No earlier lesson in this book scans the cycle budget as the row-varying axis within a single lesson. This scan holds start count (3) and threshold (24) fixed and checks three cycle budgets instead: 2 cycles never reaches the threshold and is rejected, 3 cycles gives the doubling series exactly enough room to cross, and 4 cycles reproduces the book's own first threshold lesson exactly, crossing at the same cycle as the 3-cycle row. The cycle budget is not a fixed background setting; running out of cycles before the threshold is crossed is its own way to reject a source row.
No earlier lesson scans the cycle budget as the row-varying axis within a single lesson; this scan holds start count and threshold fixed and scans the cycle budget itself. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Too few cycles never reaches the threshold
Start count stays 3 and threshold stays 24. Running only 2 cycles gives counts 3,6,12, which never reaches the threshold, so the crossed bit is 0.
3,6,12
The cycle budget itself decides whether the source can cross
Start count and threshold never change across these rows. The cycle budget alone decides whether the same doubling series is given enough room to reach the threshold.
3,6,12,24
Extra cycles beyond the crossing do not change the result
With 4 cycles budgeted, the counts extend to 3,6,12,24,48, but the threshold cycle is still 3, the same crossing point as the row above. Three rows separate the cycle budget from the start-count factor already scanned elsewhere in this book.