Newman's Quantized Omega Contraction Theorem

This paper proves an exact residual decomposition for a floor-quantized contraction determined by the Omega parameter λΩ = φ/(φ + π), where φ = (1 + √5)/2. For the quantizer Qτ(y) = τ⌊y/τ⌋ and the orbit zₖ₊₁ = Qτ(λΩzₖ), the continuous iterate λΩⁿx is expressed exactly as a lattice-valued state zₙ plus a weighted sum of quantization remainders. The resulting identity yields a uniform residual bound and an exact backward reconstruction formula, so the augmented state uniquely determines the initial value. The long-term dynamics are also classified: nonnegative initial data reach 0 after finitely many iterations, while negative initial data reach the fixed lattice state −τ. An affine-coordinate corollary applies the result to Ω(v) = 1 + λΩ(v − 1), and the paper also records the corresponding extension to a general contraction parameter satisfying 0 < α < 1.