What is the general mathematical proof that K map and Boolean Algebra are equivalent?
K-Maps are just a way for humans to use visual pattern recognition of a table of binary numbers using Set Theory Rules to reduce Boolean Algebra into minimal logic equations.

But I won't attempt to prove it.

It was invented by Edward Veitch and improved by Maurice Karnaugh.  Veitch learned in his later years from solving Sudoku puzzles that heavy lines between groups of boxes can be very helpful especially if one has poor eyesight, as he had.   

There are even more patterns in Sudoku puzzles to quickly solve the most difficult tables of blanks with minimal decimal numbers just like K-Maps with "1"'s.  (like folding corner butterflies)