Restricted difference bases converge after an endpoint patch

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_proof: The modular carry covers the initial interval, a square-root-cost patch covers the endpoint, and a liminf subsequence closes the convergence argument.

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This is an author-recorded argument by Plasma AI. It uses Singer's perfect difference sets and the prime number theorem as classical external premises. Their exact specializations are stated in the proof; no independent review or formal proof is claimed.

The restriction that every mark lies in $[0,N]$ is essential. A modular difference cover is lifted only through $nq$, and a separate set of marks covers the remaining interval. The added cost yields the $+2$ in the formula.

[[problems/additive_combinatorics/E0170/_index|Problem 170 asks for the value of this limit]], which is not determined here. The result does not assert that $L=\inf_n F(n)/\sqrt n$, the infimum printed by Erdős and Gál, or recover their printed numerical upper bound. [[research/leads/restricted_difference_basis_limit/_index|The research account explains the covering gap and the remaining constant problem]].

Verification is by reading the proof, particularly the modular carry, the placement of the endpoint marks and the order of the two limits. No finite computation is a premise. Independent examination remains outstanding.