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Annex — The structural skewness law at the World Cup

Live spin-off of the main study. Block analysis/52_worldcup.py, library skewlib/worldcup.py. Updates during the 2026 tournament.

The question

The study proves, on clubs and with market odds, that the return-skewness of a 1X2 bet on the favourite is fixed by a single quantity — the favourite's probability p — through the closed form (1−2p)/√(p(1−p)). This annex takes the law out of its own population: from clubs to national teams, from league to knockout tournament, and from odds to no odds at all.

It is an out-of-sample test in three senses at once (population, structure, probability source) — and, with the 2026 World Cup under way, a live one.

How

  • Data: martj42/international_results (public, ~49k international games, 1872→today). No odds — only scores. Includes the 2026 World Cup in near real time.
  • Model (no odds): the skewlib/elo.py engine from Block W2 — a chronological Elo over all internationals (each team's rating travels across friendlies, qualifiers and World Cups) + an MNLogit map rating-diff → (P_home, P_draw, P_away), calibrated on results. Home advantage is zeroed at neutral venues (neutral=True), the norm at the World Cup.
  • The bet: 1 unit on the Elo favourite (the most likely outcome). With no market we use fair odds o = 1/p (zero EV) — so the predicted skewness is purely the two-point shape, and the realised one is the empirical skew of the return.

The result (the main validation)

Predicted skewness (structural, via the Elo p_fav) × realised, by p_fav bucket, over all World Cups (large N per bucket → the 3rd moment stabilises):

favourite p games win rate predicted skew realised skew
0.39 200 43% +0.46 +0.29
0.45 170 49% +0.22 +0.06
0.51 151 54% −0.03 −0.16
0.58 200 59% −0.30 −0.33
0.71 316 71% −0.83 −0.84

corr(predicted, realised) = +0.998. Monotonic, crossing zero at p ≈ ½ exactly where (1−2p)=0, and in the most lopsided bucket −0.827 predicted against −0.837 realised. No fitting, no odds. The win rate ≈ p column attests that the odds-free Elo is well calibrated on real World Cup outcomes — that is what sustains the skewness prediction.

Overall pool: predicted +0.129, realised +0.051 (bootCI [−0.056, +0.152] — contains the predicted value). Across editions, corr(p_fav, predicted skew) = −0.688 (the more lopsided the tournament, the more negative the skew). Skewness by edition is noisy by construction (N=17–73 per cup → the 3rd moment is dominated by sampling); that is why the serious validation is by p_fav bucket, not by edition.

The forecast (out-of-time)

The honest test: train the Elo only on games before the knockout and predict the bracket's skew without seeing a single result. In the 2022 hold-out (train through 2022-12-03, 16 knockout games): predicted +0.065 × realised −0.125, with 44% upsets — Qatar 2022 was an extremely high-variance tournament, and the prediction is the structural baseline the chaos departs from (N=16, noisy). It is exactly the mechanism the cron applies to the 2026 knockout as soon as it begins.

The live demonstration

analysis/predict_worldcup.py predicts the next World Cup games (fixtures with no score in the dump) and freezes each into an append-only, pre-registered ledger (site/src/data/wc_predictions.json): favourite, p_fav, predicted skewness, fair odds. When the score arrives the row is reconciled (realised filled in, prediction never rewritten). The /worldcup page renders an interactive "next match" card + a live scorecard from wc_live.json.

Market overlay — model vs market (Box A / Box B), page /odds

The odds-free Elo gives a probability for each favourite bet without ever seeing a price. A real bookmaker gives one too. /odds puts them side by side:

  • Box A — observer-invariance (descriptive, on-thesis). De-vig a real published 3-way line (Shin) → the market's fair p for the same bet (the model's favourite pick). Apply the law (1−2p)/√(p(1−p)) to each. The claim: both the model's (p, skew) and the market's land on the same curve — they differ only in where. Agreement (rank corr of the two p's, 0.998 over the first three) is the headline: the odds-free engine reconstructs what a sportsbook priced. The WC analogue of the realised-skew +0.998, now against a second observer, not frequency.
  • Box B — calibration (honest, weak at small n). Whose p was better calibrated vs the outcome: model vs book Brier / log-loss (model 0.272 / 0.740 vs book 0.300 / 0.813 so far — our humility on the upsets paid). With n = 3, two of them penalty lotteries, it is noise; real at n = 16.
  • Box C — value/edge — deliberately NOT crossed. The gap p_model − p_book is a betting signal; we display it as a diagnostic and stop. Staking it would turn a structural law into an alpha claim with a different burden of proof. The thesis's credibility rests on it being descriptive history, not a tip sheet.

Book odds are hand-recorded, sourced, frozen in study/wc_book_odds.json (American moneyline → decimal at load) — a pre-match snapshot, same discipline as wc_manual_results.json. No API, no feed. skewlib/worldcup.py: load_book_odds

  • book_pfav; predict_worldcup.py freezes a book block per ledger entry (never rewritten) and emits triangulation + model_vs_book into wc_live.json.

Reproduce & provenance

python analysis/52_worldcup.py            # full analysis (stdout + outputs/wc_*.csv)
python analysis/export_worldcup_data.py   # regenerate site/src/data/worldcup.json
python analysis/predict_worldcup.py       # predict next games + reconcile the ledger

No API key, no scraping. Each run stamps outputs/_provenance/52_worldcup.json with the git sha + the dump's sha256 + the date of the last game — provenance that travels with the live data.

Versioning note. This is a live annex: the dataset changes while the cup runs. So it stays out of the study's frozen drift ledger (build_lineage.py --check, which is for the frozen matches.csv) — instead it carries its own fingerprint (sha256 + date) embedded in every regeneration and on the /worldcup page. The evidence/worldcup tag freezes the code + JSON snapshot of the publication day. It reuses the football elo/exante path (still a football 1X2 contract, just national teams and results-not-odds), not the multi-sport canonical adapter layer.