Stability of Skewness in Football Betting Markets

In-depth results — 6 fronts

Sample: 205,435 matches · 38 leagues · 2005–2025 · football-data mirror Baseline strategy: unit bet on the favourite (shortest odds), ex-post return.

CENTRAL THESIS (revised and strengthened)

The skewness of the betting market is not a dynamic process — it is a structural invariant. Each league has its own level of skewness, fixed in time, determined by its competitiveness. The observed fluctuation is pure sampling noise, not market memory. The asymmetry is a property of the sporting generator, exogenous to the bookmakers' pricing.

It evolved from "stable-with-memory" (draft v1) → "structural constant + white noise" (confirmed by 3 independent tests). And from skewness invariance → SHAPE invariance (Phases B1/B2): not only the 3rd moment, but the entire implied distribution (var/skew/kurtosis) is a unique function of competitiveness — it collapses across leagues once conditioned on it.

BLOCK A — Robustness (does the finding survive the confounds?)

Configuration	ADF	KPSS	Ljung-Box	ACF(1)
baseline (overlap)	<0.001	>0.10	<0.001	0.74
league-demeaned (overlap)	<0.001	>0.10	<0.001	0.74
non-overlap	<0.001	>0.10	0.71	−0.08
league-demeaned non-overlap	<0.001	>0.10	0.70	−0.08

Stationarity: armoured. Survives demeaning, non-overlap and windows of 500 to 3000 matches. ADF<0.001 and KPSS p>0.10 (the test's tabulated ceiling; stationarity is not rejected) in all 10 configs.
Persistence: it was an artefact. The ACF=0.74 came 100% from window overlap. Without overlap → white noise (LB p=0.71). An honest correction vs v1.
League composition is not a confound: demeaning moves almost nothing.
sd falls monotonically with window size → the variation is sampling error.

BLOCK B — Where does the skewness come from?

B1 — by strategy: skewness monotone in the improbability of the bet. favourite +0.23 · draw +1.27 · longshot +2.26 · "always away" +2.40. Mean return follows along (favourite −4.8%, longshot −10.2%) = classic FLB.

B2 — favourite-longshot bias (the key table): skewness falls monotonically as the favourite strengthens, crossing zero at p≈0.50:

p_favourite	mean ret	skewness	win%
(0.40]	−7.5%	+0.59	36%
(0.45,0.50]	−4.7%	+0.19	45%
(0.55,0.60]	−3.5%	−0.22	56%
(0.70,1.0]	−1.6%	−1.19	77%

B3 — the mechanism of stability: corr(favourite strength, variance)=−0.90 but corr(strength, skewness)=−0.21 and corr(variance, skewness)≈0. The variance is sensitive to composition; the skewness is not, because the contributions of the two tails (weak favourites→+, strong→−) cancel in a stable way.

BLOCK C — Is the series really i.i.d.?

Variance Ratio(2,4,8) = 0.94, 0.94, 1.00 (=1 → i.i.d.)
AR(1) φ = −0.06, non-significant (p=0.39), half-life ≈ 0
Conclusion: white noise confirmed by 3 tests. No temporal dynamics.

BLOCK D — Cross-bookmaker (average odd vs best odd) — 2nd finding

metric	average odd	best odd
skewness	0.229	0.242
mean return	−4.75%	−0.16%
overround	1.067	1.009

The bookmaker's margin is mostly spread across bookmakers. Arbitraging odds recovers 4.6 p.p. of return; the "irreducible" margin is ~0.2%.
Skewness is invariant to the bookmaker: temporal corr of the two series = 0.984. Bookmakers compete on the margin (level); the skewness is exogenous to that competition.
Separates pricing (margin, varies) from structure (skewness, fixed).

BLOCK E — Heterogeneity across leagues (the richest finding)

Skewness is not universal: it ranges from 0.10 (Netherlands) to 0.33 (Italy B), sd=0.06.
corr(league predictability, skewness) = −0.83. Almost deterministic.
Leagues with strong favourites (a few dominate) → low skewness; balanced leagues (2nd divisions) → high skewness.
Generalisation: each league has a skewness-invariant = f(competitiveness). The same mechanism as B3 (within-window scale) reappears across leagues (cross-sectional scale). A single principle explains both scales.
BRA +0.18, ARG +0.31 — South American leagues at the competitive/asymmetric extreme.

BLOCK F — Forensics of the 2012/13 blip

The only "structural break" in 20 years was a sampling artefact: the dataset jumped from 21→37 leagues in 2012 (ARG, MEX, BRA, JAP, POL, ROM... came in), many high-skewness, temporarily inflating the window mean.
There was no market event. It reinforces the real stability and validates the composition control.

SYNTHESIS FOR THE PAPER

Three levels of result, cohesive under one principle:

Cross-sectional: FLB confirmed; skewness monotone in improbability.
Temporal: skewness is constant + white noise (no dynamics) — robust.
Structural: the level is f(league competitiveness); temporal stability follows from competitiveness being a slow property (decadal scale).

Unifying principle: structural competitiveness generates asymmetry; since it changes slowly, skewness is a temporal invariant. The bookmakers' margin is orthogonal to this (it affects the level of return, not the asymmetry).

Next fronts (not yet explored)

Opening→closing drift: the skewness of the price movement (needs opening odds; the dataset has partial coverage).
Intra-season seasonality controlled per league (start vs end).
Testing the invariance in another sport (basketball/tennis) — external generalisation.

PHASE LOG (towards the paper)

Dataset frozen in data/PROVENANCE.json (sha256 6905ca53…, 205,435 matches ≥2005, 38 leagues, 2005-01→2025-06). Baseline reproduced in outputs/phase0_baseline.log. Each phase below adds a block to the paper.

Phase 0 — Reproduction + freezing (2026-06-23)

Blocks 00–06 reproduce FINDINGS exactly against the full 38-league archive. Bug fixed in 06_league_hetero (tab.skew collided with DataFrame.skew()). Dataset frozen by hash.

Phase 1 / W1 — Ex-ante skewness and the mechanical core (2026-06-23)

Primary object redefined: implied (de-vigged) skewness of the bet on the favourite, via the Shin model (mean z 3.4% of informed money; overround 1.068). Closed-form, free of sampling noise.

Ex-ante ≈ ex-post. Global ex-ante skew +0.236 vs realised +0.230 → the implied object reproduces the realised; odds well calibrated in the aggregate.
Decomposition (law of total cumulants) — establishes the mechanical core: M3 = +102.6% within-match (Bernoulli/FLB) − 2.6% covariance − 0.0% between-match. The market skewness is, at ~100%, the within-match asymmetry of the distribution of p — not a pooling artefact. within_frac ≈ 1.00 in all p_fav bands. It turns the "tautology" criticism into the thesis itself: the risk asymmetry IS the algebraic image of the FLB.
Cross-league (clean object): corr(ex-ante, ex-post)=+0.872; corr(p_fav, skew)=−0.900 — but still circular (p comes from the odds). Target of W2: reproduce with odds-free competitiveness.

Artefacts: skewlib/devig.py, skewlib/exante.py, analysis/07_devig_exante.py, outputs/exante_by_league.csv.

Phase 2 / W2 — Odds-free mechanism: the law is not tautological (2026-06-23)

Competitiveness measured by Elo built from results only (no odds; chronological multi-league step + rating-diff→(P_H,P_D,P_A) map via MNLogit calibrated on the results — P(H) 0.444 = real 0.444, P(D) 0.264 = real 0.264).

Odds and Elo measure the SAME structure: corr(elo_pfav, p_fav_dv)=+0.909 [0.83,0.97]. The odds merely read off sporting competitiveness (structural efficiency) — they do not create it.
The law skewness=f(competitiveness) SURVIVES without odds: skew ~ upset_rate +0.826 [0.71,0.91] · skew ~ elo_entropy +0.719 [0.50,0.89] · skew ~ elo_pfav −0.748 · skew ~ elo_disp −0.731. All with CI95 far from zero. Circular reference (odds p_fav): −0.900.
Attenuation −0.90→−0.75/+0.83 = measurement noise in the Elo proxy (errors-in-variables), not evidence against the law: corr(elo,odds)=0.91 shows they measure the same latent. The risk asymmetry is inherited from the competitive structure of the league, not from the pricing.

Artefacts: skewlib/elo.py, skewlib/stats.py (bootstrap_corr, ols), analysis/08_mechanism_elo.py, outputs/mechanism_elo.csv.

Phase 3 / W3 — Temporal invariance: league×season panel (2026-06-23)

Treating (league,season) as a unit dissolves the composition confound of Block F by construction. 638 obs, 38 leagues, 2005–2025.

No secular trend: league FE + linear year (cluster SE by league): β=+0.00015/year (p=0.73, CI95 [−0.0007,+0.0010]). Drift over 20 years ≈ +0.003 vs between-league sd 0.052 → null.
Structural dominance: between-league sd 0.052 vs within-league 0.034; ICC=0.70. Netting out the sampling noise (bootstrap SE 0.019), the real temporal fluctuation is sd≈0.028 — small, with no trend and mean-reverting (consistent with the white noise of Block C). The league invariant dominates ~2:1.
Per league: idiosyncratic and tiny deviations (mean |slope| 0.0024/year; breaks rare, likely PELT over-segmentation in series of ~20 points). No market-wide regime.
COVID vignette (natural experiment): empty stadiums in 2020 lowered the home win (0.447→0.417). The law predicts: HFA↓ → more parity → skewness↑. Observed: mean z +0.42 SD, 21/33 leagues with z>0. The only exogenous competitiveness shock in 20 years moved the skewness in the predicted direction — corroborates the cause without violating the secular invariance.

Artefacts: skewlib/panel.py, analysis/09_panel_temporal.py, outputs/panel_league_season.csv.

Phase 4 / W5 — Binary over/under 2.5 market: identity beyond 1X2 (2026-06-23)

148,261 matches with O/U 2.5 odds (Shin de-vig; overround 1.067, z 0.067 — more informed money than in 1X2). Impeccable calibration: real over 0.490 = de-vigged p_over 0.492.

Same identity, different market: bet on the favourite side → ex-ante skew −0.210 (within-match 99.6%), ex-post −0.217. The closed form (1-2p)/√(p(1-p)) matches exactly per-match (max|diff|=0). By p band, ex-ante and ex-post move in lockstep (p=0.52→−0.09; p=0.74→−1.05).
Conclusion: the mechanical core of W1 is not an artefact of the 3-way structure of 1X2 — it holds in a binary goals market. within≈100% replicates.

Artefacts: skewlib/overunder.py, analysis/10_overunder.py, outputs/overunder.csv.

Phase 5 / W4 — Orthogonal margin + de-vig robustness (2026-06-23)

Margin vs structure (average vs maximum odds, 202,760 matches): taking the best market price collapses the overround 1.067→1.009 (return −4.8%→~0) but the ex-ante skewness barely moves: +0.236→+0.254 (per-match corr p_fav 0.996). The bookmaker competes on the MARGIN (level), not on the asymmetry — the margin is largely orthogonal to the skewness.
Robustness to the de-vig method: global skew 0.224 (power) / 0.236 (shin) / 0.263 (mult) — ±8% in level; but the cross-league law is invariant: corr(p_fav,skew) = −0.906 / −0.900 / −0.891. The structural finding is not a de-vig-choice artefact.

Artefacts: skewlib/exante.py (market_skew), analysis/11_margin_robustness.py.

Phase P1 — Reframe: INTRA-REGIME invariance (2026-06-23)

The literature (Lee & Fort 2012; Basini 2023; see docs/LITERATURA.md) finds real regime breaks in EPL competitiveness tied to institutional shocks (Champions League 94/95, Bosman 95, revenue inequality ~2003) — all prior to the ≥2005 cut-off. We reposition the thesis from a "timeless constant" to a league-specific structural baseline, stable WITHIN the competitive regime.

Intra-window break test (conservative PELT): only 1 break across 38 leagues (F1/France 2020 = COVID, jump −0.064); no common break year (max 1 league/year) → no market-wide regime in 2005–2025.
EPL (E0): 0 breaks in the cut-off; mean 0.165, sd 0.027 — stable intra-regime, consistent with the EPL's regime shocks being pre-2005.
Conclusion: 2005–2025 ≈ a single modern regime; "no trend" (β≈0) is intra-regime invariance, not absolute timelessness. It confronts Lee & Fort/Basini head-on and uses them as a frame — the counter-evidence becomes an ally.

Artefacts: skewlib/panel.py (league_breaks), analysis/13_regimes.py.

Phase P2 — Canonical odds-independent CB from standings (2026-06-23)

Hardening of W2 with the size-robust indices from the literature (Gini out, Utt & Fort 2002), computed from the final standings (results, no odds or Elo) — a stronger attack on circularity.

Law reproduced, sign predicted (imbalance → skewness↓): skew ~ Noll-Scully −0.625 [−0.83,−0.36] · ~ HHI* (Owen 2007) −0.593 · ~ Theil/GE1 (Borooah-Mangan) −0.478. All CI95 exclude zero.
A clean errors-in-variables ladder: odds (circular) −0.90 > Elo (match-level) +0.83 > standings (season-level) 0.48–0.63. The closer to per-match p, the stronger the corr — the latent is strong, measured by proxies of varying fidelity.
Team by team it holds: N1/EPL (NS~1.84, dominance) low skew; MLS (NS 1.13, salary cap) and Argentina (NS 1.20) high skew. The mechanism is visible in the standings.

Artefacts: skewlib/balance.py, analysis/14_balance_indices.py, outputs/balance_indices.csv.

Phase P3 — Formal derivation: skewness = f(force dispersion) (2026-06-23)

The law becomes a consequence of a model, not a fit. Ordered-probit (Goddard-Asimakopoulos 2004; Koning 2000): force r~N(0,σ_L²), latent margin y*=d+h+ε, cutoffs ±c → (A,D,H); favourite p=max; under fair odds the pooled skewness S(σ_L)=E[m₃(p)]/E[σ²(p)]^{3/2}.

Calibration (pooled rates H 0.444 / D 0.264 / p_fav 0.499): h=0.220, c=0.373, σ_ref=0.291.
1st→3rd order validation: the model predicts the skewness of each league from the mean p_fav alone: corr(predicted, observed)=+0.904, RMSE 0.024 (< half the between-league sd 0.051). The 38 leagues fall on the derived curve (F5).
Reading: the law skewness~competitiveness is an analytic consequence of the sport's force model + the FLB identity — it closes "the gap" (no one had tied the 3rd moment of odds to a force model). The theoretical curve covers skew −0.03..+0.30 (p_fav 0.44..0.76), bracketing the empirical range.

Artefacts: skewlib/model.py, analysis/15_model.py, outputs/fig/f5_model.png.

Phase P4 — FLB stability over time (Angelini confound) (2026-06-23)

Angelini & De Angelis (2019) find the FLB weakening in recent European data; a bias in motion could fake skewness invariance. Tested 2005–2025:

FLB with no significant trend: ret_dog (barometer) corr(year)=+0.27 [−0.23,+0.67] (CI includes 0; a mild hint in Angelini's direction, non-significant); flb_spread corr −0.02; calib_err corr −0.13. Δ20yr of the spread ≈ −0.002.
Year-by-year calibration intact: mean |skew_exante − skew_expost| = 0.015; the favourite's calibration error ∈ [−0.004,+0.012] every year.
Conclusion: the skewness invariance is not an artefact of a drifting FLB — the bias is stable and the skewness is mechanical in the distribution of p, robust to calibration micro-drift.

Artefacts: skewlib/decompose.py (flb_by_year), analysis/16_flb_stability.py, outputs/flb_by_year.csv.

Phase B1 — SHAPE invariance: not only the 3rd moment (2026-06-23)

Generalisation of the mixture's moment decomposition (law of total moments) to var/skew/kurtosis/5th–6th order, and a test of each against the curve derived from the ordered-probit (P3 did only the skew). Object: is the entire implied distribution invariant after controlling for competitiveness?

The whole shape is MECHANICAL (within-match), not pooling: the within fraction (the part coming from within-match asymmetry/shape, the FLB, vs between-match dispersion) is ≈1 at all orders — m2 +1.000, m3 +1.026, m4 +1.006, m5 +1.026, m6 +1.016. The whole shape is the algebraic image of the distribution of p, not a mixture artefact.
The force model predicts the entire shape from p_fav: corr(predicted, observed) across the 38 leagues = var +0.987 · skew +0.904 · exkurt +0.890. Skew and exkurt (standardised, scale-free) match in level and ordering; the var follows the ordering (r=0.99) with an overround scale offset (real odds o<1/p). Global: skew +0.236 (boot SE 0.001), exkurt −1.683 (strongly short-tailed, expected from a mixture of Bernoullis), std5 +0.85, std6 +2.18.
Conclusion: "skewness invariance" strengthens to SHAPE invariance — the entire implied distribution is a unique function of league competitiveness.

Artefacts: skewlib/exante.py (pooled_moments, per_match_central_moments), skewlib/model.py (league_moments, curve_moments), analysis/17_moments.py, outputs/moments_by_league.csv, outputs/fig/f6_moments.png.

Phase B2 — Distribution collapse: shape = f(competitiveness) (2026-06-23)

A "data collapse" test (statistical physics): is the shape universal, or does it collapse under competitiveness? KS on the favourite's return (effect = KS statistic, since the p-value saturates with large n).

Without controlling for competitiveness (per-league z-scored returns, 38 leagues): pairwise KS with median statistic 0.474, 100% of pairs reject — the standardised shape differs across leagues (the skew varies), so it is not universal.
Controlling for competitiveness (one-vs-rest within 8 p_fav bands, 264 tests): median KS statistic 0.059 — an 87% drop. Within each band the leagues are nearly indistinguishable; the league identity adds nothing beyond competitiveness.
Conclusion: the distribution collapses when conditioned on competitiveness — a stylised fact that the shape is a (unique) function of competitiveness, not of the league.

Artefacts: skewlib/collapse.py, analysis/18_collapse.py, outputs/collapse_ks.csv, outputs/fig/f7_collapse.png.

Phase C1 — Skewness premium: nothing beyond the mechanical FLB (2026-06-23)

Decomposition of the favourite's return (exact identity) into margin + mechanical FLB level + residual, and a test of the residual against the per-league implied skewness.

Return = margin + FLB (calibration): global ret −4.82% = vig −4.97% + FLB +0.15%. The loss is almost all margin; the favourite's FLB is small and positive (favourites slightly underpriced). The mechanical FLB curve is monotone in p_fav (weak favourites contribute −, strong +).
No per-league skewness premium BEYOND the mechanical: corr(residual, skew) = +0.11 [−0.20,+0.38] (CI includes 0); corr(total FLB, skew) −0.04; corr(vig, skew) −0.29. The mispricing residual does not track the league's skewness — the bookmaker leaves no extra premium attached to the asymmetry. Consistent with the orthogonal margin (W4).
Conclusion: the "skewness premium" is entirely the mechanical FLB (across bet types, already in W1/Block B); at the league level no pure premium remains — the pricing is efficient up to the margin + the mechanical bias.

Artefacts: skewlib/premium.py, analysis/19_premium.py, outputs/return_decomp.csv, outputs/fig/f8_premium.png.

Phase C2 — The preference (CPT) is itself invariant (2026-06-23)

Fit of the Tversky-Kahneman probability weighting w(p)=p^γ/(p^γ+(1−p)^γ)^{1/γ} to the calibration curve (PROPORTIONAL implied q vs objective hit π; the Shin de-vig would erase the bias to be measured, so the proportional is used).

Inverse-S confirmed (γ<1 = FLB): global γ 0.958; the calibration reveals the bias (longshot q 0.101 vs π 0.086 = overweighted; favourite q 0.711 vs π 0.743 = underweighted).
γ is a TEMPORAL invariant: by season mean γ 0.955, sd 0.020, trend β=+0.0003/year (r=+0.08, Δ20yr ≈ +0.006) — no drift over 20 years. The weighting preference is stable over time, mirroring the skewness invariance (and the stable FLB of P4).
Nearly invariant across leagues: mean γ 0.945, sd 0.040, range [0.85,1.00] — tight. It shows a mild association with competitiveness (corr(γ,p_fav) −0.45 [−0.74,−0.10]), an honest nuance (it may reflect the range of p sampled per league), not a break in temporal stability.
Conclusion: the preference parameter behind the FLB is a stable structural constant (not a process) — the invariance holds also on the preference side, not only on the risk signature.

Artefacts: skewlib/cpt.py, analysis/20_cpt.py, outputs/cpt_by_league.csv, outputs/cpt_by_season.csv, outputs/fig/f9_cpt.png.

Phase E1 — Closed form of S(σ_L): the derivation leaves Monte Carlo (2026-06-23)

P3/block 15 traces the law skewness=f(competitiveness) by SIMULATION over the force d. Here we show that the expectation is a 1-D Gaussian INTEGRAL in d and evaluate it by QUADRATURE — the closed form of S(σ_L)=E[m₃(p_fav(d))]/E[σ²(p_fav(d))]^{3/2}, d~N(0,2σ_L²), deterministic and free of MC noise.

The quadrature reproduces the MC, without noise: max|MC−exact| = 0.0015 (with n=4·10⁵; this is the magnitude of the MC noise itself), mean 0.0006, over the whole σ_L grid. The theoretical curve becomes exact and smooth — the "derivation by simulation" becomes a closed-form derivation.
Balanced limit in closed form: S(σ_L→0) = (1−2p₀)/√(p₀(1−p₀)) = +0.2449, with p₀=Φ(h−c)=0.4392 (the equilibrium favourite = home team). It is the per-match identity evaluated at p₀ — the law's intercept comes out analytically. The leading curvature S₂=+8.44>0 (the skew RISES on leaving equilibrium), valid for σ_L≲0.1.
The curve is NOT monotone (exact characterisation): concave, with a peak at σ=0.123 (S_max=+0.304, p_fav=0.446)** and crossing zero at σ_L≈1.09 (a strong favourite ⇒ skew→0 and turns negative). It corrects the "monotone" of the old docstring.
Mathematical honesty: p_fav(d)=max(p_H,p_D,p_A) has KINKS where the favourite switches → S(σ_L) is C^∞ but globally non-analytic (the Taylor series diverges beyond the near-balance regime, confirmed numerically). The legitimate closed form is the integral (quadrature), not an elementary series; the expansion S₀+S₂σ² is the local analytic anchor.
Predicts the 38 leagues from the closed curve: corr(predicted,observed) = +0.903, RMSE 0.024 — identical to block 15 by MC (r=0.904), now without resampling.
Conclusion: the law skewness=f(competitiveness) is a closed-form consequence of the force model + FLB, derived from the Gaussian integral, not a fit nor a simulation artefact.

Artefacts: skewlib/model.py (league_moments_exact, league_skew_exact, mean_pfav_exact, smallsigma_coeffs/skew, fav_switch_points, curve_exact), analysis/21_closed_form.py, outputs/closed_form_curve.csv, outputs/fig/f10_closed_form.png.

Phase E2 — Robustness of the force distribution (2026-06-23)

The model assumes Gaussian force, r~N(0,σ_L²). Does the law survive if the force is heavy-tailed (Student-t), skewed (skew-normal) or bounded-support (uniform)? Theoretical prediction: the force difference d=rᵢ−rⱼ is symmetric for any iid force — the asymmetry of the force cannot bias the law; only the TAIL (kurtosis of d) can move anything.

The theory holds: exc.kurt(d) = normal 0.0, t₅ +2.8, t₃ +42.6 (very heavy tail), skew-normal ±0.3, uniform −0.6. skew(d)≈0 in ALL (incl. the skew-normals) — the skewed force generates a symmetric difference.
The skew×competitiveness curve barely moves: reparametrising by observable competitiveness (mean p_fav) and comparing to the Gaussian, max|ΔS| = t₅ 0.017, t₃ 0.032, skew-normal ±0.012, uniform 0.011 — all below the between-league sd (0.051). The shift scales with the TAIL of d (t₃ is the largest), not with its asymmetry (skew-normal sticks to the Gaussian, as predicted).
At football's operating point (p_fav=0.499): skew ∈ [+0.223,+0.250], amplitude across families = 0.027 (small relative to the competitiveness effect, which sweeps +0.30→−0.02).
Conclusion: the law is mixture geometry, not the Gaussian hypothesis — robust to heavy tails and force asymmetry. Gaussianity is convenience, not a premise carrying the result.

Artefacts: skewlib/model.py (force_diff, curve_family), analysis/22_force_robustness.py, outputs/force_robustness.csv, outputs/fig/f11_force_robustness.png.

Phase G1 — Reliable and invariant de-vig (2026-06-23)

Adversarial robustness: is the Shin de-vig reliable and the skewness not a method artefact? Reliability diagram + Brier decomposition (Murphy: BS=REL−RES+UNC) of the favourite by league/year, and skewness under 5 de-vigs/bookmakers.

De-vig calibrated almost perfectly: favourite hit 0.501 vs mean prob 0.499; global REL (calibration error) = 0.0000. Brier 0.236 = REL 0.000 − RES 0.014 + UNC 0.250.
REL small and homogeneous: across 32 leagues mean 0.0005 (sd 0.0003, max 0.0014); across 21 seasons mean 0.0002 (sd 0.0001). No league/year miscalibrated — the de-vig residual is stable (no hidden bias producing the asymmetry).
Skewness invariant to method/bookmaker: shin·odd +0.236, shin·max +0.254, mult +0.263, power +0.224, multi-bookmaker consensus +0.252 — amplitude 0.039, all positive. Extends W4: the finding depends on neither the de-vig nor the bookmaker.
Conclusion: the skewness is not manufactured by the de-vig; the implied asymmetry is well calibrated against the results and robust to the method choice.

Artefacts: skewlib/adversarial.py (fav_won, reliability, brier_decomp, reliability_by, skew_by_devig), analysis/23_devig_reliability.py, outputs/reliability_by_league.csv, outputs/fig/f12_reliability.png.

Phase G2 — Strictly balanced panel (composition killed) (2026-06-23)

The GLOBAL skewness series rebuilt using ONLY the leagues present in all 21 seasons (15 leagues: B1,D1,D2,E0–E3,F1,F2,I1,I2,N1,SP1,SP2,T1) — killing 100% of the composition confound that P1 attacked per-league.

No trend with a fixed basket: β = −0.00013/year (r=−0.06, Δ20yr −0.003) in the balanced series vs −0.00009 in the full one; KPSS p>0.10 (stationary). Mean level +0.243 (sd 0.014) — extremely tight.
Conclusion: the temporal invariance does NOT come from the league basket changing year on year; with the core fixed the global series stays flat. The "no drift" is real.

Artefacts: skewlib/adversarial.py (balanced_leagues, global_series_balanced), analysis/24_balanced_panel.py, outputs/balanced_global_series.csv, outputs/fig/f13_balanced_panel.png.

Phase G3 — CI by block-bootstrap over seasons (2026-06-23)

Honest CIs resampling whole SEASONS (with replacement), respecting the intra-year dependence that match resampling would break.

Global skewness +0.236, CI95 [+0.232, +0.239] (SE 0.0019) — excludes 0 comfortably.
Structural law corr(skew_league, p_fav_league) = −0.900, CI95 [−0.922, −0.876] (SE 0.011) — the skewness↔competitiveness relation survives the year resampling.
Favourite return −4.82%, CI95 [−5.37%, −4.43%].
Conclusion: the headline numbers carry a CI from season resampling; the sign and magnitude do not depend on a specific window of years.

Artefacts: skewlib/adversarial.py (season_block_bootstrap, stat_global_skew, stat_league_corr), analysis/25_block_bootstrap.py.

Phase D2 — Sharp vs soft: the margin is orthogonal also in the best odd (2026-06-23)

Does the skewness diverge between the market's AVERAGE odd (Odd*, soft) and the BEST odd (Max*, ~sharp/arb)? By league.

Best price almost zeroes the margin: overround soft 1.069 → sharp 1.008.
Skew barely moves, and uniformly: soft +0.218 → sharp +0.238 (mean Δ +0.020, sd 0.006). corr(skew_soft, skew_sharp) across leagues = +0.993 — the ordering of the leagues is identical; the structural law survives in the sharp (corr(skew_sharp, p_fav) = −0.876).
Conclusion: removing the margin shifts the skew little and uniformly; the bookmaker competes on margin, not on asymmetry (deepens W4) — the law is invariant to the book.

Artefacts: skewlib/microstructure.py (skew_by_book_league), analysis/26_sharp_soft.py, outputs/sharp_soft_by_league.csv, outputs/fig/f14_sharp_soft.png.

Phase D3 — Shin's z (informed money) as a series (2026-06-23)

z is a by-product of the Shin de-vig: the fraction of the book attributed to insiders. z by league/year, its stability and its relation to competitiveness/overround.

z low and tight: global 0.034 (3.4% of informed money in 1X2); across 38 leagues mean 0.035, sd 0.004, range [0.023, 0.042].
z is essentially the margin reparametrised: corr(z, overround) = +0.999 (almost tautological in the Shin model — z is monotone in the booksum). The useful part is the orthogonality to competitiveness: corr(z, p_fav) = −0.04 [−0.37, +0.30] — the informational content does not drive the skewness law.
Over time: a slight compression (β=−0.0009/year, Δ20yr −0.019), mirroring the smooth decline in the margin; small magnitude.
Conclusion: the priced-in informed money is a low structural constant, tied to the margin and orthogonal to competitiveness — consistent with the invariance.

Artefacts: skewlib/microstructure.py (shin_z_frame, z_by), analysis/27_shin_z_series.py, outputs/shin_z_by_league.csv, outputs/fig/f15_shin_z.png.

Phase D4 — Asian handicap: the identity in a 3rd market (2026-06-23)

Beyond 1X2 (W1) and O/U 2.5 (W5), the AH is a 2-way market with a MOVING line that balances the match to ~50/50. The sharpest test of the identity in a different p_fav regime.

The line balances to ~0.5: 150,003 matches with a valid AH, mean p_fav 0.533 (vs 0.44 in 1X2). AH overround 1.044.
Same identity, opposite sign: pooled ex-ante skew = −0.104 (within-match 102.7% = mechanical), because p_fav>0.5 (the favourite covers frequently ⇒ negative skew) — a mirror of 1X2 (+0.236, p_fav<0.5). Ex-post (70,965 settled) −0.117 ≈ ex-ante −0.112.
By league on the curve: skew_ah vs the identity (1−2p)/√(p(1−p)) at the AH's p_fav → r=+0.80.
Conclusion: a THIRD independent market confirms the mechanical core — the skewness is a function of p (the sign is fixed by which side of 0.5 the favourite falls), not an artefact of the 3-way structure of 1X2.

Artefacts: skewlib/microstructure.py (prep_ah, ah_league), analysis/28_asian_handicap.py, outputs/asian_handicap_by_league.csv, outputs/fig/f16_asian_handicap.png.

Phase F1 — Intra-season seasonality: the invariance holds within the year (2026-06-23)

Does the skewness move from the start to the end of the season (Aug→Jul, thirds by date)?

MILD and predicted drift: global by phase +0.243 → +0.235 → +0.229 (amplitude 0.015); p_fav rises 0.494 → 0.503 (favourites a bit stronger at the end, as the standings crystallise). Δskew(end−start) by league: mean −0.008, CI95 [−0.013, −0.0015] (just excludes 0).
Conclusion: there is a small crystallisation — but ~3–4× smaller than the between-league sd (0.05) and predicted by the law itself (more p_fav ⇒ less skew). The invariance holds also WITHIN the season, up to this minimal drift.

Artefacts: skewlib/intraleague.py (add_season_phase, skew_by_phase, phase_shift_by_league), analysis/29_intraseason.py, outputs/intraseason_shift_by_league.csv, outputs/fig/f17_intraseason.png.

Phase F2 — Which matches carry the skewness (tail cancellation) (2026-06-23)

Decomposition of the pooled 3rd moment by MATCH competitiveness band (p_fav): which matches contribute the asymmetry.

Law at the match level: skew by band ranges from +0.465 (p_fav 0.39, weak favourite) to −1.055 (p_fav 0.73, strong favourite) — exactly the identity (1−2p)/√(p(1−p)). WEAK-favourite matches (p<0.5) sum +126% of M₃; STRONG-favourite (p>0.5) −26%.
Conclusion: the league skewness is the net sum of contributions that tail cancellation nearly zeroes; competitiveness at the MATCH level fixes the sign and magnitude of each contribution — the macro law emerges from the micro.

Artefacts: skewlib/intraleague.py (m3_contribution_by_bin), analysis/30_game_contribution.py, outputs/m3_contribution_by_bin.csv, outputs/fig/f18_game_contribution.png.

Phase F3 — Per-team decomposition: the signature comes from force dispersion (2026-06-23)

By club: dominance (mean Elo) vs the mean skewness of the matches it plays.

Dominant clubs pull towards negative skew: Barcelona (Elo 1983, favourite 97%) match skew −1.10, Bayern −1.09, Real Madrid −0.90; weak clubs (Lahti Elo 1182) +0.06. corr(Elo, match skew) = −0.44 [−0.53,−0.34].
The law, seen from inside: corr(league Elo dispersion, league skew) = −0.60 [−0.77,−0.42] — leagues with more super-clubs have lower skew.
Conclusion: the league's skew signature is a function of its force dispersion at the TEAM level — the micro version of skewness=f(competitiveness).

Artefacts: skewlib/intraleague.py (team_long, team_dominance), analysis/31_team_decomposition.py, outputs/team_dominance.csv, outputs/fig/f19_team_decomposition.png.

Phase H2 — Open vs closed league: the MLS on the law (2026-06-23)

The MLS (USA) is the only CLOSED league in the sample (salary cap, draft, no relegation), designed to compress force dispersion; the European ones are open. Prediction: a closed structure ⇒ more competitiveness ⇒ balanced skew.

MLS is the most balanced by structural measure: Noll-Scully 1.13, rank 1/38 (the most competitive in the sample) — exactly what a cap + no-relegation predict. Ex-ante skew +0.162, below the open mean (+0.219); p_fav 0.503.
On the curve, with an honest nuance: residual vs the open-league law −0.06 (~1 sd) — the MLS sits at the competitive/balanced extreme, consistent with the open-vs-closed theory.
Conclusion: the closed league does not break the law — its structure tightens competitiveness and the skewness moves towards the balanced. It is not a sharp test (only 1 closed league in the sample; a full test needs more closed leagues = external data).

Artefacts: analysis/32_open_vs_closed.py, outputs/open_vs_closed.csv, outputs/fig/f20_open_vs_closed.png.

Phase C3 — Kelly/staking: growth under the skewness structure (2026-06-23)

What does the asymmetry imply for optimal bankroll growth?

Kelly says DON'T bet: under the real margin, 0.0% of bets have EV>0 (f*=0 in all) — after the vig there is no growth to extract (echoes the C1 efficiency).
Skewness is the FLB channel: decomposing the log-growth (g ≈ μ − σ²/2 + m₃/3) at a fixed fraction, the longshot SKEWNESS term is +0.60 (×1e3) vs +0.01 for the favourite — the positive asymmetry offsets part of the negative EV in growth/utility. It is the channel through which the preference for skew (FLB) survives being EV-negative.
Conclusion: the skewness structure does not open growth (efficient market), but it quantitatively explains why the longshot bettor pays EV in exchange for asymmetry — the skewness premium in growth/utility terms.

Artefacts: skewlib/staking.py (kelly_fraction, growth_rate, moment_growth_terms), analysis/33_kelly_staking.py, outputs/fig/f21_kelly.png.

Phase E3 — Per-league calibration (endogenous draw cutoff) (2026-06-23)

Calibration of (h, c, σ_L) PER league (vs the global of P3/block 15): endogenous home advantage, draw cutoff and force dispersion.

Plausible endogenous parameters (32 leagues): h [0.085, 0.350], c [0.297, 0.449] (per-league draw cutoff), σ_L [0.137, 0.436]. corr(c, draw rate) = +0.906 — c captures the league's "draw-proneness"; corr(σ_L, p_fav) = +0.874 — σ_L recovers observable competitiveness.
The law survives: skew predicted by each league's OWN model vs observed r = +0.905, RMSE 0.026 — equal to the global (r=+0.90). Calibrating (h,c,σ) per league does not change the story.
Conclusion: the invariance survives the endogenous draw cutoff; σ_L (competitiveness) keeps governing the skewness, league by league.

Artefacts: skewlib/model.py (calibrate_by_league), analysis/34_per_league_calibration.py, outputs/per_league_calibration.csv, outputs/fig/f22_per_league_calib.png.

1st round on the frozen dataset EXHAUSTED (2026-06-23): W1–W5 · P1–P5 · B1–B2 · C1–C3 · E1–E3 · D2–D4 · F1–F3 · G1–G3 · H2. The 2nd round (I…) follows, exploring untouched veins of the SAME dataset. Lineage in lineage.json / docs/LINEAGE.md.

Phase I — Cross-validation of the mechanism: goals model (Poisson) (2026-06-23)

The law skewness=f(competitiveness) was derived from an ordered-probit over the latent margin. Here a COMPLETELY different model — a Poisson of GOALS (attack/defence + home advantage by league-season, result via Skellam) — generates the probabilities and the skewness. 617 league-seasons fitted.

The goals model recovers competitiveness: corr(p_fav Poisson, empirical p_fav) across 38 leagues = +0.972 [+0.95,+0.99].
And reproduces the skewness: corr(skew Poisson, empirical skew) = +0.925 [+0.85,+0.97]; the Poisson falls on the ordered-probit curve with r=+0.85 (vs empirical +0.90). The level is slightly lower (mean skew Poisson +0.177 vs +0.215) because the Poisson slightly underdisperses p_fav — the LAW (ordering) is what matters.
Conclusion: three independent models — latent margin (ordered-probit), goals (Poisson) and the market (empirical) — fall on the SAME curve. The law is independent of the generating model; it is not an artefact of a chosen functional form.

Artefacts: skewlib/goals.py (fit_match_probs, league_season_table, by_league), analysis/35_poisson_crossmodel.py, outputs/poisson_crossmodel_by_league.csv, outputs/fig/f23_poisson_crossmodel.png.

Phase J — Information arrival (HT→FT): the mechanical core is dynamic (2026-06-23)

Without opening odds (D1 out), the HALF-TIME RESULT is the information shock. The favourite's pre-match win probability updates with the HT scoreline, and the skewness of the "rest of the match" is again the identity (1−2q)/√(q(1−q)) at the conditional probability q. 150,950 matches with HT.

The asymmetry RESOLVES with information: the favourite's HT state → the skew of the rest of the match: behind (20.6%) q=0.139, skew +2.08 (turned lottery-like); level (42.4%) q=0.402, skew +0.40; +1 (26.1%) q=0.757, skew −1.20; +2 or more (10.9%) q=0.945, skew −3.91 (almost certain). The mechanical identity holds at EVERY info state, not only at kick-off.
Dynamic calibration (martingale): E[q conditional on the HT | p0 band] ≈ p0, mean error |p0−q| = 0.0035 — the pre-match probability is well calibrated and the HT refines it without bias.
Conclusion: the FLB/identity is a DYNAMIC fact — the implied skewness tracks the win probability at any instant; it does not "discover" an efficient value over time, it already IS the algebraic image of the current probability. A temporal extension of W1.

Artefacts: skewlib/inplay.py (fav_state, conditional_table, martingale_check), analysis/36_inplay_resolution.py, outputs/inplay_conditional.csv, outputs/fig/f24_inplay.png.

Phase K — Diversification: skewness is a single-bet phenomenon (2026-06-23)

The standardised skewness of the MEAN return of N (nearly) independent bets scales as skew(X)/√N. A diversified bankroll tends to the Gaussian; the isolated bet is strongly asymmetric.

Single bet: realised skew favourite +0.230, longshot +2.254 (lottery-like). The mean return of N bets decays as skew/√N (empirical ≈ predicted).
Diversifying kills the asymmetry: the favourite becomes ~Gaussian (skew<0.1) in ~6 bets; the longshot needs ~509 (much more skewed). The diversified syndicate sees ~Gaussian returns — only the negative EV.
Conclusion: the asymmetry the bettor "loves" (Golec-Tamarkin) is that of the ISOLATED bet; it vanishes under diversification. The FLB survives because the RECREATIONAL bettor concentrates a few lottery-like bets — the microeconomic channel that sustains the bias being EV-negative (complements C3).

Artefacts: skewlib/portfolio.py (skew_decay, n_to_gaussian), analysis/37_diversification.py, outputs/diversification.csv, outputs/fig/f25_diversification.png.

Phase L — Secular home advantage vs skewness invariance (2026-06-23)

Home advantage (HFA) has fallen in recent decades; does the skewness follow?

HFA falls, skew does not: home win rate 0.449 (2005) → 0.431 (2025), β=−0.00133/year (Δ20yr −0.027, a marked fall); skewness β=−0.00009/year (Δ20yr −0.002, flat). corr(HFA, skew) year by year = −0.24 [−0.72,+0.55] (CI includes 0).
Conclusion: the asymmetry depends on the DISPERSION of p_fav (competitiveness), not on the level of home advantage — closes the confound on the home-advantage side.

Artefacts: skewlib/extras.py (hfa_and_skew_by_year), analysis/38_home_advantage.py, outputs/hfa_by_year.csv, outputs/fig/f26_home_advantage.png.

Phase M — Realised tail risk of the strategies (2026-06-23)

The quant side: realised moments, VaR/CVaR and max drawdown of the cumulative P&L (unit bet, chronological order).

Favourite vs longshot: favourite ret −4.82%, skew +0.23, exkurt −1.7, maxDD −9.9k units; longshot ret −10.2%, skew +2.25, exkurt +8.2, maxDD −20.9k. Both bleed the margin (negative final P&L), but the longshot is the TAIL — drawdown ~2× deeper, long losing streaks punctuated by rare prizes.
Conclusion: the skewness structure translates into concrete bankroll risk — the longshot "lottery" is a deep drawdown + thick tails, not only a moment.

Artefacts: skewlib/extras.py (tail_metrics, max_drawdown), analysis/39_tail_risk.py, outputs/fig/f27_tail_risk.png.

Phase N — Entropy + cross-market co-moment (2026-06-23)

(1) Shannon entropy of the 1X2 distribution as an odds-based competitiveness index; (2) do the 1X2 skew and the O/U skew share a common factor?

Entropy is competitiveness: mean 1.004 nats (3-way max 1.099); corr(entropy, skew) = +0.827 [+0.70,+0.91] — a robust alternative index (more entropy/more balanced ⇒ more positive skew).
An honest NULL in the co-moment: corr(skew 1X2, skew O/U 2.5) = +0.15 [−0.39,+0.65] (CI includes 0). The two asymmetries are NOT a single factor: 1X2 measures the dispersion of who-wins (competitiveness), O/U measures the GOALS environment — largely orthogonal dimensions. Each market prices a different structural feature.
Conclusion: entropy confirms the law via one more index; but the asymmetry is not a single latent across markets — it is specific to the dimension each market measures.

Artefacts: skewlib/extras.py (shannon_entropy, entropy_by_league), analysis/40_entropy_comoment.py, outputs/entropy_comoment.csv, outputs/fig/f28_entropy_comoment.png.

Phase O — Battery of generating models: the law is model-independent (2026-06-23)

Front I validated the law with ONE alternative generator (a Poisson of goals). Here we subject it to a BATTERY of genuinely distinct generators, each producing (pH,pD,pA) per match per league-season, and measure whether they all reproduce the law and fall on the ordered-probit curve S(σ_L). 617 league-seasons, 38 leagues.

Five families + the market, one curve. corr(model skew, empirical skew) across 38 leagues: Poisson +0.925 [+0.85,+0.97] · Dixon-Coles +0.874 [+0.77,+0.94] · Bradley-Terry-Davidson +0.840 [+0.71,+0.93] · results Elo (odds-free) +0.786 [+0.59,+0.90]. All recover competitiveness (corr p_fav +0.91…+0.97) and fall on the derived curve (r on the curve +0.85…+0.96 vs empirical +0.90).
Dixon-Coles approximates the market LEVEL: the dependence correction at low scores (mean ρ −0.05, pushes 0-0/1-1 draws) raises the skew to +0.199 (vs Poisson +0.177), closer to the empirical +0.215 — the canonical football model best matches the level, not only the ordering.
A family WITHOUT goals also falls on the law: Bradley-Terry-Davidson (multiplicative forces + draw, logistic pairwise comparison — no goals) reproduces the ordering at +0.84. And the ODDS-FREE generator (results Elo → ordinal map, the W2 machine) falls exactly on the curve (r=+0.96).
Models designed to deviate COLLAPSE to the Poisson: football goals are nearly pure independent Poisson — median home×away covariance −0.07 (bivariate Poisson λ₃≈0) and over-dispersion ≈0 (Negative-Binomial α≈0). Both collapse to the Poisson and add nothing — reported as robustness, not as a series.
Reproduction note (hardening): under the new stack (pandas 3 / numpy 2 / current statsmodels) the GLM of a pathological league-season (JAP 2017) suffers quasi-complete separation (p_fav≈1 in every match vs ~0.48 empirical), exploding the fair-odds skewness. Added goals.degenerate_fit (a separation guard); Front I returns exactly to the ledger (corr_skew +0.925) and the battery is armoured.
Conclusion: the law skewness=f(competitiveness) is not an artefact of any functional form — margin-probit, goals-Poisson, goals-with-dependence, logistic-forces and results-ratings all converge to the SAME curve. It is the geometry of the mixture of two-point bets over the league's competitiveness distribution.

Artefacts: skewlib/crossmodel.py (dc_probs, dc_rho, btd_probs, elo_by_league, battery_table), skewlib/goals.py (fit_rates, degenerate_fit), analysis/41_model_battery.py, outputs/model_battery_by_league.csv, outputs/fig/f29_model_battery.png.

2nd round on the dataset (I–N) completed (2026-06-23): cross-model Poisson, HT→FT dynamics, diversification, secular HFA, realised tail, entropy+co-moment. Phase O (2026-06-23): battery of generating models — 5 independent families + the market on the same curve (model independence).

3rd ROUND — EXTERNAL-DATA fronts (canonical football-data.co.uk)

The 1st/2nd rounds exhausted the frozen mirror. These fronts use the canonical football-data.co.uk (data/canonical/, downloaded by analysis/50_fetch_canonical.py), which brings what the mirror lacks: OPENING vs CLOSING odds and pre-2005 depth. VAR (H1) uses the frozen mirror. Provenance of the canonical data in canonical_hash() (stamped in each phase).

Phase D1 — Price discovery: the asymmetry is born at the open (2026-06-23) [canonical]

The mirror has only the close; the canonical brings OPENING odds (Avg*) and CLOSING (Avg*C) for the same match (2019/20–2023/24, 21 leagues, 34,659 matches). A test of the thesis on the TEMPORAL axis of price formation: is the skewness inherited from the structure (present already at the open) or produced by trading (built up to the close)?

The asymmetry is already in the opening price: global skew opening +0.248 → closing +0.249 (within ~1.03 in both); corr(skew_open, skew_close) across 21 leagues = +0.998 [+0.99,+1.00]. Mean Δskew per league +0.0005 (sd 0.0045).
The market refines the MARGIN, not the asymmetry: overround opening 1.0609 → closing 1.0597; favourite Brier 0.2344 → 0.2330 (the close is sharper). But the structural law is present already at the open: corr(skew, p_fav) = −0.866 (open) / −0.874 (close). Favourite drift +0.0001 (no systematic steam).
Conclusion: the asymmetry is not built up by trading — it is in the first price. The close tightens the margin and the calibration and leaves the skewness intact. A TEMPORAL extension of the margin orthogonality (W4/D2, which was across bookmakers).

Artefacts: skewlib/fdcanon.py, analysis/42_open_close.py, outputs/open_close_by_league.csv, outputs/fig/f30_open_close.png.

Phase H1 — VAR: an institutional shock does not move the skewness (placebo) (2026-06-23)

VAR is an institutional shock that does NOT alter the force dispersion of the teams. Staggered difference-in-differences (frozen mirror): leagues adopt in 2018/2019/2020; lower English/Scottish divisions (no league VAR) are the never-treated control. 321 league-years, 10 treated + 6 controls.

NULL effect on the skewness: β=−0.0066 [−0.035,+0.022], p=0.65 (=−0.14 SD of the league, CI includes 0). The favourite win rate β=+0.0024 (p=0.80) and the market p_fav β=+0.0050 (p=0.45) — also null. Event-study with no jump at adoption.
Contrast with COVID (W3): the only shock that moved the skewness was COVID (+0.42 SD, via the HFA drop) — a REAL competitiveness shock. VAR, which does not touch the force structure, leaves the skewness invariant.
Conclusion: a competitiveness placebo — only COMPETITIVENESS shocks move the skewness, not institutional factors. Confirms skewness = f(force dispersion).

Artefacts: skewlib/var.py, analysis/44_var.py, outputs/var_panel.csv, outputs/fig/f31_var.png.

Phase P6 — Pre-2005: the modern regime already holds since ~2000 (2026-06-23) [canonical]

The paper predicts that the league baseline shifts at the regime shocks (Bosman 95, Champions 94/95, revenue ~2003). We extend backwards with WILLIAM HILL — the only bookmaker continuous over 2000–2025 (a consistent bookmaker; skewness is bookmaker-invariant, G1/D2). 158,323 matches, 21 leagues, 2000–2023.

NO break in 2005: the study's ≥2005 cut-off is NOT a regime boundary — PELT finds breaks scattered (2005/2007/2010×2/2014/2016), none common. The per-league STRUCTURE is the same since 2000: corr(pre-2005 baseline, modern) across 17 leagues = +0.76.
LEVEL drift weak and marginal: modern +0.232 vs pre-2005 +0.214 (Δ +0.018); league FE β=−0.019 [−0.037,−0.002] p=0.03, but paired p=0.10 and endpoint-sensitive (with partial 2024 included, p=0.25). The magnitude is well below the between-league variation (sd 0.047) — consistent with INTRA-regime invariance + slow balance evolution, not timelessness.
Honest limit: 1X2 odds only exist since ~2000 — the 1990s shocks (Bosman/Champions) lie BEFORE the odds and remain untestable. We extend the invariance window by 5 years; the paper's full prediction requires pre-odds data.

Artefacts: skewlib/fdcanon.py (WH), analysis/43_pre2005.py, outputs/pre2005_by_league.csv, outputs/fig/f32_pre2005.png.

3rd round (external data) completed (2026-06-23): D1 (opening→closing), H1 (VAR), P6 (pre-2005). 38 phases in the ledger.

4th ROUND — synthesis: SIMILARITY OF ASYMMETRIES (root objective)

The project's original objective was to measure the similarity of asymmetries. All the evidence above becomes a single APPARATUS (skewlib/skewmeter.py) that measures the asymmetry signature of an entity and the distance between the asymmetries of two.

Phase Q — skew-meter: the similarity of asymmetries, operationalised (2026-06-23)

An apparatus that measures, by league/era/market/window: the signature (skew/shape

competitiveness), the RAW distance, the RESIDUAL distance (competitiveness netted out by the closed-form law), the sampling floor and an equivalence verdict. 38 leagues.

Similarity of asymmetries = similarity of competitiveness. The raw distance |Δskew| has a median across leagues of 0.051; netting out competitiveness by 1 parameter (mean p_fav), it falls to a residual 0.023 — 1 number explains R²=0.82 of the variance.
Sufficiency ladder (a new result, correcting the 1st reading): 1 parameter R²=0.82 → 2 moments (mean+variance of p_fav) R²=0.98 → the ENTIRE distribution R²=0.99 (residual = sampling floor). The minimal sufficient statistic is the distribution of p_fav; the mean alone leaves a stable residual (temporal split-half r=0.98), which is the curvature of the law (captured by the 2nd moment), not noise.
A FEW-parameter apparatus: without Shin (inverse-odds, ~0 cost) corr 0.997 · 1 parameter 0.90 · odds-free (W/D/L only) 0.83. Real-time convergence: SE 0.026 in 200 matches, 0.018 in 400 (< between-league sd 0.051 → already ranks).
Verdict by EQUIVALENCE (TOST), not significance (with huge n everything rejects): margin ½·sd=0.026. E0 vs E3 (Premier League vs the English 4th division): raw skew 0.167 vs 0.294 (very different) → residual 0.009 → EQUIVALENT (competitiveness explains 93%). E0 vs SP1 equivalent; N1 vs I2 and BRA vs ARG distinct (a real residual).
Conclusion: "how similar are the asymmetries of A and B?" reduces to "how similar are the competitivenesses of A and B?" — measurable with 1 number, in real time, even without odds. The distribution of p_fav is what makes the reduction EXACT.
Hardening (rigour): SE by season block-bootstrap 0.0070 vs i.i.d. 0.0042 (×1.7, intra-year dependence); Mahalanobis shape distance (skew+ exkurt) corr 0.74 with the scalar; the law out-of-sample (calibrate even years, predict odd) R²=0.80 ≈ in-sample 0.82 — the residual ruler is not overfit.

Artefacts: skewlib/skewmeter.py (measure, distance, residual, sufficiency_ladder, tost, skew_se_block, shape_distance, law_oos_r2), analysis/45_skewmeter.py, outputs/fig/f33_skewmeter.png. Product: interactive widget site/src/components/ SkewMeter.astro (gauge + law/residual + similarity) in section §07 of the site, with data exported by export_site_data.py (blocks skewmeter/convergence). Paper §4.8.

Phase R — Bet type: the law mirrored across the whole book (block 46)

Each match offers three two-point bets: favourite (argmax p), draw (outcome D) and longshot (argmin p). Measuring the pooled ex-ante skewness of the three per league (38 leagues, market odds):

All positively skewed (lottery-like): global skew +0.236 / +1.294 / +2.349 (favourite / draw / longshot). The FLB is a single-bet phenomenon, everywhere.
All three governed by competitiveness, in opposite directions: less balance (p_fav ↑) lowers the favourite's skew (corr −0.90) and raises that of the draw and the longshot (corr +0.95 / +0.91), which become bigger longshots. It is not a law of the favourite — it is the SAME structural law, mirrored across the whole book.
Diversification (block 37, now exposed): the skew of the mean return of N bets ≈ skew/√N. A favourites portfolio turns Gaussian in ~6 bets; the longshot's survives up to ~509 — which is why the bias bites the recreational bettor, not the syndicate.

Artefacts: skewlib/exante.py (fav_dog_draw, bettype_by), analysis/46_bettype.py, outputs/fig/f34_bettype.png. Product: "Every side of the book" + "What survives diversification" panels in the SkewMeter.astro widget; Mahalanobis shape column (shape_cov_inv). Service: API study/api/ (POST /measure, modes with-odds/odds-free, monitor /integrity). Data in export_site_data.py (blocks bettype/diversification).

5th round (product) (2026-06-23): bet-type — the law skew=f(competitiveness) holds across the whole book (favourite/draw/longshot), with diversification ~6 vs ~509. The apparatus became a product: API /measure (with-odds + odds-free + integrity) and a V2 widget (bet-type radar + diversification curve + Mahalanobis). 40 phases in the ledger. Remaining frontiers: other sports (decision: football exclusively) and pre-2000 odds (non-existent). Lineage in lineage.json/LINEAGE.md.

Phase S — External validity: tennis (block 48)

The CANONICAL layer (skewlib/canonical.py + adapters/) made the core sport-agnostic — it only needs (p, o, won) per bet. Plugging in tennis (tennis-data.co.uk, ATP+WTA 2005–2025, frozen snapshot in data/PROVENANCE-tennis.json), a sport with a 2-outcome market (no draw) and an independent odds source, over 62,865 matches and with ZERO new science:

Calibration: mean p_fav 0.688 ≈ real favourite win 0.692 — the de-vig is reliable beyond football.
The law reappears: the favourite's skew is more negative where the tournament is more imbalanced — corr(skew_fav, p_fav) by tier = −1.00 (ATP) / −0.98 (WTA) (football −0.90). Grand Slam (more imbalanced) has the most negative favourite in both tours.
The longshot is lottery-like: skew +2.314 ≈ football +2.349.

The structural invariance is not an artefact of 1X2 nor of football: it is a property of the SPORT as a competitive system. External validity for §7 (the "one sport" limitation).

Artefacts: skewlib/adapters/tennis.py, analysis/00b_fetch_tennis.py, analysis/48_tennis.py, outputs/fig/f35_crosssport.png, outputs/tennis_by_tier.csv. The core (canonical/skewmeter) unchanged.

6th round (external validity) (2026-06-23): tennis — the law skew=f(competitiveness) and the lottery-like longshot reappear in a 2nd sport (ATP+WTA, 2-outcome market, independent odds), via the canonical layer without changing the core. 41 phases in the ledger. Adding a sport = one adapter (docs/DATA-SCHEMA.md). Lineage in lineage.json/LINEAGE.md.

Phase T — External validity: basketball (block 49)

A 3rd sport on the same canonical layer, without touching the core. Plugging in the NBA (sportsbookreviewsonline.com, 16 seasons 2007–08…2022–23, frozen snapshot in data/PROVENANCE-basketball.json), the moneyline market (2 outcomes, no draw) and an odds source independent of football and tennis, over 19,621 matches and with ZERO new science:

Calibration: mean p_fav 0.694 ≈ real favourite win 0.685 — the de-vig is reliable also in the NBA moneyline (the ~0.9pp gap is the favourite-longshot bias itself, with the favourite slightly overpriced).
The law reappears: the favourite's skew is more negative in the more imbalanced seasons — corr(skew_fav, p_fav) by season = −0.95 (16 seasons; football −0.90, tennis −1.00/−0.98). The most lopsided NBA (2007–08, p_fav 0.71) has the most negative favourite; the most balanced (2022–23, p_fav 0.66) the least.
The longshot is lottery-like: skew +2.609 ≈ football +2.349, tennis +2.314.

On the cross-sport curve (f36), tennis and basketball overlap in the same lopsided region (~0.66–0.75), both on the descending favourite curve and the ascending longshot curve that football traces at lower competitiveness. Three sports, three markets (1X2, match-odds, moneyline), three odds sources, one structural law.

Artefacts: skewlib/adapters/basketball.py, analysis/00c_fetch_basketball.py, analysis/49_basketball.py, outputs/fig/f36_crosssport.png, outputs/basketball_by_season.csv. The core (canonical/skewmeter) unchanged; no new dependencies (the fetch uses the stdlib html.parser).

7th round (external validity, 3rd sport) (2026-06-23): basketball — the law skew=f(competitiveness) and the lottery-like longshot reappear in a 3rd sport (NBA, moneyline market, independent odds), via the canonical layer without changing the core. 42 phases in the ledger. f36 puts football+tennis+basketball on a single curve. Lineage in lineage.json/LINEAGE.md.

Phase U — Temporal equivalence: "no drift" as a positive claim (block 51)

W3 showed β_year≈0 with p=0.73 and a CI crossing zero. But a high p is only non-rejection of a trend — not evidence of absence (an underpowered test also gives a high p). Here we close the logical gap in §4.3 with an equivalence test (TOST):

Pre-registered margin Δ = ½ the between-league SD (0.026), the SAME as in §4.8 — read as the largest 20-year drift we would treat as negligible.
Result: the 90% CI of β falls entirely within [−Δ,+Δ] → we reject any drift larger than Δ, p_tost = 0.006. Robust to per-league bootstrap (p=0.005) and to the balanced panel (fixed basket of 15 leagues; β=−0.0003/year, p=0.043).
Sensitivity: equivalent at ½ and 1× the SD; inconclusive only at a severe Δ of ¼ SD (drift < 0.013 over 20 years — below that the data neither affirms nor denies).

Conclusion: the skewness is not merely not shown to drift — it is statistically equivalent to not drifting, within a fraction of a single between-league difference accumulated over the whole history. It converts the temporal invariance from absence-of-evidence into evidence-of-absence.

Honest check (γ, C2): the same TOST applied to the preference parameter γ (annual series of 21 points, Δ=½ the between-league SD of γ ≈ 0.020) comes out INCONCLUSIVE (β_γ=+0.0003/year, drift +0.006 over 20yr, p_tost=0.19): the point drift is small, but the annual series lacks the power to certify equivalence at that margin. We record this deliberately — the test is not rigged to pass; only the skewness (n=638, league×season panel) has the power for the verdict. That is why the paper claims equivalence only for the skewness; the other "no drift" results (P4/FLB, C2/γ, L/HFA) stand as non-rejection.

Artefacts: skewlib/stats.py:tost, skewlib/panel.py:trend_boot, analysis/51_temporal_equivalence.py, outputs/fig/f37_temporal_equivalence.png.

8th round (inferential rigour) (2026-06-23): temporal equivalence — the "no secular drift" (W3) becomes a positive claim by TOST (p_tost=0.006, margin ½ between-league SD; robust to bootstrap and balanced panel). 43 phases in the ledger. f37 (forest) enters as Figure 20; §4.3 and the abstract updated.

Phase V — The third observer: player-informed p (Front O, block 55)

The /odds triangulation showed two observers — results-only Elo and a real bookmaker — landing on the same curve. Phase V adds a third with a disjoint information set: team strength = log market value of the actual starting lineup (Transfermarkt, CC0 mirror; docs/SPIKE-O0.md), mapped to (P_A,P_D,P_H) by the same MNLogit link the Elo observer uses (const+z+z², z = log value ratio), fitted on 2013–2018 and evaluated on 2018–2025 — the state never reads a result or a price; results touch only the low-dimensional link.

Three engines price the identical favourite bet at fair odds 1/p on the same 17,356 evaluation matches (top European leagues; the TM source has no valued lineups for the non-European first tiers). Bucketed by each observer's own p_fav:

observer	information set	Brier	calib \|p−f\|	shape RMSE	corr law
elo	results only	0.234	0.016	0.005	+0.997
book	the market	0.230	0.012	0.006	+0.994
players	player values only	0.235	0.032	0.005	+0.994

The decomposition is the finding. The three are not equally good forecasters — the market is sharpest, the player observer (which never saw a result) is the most miscalibrated, 2.7× the book's level error. But the shape residual — realised skew minus the law evaluated at the bucket's realised frequency — is ~0.005 for all three, indistinguishable from zero. Calibration differs freely; the shape does not. Knowing more moves an observer along the law, never off it. This is the strong form of observer-invariance: the law holds for ANY p-generator, including one built from a completely different way of knowing the game.

WC overlay: each ledger fixture also carries the players' reading (log squad- value ratio through the club-fitted link, values frozen in wc_squad_values.json). On the three resolved R32 games with book odds the players observer was the most seduced by Germany (p=0.790, skew −1.42 — the loudest catastrophic-tail warning of the three; Germany fell on penalties).

Artefacts: skewlib/players.py (+ committed skewlib/data/{tm_club_map, value_link}.json), analysis/55_third_observer.py, outputs/third_observer_buckets.csv, site /odds §"The third observer", wc_squad_values.json. Tag evidence/observer-3.

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