Let \(B_8\) denote the requested number. The certified result is \[ 177{,}024{,}301{,}925{,}259{,}284 \leq B_8\leq 18{,}161{,}310{,}923{,}858{,}378{,}752. \]

For the lower bound, either long diagonal is a spanning set. A full occupied row together with a full occupied column also spans: each of the four resulting rectangles fills by induction on distance from the row-column cross. Inclusion-exclusion over these 66 sufficient patterns gives the displayed lower endpoint.

For the upper bound, an entirely vacant outer row or outer column remains vacant forever. Inclusion-exclusion over the four boundary lines shows that 285,433,149,851,172,864 initial sets have at least one such obstruction. Subtracting these from \(2^{64}\) gives the upper endpoint.

The exact value remains undetermined by this certificate.

For the main diagonal, every cell at diagonal distance \(d>0\) has two neighbors at distance \(d-1\). Induction fills the board. Reflection gives the other long diagonal.

If row \(r\) and column \(c\) are initially occupied, a cell away from their union has two neighbors whose coordinate distances to that union are each one step smaller. Induction fills all four rectangular regions. Every superset of any of these 66 patterns therefore spans.

If an outer line is vacant and all other cells are occupied, each vertex of that line has at most one occupied neighbor. The line stays vacant. Removing more initially occupied vertices cannot create a second occupied neighbor, so every initial set disjoint from that line fails to span.

Aizenman and Lebowitz introduced the rigorous finite-volume study of bootstrap percolation and its internally spanned rectangles. Holroyd later obtained the sharp two-dimensional metastability threshold. Morris studied minimal percolating sets on square grids. These sources address structural and asymptotic questions.

The targeted search found no primary source giving the exact number of spanning subsets of the open \(8\times8\) board. This records the search outcome and does not prove novelty.

An exact projected row transfer was also tested. Its nondeterministic state records the last two rows of a possible stable vacant fort, then determinizes over each incoming vacancy row. It had 256 projected states after one row and 5,968 after two rows. The computation was stopped at the task's time cap before producing a count. A future exact certificate could continue this transfer with packed bit sets or compile the fort constraints into a decision diagram.

The closure routine scans every vacant vertex, adds all vertices with at least two occupied grid neighbors, and repeats to a fixed point. It checks both long diagonals and all 64 row-column crosses on the eight board. It also checks that removing any one of the four outer lines from the full board produces a fixed point.

For a forced occupied set \(F\), the function `cross_with(F)` counts configurations that contain \(F\), at least one complete row, and at least one complete column. It evaluates \[ C(F)=\sum_{\varnothing\ne A\subseteq[8]} \sum_{\varnothing\ne D\subseteq[8]} (-1)^{|A|+|D|}2^{64-|F\cup R_A\cup C_D|}. \] The lower endpoint is the union count for the cross family and the two diagonal supersets. The upper endpoint uses a separate four-event inclusion-exclusion calculation for vacant boundary lines.

As a closure-routine check, full enumeration gives \(B_1=1\), \(B_2=7\), \(B_3=312\), and \(B_4=50{,}637\), agreeing with the candidate record.