Graded Layers and Pair Sources in Rule 30

Marking the quadratic term of Rule 30 with a formal variable produces a triangular hierarchy of driven additive layers. Each layer uses the same trinomial propagator and depends only on lower layers. Every finite binary tower is a cellular automaton. For each fixed integer layer, the space–time generating series is D-finite and the central column is P-recursive. Explicit support formulas, degree bounds and integer layer counts describe the hierarchy, which also resolves the reflected Rule 86 recurrence by successive construction. Summing the layers expresses Rule 30 as additive cones emitted by the seed and adjacent pairs of ones. Fibres of the pair projection on finite words are counted explicitly; the projected field admits no autonomous update on all configurations. These results locate the information needed to pass from fixed-layer formulas to the complete orbit.