I have a problem.
We have a document (page 4 - table) or this book (page 314 in PDF / page 293 in book numeration - table).
The question is: How should we read the decimal values in this table? From left to right? Or from left to right?
For example, the resulting $k$ (searched value) is $111...33$ or $33...111$ in decimal?
I checked the example from the document but the sum exists in both cases. I need total certainty.
Sipser's description would be weirdly worded if he meant the numbers to be read right-to-left. See, for instance, "The left-hand part comprises a $1$ followed by $l−i$ $0$s." This way of describing the numbers associated to each row of the table hints to the irrelevance of the zeros before the one, which should therefore be understood to be the most significant digit.
However, the key observation is that no sum involving the table's numbers produces a carry. Hence, the order of the digits can be chosen arbitrarily. Sipser just chose to go from left to right.
Nothing in the proof depends on the order of the columns. Sipser could have elected to introduce a permutation of the columns to be chosen arbitrarily. That, however, is not needed, and it would have made the proof less clear.
It makes sense that the order doesn't matter, if you think that relabeling variables and renumbering clauses, and hence permuting the table's columns, has no effect on the satisfiability of a 3CNF formula.
Another observation is that any base greater than $3$ would work because no digit ever exceeds $3$. Base $10$ just happens to be a popular one.