Major Conjectures and Cosmological Tests G; The Cosmological Falsifiability of the Twin Prime Conjecture

This paper explores a new paradigm connecting number theory with observational cosmology, aiming to provide a physically principled and falsifiable pathfor the Twin Prime Conjecture. Based on the two fundamental axioms of Information Conservation and Cosmological Computability, we construct an ”InformationalStructure Cosmology” framework. Within this framework, cosmic history is modeled as a binary tree, whose irreducible paths are isomorphic to the distributionof prime numbers. Theoretical derivation shows that if the Twin Prime Conjecture holds (i.e., there exist infinitely many prime pairs with a gap of 2), then thecorresponding ”twin pairs” in the information paths will imprint a characteristicsignal in the matter distribution of the universe’s large-scale structure. This signalmanifests as periodic oscillations with a logarithmically decaying envelope in theresidual of the two-point correlation function over a specific sequence of scales. Thisstudy utilizes simulated data from the DESI and Euclid projects to search for thissignal, finding preliminary agreement but with insufficient statistical significance.The study further clarifies the quantitative criteria for confirming or excluding thisprediction using data from next-generation surveys (e.g., SKA-2). This work establishes, for the first time, an operational, cosmology-observation-based test schemefor this pure mathematical conjecture.