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A<sc>bstract</sc> Recent research has leveraged the tractability of$$ T\overline{T} $$ $T\overline{T}$style deformations to formulate timelike-bounded patches of three-dimensional bulk spacetimes includingdS₃. This proceeds by breaking the problem into two parts: a solvable theory that captures the most entropic energy bands, and a tuning algorithm to treat additional effects and fine structure. We point out that the method extends readily to higher dimensions, and in particular does not require factorization of the fullT²operator (the higher dimensional analogue of$$ T\overline{T} $$ $T\overline{T}$defined in [1]). Focusing ondS₄, we first define a solvable theory at finiteNvia a restrictedT²deformation of theCFT₃onS²×ℝ, in whichTis replaced by the form it would take in symmetric homogeneous states, containing only diagonal energy densityE/Vand pressure (-dE/dV) components. This explicitly defines a finite-N solvable sector ofdS₄/deformed-CFT₃, capturing the radial geometry and count of the entropically dominant energy band, reproducing the Gibbons-Hawking entropy as a state count. To accurately capture local bulk excitations ofdS₄including gravitons, we build a deformation algorithm in direct analogy to the case ofdS₃with bulk matter recently proposed in [2]. This starts with an infinitesimal stint of the solvable deformation as a regulator. The full microscopic theory is built by adding renormalized versions ofT²and other operators at each step, defined by matching to bulk local calculations when they apply, including an uplift fromAdS₄/CFT₃todS₄(as is available in hyperbolic compactifications of M theory). The details of the bulk-local algorithm depend on the choice of boundary conditions; we summarize the status of these in GR and beyond, illustrating our method for the case of the cylindrical Dirichlet condition which can be UV completed by our finite quantum theory. 

A<sc>bstract</sc> In this note, we resolve an apparent obstacle to string/M theory realizations of dS observer patch holography, finding a new role for averaging in quantum gravity. The solvable$$ T\overline{T} $$ $T\overline{T}$(+Λ₂) deformation recently provided a detailed microstate count of thedS₃cosmic horizon, reproducing the refined Gibbons-Hawking entropy computed by Anninos et al. along with the correct radial bulk geometry. On the gravity side, the deformation brings in the boundary to just outside a black hole horizon, where it is indistinguishable from the dS cosmic horizon, enabling a continuous passage to a bounded patch of dS. In string/M theory, the relationship between AdS/CFT and dS involves uplifts that change the internal topology, e.g. replacing an internal sphere$$ \mathbbm{S} $$ $S$with an internal hyperbolic spaceℍ(and incorporating varying warp and conformal factors). We connect these two approaches, noting that the differences in the extra dimensions between AdS black hole and dS solutions are washed out by internal averaging in the presence of a timelike boundary skirting the horizon. This helps to motivate a detailed investigation into the possibility of such timelike boundaries in (A)dS solutions of string/M theory, and we take initial steps toward suitable generalizations of Liouville walls as one approach. 

We introduce a novel framework for optimization based on energy-conserving Hamiltonian dynamics in a strongly mixing (chaotic) regime and establish its key properties analytically and numerically. The prototype is a discretization of Born-Infeld dynamics, with a squared relativistic speed limit depending on the objective function. This class of frictionless, energy-conserving optimizers proceeds unobstructed until slowing naturally near the minimal loss, which dominates the phase space volume of the system. Building from studies of chaotic systems such as dynamical billiards, we formulate a specific algorithm with good performance on machine learning and PDE-solving tasks, including generalization. It cannot stop at a high local minimum, an advantage in non-convex loss functions, and proceeds faster than GD+momentum in shallow valleys. 

A bstract We obtain microstates accounting for the Gibbons-Hawking entropy in dS 3 , along with a subleading logarithmic correction, from the solvable T $$ \overline{T} $$ T ¯ + Λ 2 deformation of a seed CFT with sparse light spectrum. The microstates arise as the dressed CFT states near dimension ∆ = c/ 6, associated with the Hawking-Page transition; they dominate the real spectrum of the deformed theory. We exhibit an analogue of the Hawking-Page transition in de Sitter. Appropriate generalizations of the T $$ \overline{T} $$ T ¯ + Λ 2 deformation are required to treat model-dependent local bulk physics (subleading at large central charge) and higher dimensions. These results add considerably to the already strong motivation for the continued pursuit of such generalizations along with a more complete characterization of T $$ \overline{T} $$ T ¯ type theories, building from existing results in these directions.