DoT: The Duality of Time Postulate

A theory that starts from one source should seek relations among its physical scales rather than fit a separate number in every sector. That goal is demanding because a structural count, a dynamical response, and a measured constant are different kinds of quantity.

For example, four independent components have ten symmetric pair modes:

One possible choice is $V=\mathbb{C}\oplus W_{\mathbb{C}}$, where $W$ is the real three-direction readout of Eq. (2.5). Its extra scalar channel represents the identity operation, not a fourth spatial direction. A basis of $\operatorname{Sym}^2V$ contains four diagonal pairs and six unordered mixed pairs.

This count assumes that the receiver retains all free symmetric pairs. Quaternion multiplication alone maps products back into a four-dimensional carrier and cannot retain ten independent pair labels. Thus a ten-mode receiver is an additional memory structure, not an automatic consequence of writing a quaternion.