Forecasting the Winner of a Live Tennis Match

We present two approaches for predicting tennis match outcomes using topological data analysis and graph theory on ATP singles matches from 2000-2025. The first method applies lower-star filtration to player competitive networks, extracting topological features through persistent homology using four summary methods (VAB, HNAV, HWNAV, OW-HNPV) combined with Modified Band Depth analysis. Algorithmic optimizations including ego graph approximations and triangle elimination enable analysis of about 66k matches. Our Random Forest model achieves 66.2% accuracy (AUC = 0.719) using topological, graph-theoretic, and ranking features. Feature importance analysis reveals that rankings contribute 36.3%, centralities 25.5%, and TDA features 24.0%, with topological features providing complementary signal. When rankings are unavailable, the topology-only model maintains 63.56% accuracy, demonstrating that network-derived features alone capture meaningful competitive structure. The second method uses a modified Katz similarity index with temporal edge weighting, achieving 62.48% accuracy on held-out test data. This work represents the first application of lower-star filtration to tennis prediction, provides systematic comparison of four topological summary methods in sports analytics, and demonstrates that TDA can achieve above-chance prediction using network topology alone while providing additional value when combined with traditional features.

Benchmarks that measure the forecasting ability of large language models are almost always retrospective: the event has happened, the answer is somewhere on the Web, and the evaluation must defend itself against memorisation. We report the opposite design. Over the 39 days of the 2026 FIFA World Cup, six frontier LLMs -- all with extended thinking and native server-side web search -- were asked before every kickoff, one match at a time, to fill in a seven-market prediction card for all 104 matches, plus 12 group winners and a pre-tournament outright pool; no answer existed when the question was asked, so the evaluation is leakage-free by construction rather than by filtering, and the frozen archive holds 4,494 scored predictions. What the tournament establishes is a set of behaviours the six systems share. On match outcome they average 63.9%, level with backing the bookmaker's favourite -- which is in fact what they usually do. They agree with one another far more often than they are right, so a majority vote adds nothing. They under-commit to draws and to goals, and crowd their scoreline picks onto a single prototypical result. Accuracy tracks how lopsided a fixture is rather than how much is known about it: it collapses in the closest ties, where the dossiers are richest, while questions about the tournament as a whole are answered well. On this task the current generation of frontier systems is not sharply differentiated: the standings hold up at the top and the bottom across the run and churn in the middle, and the margins stay narrow throughout. The briefing dossiers, fixtures and official results are released as a benchmark, together with the scoring code.

Predicting a football match before kickoff requires more than knowing past results: a model must use changing information and make a clear prediction before the answer is available. We present WorldCupArena, a dynamic benchmark for language models and deep-research agents. The 2026 FIFA World Cup is its first evaluation, and the same process can be reused for future leagues and cups. Before each match, a model either receives a common evidence package or searches for information itself. It predicts the result and score, likely players and events, match statistics, and the outcome of the competition. After the match, these predictions are compared with the recorded result. We report result accuracy, exact-score accuracy, and a scoreline score that gives some credit when a predicted score is close but not exact, together with scores for the other prediction tasks. Across systems, similar result accuracy can mask larger differences in detailed predictions. Four systems predicted champion Spain, and two of them also recovered the exact final pairing. Compared with betting-market and human-fan baselines, the best system shows only small gains in result and exact-score accuracy, but a clearer gain in Scoreline. New schedules can be added as they begin, allowing the benchmark to evaluate future models without using outcomes that are already known. Code, predictions and evaluation scripts will be publicly released.