J

There is probably a way to write this more point-free. You can definitely see here the friction involved in the way J wants to regard lists as arrays: short rows of the input matrix are zero padded, so you have to snip off the padding before you process each row, and that means you can’t lift some of the operations back up to the parent matrix because it will re-introduce the padding as it reshapes the result; this accounts for a lot of the "1 everywhere (you can interpret v"1 as “force the verb v to operate on rank 1 subarrays of the argument”).

data_file_name =: '2.data'
data =: > 0 ". each cutopen toJ fread data_file_name

NB. {. take, i. index of; this removes trailing zeros
remove_padding =: {.~ i.&0

NB. }. behead, }: curtail; this computes successive differences
diff =: }. - }:

NB. a b in_range y == a <: y <: b
in_range =: 4 : '(((0 { x) & <:) * (<: & (1 { x))) y'

NB. a row is safe if either all successive differences are in [1..3] or all in [_3.._1]
NB. +. or
ranges =: 2 2 $ 1 3 _3 _1
row_safe =: (+./"1) @: (*/"1) @: (ranges & (in_range"1 _)) @: diff @: remove_padding

result1 =: +/ safe"1 data

NB. x delete y is y without the xth element
delete =: 4 : '(x {. y) , ((>: x) }. y)'"0 _
modified_row =: 3 : 'y , (i.#y) delete y'

modified_row_safe =: 3 : '+./"1 row_safe"1 modified_row"1 y'
result2 =: +/ modified_row_safe data