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Smolin's generally covariant $G_{\mathrm{Newton}}\rightarrow0$ limit of 4dEuclidean gravity is a useful toy model for the study of the constraint algebrain Loop Quantum Gravity. In particular, the commutator between its Hamiltonianconstraints has a metric dependent structure function. While a prior LQG likeconstruction of non-trivial anomaly free constraint commutators for the modelexists, that work suffers from two defects. First, Smolin's remarks on theinability of the quantum dynamics to generate propagation effects apply.Second, the construction only yields the action of a single Hamiltonianconstraint together with the action of its commutator through a continuum limitof corresponding discrete approximants; the continuum limit of a product of 2or more constraints does not exist. Here, we incorporate changes in the quantumdynamics through structural modifications in the choice of discreteapproximants to the quantum Hamiltonian constraint. The new structure ismotivated by that responsible for propagation in an LQG like quantization ofParamaterized Field Theory and significantly alters the space of physicalstates. We study the off shell constraint algebra of the model in the contextof these structural changes and show that the continuum limit action ofmultiple products of Hamiltonian constraints is (a) supported on an appropriatedomain of states (b) yields anomaly free commutators between pairs ofHamiltonian constraints and (c) is diffeomorphism covariant. Many of ourconsiderations seem robust enough to be applied to the setting of 4d Euclideangravity.

The constraint algebra in Smolins' $G\rightarrow 0$ limit of 4d Euclidean Gravity