Anchor–Ternary Covers for the Collatz Map: Lattices, No-Skipping, and Transfer Epochs

The odd Collatz dynamics is described in the coordinate of a fixed anchor n0 by the exact balance \( n_0+C_i=\lambda_i n_i \). On the levels (4/3)i with dyadic refinement, we distinguish a stable scale factor from stability of the full node. The minimal cover and the no-skipping rule give exact criteria for recognizing a terminal position. Transferring binary blocks of the starting value into a ternary remainder defines finite epochs. We prove a criterion for encoding an epoch by its complete valuation word and a stitching rule using two moduli. For a uniform layer of starts of fixed binary length, we establish a binomial law for the number of reading steps and a positive mean decrease in length over the first epoch: analytically for large layers and by finite exact computation for the remaining small layers. We also show that the proportion of starts without descent at the first epoch boundary tends to zero exponentially with binary length. An example shows why this mean cannot automatically be transferred to subsequent epochs of a single trajectory. Universal termination and finiteness of the number of epochs remain unproved. We identify an exact criterion for descent below the starting value and specify the scope of the geometric, coding, and statistical results.