Oh dear, oh my, how blatant

Question

Looking at the Puzzling Stack Exchange recently, I found a surfeit of number sequence riddles that all look the same! I mean, there's no difference between any of them. I'm sick and tired of all these uninspired riddlers cluttering up our precious site with all of their garbage!

Well anyhoo, what comes next?

0 0 1 3 4 5 5  5  6  6  8  9  11
0 2 3 3 6 7 10 11 14 17 17 18 19
0 0 1 1 2 3 4  5  6  7  7  8  9
0 3 4 4 8 8 8  9  11 11 11 14

Hint:

Before you consider my riddle to shelve,
know that its message has characters twelve.

Hint 2:

Maybe it's the math, which you are missing,
CRT you should not be dismissing.

@North This was an attempt at ironic humor. The riddle is genuine. — Austin Weaver, Mar 26 '18 at 03:10
Looks interesting! Are you looking for just the missing number in the bottom-right corner, or a continuation of each of the 4 sequences, or for a 5th sequence? — Phylyp, Mar 26 '18 at 03:21
@Phylyp I'm looking for specifically the number that is missing. — Austin Weaver, Mar 26 '18 at 03:23
Does the missing number have to go in the bottom right hand corner, or can it go anywhere on the bottom row thereby shifting all the numbers to its right to the right? — NL628, Mar 26 '18 at 03:26
@NL628 Given other details, the answer to that will become clear. — Austin Weaver, Mar 26 '18 at 03:28
To your question of Well anyhoo, what comes next?, I posit: "A number" :-P — Phylyp, Mar 26 '18 at 05:25
@Phylyp to confirm that would give you simply too much information. — Austin Weaver, Mar 26 '18 at 14:26
I might suggest rolling back your latest edit, as our fellow puzzlers are astute enough to have picked up on the subtle formatting used in your original question, and are probably only struggling in applying it. — Phylyp, Mar 27 '18 at 03:42

score 13 · Accepted Answer · answered Mar 30 '18 at 16:51

The next number is:

14

We can namely do the following:

If for each line we take the difference between all pairs of consecutive integers, we get the following:
0 1 2 1 1 0 0 1 0 2 1 2
2 1 0 3 1 3 1 3 3 0 1 1
0 1 0 1 1 1 1 1 1 0 1 1
3 1 0 4 0 0 1 2 0 0 3
The second hint refers to the Chinese remainder theorem, which seems to mean that these four sequences are actually the same sequence, just modulo different numbers. These moduli appear to be 3, 4, 2 and 5 respectively, resulting in the following sequence modulo 60:
18 1 20 19 25 15 21 7 15 20 13, and some number that is 5 mod 12.
Replacing these numbers by letters (1 -> A, 2 -> B, etc.), we get the message RATSYOUGOTM?, where the ? is either an E or a Q (corresponding to either 5 or 17). This should be an E, so the last number is 5. From this we see that the last difference should be 0, and the last number of the riddle is 14.

Oh dear, oh my, how blatant

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