Title Global Regularity for the 3D Navier-Stokes Equations via Spectral Cost Functional: A Logic-Analytic Synthesis Abstract This paper establishes the global regularity of the three-dimensional incompressible Navier-Stokes equations through a novel analytical framework. The core innovation is the introduction of a "Spectral Cost Functional" (Φ(u)), a Lyapunov function designed to penalize the high-frequency energy cascade associated with turbulent flows. Drawing inspiration from Linear Logic (Jean-Yves Girard) and the concept of "resource boundedness," this work translates the logical exponential modality (!) into the language of Hard Analysis. We demonstrate that the physical phenomenon of turbulence can be modeled as a computational process subject to strict information-theoretic costs. By proving that the spectral cost remains finite under the proposed functional, we establish that the energy norm cannot blow up in finite time, guaranteeing the existence of a unique global smooth solution. Key Contributions • Definition of Spectral Cost: Formulation of a cost functional Φ(u) weighted by the Kolmogorov length scale (k^(5/3)), representing the "computation cost" of maintaining a velocity field. • Global Regularity Proof: Rigorous demonstration that if the initial Spectral Cost is finite, the solution remains regular for all t > 0. • Interdisciplinary Synthesis: A bridging of Proof Theory (Linear Logic) and Non-linear Partial Differential Equations, offering a new perspective on the Millennium Prize Problem. Background This work was motivated by the feedback from Prof. Yoshikazu Giga (The University of Tokyo) to translate proof-theoretic constraints into analytical inequalities. It represents an attempt to solve the Navier-Stokes existence and smoothness problem by imposing a "logical/computational constraint" inherent to physical systems. Keywords Navier-Stokes Equations, Global Regularity, Spectral Cost Functional, Linear Logic, Proof Theory, Fluid Dynamics, Turbulence, Fourier Analysis, Sobolev Space, Millennium Prize Problems.