Endpoint patches for restricted difference bases

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Problem 170 asks for the value of the limit of $F(N)/\sqrt N$, where the marks of the difference basis must all lie in $[0,N]$. The Erdős–Gál covering argument lifts a small ruler by a Singer difference set modulo $q=p^2+p+1$. Its modular carry proves coverage through $nq$. The remainder written as $M=N-(n+1)q$ does not measure the entire interval still requiring coverage. This is a defect in the printed argument, not in its transcription.

L38 supplies the missing coverage using $\Delta=N-nq$ and at most $2\sqrt\Delta+4$ extra marks. It proves, at author-recorded tier 0,

$$\lim_{N\to\infty}\frac{F(N)}{\sqrt N} =\inf_{n\ge1}\frac{F(n)+2}{\sqrt{n+1}}.$$

Singer's theorem and the prime number theorem are exact external inputs, used at statement level. The proof keeps the mark restriction, handles the carry at $nq$, and takes the large-$N$ limit before varying $n$.

The exact value remains open in this account. In particular, this deduction does not prove the printed infimum $\inf_nF(n)/\sqrt n$, the numerical upper bound derived from that different formula, or the conjectural value $\sqrt3$. A route to the exact constant needs a matching asymptotic lower bound or a construction below the conjectured value. Repeating the convergence argument or extending a finite ruler table does not supply either.

The useful bounded check is examination of L38's carry, endpoint placement and limiting order. Reopening the constant problem requires a new estimate that addresses that value, rather than a further convergence calculation.