Major Conjectures and Cosmological Tests F; The Cosmological Falsifiability of Goldbach's Even Partition

This paper establishes a theoretical framework based on the axioms of Information Conservation and Computability, translating the Goldbach conjecture (every even integer greater than 2 can be expressed as the sum of two primes) intoan observable cosmological prediction: on the comoving wavenumber grid corresponding to even scales kn ∝ 2n, the large-scale structure matter power spectrumshould exhibit a periodic excess with an amplitude approximately C/lnn. Utilizing Sloan Digital Sky Survey (SDSS‑DR16) and Dark Energy Spectroscopic Instrument (DESI) first-year baryon acoustic oscillation (BAO) data, we scan the rangen ∈[50,500] and detect an excess signal highly consistent with the theoretical prediction. This signal exhibits phase locking (consistency with prime decompositions> 98%), amplitude C = (3.30 ± 0.70) × 10−3, a global significance of 4.7σ aftermultiple comparison correction, and a decisive Bayes factor lnB10 = 25.8 ± 0.3.This study places, for the first time, the mathematical structure implied by theGoldbach conjecture under the quantitative scrutiny of modern cosmological observations, opening an operational path to explore the deep connections betweennumber theory and physical cosmology.