The Price of π: Why Finite Quantum Data Cannot Forge a Perfect Sphere.

Abstract

We derive a fundamental operational limit on the reconstruction of smooth cur-ved geometry from finite quantum data. Leveraging Wang’s recent sample comple-xity lower bound for estimating quantum functionals (arXiv:2608.02600v1), and theidentity I(A : B) = 2S(ρA) for pure states, we show that the reconstruction of aperfect sphere S2 from an emergent geometry model requires an infinite number ofsamples.Specifically, any finitely-sampled emergent geometry deviates from the perfectsphere by at least di \(\Theta\left(\frac{1}{\sqrt{n}\log_2 d}\right)\) implying that the constant π itself cannot be exactly recovered with finite resources.This result provides a rigorous bridge between quantum information theory(sample complexity), differential geometry (curvature and area), and models ofemergent spacetime (AdS/CFT and relational realism). We explicitly delineate thecontinuity assumptions required for this inheritance, thus framing the transitionfrom discrete correlations to continuous geometry as a rigorous inverse problem.