Manuscript VII-A-ST@1 · VII-A-ST.tex · theorem · c39a67a3

Theorem: Second-order plateau descent

Density covariance and the second-order intertwiner › An operator audit of the density terms · lines 1055–1085 · proof included or linked

VII-A-ST@1#thm:second

For arbitrary smooth $h, k$ , the pointwise relational scalar and its fixed-label smearing satisfy

Φ_{g} [F_{κ}^{*} γ; ϕ_{0}] = Φ_{g} [γ; ϕ_{0}] (mod κ^{3}), O_{f} [g] (F_{κ}^{*} γ, ϕ_{0}) = O_{f} [g] (γ, ϕ_{0}) (mod κ^{3})

under $G_{U}$ . In particular, $C_{1} [g]$ and $C_{2} [g]$ descend under this restricted formal action.

Proof. The coordinate intertwiner implies

Y_{F_{κ}^{*} γ} = F_{κ}^{- 1} \circ Y_{γ} (mod κ^{3}) .

Composing with the scalar intertwiner gives

ϕ_{g} [F_{κ}^{*} γ; ϕ_{0}] \circ Y_{F_{κ}^{*} γ} = (ϕ_{g} [γ; ϕ_{0}] \circ F_{κ}) \circ F_{κ}^{- 1} \circ Y_{γ} = Φ_{g} [γ; ϕ_{0}] (mod κ^{3}) .

The fixed label density gives the smeared identity. Appendix app:direct checks the second coefficient directly.

Displayed formulas in this passage

unlabeled · line 1059
$Φ_{g} [F_{κ}^{*} γ; ϕ_{0}] = Φ_{g} [γ; ϕ_{0}] (mod κ^{3}), O_{f} [g] (F_{κ}^{*} γ, ϕ_{0}) = O_{f} [g] (γ, ϕ_{0}) (mod κ^{3})$
unlabeled · line 1071
$Y_{F_{κ}^{*} γ} = F_{κ}^{- 1} \circ Y_{γ} (mod κ^{3}) .$
unlabeled · line 1076
$ϕ_{g} [F_{κ}^{*} γ; ϕ_{0}] \circ Y_{F_{κ}^{*} γ} = (ϕ_{g} [γ; ϕ_{0}] \circ F_{κ}) \circ F_{κ}^{- 1} \circ Y_{γ} = Φ_{g} [γ; ϕ_{0}] (mod κ^{3}) .$