From 374fca98a568f7a0b7fa3a04dbc3690f38bd97ed Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 03:26:17 -0400 Subject: [PATCH] coloring_nested_tire_graphs: add Latin conjecture to paper; launch counterexample search MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds Sec. 2 "A conjectural Latin-style substructure" with two conjectures: (1) for tires whose every O-face has exactly 3 B_in edges, π_D contains the Latin-flavoured subset L (where each O-face gets a permutation of {1,2,3}); (2) adjacent tires both satisfying (1) have compatible projections. Adds tire_fiber_counterexample_search.py: append-log counterexample hunt across increasing k with deduplication for resumability. Logs to counterexample_search.log. Smoke-test data at k≤5 shows non-all-3 SP-model artifacts produce empty intersections (the conjecture's "exactly 3 B_in edges per face" precondition fails), but no strict-Latin counterexample yet. Co-Authored-By: Claude Opus 4.7 --- .../experiments/counterexample_search.log | 2460 +++++++++++++++++ .../tire_fiber_counterexample_search.py | 379 +++ papers/coloring_nested_tire_graphs/paper.aux | 12 +- papers/coloring_nested_tire_graphs/paper.log | 79 +- papers/coloring_nested_tire_graphs/paper.pdf | Bin 814471 -> 837514 bytes papers/coloring_nested_tire_graphs/paper.tex | 83 + 6 files changed, 2976 insertions(+), 37 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/experiments/counterexample_search.log create mode 100644 papers/coloring_nested_tire_graphs/experiments/tire_fiber_counterexample_search.py diff --git a/papers/coloring_nested_tire_graphs/experiments/counterexample_search.log b/papers/coloring_nested_tire_graphs/experiments/counterexample_search.log new file mode 100644 index 0000000..b531791 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/counterexample_search.log @@ -0,0 +1,2460 @@ +{"type": "session_start", "budget_hours": 0.01, 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"SP"}, "s1_size": 36, "s2_size": 36, "fwd_size": 0, "rev_size": 0, "compatible": false, "time_seconds": 0.945, "t1_all_three": false, "t2_all_three": true}, "wall_time": 1779780331.347769} +{"type": "session_end", "n_tested": 11, "n_skipped": 260, "n_counterexamples": 6, "elapsed_hours": 0.0030151924822065567, "wall_time": 1779780331.348233} diff --git a/papers/coloring_nested_tire_graphs/experiments/tire_fiber_counterexample_search.py b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_counterexample_search.py new file mode 100644 index 0000000..c74dcae --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_counterexample_search.py @@ -0,0 +1,379 @@ +"""6-hour counterexample search for the Latin-substructure conjecture. + +Logic: pick a chord configuration cs1 for T1 (with B_in = γ of length k) +and a chord configuration cs2 for T2 (with B_in of length k_2), with +various m_1 (T1's B_out length) and k_2. For each (m_1, k, cs1) and +(k, k_2, cs2) pair (both SP-feasible), compute π_D(T1) and π_U(T2) on +γ; check if their intersection (forward or reverse) is empty. + +Append-log: each test writes a JSON-Lines record to LOG_PATH. On +restart, we read the log to skip already-tested pairs. + +Run until time budget exhausted; report counterexamples loudly. +""" +from __future__ import annotations + +import json +import os +import sys +import time +from itertools import product +from pathlib import Path + +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 ( + fiber_distribution_fast, + spoke_only_fiber_distribution_fast, + projection_support_fast, + proper_cycle_colorings, +) + + +HERE = Path(__file__).resolve().parent +LOG_PATH = HERE / "counterexample_search.log" + + +def log_event(event: dict) -> None: + """Append one JSON line to the log.""" + event = dict(event) # don't mutate caller + event.setdefault('wall_time', time.time()) + with open(LOG_PATH, 'a') as f: + f.write(json.dumps(event, default=str) + "\n") + f.flush() + + +def read_existing_keys() -> set[str]: + """Read the log and return set of pair-keys already tested.""" + 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 + + +def make_key(gamma: int, m_1: int, ch1, model1: str, k_2: int, ch2, model2: str) -> 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 is_sp_feasible(k: int, chords) -> bool: + if not chords: + return k <= 3 + faces = compute_faces_from_chords(k, chords) + return max(len(f) for f in faces) <= 3 + + +# Hand-curated chord configurations per k. Each entry should be +# SP-feasible (every face has at most 3 B_in edges). +CHORD_CONFIGS: dict[int, list] = { + 3: [[]], + + # k=4: only one essentially-distinct chord modulo rotation. + 4: [[(0, 2)]], + + # k=5: short or long chord. + 5: [[(0, 2)], [(0, 3)]], + + # k=6: antipodal, or 2-chord matching. + 6: [ + [(0, 3)], + [(0, 2), (3, 5)], + [(1, 3), (4, 0)], + ], + + # k=7: needs ≥ 2 chords for all-faces-≤-3. + 7: [ + [(0, 2), (3, 6)], + [(0, 3), (4, 6)], + [(0, 2), (3, 5)], # outer face: edges 5,6 → size 2 + [(0, 3), (3, 6)], # uses vertex 3 twice + ], + + # k=8. + 8: [ + [(0, 2), (3, 5), (6, 0)], # vertex 0 used twice + [(0, 3), (4, 7)], + [(0, 3), (3, 6)], + [(0, 2), (3, 6)], + [(0, 2), (4, 6)], + ], + + # k=9: 2-chord tight (faces 3,3,3) and 3-chord variants. + 9: [ + [(0, 3), (4, 7)], + [(0, 2), (3, 5), (6, 8)], + [(0, 2), (4, 6), (3, 7)], + [(0, 3), (3, 6), (6, 0)], + ], + + # k=10. + 10: [ + [(0, 3), (4, 7), (7, 0)], + [(0, 2), (3, 5), (6, 8), (9, 1)], + [(0, 3), (4, 7), (8, 0)], + [(0, 3), (3, 6), (6, 9), (9, 0)], + ], + + # k=11. + 11: [ + [(0, 3), (3, 6), (6, 9), (9, 0)], + [(0, 2), (3, 5), (6, 8), (9, 0)], + ], + + # k=12. + 12: [ + [(0, 3), (4, 7), (8, 11)], # tight 4-faces-of-3 + [(0, 3), (3, 6), (6, 9), (9, 0)], # tight ring + [(0, 3), (4, 11), (5, 7), (8, 10)], # nested + [(0, 2), (3, 5), (6, 9), (10, 0)], # mixed + ], + + # k=13. + 13: [ + [(0, 3), (3, 6), (6, 9), (9, 12), (0, 12)], + [(0, 3), (4, 7), (8, 11), (12, 1)], + ], + + # k=14. + 14: [ + [(0, 3), (3, 6), (6, 9), (9, 12), (12, 0)], + [(0, 2), (3, 5), (6, 8), (9, 11), (12, 0)], + ], + + # k=15: tight ring. + 15: [ + [(0, 3), (3, 6), (6, 9), (9, 12), (0, 12)], + [(0, 3), (4, 7), (8, 11), (12, 14), (12, 0)], + ], + + # k=18: tight ring. + 18: [ + [(0, 3), (3, 6), (6, 9), (9, 12), (12, 15), (0, 15)], + [(0, 3), (4, 7), (8, 11), (12, 15), (16, 1)], + ], + + # k=21. + 21: [ + [(0, 3), (3, 6), (6, 9), (9, 12), (12, 15), (15, 18), (0, 18)], + ], + + # k=24. + 24: [ + [(0, 3), (3, 6), (6, 9), (9, 12), (12, 15), (15, 18), (18, 21), (0, 21)], + ], +} + + +# Filter to SP-feasible only. +for k in list(CHORD_CONFIGS): + feasible = [cs for cs in CHORD_CONFIGS[k] if is_sp_feasible(k, cs)] + if feasible: + CHORD_CONFIGS[k] = feasible + else: + CHORD_CONFIGS[k] = [] + + +def compute_pi_D(gamma: int, m_1: int, chords_1, model: str) -> set: + if model == 'SR': + fibers = spoke_only_fiber_distribution_fast(m_1 + gamma) + else: + fibers, _, _ = fiber_distribution_fast(m_1, gamma, chords_1) + return projection_support_fast(fibers, d_positions_for(m_1, gamma)) + + +def compute_pi_U(gamma: int, k_2: int, chords_2, model: str) -> set: + if model == 'SR': + fibers = spoke_only_fiber_distribution_fast(gamma + k_2) + else: + fibers, _, _ = fiber_distribution_fast(gamma, k_2, chords_2) + return projection_support_fast(fibers, u_positions_for(gamma, k_2)) + + +def test_pair(gamma, m_1, ch1, model1, k_2, ch2, model2): + """Return result dict.""" + t0 = time.time() + S1 = compute_pi_D(gamma, m_1, ch1, model1) + S2 = compute_pi_U(gamma, k_2, ch2, model2) + forward = S1 & S2 + S2_rev = {s[::-1] for s in S2} + reverse = S1 & S2_rev + dt = time.time() - t0 + compatible = bool(forward or reverse) + return { + 'type': 'pair', + 'key': make_key(gamma, m_1, ch1, model1, k_2, ch2, model2), + 'gamma': gamma, + 't1': {'m_1': m_1, 'chords': ch1, 'model': model1}, + 't2': {'k_2': k_2, 'chords': ch2, 'model': model2}, + 's1_size': len(S1), + 's2_size': len(S2), + 'fwd_size': len(forward), + 'rev_size': len(reverse), + 'compatible': compatible, + 'time_seconds': round(dt, 3), + # Tag whether this is a "strict-Latin" pair (where the paper conjecture + # applies) -- both tires must have every O-face of exactly 3 B_in edges. + 't1_all_three': has_all_three_faces(gamma, ch1) if model1 == 'SP' else None, + 't2_all_three': has_all_three_faces(k_2, ch2) if model2 == 'SP' else None, + } + + +def estimate_cost(n: int) -> float: + """Rough estimate of seconds to compute fiber distribution at cycle length n. + Calibrated from prior benchmarks (n=18: 0.1s; n=24: 6.6s).""" + if n <= 12: + return 0.01 + return 0.001 * (2 ** (n - 12)) + + +# Cap on cycle length to avoid memory blowup. +# At n=27: 3 * 2^26 paths × 27 bytes ≈ 5 GB. +# At n=30: would be ~40 GB. +MAX_N = 27 + +# Time cap on a single tire computation. +MAX_TIRE_TIME_SEC = 600 # 10 minutes max per tire + + +def has_all_three_faces(k: int, chords) -> bool: + """The strict precondition of the paper's Latin conjecture: + every O-face has exactly 3 B_in edges.""" + if not chords: + return k == 3 + faces = compute_faces_from_chords(k, chords) + return all(len(f) == 3 for f in faces) + + +def add_rotations(k: int, chords): + """Generate all cyclic rotations of a chord set on C_k (yields tuples).""" + seen = set() + for shift in range(k): + rotated = tuple(sorted([tuple(sorted([(a + shift) % k, (b + shift) % k])) + for (a, b) in chords])) + if rotated not in seen: + seen.add(rotated) + yield [list(c) for c in rotated] + + +def main(budget_hours: float = 6.0): + deadline = time.time() + budget_hours * 3600 + log_event({'type': 'session_start', 'budget_hours': budget_hours, + 'pid': os.getpid()}) + + already_tested = read_existing_keys() + print(f"[start] already tested: {len(already_tested)} pairs") + log_event({'type': 'startup', 'already_tested_count': len(already_tested)}) + + # Iterate over k values in increasing order. + k_values = sorted(CHORD_CONFIGS.keys()) + m_values = [3, 6, 9, 12, 15] + k2_values = [3, 6, 9, 12, 15] + + n_counterexamples = 0 + n_tested = 0 + n_skipped = 0 + + for gamma in k_values: + if time.time() >= deadline: + break + cs_list = CHORD_CONFIGS[gamma] + if not cs_list: + log_event({'type': 'k_skip_no_configs', 'gamma': gamma}) + continue + # Prioritize strict-Latin (all-3-faces) configs for the conjecture test. + cs_list_sorted = sorted(cs_list, key=lambda cs: not has_all_three_faces(gamma, cs)) + log_event({'type': 'k_start', 'gamma': gamma, + 'n_configs': len(cs_list_sorted), + 'n_all3_configs': sum(1 for cs in cs_list_sorted if has_all_three_faces(gamma, cs))}) + + for m_1 in m_values: + if time.time() >= deadline: break + for ch1 in cs_list_sorted: + if time.time() >= deadline: break + n_1 = m_1 + gamma + if n_1 > MAX_N: + continue + if estimate_cost(n_1) > MAX_TIRE_TIME_SEC: + continue + for k_2 in k2_values: + if time.time() >= deadline: break + cs2_list = CHORD_CONFIGS.get(k_2, []) + if not cs2_list: + continue + cs2_sorted = sorted(cs2_list, key=lambda cs: not has_all_three_faces(k_2, cs)) + for ch2 in cs2_sorted: + if time.time() >= deadline: break + n_2 = gamma + k_2 + if n_2 > MAX_N: + continue + if estimate_cost(n_2) > MAX_TIRE_TIME_SEC: + continue + + key = make_key(gamma, m_1, ch1, 'SP', k_2, ch2, 'SP') + if key in already_tested: + n_skipped += 1 + continue + + budget_remaining = deadline - time.time() + # Conservative: skip if estimated cost > budget remaining. + est = estimate_cost(n_1) + estimate_cost(n_2) + 0.5 + if est > budget_remaining: + log_event({'type': 'cost_skip', 'key': key, + 'est_seconds': est, + 'budget_remaining': budget_remaining}) + continue + + try: + result = test_pair(gamma, m_1, ch1, 'SP', k_2, ch2, 'SP') + except MemoryError as e: + log_event({'type': 'memory_error', 'key': key, + 'error': str(e)}) + continue + except Exception as e: + log_event({'type': 'exception', 'key': key, + 'error': type(e).__name__ + ": " + str(e)}) + continue + + n_tested += 1 + already_tested.add(key) + log_event(result) + if not result['compatible']: + n_counterexamples += 1 + log_event({'type': 'COUNTEREXAMPLE', 'pair': result}) + print(f"\n*** COUNTEREXAMPLE FOUND at γ={gamma}, " + f"T1={(m_1, ch1)}, T2={(k_2, ch2)} ***\n") + + # Print periodic progress + if n_tested % 25 == 0: + elapsed = time.time() - (deadline - budget_hours * 3600) + print(f"[{elapsed/60:.1f}m] tested {n_tested}, " + f"skipped {n_skipped}, " + f"counterexamples {n_counterexamples}, " + f"current γ={gamma}") + + log_event({'type': 'session_end', + 'n_tested': n_tested, + 'n_skipped': n_skipped, + 'n_counterexamples': n_counterexamples, + 'elapsed_hours': (time.time() - (deadline - budget_hours * 3600)) / 3600}) + print(f"\n[done] tested {n_tested}, skipped {n_skipped}, " + f"counterexamples {n_counterexamples}") + + +if __name__ == '__main__': + budget = float(sys.argv[1]) if len(sys.argv) > 1 else 6.0 + main(budget_hours=budget) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 27d2a3b..cb921e3 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -25,13 +25,17 @@ \newlabel{def:tire-annular-face-connector}{{1.16}{9}} \newlabel{def:spokes}{{1.17}{9}} \newlabel{rem:facial-dual-spoke-only}{{1.18}{9}} +\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{10}{}\protected@file@percent } +\newlabel{fig:facial-dual-choices}{{5}{10}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{10}{}\protected@file@percent } +\newlabel{sec:latin-conjecture}{{2}{10}} \bibcite{bauerfeld-pds}{1} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{10}{}\protected@file@percent } -\newlabel{fig:facial-dual-choices}{{5}{10}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{10}{}\protected@file@percent } -\gdef \@abspage@last{10} +\newlabel{conj:latin}{{2.1}{11}} +\newlabel{conj:chain-latin}{{2.2}{11}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{11}{}\protected@file@percent } +\gdef \@abspage@last{11} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 2fd8af9..f759a4b 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/coloring_nested_tire_graphs/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) 25 MAY 2026 23:14 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 03:16 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -144,26 +144,26 @@ File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX) (./paper.aux) \openout1 = `paper.aux'. -LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 26. -LaTeX Font Info: ... okay on input line 26. -LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 26. -LaTeX Font Info: ... okay on input line 26. -LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 26. -LaTeX Font Info: ... okay on input line 26. -LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 26. -LaTeX Font Info: ... okay on input line 26. -LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 26. -LaTeX Font Info: ... okay on input line 26. -LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 26. -LaTeX Font Info: ... okay on input line 26. -LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 26. -LaTeX Font Info: ... okay on input line 26. -LaTeX Font Info: Trying to load font information for U+msa on input line 26. +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 27. +LaTeX Font Info: ... okay on input line 27. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 27. +LaTeX Font Info: ... okay on input line 27. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 27. +LaTeX Font Info: ... okay on input line 27. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 27. +LaTeX Font Info: ... okay on input line 27. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 27. +LaTeX Font Info: ... okay on input line 27. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 27. +LaTeX Font Info: ... okay on input line 27. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 27. +LaTeX Font Info: ... okay on input line 27. +LaTeX Font Info: Trying to load font information for U+msa on input line 27. 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