Independence of the polynomial choice in defining A^{alg}(T) (odd case)

Determine whether the algebraic K-theory class A^{alg}(T) ∈ K_1^{alg}(B M) assigned to an odd summable Fredholm module T via a polynomial function φ satisfying conditions (i)–(iii) is independent of the choice of the polynomial φ at the level of algebraic K-theory.

Background

In the odd case, the algebraic assembly A^{alg}(T) is defined using a polynomial φ applied to a projection P derived from a truncated operator. While different choices of φ yield the same class after passing to topological K-theory or Weibel’s homotopy K-theory (and hence after applying the algebraic Chern character), the independence at the level of algebraic K-theory itself is not established.

Clarifying this independence would strengthen the robustness of A^{alg} and remove reliance on passage through comparison maps when working purely in algebraic K-theory.