For a polynomial exponential period, repeated differentiation can generate a differential system. This release asks how much of that system is generated by one fixed starting form, dx. The answer is encoded at infinity. Normalize the inverse series of the polynomial and split it into congruence channels. The candidate proves, without assuming stationary phase, that the number of nonzero channels is exactly the fixed-seed cyclic rank. It is also exactly the dimension of the reduced constant span of the polynomial's inverse roots. This replaces the analytic condition inherited by the preceding release with a direct formal-moment isomorphism. The analytic story is then rebuilt separately: fixed rapid-decay contours on a narrow sector give a Fourier-Gamma period matrix with a nonzero determinant. The second contribution concerns polynomial composition. If P is R composed with Q, the inverse roots split into Q-blocks. A finite Fourier-submatrix formula determines the dimension of every sum of those block spans. It therefore distinguishes no fusion, partial fusion, and complete equality of blocks. When blocks are equal, their classes define an actual coarser polynomial right factor. The construction works along complete decomposition chains and survives Ritt moves, the basic transformations relating different polynomial decompositions. At global rank at most three, the candidate lists exactly eleven possible unordered Ritt rank diamonds. The degree-twelve collision, P equals the cube of x to the fourth plus x, becomes transparent: the quartic blocks coincide in one decomposition, while four cubic-block lines span the same three-dimensional space after the Ritt move. The third contribution is a coefficient atlas. Direct arguments settle degrees two through four. For degrees five through twenty, an exact-rational Groebner sweep checks every support set of size at most three: 1,152 cases in total. Forty-five are full-family cases, 884 have power-only certificates, and 223 have union certificates. No anomaly and no missing certificate is recorded. Ordinary and optimized Python must regenerate identical receipts, and deliberate semantic corruptions must fail. This is a finite computer-assisted theorem candidate, not an all-degree proof. The list of powers, Dickson polynomials, quadratic and cubic compositions over a power, quartics, and one exceptional degree-twelve orbit is proved only through degree twenty. The first uncovered quartic lift appears in degree thirty-two, where a proposed descent uses the conclusion it needs to establish. The generic differential-equation order is a corollary with a non-annihilation condition. Zero-cycle and Laurent-polynomial extensions remain research programmes, not results of this release. The package contains a thirty-two-page tagged paper, canonical Markdown, machine-readable claim and assurance maps, complete decomposition and coefficient data, exact code, per-case receipts, negative controls, historical sources, internal reviews, and deterministic archive tools. These are producer-side checks. No unaffiliated specialist has reconstructed the proof or implementation, no proof assistant has checked it, and novelty and priority remain open. The scholarly creator is Anonymous; Ian Pitchford is the repository maintainer and publisher. This synthetic-voice Evidence Press briefing was released on 20 August 2026. It is a communication aid, not additional mathematical evidence.