Let \(C_8\) be the largest cover time under the convention in the problem. The present certified interval is \[ \boxed{1282\leq C_8\leq3136}. \]

The lower bound comes from the row-major rotor vector in the replay record. Start at vertex 56. Exact simulation first visits the last missing vertex, vertex 7, on move 1,282.

For the upper bound, replace each of the 112 undirected grid edges by its two orientations. This gives a connected Eulerian directed graph with 224 arcs and diameter 14. Theorem 6.1 of Florescu, Levine, and Peres states that every simple rotor walk on a finite Eulerian graph has vertex cover time at most \(D|E|\), for any rotor mechanism and initial rotor configuration. Therefore \[ C_8\leq14\cdot224=3136. \] The problem's clockwise order is one of the mechanisms covered by that theorem. Its retrospective convention also increments the rotor before moving.

Holroyd and Propp describe the retrospective rotor convention on pages 2 and 3: the rotor advances to the next arc and the particle follows that arc. This matches the problem after each boundary vertex receives its shortened clockwise neighbor list.

Florescu, Levine, and Peres prove \(t_{\mathrm{vertex}}\leq D|E|\) for every finite Eulerian directed graph, arbitrary rotor mechanism, and arbitrary initial configuration. Their proof decomposes the walk into excursions. Applied to the bidirected grid, it gives 3,136. They credit Yanovski, Wagner, and Bruckstein with the earlier edge-cover bound \(2D|E|\). Friedrich and Sauerwald give other general vertex and edge cover-time techniques and work out several graph families.

Focused searches used `rotor-router cover time grid`, `clockwise rotor walk finite square`, and the exact dimensions 8 by 8. The sources found asymptotic grid results and general finite-graph bounds. None supplied the exact extremum under this boundary-order convention. Novelty of the finite question remains unverified.

A deterministic C++ search began with the candidate's 807-move state and used `mt19937_64` seed 2026072508. Each proposal changed one rotor. One proposal in 64 changed two rotors, one in 1,024 changed four, and one in 2,048 also changed the start vertex. Nondecreasing moves were accepted, with a rare random escape. Random restarts were interleaved with restarts at the incumbent. The run produced the 1,282-move configuration preserved in the replay record.

This search certifies only the lower bound supplied by its saved state. The raw search space has \[ 64\cdot2^4\cdot3^{24}\cdot4^{36} =1{,}365{,}745{,}435{,}494{,}690{,}943{,}435{,}859{,}428{,}192{,}026{,}624 \] start-and-rotor states. A complete computation can quotient the four rotations of the square, branch on the exit sequences actually used before a proposed deadline, and prune at 3,136 using the excursion bound. Per-target rotor-potential bounds may lower that deadline before branching.

Vertices are numbered row-major from 0 through 63. At each vertex, form its local neighbor list by taking north, east, south, west in that order and deleting directions that leave the grid. A rotor index points into this shortened list. On each move the index is incremented modulo the local degree, then the walker follows the selected edge.

Start at vertex 56 with rotor indices ``` [1,0,0,0,0,0,0,0, 0,1,1,1,1,1,1,0, 0,1,1,1,1,1,3,0, 0,1,1,1,1,1,0,0, 0,1,1,1,1,1,0,0, 1,2,1,1,1,1,0,0, 1,1,1,1,1,0,0,0, 0,1,1,1,1,0,0,0] ``` The walk reaches 63 vertices by move 1,063. Vertex 7 remains unseen until move 1,282. The trajectory uses 221 of the 224 directed arcs. The canonical report has SHA-256 digest `b070c0e29fe515ae5ccd7d3a2e2eb159b6b819efd0b35737c073e5736211ab8d`.