Term growth along the diagonals of Rule 30: integer lifts, the doubling coin, and where the centre's randomness lives

We follow every algebraic term of the Rule 30 cellular automaton, started from three free cells, along the diagonals of the space-time triangle. Along every down-right diagonal ("line") each term's count is an exact integer polynomial, and one recursion produces each line from the previous two. We give exact laws for the degrees, leading coefficients, 2-adic valuations and parities. New results: the periods of the down-right lines never decrease (proved); the first line of every period class is exactly half black, is orthogonal over its period to every combination of earlier lines, and such lines are exactly jointly independent under arbitrary time shifts (proved); an exact jump-rate law; and an exact formula for the single bit per period class that decides each period doubling (the "doubling coin"). The centre column reads each line only in the first sliver of its period, and none of these theorems reaches it. We show why: toy initial rows with a periodic centre satisfy every per-line theorem, and Rule 90 satisfies them with a white initial row, so a proof that the centre is aperiodic must use the white initial row and Rule 30's OR together. Prior work (Nersissian 2026, Rowland 2006, Jen 1990, Kopra 2022, Brunnbauer 2019) is compared claim by claim. Every statement carries a status label: proved, exact (verified on stated ranges), certified, measured, or open. Not peer reviewed. Computational and editorial assistance from Claude (Anthropic). A two-page sheet of all exact equations is included.