A portfolio of 1,000 loans. Each has its own default probability, loss given default, exposure at default. You need the distribution of total losses — not the mean (easy) and not the variance (doable), but the full shape, including the tail where catastrophic losses live. The exact distribution is a convolution of 1,000 individual distributions. Computing it directly requires a grid fine enough to resolve the tail, which at high resolution means billions of grid points.
Rodríguez-Aldavero and García-Ripoll (arXiv:2603.23106) observe that the characteristic function — the Fourier transform of the distribution — factorizes when the variables are independent. Each loan contributes a multiplicative term. The product of 1,000 smooth functions in Fourier space has structure that tensor networks can exploit: quantized tensor train (QTT) representations achieve exponential compression of the characteristic function, enabling discretization at N = 2³⁰ frequency modes with polylogarithmic cost.
The compression doesn't appear immediately. For small portfolios (few components), the characteristic function is incompressible — the tensor has no low-rank structure to exploit. But above roughly 300 components, a sharp bond-dimension collapse occurs: the QTT rank drops dramatically, and the representation becomes efficient. The compression is a phase transition in the information geometry of the aggregate distribution.
This is counterintuitive. More components, easier computation. The aggregate of many independent variables develops internal structure — the central limit theorem ensures the characteristic function becomes smoother and more compressible as components accumulate. The difficulty lives in the small-portfolio regime, where each component's idiosyncratic shape matters. At scale, universality takes over, and the tensor network finds the low-rank structure that universality creates.