Winding Rigidity for Finite Recurrences: From geometric cancellation to sharp recovery

A sequence draws a polygon; the polygon’s winding can remember the sequence. This paper determines when that memory is complete and how few geometric observations suffice to recover it. An exact cancellation theorem provides the foundation: constant loops and equal opposite-edge pairs form the entire winding kernel. It yields reconstruction of visible deterministic cycles and their admissible ensembles. A three-phase clock removes the stationary-block ambiguity, and the resulting hierarchy determines every stationary law. For predictive rank at most r ≥ 2, the identifying window 2r − 1 is sharp, even under strict positivity and rapid mixing. For a fixed alphabet, a quadratic number of weighted observations suffices for global recovery. Exact local counts and sharp dimension-only and Gaussian thresholds establish complementary forms of optimality under the stated hypotheses, while a geometric separation margin governs stability. Affine dynamics, Hankel factorization, moment cancellation, and secant geometry unite the proofs. Triangle encodings and exponential obstructions for native point sensors at shorter windows reveal how observation costs depend on the chosen geometry. Arithmetic applications recover modular coefficients through gcd and Cassini invariants and give an exact complete-orbit area formula. The best specially designed global bank remains open in general.