Write \(N_2\) for the number of initial configurations that reach a genuine two-cycle. The current certificate proves \[ 128\leq N_2\leq 2^{64}-510 =18{,}446{,}744{,}073{,}709{,}551{,}106. \] The lower bound comes from the 64 translations of the displayed witness and their color complements. The verifier checks that these 128 initial configurations are distinct. Majority update commutes with translations and complementation, so each reaches a genuine two-cycle.

For the upper bound, every configuration whose rows are individually monochromatic is fixed. East and west already give two neighbors of the cell's own color, so the vertical neighbors can never force a strict opposing majority. The same argument applies to monochromatic columns. These two families each contain \(2^8=256\) configurations and intersect in the two constant configurations. Their union therefore contains 510 distinct fixed points.

Color complementation pairs the two-cycle basin without fixed configurations under the pairing, so \(N_2\) is even. Determining it exactly still requires an exhaustive symbolic basin count over all \(2^{64}\) starting states.

Retention on a two-against-two neighbor tie can be written as ordinary majority among five inputs: the four neighbors and the vertex's current color. Thus the update is a binary threshold network with a symmetric interaction matrix, including a diagonal self-weight of one.

Goles and Olivos proved that iteration of a finite symmetric binary threshold map reaches a fixed point or a cycle of length two. Their theorem applies to this torus and this tie convention. It justifies the fixed-point versus genuine-two-cycle dichotomy used by the candidate and the verifier. The theorem controls the possible eventual periods. It does not count either basin.

The candidate-generation record reports a run with seed 20260724. Of 50,000 sampled initial states, 41,609 reached fixed points and 8,391 reached genuine two-cycles. The observed two-cycle proportion was \[ \frac{8391}{50000}=0.16782. \] Every sampled trajectory settled within 17 synchronous updates.

The generation record did not preserve the pseudorandom generator or a digest of the sampled states. These figures are therefore reported evidence rather than a replayable exact certificate. They do not determine the numerator over \(2^{64}\).

The audit searched for synchronous majority dynamics on finite graphs, period-two behavior of symmetric threshold functions, toroidal majority cellular automata, and exact basin counts. The 1981 paper of Goles and Olivos supplies the general period-one-or-two theorem. Benjamini, Chan, O'Donnell, Tamuz, and Tan cite the finite period-two property while studying convergence on infinite unimodular graphs and random finite graphs.

No located primary source reports the basin numerator for \(C_8\square C_8\) with retention on a four-neighbor tie. This is a search result rather than a proof of novelty. A complete entry needs a replayable symbolic count, such as a reduced decision diagram or transfer computation, whose fixed and two-cycle basin counts sum to \(2^{64}\).

Rows and columns are indexed modulo eight. The update routine counts the four neighbors separately and retains the center bit exactly when that count is two. The classifier compares consecutive states for a fixed point and states two steps apart for a genuine two-cycle.

The program checks the candidate's displayed starting state against its reported alternating pair. It then forms every torus translation of the start and every complement, verifies that all 128 states are distinct, and classifies each one as reaching a genuine two-cycle. A final exhaustive loop constructs the union of the monochromatic-row and monochromatic-column families, checks that it has size 510, and verifies every member is fixed.