Ordered Euclidean Remainders for Exact Orbit Reconstruction and Parameter Intervals in Rational Floor Maps

We study finite orbit segments of the rational floor map F_q(d) = ⌊qd⌋, where q = Q/S ∈ (0,1) has a fixed integer presentation. For an orbit dₖ₊₁ = F_q(dₖ), define the ordered Euclidean remainders ρₖ = Qdₖ − Sdₖ₊₁. The terminal state together with this ordered remainder sequence reconstructs the entire finite orbit exactly by reverse Euclidean division. The same remainder data also determine the exact maximal two-sided parameter interval containing q on which every transition of the orbit segment is preserved. Thus a single ordered Euclidean remainder record jointly determines both exact finite-orbit reconstruction and exact parameter stability.

The paper also gives an exact phase-winding realization of the one-step floor law: for two phase-coherent oscillators with frequency ratio α, ⌊αd⌋ is the number of completed cycles of one oscillator during d cycles of the reference oscillator, while the normalized Euclidean remainder is the residual phase fraction in the rational case.

As a concrete specialization, the paper examines λ_Ω = φ/(φ + π) and the rational approximation q_Ω =339949771344778/10¹⁵, proving exact agreement on the 15-bit domain 0 ≤ d ≤ 32767, determining the first disagreement at d* = 42189065, and recording finite predecessor-tree invariants. The Omega specialization is illustrative; no physical privilege for λ_Ω is claimed.