Toward a Full-Rank Effective Metric: No Central Coefficient from the Heisenberg Commutator, and Scale-Free Casimir Rigidity

Abstract

The effective operator of Q5b, conditional on its spatial limit and lifting hypotheses, has a spatial principal symbol of rank two with an empty central slot; completing it to a four-dimensional Lorentzian co-metric requires a positive coefficient $A_Z$ of $k_Z^2$ (open problem Q5b-O2). We prove that the Heisenberg commutator $[X, Y] = Z$ does not supply it. In exponential coordinates the sub-Laplacian $X^2 + Y^2$ has no first-order part, the principal symbol of its negative is a sum of two squares of rank two at every point, and its $k_Z^2$ coefficient is position-dependent and vanishes at the identity; no term of lower differential order changes this. Under the Carnot dilations the sub-Laplacian is homogeneous of degree two while $Z^2$ is homogeneous of degree four, so a constant central coefficient requires a new operator, and the ratio of central to horizontal coefficients is not dilation-invariant: equality of the two selects a length scale. If a positive rank-three spatial form is supplied together with the compatibility conditions of Q7 and is invariant under the full $\mathfrak{su}(2)$ action, Schur's lemma gives $A_H = A_Z = 2\lambda$ with $\lambda > 0$ free; rotation invariance about one axis does not suffice. The Casimir acts as 2 on the spin-one module, but this fixes the normalisation of one invariant form, not a coefficient. No value of $A_Z$ is derived, the passage from the rank-two symbol to a positive rank-three form is not made, and Q5b-O2 remains open. Interpretive outlook: the central direction of the emergent space is not generated by the non-commutativity of its horizontal directions at the level of the metric, and isotropy between the two kinds of direction is a statement about a physical length scale, which the substrate would have to supply.