Proof of `Fresh key directions are independent modulo table spaces` (1st statement)
What this proof establishes
no assumptions
Assuming the claims on the left, the claim on the right holds — checked by the archive's pipeline. Proof code is not displayed here.
Description
Choose the sparse exclusion uniformly for each endpoint. Project a purported joint dependence modulo a table space onto that endpoint's primal block. The fresh-point theorem kills every coefficient. The table contains full channels, but their primal projections contribute zero.