planeH
plain-language theorem explainer
Axis plane wave on the discrete 3-torus: a 1D Fourier mode along the first lattice coordinate, tensored with a constant 3×3 polarization matrix. Gravity and spectral analysts cite it as the test field for axis-sector discrete Lichnerowicz identities. The body is a one-line scalar multiplication of the 1D mode by the polarization.
Claim. For lattice size $N\in\mathbb{N}$, wavenumber $k\in\mathbb{Z}$, and constant polarization $\varepsilon\in M_3(\mathbb{C})$, the axis plane wave is the lattice tensor field $x\mapsto e^{2\pi i k x_1/N}\,\varepsilon$ on sites $x=(x_1,x_2,x_3)\in\mathbb{Z}^3$.
background
Module setting is Seven-Gaps Lane 4: connect the discrete perturbation spectrum on a lattice to the continuum Lichnerowicz operator on the flat 3-torus. All results here are axis-sector only (plane waves with wavevector $(k,0,0)$ under the componentwise axis-stencil Laplacian). Axis stencils are blind to Freudenthal anisotropy, so these are not isotropic full-spectrum recovery.
Lattice sites are $\mathbb{Z}^3$ with $N$-periodicity along each axis; fields are $N$-periodic maps $\mathbb{Z}\to\mathbb{C}$ rather than functions on $\mathrm{ZMod},N$. A lattice tensor field assigns a $3\times 3$ complex matrix to every site. The 1D Fourier mode is $j\mapsto\exp(2\pi i k j/N)$, already known to be $N$-periodic on the discrete circle.
proof idea
Definition, not a theorem. At site $x$, evaluate the 1D Fourier mode at the first coordinate $x_1$ and left-scale the constant polarization matrix by that complex scalar. No lemmas are invoked in the body; downstream lemmas unfold this definition and apply Fourier-mode periodicity or the 1D discrete-Laplacian eigenvalue identity.
why it matters
Canonical test section for the axis-sector discrete Lichnerowicz package. It is the input field for the 3D eigenvector identity (axis plane wave is an eigenfield of the 3D discrete Laplacian with the same eigenvalue as the 1D mode, transverse stencils acting trivially), for axis periodicity and $y,z$-shift invariance, and for the packaged convergence theorem that discrete TT axis modes stay discrete-transverse at every $N$ and converge to the flat Lichnerowicz spectrum along the axis sector. Downstream, the curved countermodel operator is evaluated on these modes. Closes the representation step of Lane 4 operator convergence on the flat torus; does not touch the direction-resolved isotropy probe (C10).
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