Theorem. Neutral Gauss's law as an averaging constraint [fdip-008A]
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  • November 1, 2026

For the residual action, the neutral Gauss constraint requires \[R_gf=f\qquad \text {for every }g\in K.\] Its solution space is the physical Hilbert space \(\mathcal H_{\mathrm {phys}}=\mathcal H_{\mathrm {link}}^K\). The orthogonal projector onto it is \[P=\int _KR_g\,dg,\] with normalized Haar measure. For a finite group \(G\), this is \[P=\frac 1{|G|^{|V_s|}}\sum _{g\in G^{V_s}}R_g.\] For the positive gauge transfer operator, \(TP=PT\); hence time evolution preserves the constraint.

Proof. 

  • November 1, 2026

For finite \(G\), left translation permutes the sum, so \(R_hPf=Pf\) for every \(h\). If \(f\) is invariant, every summand is \(f\), so \(Pf=f\). These facts prove \(P^2=P\) and identify its image. Unitarity gives \(R_g^*=R_{g^{-1}}\); inversion permutes the sum, proving \(P^*=P\). Thus \(P\) is the orthogonal projector.

For compact \(G\), the same identities use normalized Haar invariance, inversion and Fubini. The integral on \(L^2\), its boundedness and its image were proved in Theorem [fdip-0083]; that theorem applies here without a finite-group assumption. The transfer operator commutes with every \(R_g\) by Theorem [fdip-0080], so it commutes with the average.

The finite algebraic projection statement is Representation.isProj_averageMap. It assumes an invertible group order in the coefficient ring, as holds over \(\mathbb C\). Unitarity gives the orthogonality used here; the compact Haar and transfer assertions have the separate proofs just cited. Maschke averaging uses a related finite average to produce invariant complements.

For a finite configuration set \(X\), an invariant function is exactly a function constant on each gauge orbit. Consequently \[\dim _{\mathbb C}\mathcal H_{\mathrm {phys}}=|X/K|.\] Indeed, the orbit indicators have disjoint support and form a basis. For Haar probability on a finite set, dividing each indicator by the square root of its orbit's probability makes this basis orthonormal.