Exact Signs and Sharp Boundaries for Product-Binomial Dold Coefficients

Abstract

Let X be a rational exterior space and let f: X → X. Let A = Q(f*) be the graded action on the indecomposable quotient of the positive-degree rational cohomology of X. Suppose that χ_A(t) = t^s ∏_j (t^{n_j} − a_j)^{k_j}, with a nonempty product and every a_j ≥ 3. Powering the binomial roots yields a closed formula for each Lefschetz number L(f^q), and every unnormalized Dold coefficient is nonzero, with exact sign determined by ∑_j k_j gcd(n_j,m). In the single-factor toral case this proves the stated all-dimensional Dold-nonvanishing conjecture of 2021 and gives the parity correction to a later all-negative formulation. The parameter threshold is sharp: for t^n − 2, the coefficient at order m vanishes exactly when m ≥ 3 and m divides n. For affine toral maps with these characteristic polynomials, the unsigned Möbius transform counts least-period points and determines the complete period set; for arbitrary maps in the same toral homotopy class, the signed Dold coefficient remains a homological invariant. This deposit contains the final preprint, LaTeX source, a self-contained exact-integer reproducibility archive, and SHA-256 checksums.