A normal--inverse-gamma (NIG) latent hierarchy has four parameters, but its induced latent law does not identify all four.
For $σ^2\sim\mathrm{InvGamma}(α,β)$, $μ\midσ^2\sim\mathcal{N}(γ,σ^2/ν)$, and $z\midμ,σ^2\sim\mathcal{N}(μ,σ^2)$, the marginal law of $z$ depends on $(ν,β)$ only through $c=β(1+1/ν)$. Hence the reconstruction-visible parameter space is the three-dimensional quotient $(γ,α,c)$, with a one-dimensional fiber degree of freedom.
For a fixed hierarchical variational objective, exact partial minimization of the forward KL divergence to a complete NIG prior selects a unique prior-relative representative on each fiber, yielding an exact three-coordinate reduction with the same optimum as the four-coordinate objective.
Writing $ρ_0=2β_0/ν_0$ and $T=c/\{α[(γ-γ_0)^2+ρ_0]\}$, we show that inverse canonical allocation $1/ν_{\rm can}$ is an explicit strictly increasing function of $T$.
For $α>1$, the ratio $u_{\rm epi}/u_{\rm var}=1/ν_{\rm can}$ is therefore determined by the quotient state and prior; for rank-only use under a common calibration, $T$ contains the same coordinatewise ordinal information.
The residual prior gauge is characterized rather than eliminated: $(γ_0,ρ_0)$ govern ordinal dependence, while $(ν_0,α_0)$ determine numerical calibration and the analytic ceiling of $1/ν_{\rm can}$.