Write \(R_{10}\) for the requested maximum. The current certified interval is \[ \boxed{\frac{18}{11}\leq R_{10}\leq\frac{11}{5}}. \] The lower endpoint comes from the graph with edges \[ 02,03,17,18,19,23,25,26,27,45,48,56,78,89. \] Its deterministic nearest-neighbor order is \[ 0,2,3,5,4,8,1,7,6,9, \] with leg lengths \(1,1,2,1,1,1,1,2,4,4\), hence total length 18. Exhaustion of all \(9!/2=181{,}440\) unoriented tours gives optimum 11.

For the upper bound, let \(O\) be the optimal tour length and associate to every vertex the length of its outgoing edge in the nearest-neighbor tour. Sort these ten integer lengths as \(l_1\geq\cdots\geq l_{10}\). The proof of Lemma 1 in Rosenkrantz, Stearns, and Lewis gives \[ O\geq2\sum_{i=k+1}^{\min(2k,10)}l_i\qquad(1\leq k\leq10), \] and each \(l_i\leq O/2\). These facts hold for arbitrary tie resolution, so they cover the specified label rule.

Every distinct pair in a connected unweighted graph has positive integral distance. Thus \(O\geq10\). Doubling a nine-edge spanning tree and shortcutting its Euler tour gives \(O\leq18\). The exact artifact enumerates every nonincreasing positive integer sequence with \(l_1\leq\lfloor O/2\rfloor\) for each integer \(O\) between 10 and 18. The largest permitted sums are respectively \[ 22,22,26,26,30,30,34,34,39. \] The largest quotient is \(22/10=11/5\). This integer relaxation covers every graph in the stated family, although some sequences admitted by it may have no graph realization. Closing the interval requires a graph-level exclusion certificate.

Rosenkrantz, Stearns, and Lewis allow arbitrary tie resolution, so their theorem applies directly to the smaller-label rule in this problem. Their paper proves the general logarithmic guarantee and constructs asymptotic bad examples. It does not tabulate the worst ten-city unweighted graph metric.

A focused search for the exact phrases "nearest-neighbor graph metric" and "ten-city nearest neighbor", together with searches around the 1977 theorem, found no source settling this finite labeled maximum. This is a dated status check rather than a novelty proof.

An exact computation can enumerate connected graph realizations while deduplicating equal distance matrices. Since labels control tie resolution, ordinary unlabeled-graph reduction is insufficient by itself. A valid symmetry reduction must retain vertex 0 and transport the full label order. For each remaining metric, breadth-first search determines the distances, the label rule determines the nearest-neighbor tour, and Held-Karp determines the optimum. A final certificate should publish the metric hashes, the number rejected at each canonicalization stage, and every maximizing labeled orbit. This record stops at the universal integer relaxation.

The program builds the graph directly from its fourteen edges and computes all-pairs distances by breadth-first search. It then applies the stated nearest-neighbor rule, including the smaller-label tie break.

For the optimum it fixes vertex 0, identifies a tour with its reversal, and checks the remaining \(9!/2=181{,}440\) orders. The nearest-neighbor length is 18 and the optimum is 11. There are four optimal unoriented tours. The canonical report has SHA-256 digest `a1d0820ab4475898cee1a7eea91a8e73cd82003b4031aea2dd1c8f2e84f93801`.

For each \(O\in\{10,\ldots,18\}\), the program enumerates every nonincreasing sequence of ten positive integers satisfying \(l_1\leq\lfloor O/2\rfloor\). It retains precisely those sequences that satisfy \[ O\geq2\sum_{i=k+1}^{\min(2k,10)}l_i \] for every \(k\). Any nearest-neighbor tour in the target family supplies such a sequence.

The nine maximum sums are 22, 22, 26, 26, 30, 30, 34, 34, and 39. Dividing by the corresponding values of \(O\) gives a largest relaxed ratio of \(22/10=11/5\). The program checks 106,678 candidate sequences with exact integer arithmetic. Its canonical report has SHA-256 digest `fefff78fe87f7c59f52986f6366ec0a11746ecc9dbc1c0ea3fd0d7b38d843a97`.