Toward a Full-Rank Effective Metric: No Central Coefficient from the Heisenberg Commutator

Current Zenodo release: version 2.0 (2026-09-27). Official title: Toward a Full-Rank Effective Metric: No Central Coefficient from the Heisenberg Commutator, and Scale-Free Casimir Rigidity.

Overview

Paper Q5b extracts, under its spatial limit hypothesis [H-L] and the open lifting hypothesis [H-lift], an effective co-metric $\mathrm{diag}(-A_\tau, A_H, A_H, 0)$ on $\mathbb{R}_\tau \times \mathrm{Heis}_3(\mathbb{R})$. Its central slot is empty. Completing it to a four-dimensional Lorentzian co-metric (open problem Q5b-O2) would need a positive coefficient $A_Z$ of $k_Z^2$.

Q8 asks two questions. Can the Heisenberg commutator $[X, Y] = Z$ supply that coefficient? And if a positive three-dimensional spatial form is supplied by some other construction, what does $\mathfrak{su}(2)$ representation theory fix about its coefficients?

Result. The commutator supplies no central coefficient; a constant one requires a term of homogeneous degree four, whose presence makes isotropy a statement about a length scale. For a supplied fully invariant form, $A_H = A_Z = 2\lambda$ with $\lambda$ free. No value of $A_Z$ is derived.

Quick facts

Author

Jérôme Beau

Work type

Preprint

Focus

Sub-Laplacian, principal symbol, Carnot dilations, invariant forms, Schur's lemma

Key result

No central coefficient from the commutator; for a supplied fully invariant target, $A_H = A_Z = 2\lambda$ with $\lambda$ free

Status

Proved: the rank and homogeneity statements concern Q5b's operator, the rigidity statement a supplied target that is not constructed; Q5b-O2 open

Depends on

Q5b, Q7

Feeds into

Q5b, Q10, Q11

Core contributions

Rank obstruction: with $X = \partial_x + \tfrac{y}{2}\partial_z$ and $Y = \partial_y - \tfrac{x}{2}\partial_z$, the sub-Laplacian $X^2 + Y^2$ has no first-order part, and the principal symbol of $-\Delta_H$ is $(k_X + \tfrac{y}{2}k_Z)^2 + (k_Y - \tfrac{x}{2}k_Z)^2$, of rank two at every point. Its $k_Z^2$ coefficient $(x^2+y^2)/4$ depends on position and vanishes at the identity. No term of lower differential order changes this.
Homogeneity: under the Carnot dilations $(x, y, z) \mapsto (rx, ry, r^2 z)$ the sub-Laplacian is homogeneous of degree two and $Z^2$ of degree four. A left-invariant operator with a positive constant central coefficient must contain a term $Z^2$.
Isotropy selects a length scale: for $A_H\,\Delta_H + A_Z\,Z^2$, a dilation multiplies $A_Z/A_H$ by $r^2$, so the equality $A_H = A_Z$ holds at one scale only.
Scale-free rigidity: if a positive rank-three spatial form is supplied 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. Invariance under rotations about one axis gives no relation.
The number 2: the Casimir acts as $2$ on the spin-one module, but this singles out one form on a ray of invariant forms; it is a normalisation, not a coefficient.

Casimir rigidity and scale

Every $\mathfrak{su}(2)$-invariant Hermitian form on the spin-one module $\mathrm{Sym}^2(V_\rho)$ is a real multiple of one invariant inner product. Representation theory therefore fixes the shape of an invariant form and leaves its size free. The Casimir form is the point of that ray whose factor is $2$ for the standard normalisation of the generators; rescaling the generators rescales the Casimir.

A value of $A_Z$ would require a datum that is not representation-theoretic: a dimensionful scale of the construction, obtained independently of $A_H$. Q10 treats such a scale as an explicit input, so Q8 and Q10 together fix no value.

Interpretation. The non-commutativity of the horizontal directions makes the central direction reachable; it does not assign it a length. Giving it a length comparable to the horizontal ones is an additional physical input, which the substrate would have to supply.

Relation to the Cosmochrony programme

Q8 belongs to the analysis of the effective co-metric coefficients in the emergent geometry sub-programme:

Q5b: Carnot convergence and a conditional rank-three co-metric with an empty central slot.
Q7: requirements for an equivariant bridge to a supplied positive rank-three target.
Q8: no central coefficient from the commutator; scale-free rigidity of a supplied invariant form.
Q10: conditional isotropy of the coefficient ratio under named inputs; no value derived.
Q11: version 2.0 proves that spatial $\mathrm{SU}(2)$ invariance leaves $A_\tau$ independent of the spatial coefficients and derives the absence of mixed temporal–spatial terms for an invariant symbol; no value of $A_\tau$ is established.

The co-metric $\mathrm{diag}(-2, 2, 2, 2)$ is not established; the Lorentz-capacity papers use the conditional form of Q11 with two independent coefficients, $g = 2\eta$ being a choice of units.

Open directions

Extended target: construct an operator or effective form with a central term of homogeneous degree four, and relate it to the horizontal geometry of Q5b.
Full invariance: derive the full $\mathfrak{su}(2)$-invariance of such a form, or show that it fails; without it no relation between $A_H$ and $A_Z$ follows.
Scale: identify the length scale that a central term carries, and a value of $A_Z$ obtained without reference to $A_H$.

References

Jérôme Beau. Toward a Full-Rank Effective Metric: No Central Coefficient from the Heisenberg Commutator, and Scale-Free Casimir Rigidity, 2026. doi:10.5281/zenodo.19879909