# Verification report

The manuscript proves equality of weave cluster positivity and Bao-He geometric positivity on the same standard-pinned flag-chain braid variety for every positive word of finite crystallographic simple type. The accepted proof is analytic; finite computations provide regression checks.

## Analytic audit

The positive-component argument uses an actual acyclic cluster-localization cover with inverted frozen variables. Nonnegative limits cannot acquire a zero mutable variable in an acyclic seed: a source or sink in the zero subquiver makes one exchange monomial strictly positive. Frozen variables stay nonzero. This gives the closedness needed in addition to the positive chart's connectedness and openness.

The root-subgroup stabilizer fixes every flag of the recursively constructed chain. Ascents use conjugated stabilizer inclusion, and descents use the full Chevalley commutator root string, including non-simply-laced terms. The rank-one limit is simultaneous for the entire chain, not an inference from individual flag positivity. The all-one seed calculation uses the stated CGGLSS recursion and its finite-type extension. Bao-He supply the geometric connected-component theorem. A common point therefore identifies the two components.

The closure discussion distinguishes the finite affine chart from the compact flag-chain space. The rank-one raw-coordinate example is attributed to the upstream research attempt. The SL3 word 121121 example includes the global equation, frozen units, positivity and six real components.

## Exact regression checks

`verify_common_point.py` performs 299,141 exact integer or rational checks on 1,742 type-A chains and finite root-system calculations. It tests endpoints, root inclusions, whole-chain stabilizers, rank-one identities and an inverse-lift negative control. Root coverage includes A1-A5, B2-B3, C2-C3, D4, F4, G2, E6, E7 and E8. A5 and E6-E8 Weyl groups are sampled; the saved report identifies exhaustive cases.

`verify_rank_two.py` performs 1,063 checks of the SL3 word 121121 identities, including coefficientwise Laurent-polynomial identities, inverse charts and rational specializations.

The runner totals 300,204 checks. Ordinary and optimized executions must agree byte-for-byte with the supplied reports. Optimized mode is propagated to each subprocess; the checkers use explicit failures rather than relying on removable Python assertions.

## Limits and presentation

These are two mathematical checkers, each run in two modes, not two independent implementations of the general proof. No proof-assistant verification or independent review is claimed. The topological limit and the imported cluster construction are not mechanically verified. The broader AIM source record remains partial.

The standalone source compiles to nine pages. The final PDF was visually inspected for legibility, mathematical layout, footnote placement and references. Author contact and affiliation appear in a footnote; AI assistance is disclosed in the introduction. The bibliography contains eight attributed sources. No absolute novelty or priority certification is claimed.