On the Reconstruction of Divisors from Orbits : Quadratic Families, Parity Obstructions and Universal Sequences

This article investigates the orbital separation reconstruction conjecture, which arose from an earlier study of quadratic orbits in remanent tomography. The conjecture asks whether a fixed finite family of maps \(x^2+c\), with positive integer constants and initial value zero, can reconstruct every divisor of every integer \(N\) from gcds of within-orbit differences with \(N\), followed by closure under gcd and lcm. We establish an exact local–global separation criterion, formulate reconstruction through canonical orbit differences, and prove the exponent-independent parameter bound \(τ(N)≤32P^+(N)^2\), where \(P^+(N)\) denotes the largest prime factor of \(N\). Further results identify infinite domains on which fixed families suffice. An analysis of period lifting and cycle multipliers yields a permanent parity obstruction. Exact computational certificates exclude every finite family whose odd constants are at most \(100\,000\). A separate theorem excludes all finite quadratic families under an explicit, unproved density hypothesis. The unrestricted quadratic conjecture remains open. The investigation also considers general integer sequences. We prove that a single sequence of independent integer-valued random variables, asymptotically uniform modulo every fixed integer, almost surely reveals every divisor of every modulus. An elementary deterministic nonquadratic orbit provides another universal construction. These results identify the arithmetic separating capacity of orbit differences as the central property, rather than quadraticity alone. Proofs, computational certificates, reproducibility programs, and numerical comparisons with random sequences are provided.