This math and the key analogy makes me think of the Reed–Solomon error correction algorithm which underlies the parchive sets used to transmit data on systems t

This math and the key analogy makes me think of the Reed–Solomon error correction algorithm which underlies the parchive sets used to transmit data on systems that are prone to errors in transmission or storage or that don't have any ECC built into the transfer protocols. That's effectively the same result. You are trying to reconstruct the full, original message/file and the checksum for certain pieces show damage so you use the additional data to reconstruct it. The math for the Reed–Solomon is similar in concept to oversampling the points on a line. You only need two points on a line to determine the actual path/slope, etc. so if you take three points, any two will allow you recreate the line.