Anton Matrix : Parametric Period-3 RATS Cycles of Length 7k: A Classification under Linear Multiplicities

We study parametric period-3 cycles of length 7k for the Reverse-Add-Then-Sort (RATS) operation. Working under a restricted setting in which the words are built from two distinct digits with multiplicities purely linear in k (no offset), and the smallest digit appearing in the cycle is 1, we establish a complete classification by carry-automaton signature. Two structural classes emerge, each completely characterized: - Class C1, signature (D, D, S): contains exactly three universal families F1, F2, F3, parametrized by m >= 2 with bases b = m+7, b = 3m+5, b = 5m+3 respectively. - Class C2, signature (DSD, DSD, S): contains a single family F4, valid for m congruent to 2 mod 3, m >= 8, b = 5 (m+1) /3, in which two of the three words use four distinct digits. We further prove a stability theorem: lifting the smallest-digit normalization produces no new family. Every parametric cycle of length 7k with global signature in C1, C2 is a cyclic rotation of a cycle in F1, F2, F3, or F4. Moreover, F4 admits no extension to s1 >= 2, the obstruction being a single diophantine equation on the leading column of the saturating transition. The base-9 cycle previously identified as Brick 3 is the case (m, b) = (2, 9) of F1. All formal results are proved by direct calculation of the carry automaton column by column.