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Why a valuation-selected label coordinate makes the kernel local

Fix a nonzero interpolation grid point t with i=v_2(t) smaller than a, and group vertices on both sides by label coordinate i. In a packet, W_xy(t) is zero unless y-x=t. Adding an integer of valuation i flips bit i, so a surviving pair has opposite slot bits there. Under the packet orientation convention, opposite bits mean the row and column symbols agree when the coordinate has unequal symbols. If its symbols are equal, they already agree. Hence every surviving interaction at T=t stays within a group. This uses coordinate i, not an arbitrary graph partition.

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Updated 2026-11-08

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Math