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

An emergent Lorentzian co-metric on the product R_τ×Heis₃(R) of an ordering line and a Heisenberg spatial factor carries a temporal coefficient A_τ and spatial coefficients, and a four-dimensional spacetime needs their relative normalisation. This paper determines what invariance under a spatial SU(2) action can contribute to that normalisation. On a cotangent fibre R_τ⊕ W on which 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 α\,k_τ² + β\,Qₛₚ with α and β independent. Invariance excludes mixed temporal–spatial terms but relates the two scalars by no equation. The ratio A_τ/AH is therefore unconstrained, and a spatial normalisation AH = 2λ yields no value of A_τ, in particular not A_τ = 2. The conclusion does not depend on the chosen action: for every linear 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 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 Δτ, 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_τ = 2 obtained from the Casimir eigenvalue is therefore not derived on this route: the result closes a derivation route, not the possibility A_τ = 2. The missing element is named: one datum coupling the temporal and spatial normalisations, such as the step Δτ 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.