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In this note we study the SYK model with one time point, recently considered by Saad, Shenker, Stanford, and Yao. Working in a collective field description, they derived a remarkable identity: the square of the partition function with fixed couplings is well approximated by a ``wormhole'' saddle plus a ``pair of linked half-wormholes'' saddle. It explains factorization of decoupled systems. Here, we derive an explicit formula for the half-wormhole contribution. It is expressed through a hyperpfaffian of the tensor of SYK couplings. We then develop a perturbative expansion around the half-wormhole saddle. This expansion truncates at a finite order and gives the exact answer. The last term in the perturbative expansion turns out to coincide with the wormhole contribution. In this sense the wormhole saddle in this model does not need to be added separately, but instead can be viewed as a large fluctuation around the linked half-wormholes. 

We consider previously derived upper and lower bounds on the numberof operators in a window of scaling dimensions [\Delta - \delta,\Delta + \delta] [ Δ − δ , Δ + δ ] at asymptotically large \Delta Δ in 2d unitary modular invariant CFTs. These bounds depend on a choice offunctions that majorize and minorize the characteristic function of theinterval [\Delta - \delta,\Delta + \delta] [ Δ − δ , Δ + δ ] and have Fourier transforms of finite support. The optimization of thebounds over this choice turns out to be exactly the Beurling-Selbergextremization problem, widely known in analytic number theory. We reviewsolutions of this problem and present the corresponding bounds on thenumber of operators for any \delta \geq 0 δ ≥ 0 .When 2\delta \in \mathbb Z_{\geq 0} 2 δ ∈ ℤ ≥ 0 the bounds are saturated by known partition functions withinteger-spaced spectra. Similar results apply to operators of fixed spinand Virasoro primaries in c>1 c > 1 theories.