Let \[ \gamma_8=\max_H\frac{\operatorname{TSP}(d_H)}{\operatorname{SUBT}(d_H)}, \] where \(H\) runs through connected simple eight-vertex graphs and \(d_H\) is its shortest-path metric. The current certified interval is \[ \boxed{1\leq\gamma_8<\frac32-10^{-36}}. \] The lower endpoint is attained by \(H=C_8\). Its metric tour optimum and subtour optimum are both 8, as checked by `gmstg8-artifact-cycle-eight-certificate`.

Karlin, Klein, and Oveis Gharan prove that there is an \(\epsilon>10^{-36}\) such that every metric TSP instance has a tour of expected cost at most \((3/2-\epsilon)\) times the subtour-LP optimum. Every graph metric in this finite family is a metric TSP instance, so their theorem gives the displayed strict upper bound.

This interval is deliberately conservative. It records only bounds whose hypotheses and certificates have been checked. The exact maximum requires the isomorph-free LP sweep described below.

If \(H\) has diameter at most two, every off-diagonal entry of \(d_H\) is 1 or 2. It is therefore an eight-city 1,2-TSP instance.

Qian, Schalekamp, Williamson, and van Zuylen generated the nonisomorphic cost-one graphs with nauty and solved the subtour LP and tour integer program. Their Table 1 reports that the largest eight-city 1,2-TSP ratio is \[ \frac{9}{8.5}=\frac{18}{17}. \] Consequently every diameter-two graph metric in the present problem has ratio at most \(18/17\). Their worst 1,2 cost matrix need not be a shortest-path metric, so the table supplies an upper bound for this subclass rather than an incumbent for the present maximum. Any graph with ratio greater than \(18/17\) must have diameter at least three.

Held and Karp's 1970 paper develops the classical LP lower-bound framework for symmetric TSP. Karlin, Klein, and Oveis Gharan give the current general metric bound used here. Qian, Schalekamp, Williamson, and van Zuylen report an isomorph-free computation for eight-city 1,2-TSP instances.

The 1,2 computation does not settle this question. Graphs of diameter three or greater produce shortest-path distances above two, while an arbitrary 1,2 cost matrix need not equal the shortest-path metric of its cost-one graph. A focused search for `graphic TSP`, `subtour LP`, `eight vertices`, `graph metric`, and `integrality gap enumeration` found no source stating the maximum over connected eight-vertex graph metrics.

For each graph6 record, the exact computation should perform these steps:

1. Decode the graph and compute its 28 integer shortest-path distances. 2. Enumerate the \(7!/2=2{,}520\) unoriented Hamilton cycles to obtain the exact tour value and a minimizing order. 3. Solve the subtour LP with eight degree equations and one representative of each complementary pair of nontrivial cut inequalities. Preserve a rational primal optimum and a rational dual optimum. 4. Verify both certificates with integer arithmetic after clearing denominators. Record every tied maximizer. 5. Recompute each maximizing graph's canonical graph6 label with an independent nauty call.

The diameter-two cases can be pruned against the sourced \(18/17\) upper bound once an incumbent exceeds it. Bridges and articulation decompositions offer further reductions, though each reduction needs a proof for this exact degree-constrained metric LP.

This record stops before the 11,117 LP solves. It makes no claim that the cycle incumbent is optimal.

Every nonzero distance in the \(C_8\) metric is at least one. Summing the eight subtour-LP degree equations gives \[ 2\sum_e x_e=16, \] so every feasible LP solution has \(\sum_e x_e=8\) and objective value at least 8. The incidence vector of the cycle is feasible and costs 8. It is therefore an optimal LP solution.

The program checks all \(7!/2=2{,}520\) unoriented tours and finds tour optimum 8. It also checks the degree equations and all 254 proper nonempty cuts for the cycle vector. The canonical report has SHA-256 digest `711b63f56d7591266ddd566d29632e6fd92bf785857c05022f193250b135fbd1`.

Running `geng -cq 8` with nauty 2.6r6 produced 11,117 newline-terminated graph6 records. The resulting file has SHA-256 digest `37010dfb9ca35c86bcbfd488c3e4cadcb3e918dc8c6acebd81ea966e79c35a84`. Brendan McKay's official graph-data page independently lists 11,117 connected isomorphism classes on eight vertices.

This manifest fixes the finite input to the eventual universal sweep. The digest is version-and-order specific. A sweep using a newer nauty release may order the same isomorphism classes differently, so it should also verify the class count and canonicalize each reported maximizer.