Document H: The Phase-Velocity Problem (PVP(H)) and the Open Integral H-R3 on the Geometric Component of the A-polynomial of the Figure-Eight Knot 4₁

Document H continues the STKWC Preprint Series by formulating the Phase-Velocity Problem (PVP(H)) and identifying the remaining open integral H-R3 = ∫_γ v du along the geometric component of the A-polynomial of the figure-eight knot 4₁ from the complete structure to the (5,2)-surgery. Using the canonical non-abelian projection pipeline (Theorem 18 of Document F) and the Kirk–Klassen three-term surgery formula, we show that the topological filling correction Δ_top = 3/20 is algebraically forced if and only if the integral I[γ] vanishes. The integral remains an open computational problem (H-R3) due to branch-cut singularities and convergence failure of floating-point Chern–Simons algorithms in SnapPy. Part of the STKWC Preprint Series • Document H v1.0 This preprint was prepared together with Document F but uploaded sequentially to allow initial visibility of the renormalization framework. *Author identifiers: ORCID 0009-0002-1330-7530 | Bluesky @yafffesh.bsky.social*