From 1594a3f58aad48bb455fa4e22e002e94e4b10da2 Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 11:44:58 -0400 Subject: [PATCH] =?UTF-8?q?coloring=5Fnested=5Ftire=5Fgraphs:=20structural?= =?UTF-8?q?=20description=20of=20surviving=20=CE=B3-partitions=20at=20k=3D?= =?UTF-8?q?9=20(positive=20result)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Investigated the 8 surviving triple-partitions of γ at k=k_2=9 (chord (0,3),(3,6) on B_in^(2)). Found a clean structural description. CLASSIFICATION of γ-edges by T_2's face structure: For each O^(2)-face F_i, 2 γ-edges are "internal" to F_i (their adjacent D-triangles are both in F_i). For each adjacent face pair (F_i, F_{i+1}), 1 γ-edge is "boundary" between them. Total: 2r internal + r boundary = 3r γ-edges = |γ| when k=k_2. STRUCTURAL DESCRIPTION (Prop face-pair-connection): Latin ⊆ π_U(T_2) iff the partition has the following form: - One block per cyclically-adjacent face pair (F_i, F_{i+1}). - Each block = 1 boundary edge δ_{i,i+1} + 1 internal of F_i + 1 internal of F_{i+1}. - For each face F_i, its 2 internal γ-edges are distributed one per block (the two blocks involving F_i). Count: 2^r partitions (each face has 2 choices of how to split its internals across its 2 adjacent blocks). AT k = k_2 = 9 (r = 3 faces): 2^3 = 8 partitions, matching the empirical survivors. WHY NAIVE CANDIDATES FAIL: The next-D and prev-D candidates from worst_case_proof_sketch.tex group BOTH internals of one face into one block (e.g., {0,1,2} = both Internal_{F_A} + δ_{AB}, no internal F_B). This violates the "one internal per face per block" rule. IMPLICATION: The König-lift approach can be RESCUED by replacing the naive candidate F~_2 with any of the 2^r face-pair-connection partitions. Apply König's theorem on bipartite face-incidence graph of F_1 vs this new F~_2. NEXT STEP: prove Prop face-pair-connection for all r, then apply König lift. This is more leveraged than re-tackling the naive construction. Files: experiments/k9_surviving_analysis.py notes/k9_surviving_partitions.tex (3 pages) Note also updates notes/induced_partition_findings.tex to point at the new structural description. Co-Authored-By: Claude Opus 4.7 --- .../experiments/k9_surviving_analysis.py | 182 +++++++++++ .../notes/induced_partition_findings.pdf | Bin 222172 -> 223659 bytes .../notes/induced_partition_findings.tex | 19 +- .../notes/k9_surviving_partitions.aux | 4 + .../notes/k9_surviving_partitions.log | 307 ++++++++++++++++++ .../notes/k9_surviving_partitions.pdf | Bin 0 -> 200885 bytes .../notes/k9_surviving_partitions.tex | 211 ++++++++++++ 7 files changed, 712 insertions(+), 11 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/experiments/k9_surviving_analysis.py create mode 100644 papers/coloring_nested_tire_graphs/notes/k9_surviving_partitions.aux create mode 100644 papers/coloring_nested_tire_graphs/notes/k9_surviving_partitions.log create mode 100644 papers/coloring_nested_tire_graphs/notes/k9_surviving_partitions.pdf create mode 100644 papers/coloring_nested_tire_graphs/notes/k9_surviving_partitions.tex diff --git a/papers/coloring_nested_tire_graphs/experiments/k9_surviving_analysis.py b/papers/coloring_nested_tire_graphs/experiments/k9_surviving_analysis.py new file mode 100644 index 0000000..08a9406 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/k9_surviving_analysis.py @@ -0,0 +1,182 @@ +"""Enumerate and analyze all surviving triple-partitions of γ at k = 9 +for T_2 = (m_2=9, k_2=9, chords (0,3),(3,6), SP). + +Look for common structural properties: + - Block sums modulo k = 9 (rotation invariance) + - Block differences modulo something + - Mapping under T_2's annular structure + - Symmetries (cyclic, reflection) +""" +from itertools import product, permutations +from collections import defaultdict, Counter + +from tire_fiber_chords import ( + fiber_distribution, + projection_support, + u_positions_for, + d_positions_for, +) +from tire_fiber_chunked import projection_support_streaming + + +def latin_set(partition, k): + L = set() + if any(len(b) != 3 for b in partition): + return None + blocks = [sorted(b) for b in partition] + for assignment in product(permutations((1, 2, 3)), repeat=len(blocks)): + sigma = [0] * k + for block, perm in zip(blocks, assignment): + for pos, color in zip(block, perm): + sigma[pos] = color + L.add(tuple(sigma)) + return L + + +def all_partitions_into_triples(elements): + elements = list(elements) + if not elements: + yield [] + return + if len(elements) % 3 != 0: + return + if len(elements) == 3: + yield [tuple(elements)] + return + first = elements[0] + rest = elements[1:] + from itertools import combinations + for combo in combinations(rest, 2): + triple = (first,) + combo + remaining = [x for x in rest if x not in combo] + for sub_partition in all_partitions_into_triples(remaining): + yield [triple] + sub_partition + + +def normalize(partition): + """Return a canonical tuple of frozensets for the partition.""" + return tuple(sorted(tuple(sorted(b)) for b in partition)) + + +def find_surviving_partitions(k, pi_U): + survivors = [] + for partition in all_partitions_into_triples(range(k)): + L = latin_set(partition, k) + if L is None: + continue + if L <= pi_U: + survivors.append(tuple(tuple(sorted(b)) for b in partition)) + return survivors + + +def analyze_block(block, k=9): + """Return structural properties of a 3-element block of γ-edges.""" + s = sum(block) % k + d = sorted((b - block[0]) % k for b in block) # differences from first + diffs_pairwise = sorted(((b - a) % k for a in block for b in block if a != b)) + return { + 'sum_mod_k': s, + 'first_diffs': d, + 'pairwise_diffs': diffs_pairwise, + } + + +def main(): + k, k_2 = 9, 9 + chords = [(0, 3), (3, 6)] + print(f'Analyzing surviving triple-partitions at k = k_2 = {k}, chords = {chords}') + print() + + # Compute π_U via chunked streaming (faster for n = k + k_2 = 18) + u_pos = u_positions_for(k, k_2) + d_pos = d_positions_for(k, k_2) + pi_U = projection_support_streaming(k, k_2, chords, u_pos) + print(f'|π_U| = {len(pi_U)}') + print(f'd_positions on T_ann_prime = {d_pos}') + print(f'u_positions on T_ann_prime = {u_pos}') + print() + + survivors = find_surviving_partitions(k, pi_U) + print(f'Number of surviving triple-partitions: {len(survivors)}') + print() + + print('=== Surviving partitions ===') + for i, p in enumerate(survivors): + print(f' {i+1}. {list(p)}') + for block in p: + props = analyze_block(block, k) + print(f' block {block}: sum mod {k} = {props["sum_mod_k"]}, ' + f'pairwise diffs = {props["pairwise_diffs"]}') + print() + + # Structural analysis + print('=== Cyclic rotation analysis ===') + print('For each partition, check if it is a rotation of any other:') + canonical_to_partitions = defaultdict(list) + for p in survivors: + # Compute canonical form under cyclic rotation of γ + rotations = [] + for r in range(k): + rotated = tuple(tuple(sorted((x + r) % k for x in b)) for b in p) + rotated_canonical = tuple(sorted(rotated)) + rotations.append(rotated_canonical) + canonical = min(rotations) + canonical_to_partitions[canonical].append(p) + print(f'# distinct cyclic-rotation orbits: {len(canonical_to_partitions)}') + for can, orbits in canonical_to_partitions.items(): + print(f' Orbit rep: {list(can)}; members: {len(orbits)}') + + # Block sum analysis + print() + print('=== Block sum mod 3 / mod 9 analysis ===') + sums_mod3 = Counter() + sums_mod9 = Counter() + for p in survivors: + s3 = tuple(sorted(sum(b) % 3 for b in p)) + s9 = tuple(sorted(sum(b) % 9 for b in p)) + sums_mod3[s3] += 1 + sums_mod9[s9] += 1 + print(f'Distribution of sorted block-sums mod 3:') + for k_, v_ in sums_mod3.items(): + print(f' {k_}: {v_} partitions') + print(f'Distribution of sorted block-sums mod 9:') + for k_, v_ in sums_mod9.items(): + print(f' {k_}: {v_} partitions') + + # Check if any survivors are "arithmetic progressions" (3-APs) + print() + print('=== Arithmetic progression check ===') + ap_count = 0 + for p in survivors: + is_ap_partition = all( + (block[1] - block[0]) % k == (block[2] - block[1]) % k + for block in p + ) + if is_ap_partition: + ap_count += 1 + print(f' AP partition: {list(p)}') + print(f'Total AP partitions: {ap_count}') + + # Check if block 1 and block 2 are "shifts" of each other + print() + print('=== Shift relationships between blocks ===') + for p in survivors: + if len(p) != 3: + continue + # Check if block[1] is a cyclic shift of block[0], etc. + shift_dets = [] + for i in range(3): + for j in range(3): + if i == j: continue + # Is block j a shift of block i? + for r in range(k): + shifted = tuple(sorted((x + r) % k for x in p[i])) + if shifted == tuple(sorted(p[j])): + shift_dets.append((i, j, r)) + break + if shift_dets: + print(f' Partition {list(p)}: shift structure {shift_dets}') + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/notes/induced_partition_findings.pdf b/papers/coloring_nested_tire_graphs/notes/induced_partition_findings.pdf index 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a/papers/coloring_nested_tire_graphs/notes/induced_partition_findings.tex +++ b/papers/coloring_nested_tire_graphs/notes/induced_partition_findings.tex @@ -135,17 +135,14 @@ Plan-step 3 from \texttt{two\_approaches\_comparison.tex} was ``prove inclusion via transfer matrix / fibre lifting,'' assuming the candidate partition was empirically correct. The candidate is \emph{not} empirically correct beyond $k = 6$, so trying to prove -the wrong statement is futile. Instead the right next move is: -\begin{enumerate} - \item Find the right induced $\widetilde{\mathcal{F}_2}$ at - $k = 9$: study the $8$ surviving triple-partitions, see if - they have a common structural description (e.g.\ via the - $T_2$ annular triangulation). - \item Or abandon the ``$\widetilde{\mathcal{F}_2}$ is a partition'' - framing entirely and look for a different structure on - $\gamma$ that $T_2$ induces and that suffices for chain - pigeonhole. -\end{enumerate} +the wrong statement is futile. Instead the right next move is to +\textbf{study the $8$ surviving triple-partitions at $k = 9$} and +look for a common structural description (e.g.\ via the $T_2$ annular +triangulation, $T'_{\mathrm{ann}}$ cyclic distance, or modular +arithmetic on $D$- vs $U$-positions). If no such description exists, +the ``$\widetilde{\mathcal{F}_2}$ is a partition'' framing should be +abandoned and a different structure on $\gamma$ sought. This is the +new step done next; see \texttt{notes/k9\_surviving\_partitions.tex}. \section*{Reassessment of Approach 2} diff --git a/papers/coloring_nested_tire_graphs/notes/k9_surviving_partitions.aux b/papers/coloring_nested_tire_graphs/notes/k9_surviving_partitions.aux new file mode 100644 index 0000000..2ad4ff9 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/k9_surviving_partitions.aux @@ -0,0 +1,4 @@ +\relax +\newlabel{prop:fp-connection}{{}{1}} +\newlabel{prop:general}{{}{2}} +\gdef \@abspage@last{3} diff --git a/papers/coloring_nested_tire_graphs/notes/k9_surviving_partitions.log b/papers/coloring_nested_tire_graphs/notes/k9_surviving_partitions.log new file mode 100644 index 0000000..f302a98 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/k9_surviving_partitions.log @@ -0,0 +1,307 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 11:44 +entering extended mode + restricted \write18 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