PBW and bubble normal form in the minimal-parabolic diagram category

Finite spanning and the downward corner. The construction in [F3] first counts intersections, clears an innermost monogon/bigon by its finite disk sweep and uses the typed curl/cancellation and KLR equations. A correction may add extrema, but it removes crossings and hence enters a smaller primary induction value. After intersections are reduced, triangles remove extra extrema, finite coefficient slides move bubbles left, and reciprocal coefficients are finite polynomials in the positive family. The output is precisely the reduced matching, open-dot and far-left bubble family of the Statement. With all boundaries downward, each open component is a colored permutation strand and the reduced crossings and dots are the chosen KLR PBW family [F2]. Thus the claimed downward tensor product follows once independence is established.

Independence and joint faithfulness. The actual flag witness of [F3] assigns distinct free framing roots to the open components of a maximal crossing-number matching. Its leading root-label transport has a nonzero crossing/cup/cap coefficient; any other term deletes a crossing, and a different maximal matching imposes different endpoint equalities. Unused complementary framing roots make the finitely many bubble coefficients a unit triangular independent family over the component roots. This remains valid when spectator factors are inseparable: only free framing roots enter the selected flag summands. For a downward PBW combination the same construction has the concrete fixed-source-weight degree witness: take background count $r$ increasing, total degree $N=h+2r$ and framing degree $N-A_{mathrmspect}$; all tested flag relations then have degree greater than the fixed nonzero polynomial output. Every finite nonzero normal-form combination therefore survives in an actual flag model. This proves independence at arbitrary boundaries and joint faithfulness, with the normalized $H_i$ and word shifts already fixed by [F1]. No unrelated cyclotomic quotient or Frobenius conclusion is used.

The family spanning in step 1.1 is independent by step 2.1, so it is a basis and its all-downward corner is the stated KLR PBW corner tensored with the polynomial positive-bubble ring. Bubble inversion expresses the opposite family by its reciprocal, and the finite straightening has unique output coefficients because this basis is independent. The assertions hold in every characteristic for all the admissible homogeneous parameters and concern the formal category before quotient transfer.