Number cells by \(7y+x\), with \(0\leq x,y<7\). Store the 49 colors in bits 0 through 48, the ant's cell in bits 49 through 54, and its direction in bits 55 and 56, using north, east, south, west as 0, 1, 2, 3. The all-white state at cell 0 facing north is therefore the integer 0. Each transition reads the bit at the current cell, turns, toggles that bit, moves with coordinates reduced modulo 7, and repacks the state.

Floyd cycle detection on this exact 57-bit transition map gives \[ \mu=0,\qquad \lambda=64{,}165{,}598. \] Thus the initial state belongs to the eventual cycle and the orbit has no transient. A second direct replay starts from state 0, asserts that every state after steps 1 through 64,165,597 is nonzero, and reaches state 0 after step 64,165,598. This proves both recurrence of the complete state and minimality of the positive return time: \[ \boxed{64{,}165{,}598}. \] Since the map is deterministic, these checks also imply that all states at times 0 through 64,165,597 are distinct. The cycle contains the labeled colors, position, and direction, with no quotient by translation or rotation.

A Mathematics Stack Exchange answer published in July 2019 gives a rectangular-torus table under the same all-white, original-position, original-direction recurrence convention. Its 7 by 7 entry is `[7] 64,165,598`, where the bracket records seven symmetry-based quasi-recurrences before the full labeled return. The value agrees with the independent certificate here.

Langton introduced the mobile automaton in 1986. Later work proves complexity and recurrence results for related Lorentz lattice-gas models, while the focused search found no peer-reviewed table of these small square-torus periods. The exact value therefore has a prior public computation. This entry adds a compact full-state encoding, cycle parameters, replay assertions, and trace digests.

The replay hashes the little-endian eight-byte encoding of each state at times 0 through 64,165,597 with standard 64-bit FNV-1a. The digest is `981fc1e057236ac9`. The final packed state is `0000000000000000`.

Selected packed-state checkpoints are \[ \begin{array}{r|c} 1&0082000000000001\\ 10&011b040000000183\\ 100&0022a15941424494\\ 1000&015359d5f9766cb9\\ 1000000&0161407e02511400\\ 3000000&002edbee6e815e65\\ 10000000&00179517efcf3c12\\ 64165597&008e000000000080\\ 64165598&0000000000000000 \end{array} \] The canonical stdout line, including its terminating line feed, has SHA-256 digest `e78091b2d9b11ecd5126acd78067e4efc1397f09aa1ee18c5afe50eb2c0a75f6`. A reproduced optimized run took about two seconds.