The Moment Ladder: Exact Integer Checksums that Certify, Locate, and Fill

A string of digits, a frame, a tile of a matrix product, any list of integers, carries a ladder of exact totals: the plain sum T, the position-weighted sum W, the quadratic sum Q, the cubic C, and onward. This paper states the laws of that ladder and proves them by algebra and by exhaustive computation. Each rung stamps damage with an exact integer signature. The first m totals cannot all be silenced by a change touching at most m cells, which is the ladder theorem, a Vandermonde argument in plain integers, and the same argument read the other way fills m holes at known positions exactly. One wrong cell names itself: T is the error, W/T the position, Q = T·c² the confirmation, and with the quadratic rung the named cell is unique. Two or three wrong cells on a line are solved by the Prony recurrence in exact integers. The identity behind every use is linear: the ladder of a matrix product is predicted from its inputs, row moments of A·B equal A·(B·w) for any fixed weight vector w, so the columns may carry any fixed sign pattern without changing a theorem, and splitting an index into residue classes gives independent witness families that are the Chinese-remainder box of the ring seen on a tile. On the Fibonacci 60-ring the ladder is the ring's own arithmetic. The split Lucien Khan noticed in 2013, the sixty digits summing to 280 with the first thirty at 128 and the second thirty at 152, is its zeroth rung, the ring's constants 280, 8,760 and 344,660 repair all 540 single-digit corruptions with no reference copy, and every ring of the shell family carries the same stamps with its own half-period. On the ring itself one struck digit is already named by its rectangle, the twin grid of the system paper, with no total consulted, and the smallest damage that keeps every rectangle relation, a swap on all four corners, is stamped by W on the even rectangles and by Q on the odd ones. Every law is re-verified by enumeration when this document is built, and every number quoted from a companion paper is checked against that paper's archived record. The architectures built on these laws, frames, the fractal tower, the cube, the compute guard and the training guard, are the companion paper The Self-Healing Tower. How much they repair, and why, is the companion paper The Scaling Law.