Holographic tilting principle for perfectoid boundary carriers

Let $X$ be a perfectoid boundary carrier equipped with a condensed sheaf/module $\mathcal{F}$ of sources and operators, and let $X^{\flat}$ be its tilt. Assume the holographic generating functional is a derived-limit object on $X$:

$$Z_{X}[J]:=\exp\!\bigl(-W_{X}[J]\bigr),\qquad W_{X}[J]\in R\!\varprojlim_{\Lambda\to\infty}\Bigl(W_{\Lambda}[J_{\Lambda}]+S_{\mathrm{ct}}[\Lambda,J_{\Lambda}]\Bigr).$$

Suppose there is a functorial correspondence

$$\mathfrak{T}\:(X,\mathcal{F},J)\longmapsto (X^{\flat},\mathcal{F}^{\flat},J^{\flat})$$

such that, for a suitable class of observables $\mathcal{O}$,

$$\langle \mathcal{O}\rangle_{X}\ \longleftrightarrow\ \langle \mathcal{O}^{\flat}\rangle_{X^{\flat}},$$

with matching scaling exponents and monodromy/duality actions transported through $\mathfrak{T}$. Holographic tilting principle. $p$-adic holography should be an arithmetic avatar of tower-completed Archimedean holography, with tilting providing a canonical bridge at the level of boundary data. In particular, for a scalar primary $\mathcal{O}_{\Delta}$, the two-point kinematic scaling should be related by

$$\langle \mathcal{O}_{\Delta}(x)\mathcal{O}_{\Delta}(y)\rangle_{X}\propto |x-y|^{-2\Delta},$$

and

$$\langle \mathcal{O}_{\Delta}(x)\mathcal{O}_{\Delta}(y)\rangle_{X^{\flat}}\propto |x-y|_{p}^{-2\Delta}.$$

The principle proposes a background-level correspondence between Archimedean and non-Archimedean holographic data that classical AdS/CFT does not intrinsically provide; the displayed two-point relation is a minimal checkable milestone, while realizing the full functorial correspondence and transporting harmonic, monodromy, and duality data remain open.