A Common Positive Point for Braid Varieties

We compare two positive subsets of a braid variety: the positive locus of its weave cluster structure and the geometric positive locus defined by the joint nonnegative closure in a twisted product of flag varieties. For every positive braid word of finite crystallographic simple type, with a fixed standard pinning, these subsets agree on the same flag-chain variety. We construct a point at which every variable of the left-inductive seed is one. A root-subgroup stabilizer fixes its entire flag chain, allowing a simultaneous rank-one positive limit that puts the point in the geometric locus. An elementary connected-component criterion for locally acyclic cluster algebras with inverted frozen variables, together with Bao-He's geometric component theorem, proves the equality. We also distinguish affine from compact closures and give explicit small-rank examples, including a six-component real braid variety. The comparison concerns positive loci, not an identification of two cluster atlases or a classification of all real components.

This is a scoped contribution to the positivity question recorded as AIM-GEOMETRY-0038 in UnsolvedMath v1.6.0, not a claim to resolve every interpretation of the broader AIM research agenda. Existing cluster constructions and geometric component theorems retain attribution. The rank-one raw-coordinate example is credited to the dataset's accompanying research attempt. No absolute priority claim is made.

AI-assisted, self-audited and unrefereed preprint. No independent review or formal proof-assistant verification is claimed. Two portable exact-arithmetic checkers accompany the source; their 300,204 checks per execution test specified finite domains and do not replace the general written proof.