No formal Labour leadership challenge mechanism will be triggered by July 4, 2026 — Andy Burnham will not publicly declare a leadership candidacy, fewer than 20 Labour MPs will submit letters to the party chair calling for a contest, and no formal no-confidence motion will be tabled in the PLP.

Step 1 — direction: all four frameworks independently predict no formal trigger, for distinct mechanistic reasons. This convergence across analytical traditions with different ontological commitments is a strong confidence signal. Step 2 — base rate: formal Labour leadership mechanisms triggered within 14 days of a single by-election loss, absent co-occurring forcing event, is effectively zero in the post-1980 record; I set base rate at 0.05 to allow for novel-situation uncertainty. Step 3 — Bayesian update from framework evidence: the Institutionalist analysis provides a near-decisive structural fact (Burnham’s non-MP status eliminates the candidacy-declaration trigger almost entirely for this window); the Austrian framework specifies precisely why the coalition-assembly discovery cycle cannot complete in 14 days; the Keynesian framework explains why aggregate discontent fails to become effective formal action without a focal-point catalyst; the Marxist framework, while explicitly weak at 14-day horizons, confirms the apparatus was redesigned to prevent bottom-up formal challenges. Together these push the no-challenge probability from the 0.95 base to approximately 0.88, reserving 12% for cascade scenarios (ministerial resignations, second major poll shock) that could compress the discovery cycle within the window. Step 4 — tradition weighting: Institutionalist carries the highest weight (0.36) because it identifies the most operationally precise barrier (Burnham’s constitutional status); Austrian and Keynesian share equal weight (0.27 each) as complementary coordination-failure accounts; Marxist carries minimal weight (0.10) given the framework’s own acknowledgment of limited short-horizon predictive power.