A depth obstruction for terminal cycles of marked polynomial families over finite fields.
Sylvain Geffroy · Zenodo (CERN European Organization for Nuclear Research) · 2026
This paper is the second autonomous article extracted from the broader developing corpus "Remanent Tomography of Multiplicative Structures by Dynamic Collisions". The programme asks how much arithmetic and multiplicative information can be reconstructed from the finite traces left by dynamical collisions when the same marked orbit is observed across different modular levels. While the first extracted article developed the Galois–Kummer branch through multiplier towers, the present work explores a complementary question: how far can finite-depth collision data actually reach when one seeks statements that remain valid uniformly over all primes? For polynomial dynamical families, finite orbit portraits can be encoded by collision equations. Their algebraic degree grows exponentially with the depth being explored, suggesting that increasingly deep information should become rapidly available. The main obstruction uncovered here comes from a different arithmetic scale. The first key idea of the proof is to replace raw algebraic degree by the average number of points that survive after reduction modulo primes. Chebotarev identifies these points with fixed points of Frobenius, and Burnside’s lemma then produces a striking simplification: an irreducible portrait packet, however large its degree, contributes on average only one root. Thus a rapidly growing algebraic object may remain arithmetically very sparse. A second obstruction is obtained by turning the problem into a question about derangements in Galois groups. Here the proof relies on a particularly economical route to large symmetric groups. Instead of computing the full Galois group directly, a few carefully chosen factorizations modulo small primes reveal the required cycle structures. Irreducibility gives transitivity, a long cycle forces primitivity, Jordan’s theorem supplies the alternating group, and a nonsquare discriminant finally promotes it to the full symmetric group. For two explicit period-3 packets of the quadratic family, this yields the groups S₆ and S₁₂. Their linear disjointness is then reduced to a much simpler comparison of the quadratic fields encoded by their discriminants. The resulting simultaneous failure set has an exact positive density of approximately 13.54%. These arguments reveal a genuine depth–degree obstruction: bounded-depth collision data can carry rich algebraic structure while still being too sparse to certify certain universal modular statements. Numerical experiments suggest that the relevant dynamical depth grows on a square-root scale, whereas the algebraic method naturally controls only a logarithmic window. The paper keeps these unconditional and heuristic layers separate, and identifies the gap between them as a structural feature rather than a technical accident. Viewed within Remanent Tomography of Multiplicative Structures by Dynamic Collisions, the result isolates one of the limitations of finite dynamical tomography and helps clarify which kinds of information can, and cannot, be reconstructed from collision traces alone.