Toward a Temporal Scale for Emergent Spacetime: Spatial SU(2) Invariance Leaves the Temporal Coefficient Free

An emergent Lorentzian co-metric on the product $\mathbb{R}_\tau\times{\mathrm{Heis}}_3(\mathbb{R})$ of an ordering line and a Heisenberg spatial factor carries a temporal coefficient $A_\tau$ and spatial coefficients, and a four-dimensional spacetime needs their relative normalisation. This paper determines what invariance under a spatial ${\mathrm{SU}}(2)$ action can contribute to that normalisation. On a cotangent fibre $\mathbb{R}_\tau\oplus W$ on which ${\mathrm{SU}}(2)$ acts trivially on the ordering line and by the real spin-one representation on the spatial sector, every invariant symmetric bilinear form is $\alpha\,k_\tau^2 + \beta\,Q_{\mathrm{sp}}$ with $\alpha$ and $\beta$ independent. Invariance excludes mixed temporal–spatial terms but relates the two scalars by no equation. The ratio $A_\tau/A_H$ is therefore unconstrained, and a spatial normalisation $A_H = 2\lambda$ yields no value of $A_\tau$, in particular not $A_\tau = 2$. The conclusion does not depend on the chosen action: for every linear ${\mathrm{SU}}(2)$ action on a four-dimensional real space, the invariant forms are either the multiples of one definite form or a family of dimension at least two, so no ${\mathrm{SU}}(2)$-invariance condition fixes a Lorentzian co-metric up to an overall factor. With the ordering parameter identified with breadth-first depth through an explicit free step $\Delta\tau$, temporal rescalings commute with every action that is trivial on the ordering line and move the ratio, so equivariance cannot fix it either. The value $A_\tau = 2$ obtained from the Casimir eigenvalue is therefore not derived on this route: the result closes a derivation route, not the possibility $A_\tau = 2$. The missing element is named: one datum coupling the temporal and spatial normalisations, such as the step $\Delta\tau$ expressed in spatial units or a saturation rule for the capacity bound. Interpretive outlook (a reading, not a result): spatial isotropy and the unit of time are separate pieces of information, and the rate at which projective ordering advances relative to spatial relational distance, which a kinematic reading would call the invariant speed, is an independent datum that the representation theory of space alone does not supply.