A polynomial can drive several kinds of mathematical behaviour at once. Its inverse branches split into channels, a polynomial amplitude can activate some channels and suppress others, and the resulting periods satisfy differential equations whose order reflects that active support. This anonymous, unrefereed candidate turns those ideas into three explicit results. First, it treats every polynomial amplitude for a nondegenerate Dickson phase. The amplitudes fall into paired modules, with one additional midpoint module in even degree. Counting the nonzero modules gives the exact induced rank. The consequence is a clean parity law: odd degree realizes only even ranks, while even degree realizes every possible rank. Second, the candidate studies the degree-twelve phase obtained by cubing x to the fourth plus x. That phase has two genuinely different polynomial decompositions. Its amplitudes split into two single composition channels and three fused modules. A fibre-trace argument proves the hidden cancellations: every nonzero element of a fused module activates exactly three channels, not four. The five pieces together realize every rank from zero to eleven. Third, the paper changes from amplitude classification to zero-cycle perturbations. For a degree-d phase, it shows that the ordinary coefficient Bautin ideal is generated by the first d minus one Melnikov orders. This gives a uniform coefficient-index bound. The manuscript calls that index b coefficient. Applying the same inequality to the terse b of the motivating paper is conditional, because the source does not fully specify whether its ambient ideal uses the identical convention. The public package includes the written proofs, a twenty-three-page PDF, accessible MathML HTML, exact programs and receipts, source and licence records, five deliberate mutation controls, and ordinary and optimized test reports with the same nine testcases. A five-role internal model-mediated review required major repairs, and one bounded confirmation found no remaining major scientific defect but required the deterministic release evidence now attached. Those reviews, computations, hashes, a DOI and Evidence Press publication are not independent mathematical validation. No unaffiliated specialist has approved the complete proofs, no proof assistant formalizes them, priority remains bounded by the searched corpus, and the release does not claim that three historically posed open problems have been solved. The most valuable next step is independent reconstruction of the support theorem, the fused trace cancellation and the Bautin ideal argument from the public definitions, followed by a separately authored implementation and a direct source-convention audit for the motivating b of m.