We have determined the asymptotic number of landmarks needed to
identify edges of subdivided complete graphs from unnamed distance
reports. For each fixed \(L\ge3\), this number has order
\(n^{1+1/\lfloor L/2\rfloor}\), whereas named reports require only
\(\Theta(n)\) landmarks. The two shortest lengths are also settled
for every \(n\ge3\): no selection resolves \(S_2(K_n)\), while
\(S_3(K_n)\) has a resolving set of size at most \(\binom n2\).
These results extend the earlier finite computations to sharp
asymptotic bounds and identify a change in existence between
lengths two and three.

The proofs explain this behavior through the information retained
by endpoint counts. Counting the possible vectors gives the lower
bound, while realizing suitable labels as degrees or matrix margins
gives the matching constructions. At odd lengths, Sidon labels
identify endpoint pairs, and the projected kernels describe the
changes that a noncentral edge can hide. The quantitative results
also show that the freedom in these profiles matters: for fixed
\(L\ge5\), using only the first two depths has cost
\(\Theta(n^{3/2})\), and symmetric profiles at length \(2k+1\),
with fixed \(k\ge3\), have cost \(\Theta_k(n^{1+2/k})\).
The constructions uniform in \(L\) and the probabilistic bounds
complement this picture with explicit finite existence ranges.

Two questions follow naturally. At length three, does
\(\edimm(S_3(K_n))/n^2\) converge, and what is its limiting value
if it does? The current bounds place such a value between
\(c_*=0.080208390067\ldots\) and \(1/2\), leaving room for sharper
counting and more economical matrix constructions. A second question
is to classify the finite pairs \((n,L)\), with \(L\ge4\), that
admit a resolving set. The criteria for realizing endpoint profiles
and the exact certificate methods provide concrete ways to
investigate the orders below the sufficient thresholds obtained here.