This introduces a bar diagonalizer (which uses the vertical line glyph, "|"), sitting beyond all whole number n in {0, n}. What it does is to expand and diagonalize with some peculiar operators.
This should beat the limit of three-entry Bashicu matrix system (BMS) at (0,0,0,0)(1,1,1,1), or equivalently, (1,2,4,8,16) in Y sequence.
{{0, {1}}} → (0,0,0)(1,1,1) | (1,2,4,8)
{{0, {1}}{0, 1}} → (0,0,0)(1,1,1)(1,1,0) | (1,2,4,8,4)
{{0, {1}}{{0, 2}, 1}} → (0,0,0)(1,1,1)(1,1,0)(2,2,0) | (1,2,4,8,4,7)
{{0, {1}}{{0, {1}}, 1}} → (0,0,0)(1,1,1)(1,1,0)(2,2,1) | (1,2,4,8,4,8)
{{0, {1}}{{0, {1}}, 1}{{0, {1}}, 1}} → (0,0,0)(1,1,1)(1,1,0)(2,2,1)(1,1,0)(2,2,1) | (1,2,4,8,4,8,4,8)
{{0, {1}}{{0, {1}} + 1, 1}} → (0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,0,0) | (1,2,4,8,5)
{{0, {1}}{{0, {1}}{0, 1}, 1}} → (0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,1,0) | (1,2,4,8,6)
{{0, {1}}{{0, {1}}{{0, {1}}, 1}, 1}, 1}} → (0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,1,0)(3,2,1) | (1,2,4,8,6,10)
{{0, {1}}{{0, {1}}{0, 2}, 1}, 1}} → (0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,2,0) | (1,2,4,8,7)
{{0, {1}}{{0, {1}}{{0, {1}}, 2}, 1}, 1}} → (0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,3,1) | (1,2,4,8,7,12)
{{0, {1}}{0, {1}}} → (0,0,0)(1,1,1)(1,1,1) | (1,2,4,8,8)
{{0, {1}}{0, {1}}{0, {1}}} → (0,0,0)(1,1,1)(1,1,1)(1,1,1) | (1,2,4,8,8,8)
{{1, {1}}} → (0,0,0)(1,1,1)(2,0,0) | (1,2,4,8,9)
{{{0, 1}, {1}}} → (0,0,0)(1,1,1)(2,1,0) | (1,2,4,8,10)
{{{{0, 1}, 1}, {1}}} → (0,0,0)(1,1,1)(2,1,0)(3,1,0) | (1,2,4,8,10,12)
{{{{0, 2}, 1}, {1}}} → (0,0,0)(1,1,1)(2,1,0)(3,2,0) | (1,2,4,8,10,13)
{{{{0, {1}}, 1}, {1}}} → (0,0,0)(1,1,1)(2,1,0)(3,2,1) | (1,2,4,8,10,14)
{{{{0, {1}}{0, {1}}, 1}, {1}}} → (0,0,0)(1,1,1)(2,1,0)(3,2,1)(3,2,1) | (1,2,4,8,10,14,14)
{{{{1, {1}}, 1}, {1}}} → (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,0,0) | (1,2,4,8,10,14,15)
{{{{{0, 1}, {1}}, 1}, {1}}} → (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,1,0) | (1,2,4,8,10,14,16)
{{{{{{0, {1}}, 1}, {1}}, 1}, {1}}} → (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,1,0)(5,2,1) | (1,2,4,8,10,14,16,20)
{{{{{0, 2}, {1}}, 1}, {1}}} → (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,2,0) | (1,2,4,8,11)
{{{{{{0, {1}}, 2}, {1}}, 1}, {1}}} → (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,2,0)(5,3,1) | (1,2,4,8,11,16)
{{{0, {1}}, {1}}} → (0,0,0)(1,1,1)(2,1,1) | (1,2,4,8,12)
{{{0, {1}}{0, {1}}, {1}}} → (0,0,0)(1,1,1)(2,1,1)(2,1,1) | (1,2,4,8,12,12)
{{{1, {1}}, {1}}} → (0,0,0)(1,1,1)(2,1,1)(3,0,0) | (1,2,4,8,12,13)
{{{{0, 1}, {1}}, {1}}} → (0,0,0)(1,1,1)(2,1,1)(3,1,0) | (1,2,4,8,12,14)
{{{{0, {1}}, {1}}, {1}}} → (0,0,0)(1,1,1)(2,1,1)(3,1,1) | (1,2,4,8,12,16)
{{{{{0, {1}}, {1}}, {1}}, {1}}} → (0,0,0)(1,1,1)(2,1,1)(3,1,1)(4,1,1) | (1,2,4,8,12,16,20)
{{{0, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0) | (1,2,4,8,13)
{{{0, {1} + 1}{0, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(2,2,0) | (1,2,4,8,13,13)
{{{1, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,0,0) | (1,2,4,8,13,14)
{{{{0, 1}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,1,0) | (1,2,4,8,13,15)
{{{{{0, 1}, 1}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,1,0)(4,1,0) | (1,2,4,8,13,15,17)
{{{{{0, 2}, 1}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,1,0)(4,2,0) | (1,2,4,8,13,15,18)
{{{{{0, {1}}, 1}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,1,0)(4,2,1) | (1,2,4,8,13,15,19)
{{{{{0, {1}}{0, {1}}, 1}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,1,0)(4,2,1)(4,2,1) | (1,2,4,8,13,15,19,19)
{{{{{1, {1}}, 1}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,1,0)(4,2,1)(5,0,0) | (1,2,4,8,13,15,19,20)
{{{{{{0, 1}, {1}}, 1}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,1,0)(4,2,1)(5,1,0) | (1,2,4,8,13,15,19,21)
{{{{{{0, 2}, {1}}, 1}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,1,0)(4,2,1)(5,2,0) | (1,2,4,8,13,15,19,22)
{{{{{{{0, {1}}, 2}, {1}}, 1}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,1,0)(4,2,1)(5,2,0) | (1,2,4,8,13,15,19,22)
... (I don't care)
{{{{0, {1}}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,1,1) | (1,2,4,8,13,17)
{{{{0, {1} + 1}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,2,0) | (1,2,4,8,13,18)
{{{{0, {1} + 2}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,3,0) | (1,2,4,8,13,19)
{{{{{0, {1} + 3}, {1} + 2}, {1} + 1}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,3,0)(4,4,0) | (1,2,4,8,13,19,26)
{{{0, {1}{1}}, {1}}} → (0,0,0)(1,1,1)(2,2,0)(3,3,1) | (1,2,4,8,13,20)
{{{0, {2}, {1}}}} → (0,0,0)(1,1,1)(2,2,1) | (1,2,4,8,14)
{{{0, {0, 1}}, {1}}} → (0,0,0)(1,1,1)(2,2,2) | (1,2,4,8,15)
{{0, 0, 1}} → (0,0,0,0)(1,1,1,1) | (1,2,4,8,16)
Or wait for a moment.