We give a polynomial-time construction for a sixth-power excluded-grid theorem. Every graph whose treewidth is at least $g^6$ times a fixed polylogarithmic factor contains a $g\times g$ grid minor that our randomized algorithm finds with constant probability. The algorithm outputs an explicit minor model, and the exponent in its running time is independent of $g$. This improves the ninth-power existence bound of Chuzhoy and Tan~\cite{CT21} and the $g^{98}$ threshold in the polynomial-time construction of Chekuri and Chuzhoy~\cite{CC16}. Quantitative bounds relating treewidth to grid size are central to bidimensionality, developed by Demaine, Hajiaghayi, and collaborators~\cite{DH05,DH08}. The main ingredient is an asymmetric local routing theorem that uses substantially fewer attachment paths than backbone paths. A balanced recursion preserves connectivity by charging deletions to the boundary of a fixed reference cluster. We construct the smaller attachment sets through a routing hierarchy: either it exposes an accessible well-linked boundary, or its quotient graphs contain the desired grid. These arguments implement cluster splitting using approximate cuts and flow computations. An explicit cut-matching construction and a constructive grid-extraction procedure complete the algorithm. We also give a general packing--covering hypothesis for trees meeting three prescribed terminal sets under which the exponent can be reduced below six. A companion Lean development verifies the new construction, its success guarantee, and the composition of its polynomial running time, using explicit interfaces for the published algorithms and standard finite-data operations.

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