  Quantum gravity in 4D asymptotically flat spacetimes features spontaneous
symmetry breaking due to soft radiation hair, intimately tied to the
proliferation of IR divergences. A holographic description via a putative 2D
CFT is expected free of such redundancies. In this series of two papers, we
address this issue by initiating the study of Quantum Error Correction in
Celestial CFT (CCFT). In Part I we construct a toy model with finite degrees of
freedom by revisiting noncommutative geometry in Kleinian hyperk\"ahler
spacetimes. The model obeys a Wick algebra that renormalizes in the radial
direction and admits an isometric embedding \`a la Gottesman-Kitaev-Preskill.
The code subspace is composed of 2-qubit stabilizer states which are robust
under soft spacetime fluctuations. Symmetries of the hyperk\"ahler space become
discrete and translate into the Clifford group familiar from quantum
computation. The construction is then embedded into the incidence relation of
twistor space, paving the way for the CCFT regime addressed in upcoming work.
