The discrete Chen–Chow theorem: Exact realization by finite complex time series

Discrete signatures describe a time series through ordered sums of its increments. The discrete Chen–Chow problem asks whether every truncated collection satisfying the quasi-shuffle identities comes from an actual finite time series. This paper proves realization at every weight over any algebraically closed field of characteristic zero. Every target requires at most 2(D − d + 1) increments, where D is the weighted Lie-algebra dimension and d is the time-series dimension. The universal discrete signature variety is identified with the quasi-shuffle variety. For scalar series through weight three, the exact universal length is five increments; generic targets require four. An algebraic obstruction proves necessity, and a degree-eighteen resultant gives a construction for every target. The proof combines the established Hoffman monomial lift with dimension growth, constructibility, and finite products in algebraic groups. Two geometric figures explain the real positivity obstruction and the signed area created by the monomial lift of a retraced path. The results concern unrestricted increments over algebraically closed fields. Real increments and fixed finite alphabets impose additional restrictions. Sharp length bounds in higher dimensions and weights, efficient reconstruction, and a characterization of real realizability remain open.