Sieve dimension and search depth for the Erdős-Straus conjecture, $n \equiv 1 \pmod{24}$
For primes $n\equiv1\pmod{24}$, we study the depth of explicit, factorization-free searches for a decomposition of $4/n$ into three positive unit fractions. Let $E_2(N;J)$ be the set of such primes $n\le N$ for which no coprime pair $1\le u,a\le J$ admits a divisor $s\mid an+u$ with $s\equiv-1\pmod{4ua}$. For every fixed $J$, we prove $|E_2(N;J)|\ll_J N/(\log N)^{1+A(J)/2}$, where $A(J)$ counts the coprime pairs in $[1,J]^2$. The proof uses a fixed-point-free involution and identifies the exact dimension of the covering. We also establish a dual Type I divisor criterion and an unconditional obstruction: at the fixed shift $c=7$, both branches of the cofactor criterion fail for $\asymp N/(\log N)^{3/2}$ primes. At growing depth, an exact reciprocal-totient asymptotic with leading constant $9/(2π^2)$ improves the explicit large-sieve bounds to $O(N\exp(-0.0942(\log N)^{2/3}))$ at cutoff $p>J^5$, and to $O_\varepsilon(N\exp(-(0.1093-\varepsilon)(\log N)^{2/3}))$ at cutoff $p>J^{4+\varepsilon}$ for fixed sufficiently small $\varepsilon>0$. Finally, combining a truncated Type I pass with a sparse residual search yields a complete, factorization-free batch decision procedure for all primes in $[N,2N]$, using $O(N^2(\log N)^2\exp(-c_0(\log N)^{2/3}))=o(N^2)$ arithmetic operations for some $c_0>0$. The conjecture itself remains open.