Remanent Tomography of Multiplicative Structures by Dynamic Collisions: Universal Reconstruction and Local–Global Separation

Abstract

We study a collision-based reconstruction problem for the ideal lattice of the residue ring Z modulo n. For the quadratic transport family $F_c(x) = x^2 + c$, we consider critical orbits and the collision ideals generated by differences of orbit terms modulo n. Closing these collision ideals under addition and intersection generates a visible sublattice of the ideal lattice, whose height measures visible multiplicative depth. The main result is an elementary universal reconstruction theorem: for every integer greater than 1, a finite initial family of polynomials reconstructs the entire ideal lattice. The proof relies on the admissible collision where the difference between the first iterate and the initial point reveals prime-power divisors directly, after which all divisor ideals are generated by lattice operations. We also prove a local-global separation theorem showing that complete reconstruction is equivalent to the existence of oriented collision witnesses separating every level from every non-forced level, thereby reducing global lattice reconstruction to finite local separation conditions.