $6$ has a clean
+abstract origin:
+
+\begin{lem}[Lower bound from $S_3$-invariance]
+Both $S_1$ and $S_2$ are $S_3$-invariant (the $S_3$ action on
+colours acts on edge $3$-colourings of $G'$).  Hence $S_1 \cap S_2$
+is $S_3$-invariant, so it decomposes into $S_3$-orbits.  Any
+non-trivial $S_3$-orbit on $\{1,2,3\}^k$ where $\sigma$ uses all
+three colours has size exactly $|S_3| = 6$.  Constant $\sigma$
+(single colour) orbits have size $3$, but constants are typically
+not in the support under SP chord constraints with $\geq 2$ faces
+of size $\geq 2$ (which forces $\sigma$ to use both ``other'' colours
+on each face).
+\end{lem}
+
+So the absolute floor is $6$ once we rule out emptiness.  The
+$\gamma$-partition / König argument is the cleanest way I know to
+rule out emptiness.
+
+\section*{Summary of what's proven, conjectured, and open}
+
+\begin{itemize}
+  \item \textbf{Proven (Prop.~\ref{prop:konig-overlap}):}
+        When both $T_1$ and $T_2$ give direct all-$3$ $\gamma$-face
+        partitions, $|S_1 \cap S_2| \geq 6$, witnessed by lifting
+        a König $3$-edge-colouring of the $3$-regular bipartite
+        ``face-incidence'' graph.
+
+  \item \textbf{Lower-bound origin:}
+        Both $S_1$ and $S_2$ are $S_3$-invariant, so any non-empty
+        intersection has size at least $6$ (the size of a
+        non-constant $S_3$-orbit).
+
+  \item \textbf{Conjectured (Conj.~\ref{conj:t2-induces-partition}):}
+        The real chain-pigeonhole setup, where $T_2$'s chord is on
+        $B_{\mathrm{in}}^{(2)}$ rather than on $\gamma$, also reduces
+        to Prop.~\ref{prop:konig-overlap} via an induced
+        $\gamma$-partition $\widetilde{\mathcal{F}_2}$.  Empirical
+        data is consistent with this conjecture but I haven't
+        constructed $\widetilde{\mathcal{F}_2}$ explicitly.
+
+  \item \textbf{Open:}
+        Find the explicit map $T_2 \mapsto \widetilde{\mathcal{F}_2}$,
+        prove the support inclusion, and combine with
+        Prop.~\ref{prop:konig-overlap} to close the worst-case
+        lower bound.
+\end{itemize}
+
+\end{document}