Mode3
plain-language theorem explainer
Mode3 supplies the index set of integer triples for Fourier wavevectors on the three-torus in a spectral Galerkin truncation of the Navier-Stokes equations. Numerical analysts and fluid dynamicists working on energy identities and enstrophy bounds in finite-mode 3D flows cite this abbreviation when constructing velocity fields and dissipation operators. The declaration is a bare type alias with no computational content or proof obligations.
Claim. Let $M$ denote the set of all triples of integers, written $M = (k_x, k_y, k_z) : k_x, k_y, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z, k_z
background
The module IndisputableMonolith.NavierStokes.Galerkin3D extends the two-dimensional Galerkin truncation to three spatial dimensions on the torus T³. It indexes Fourier modes by integer triples and records the energy identity for the projected nonlinearity together with the non-positive inner product of the velocity field against its Laplacian. The finset of admissible modes for truncation radius N is formed by the Cartesian product of three copies of the integer interval [-N, N]. The squared wave-number magnitude of any such triple is obtained by summing the squares of its three components, and this quantity multiplies the velocity coefficients to produce the viscous term. The module documentation states that the construction yields an enstrophy bound controlled by νN² and connects to the discrete sub-Kolmogorov framework.
proof idea
The declaration is a direct type abbreviation that equates the symbol to the Cartesian product of three copies of the integer type. No lemmas are applied and no tactics are executed; the abbreviation simply introduces the index type used by every subsequent definition in the file.
why it matters
Mode3 is the foundational index type for the entire 3D Galerkin construction. It is referenced by the definitions of squared wave-number magnitude, the Laplacian multiplier, spectral enstrophy, and the non-positivity lemma for viscous dissipation. These objects in turn support the energy-skew property of the nonlinearity and the discrete energy bounds that appear in the Recognition Science treatment of Navier-Stokes. The construction fills the Galerkin-approximation step described in RS_NavierStokes_BKM.tex §4 and supplies the finite-mode setting in which enstrophy remains controlled by the truncation radius.
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