The zero extension of the signature is bookkeeping, not a geometric definition

Assume AC, inherited from the signature definition and theorem. The signature $\sigma (M)$ is intrinsically defined only for closed oriented smooth manifolds of dimension divisible by four, through the middle-dimensional form $Q_M$ (The signature of a closed oriented manifold of dimension divisible by four). Declaring $\sigma (M)=0$ when $4\nmid \dim M$ is a bookkeeping extension used by the sources to make the induced map ${\mathrm{\Omega }}_{\ast }^{SO}\to \mathbb{Z}$ additive in all degrees (Unoriented and oriented bordism groups, The signature is additive under disjoint union and negates under orientation reversal); it does not extend the middle-form definition of The signature of a closed oriented manifold of dimension divisible by four beyond dimensions $4k$. The geometric identity in The Hirzebruch signature theorem applies in dimensions $4k$, while its algebraic formulation explicitly includes this zero extension: on rational oriented bordism, the extended signature equals the L-genus and is a unital $\mathbb{Q}$-algebra homomorphism. Thus the zero extension is multiplicative in all degrees. It supplies no middle-form signature or top $L_k$ evaluation outside dimensions divisible by four: in dimensions $4k+2$ the middle cup pairing is alternating rather than symmetric, so it has no inertia signature in the sense used here, and the zero value assigned to those dimensions is bookkeeping for the ring structure, not a geometric pairing.