Holographic Reconstruction from Broken Conformal Gauge Symmetry

We investigate bulk–boundary structures in ${\text{AdS}}_4\mathrm{/}{\text{CFT}}_3$ from the viewpoint of gravity as a broken phase of conformal gauge symmetry. A crucial distinction is made between the Euler and Pontryagin characteristic classes. The gauge-theoretic gravitational action contains an Euler sector, whose total-derivative representation gives a boundary form depending on both intrinsic and extrinsic geometry. This structure is different from the Pontryagin descent to the three-dimensional gravitational Chern–Simons functional, whose metric variation gives the Cotton tensor and underlies the mixed boundary conditions studied by de Haro. We perform a Fefferman–Graham expansion of the Euler boundary form in asymptotically ${\text{AdS}}_4$ spacetime. The normalizable coefficient $g_{(3)ij}$ does not contribute linearly to the finite Euler boundary term: its contribution from the cubic extrinsic- curvature sector vanishes by $\mathrm{T}\mathrm{r}{g}_{(3)}=0$, while curvature-- extrinsic-curvature terms containing $g_{(3)}$ vanish at the conformal boundary. A direct Codazzi analysis additionally shows that the Cotton tensor appears in the asymptotic mixed curvature but its contribution through the tangential-connection boundary variation vanishes for standard Dirichlet data. This indicates two distinct types of bulk–boundary structure: a global/topological Euler relation and a local Pontryagin–Cotton relation. We distinguish this kinematic Cotton appearance from a finite Cotton boundary response and discuss the implications for holographic reconstruction.