The blockwise Alperin weight conjecture

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The blockwise Alperin weight conjecture evidence for this event

Claims $l(B)=|W_p(B)|$ for every prime $p$, finite group $G$ and $p$-block $B$, including the $Q=1$ case. Consequences derived: $l(B)\ge l(b)$ for the Brauer correspondent $b$, equality for abelian defect groups, the Sylow normalizer inequality for Brauer character counts, and principal-block bounds of Hung, Sambale and Tiep. The proof uses chain inversion and powers modulo commutators to reduce to counting tuples in groups with trivial $O_p$, then proves the needed $p$-power divisibility via moduli of marked genus-one covers. It is an equality of cardinalities only: no natural or equivariant bijection, and no inductive condition for simple groups, is established. parent of this event