Major Conjectures and Cosmological Tests D; The Falsifiable Path of Hilbert's Sixth Problem

This paper proposes a derivation path based on two fundamental axioms, “Information Conservation” and “Computability”, aiming to solve the core challengein Hilbert’s sixth problem: “rigorously deriving macroscopic continuum equationsfrom microscopic particle dynamics”. By introducing computational complexityas a physical selection criterion, the framework judges “microscopic histories thatrequire infinite computational resources to distinguish” as having no contributionto macroscopic reality, thereby exponentially suppressing pathological structures(such as complex recollision diagrams) that hinder rigorous derivation. The Information Conservation axiom further uniquely couples the statistical average ofthe surviving historical set, after screening, to the macroscopic geometric and fluiddescriptions, giving rise to the structures of the Einstein field equations and theNavier-Stokes equations. This theory yields a testable quantitative prediction: theprimordial gravitational wave power spectrum contains a periodic modulation signal with an amplitude of approximately 0.08% to 3% and a characteristic scalearound 10−3 Mpc−1. This signal falls precisely within the sensitivity window of thenext generation of astronomical observation facilities (such as LISA and CMB-S4),transforming this originally abstract mathematical problem into a proposition thatmay receive empirical adjudication within the next decade.