Binomial Transforms, Geometric Scalings, and Irregular Hadamard Products of Order-3 Signatures

This paper extends the binomial-transform framework from order-2 to order-3 signatures. Let (a,b,c) be an order-3 signature with characteristic polynomial of degree 3. The binomial transform is defined at the level of roots by the map x -> x + 1, yielding an explicit transformed signature. We then define the associated geometric scaling and study the Hadamard product between the transformed signature and its scaled version. In the generic case, the expected degree 9 collapses systematically to degree 6, and the resulting signature is derived explicitly. We also discuss a repeated-root special case, namely the polygonal signature (3,-3,1), where the collapse is stronger and the final degree is 5. In addition, we record the general order-m form of the iterated binomial transform obtained from the substitution P(t-n). Representative examples include Tribonacci, a Narayana-related case, and the polygonal family.