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Thanks, Aaron. Your comment has reminded me that I have been negligent in the computational searches conducted so far, in that I have failed to report any information on minimum distances encountered. I will attend to that.

By the way, I have reversed the definitions of X and Y above as they were the opposite of what I have in all existing code and research notes. My apologies!

In terms of k the first few polynomials are

$Py_1 = 4k - 1$
$Px_1 = 4k + 1$

$Py_2 = 16k^2 - 12k + 1$
$Px_2 = 16k^2 + 12k + 1$

$Py_3 = 64k^3 - 80k^2 + 24k - 1$
$Px_3 = 64k^3 + 80k^2 + 24k + 1$

$Py_4 = 256k^4 - 448k^3 + 240k^2 - 40k + 1$
$Px_4 = 256k^4 + 448k^3 + 240k^2 + 40k + 1$

If we define the distance polynomial $D_{j,i} = Py_j - Px_i$ then $D_{2,1} = 16k^2 - 16k$ so the quadratic case is disposed of, as you say.

We can also rule out the cubic case, and in fact all odd j. We have

$D_{3,1} = 64k^3 - 80k^2 + 20k - 2$
$D_{3,2} = 64k^3 - 96k^2 + 12k - 2$

For all odd j we get even coefficients and $c_0 = -2$, so no $D_{2e+1,i}$ can have an integer root $k > 1$.

For even j we get polys like these:

$D_{4,1} = 56k^4 - 448k^3 + 240k^2 - 44k$
$D_{4,2}= 256k^4 - 448k^3 + 224k^2 - 52k$
$D_{4,3} = 256k^4 - 512k^3 + 160k^2 - 64k$

What I'm hoping to find is some magic property for even j that will tell us that all $D_{2e,i}$ are either irreducible or have a single integer root $k=1$.

Since $Y(1,j) = 3,5,7 \ldots$, all of $X(1,i) = 5, 29, 169 \ldots$ are to be found in $Y(1,j)$ so the corresponding $D_{14,2}, D_{84,3} $ etc will all have root $k=1$.

I suspect that all other D are irreducible, but these isolated exceptions are a bit of a fly in the ointment!

Oh yes, and I can tell you that a search on all pairs of sequences $Y(k,j), X(k,i)$ revealed no match for a rather staggering j up to 100,000. For a given depth limit j < J, such a search is finite, since beyond a certain k we find that all $Y(k,J) > X(k,J-1)$ and so we need look no further.

It follows then that the proposition, that all $D_{j,i}$ are either irreducible or have a single integer root $k=1$ is true for all j < 100,000.