The Scaling Law: How Much a Moment-Ladder Guard Repairs, and Why

The Self-Healing Tower repairs a matrix product, or a stored frame, by solving each checksum line with an integer moment ladder and peeling the solved wounds away until every line re-verifies. This paper states what that decoder is and how much it can repair. It is a peeling process on a graph with two kinds of vertices, the row lines and the column lines, whose edges are the wrong or missing cells, each joining its row to its column. A line is solved by one of two rules. With a given number of totals stored per line, the first rule solves up to half that many wounds at unknown seats, by the ratio and the Prony recurrence. The second rule reads the seats off the graph: a wrong cell sits where a dirty row meets a dirty column, so a line joined to no more dirty lines than it has totals is filled outright, whatever the number of its wounds, the erasure corollary of the Moment Ladder read on a tile. The storm heals if and only if the peel with both rules empties the graph. For dense scattered damage on long lines nearly every line is dirty, only the first rule can act, and the capacity is the k-core threshold of Pittel, Spencer and Wormald: 3.3509 wounds per line with four totals and 5.1494 with six, the capacity law c·s·n for an n × n tile with s witness families. Below that threshold the second rule dissolves the small cores the first rule leaves, the 3 × 3 and 4 × 4 grids among them, so the floor drops to the rarer cores that need more dirty lines in each direction than a line has totals, measured as 59 of 60 storms healed where the first rule alone heals 93%. Where families shorten a sub-line to no more cells than it has totals the second rule always applies, and a sixteen-family tile heals with 60 to 100% of its cells wrong. Measured against the decoder itself on identical fault sets, the peel with both rules agrees in 460 of 460 trials, and on the frame side the erasure hologram of the 300-ring cube is the same process with the pillars filling up to six holes each, agreeing with the cube engine in 1,300 of 1,300 wipe trials. Every number is read from its archived record or measured again when this document is built, with the trial count stated beside it.