The Millennium Prize problem regarding the global regularity of Navier-Stokes solutions is traditionally framed as a challenge of analysis within the continuous domain (ℝ3). This paper argues that finite-time singularities (blow-up) are not physical failures but logical manifestations of the "Continuum Fallacy"—the erroneous assumption of infinite spatial divisibility (C∞). We propose that the Navier-Stokes equations are incomplete without an intrinsic scale constraint dictated by the hardware resolution of reality. To resolve this, we evolve the previous heuristic framework of Version 5 into a rigorous modular formulation by introducing the Ramanujan-Hernández Packing Factor (ΦRH). Derived from the fundamental ground state (n=0) of Ramanujan's 1914 modular series for 1/π, ΦRH establishes the isotropic topological stiffness of the vacuum. This formalizes the exact physical value of the Hernández-Valdivia Limit (εHV = lP2 / ΦRH ≈ 10-70 m2). By augmenting the classical equations with a non-linear topological dissipation term governed by the Frobenius norm of the velocity gradient tensor, we enforce a mechanism of Hydrodynamic Cosmic Censorship. This adaptive hyperviscosity activates strictly at the limit of the continuum approximation, converting potential mathematical divergences into a deterministic vorticity saturation state. Furthermore, our derived limit naturally recovers the holographic Kovtun-Son-Starinets (KSS) bound and predicts quantized Landau-Ramanujan Oscillations in the vorticity of high-energy Quark-Gluon Plasma (LHC Run 4), rendering the fluid system a consistent, computable finite-state machine. Version 6 (Major Theoretical Update): Mathematical Rigor & Physical Unification Modular Formalization: Replaced the heuristic quantum derivation of the spatial cutoff with the exact Ramanujan-Hernández Packing Factor (ΦRH), providing a strict geometric foundation for the Hernández-Valdivia Limit. Isotropy Preservation: Demonstrated that the modular framework bypasses grid anisotropy vulnerabilities, preserving Lorentz invariance and rotational symmetry. Experimental Predictions: Introduced explicit empirical signatures for LHC Run 4 (ALICE/CMS), predicting quantized Landau-Ramanujan Oscillations governed by the Dedekind Eta function η(τ). Cosmological & Yang-Mills Integration: Expanded the paradigm to model the Yang-Mills Mass Gap via topological friction and identify Dark Energy as the emergent bulk viscosity of the modular vacuum.
Carlos Mariano Hernández Valdivia (Fri,) studied this question.