lichnerowiczFlatEigenvalue
plain-language theorem explainer
Names the flat-background Lichnerowicz eigenvalue on a wavenumber-k transverse-traceless plane wave as (2πk)². Anyone citing discrete-to-continuum TT spectrum convergence on the flat 3-torus uses this as the continuum target. It is a pure MODEL abbreviation: the flat reduction of Δ_L to −∇², not a proved spectral theorem.
Claim. For each natural wavenumber $k$, the flat Lichnerowicz eigenvalue on the corresponding TT plane wave is defined to be $(2\pi k)^2$. On a Ricci-flat (in particular flat) background the Lichnerowicz operator reduces to minus the Laplacian on TT tensors, and that is the continuum eigenvalue of the axis mode on the unit 3-torus.
background
Lane 4 of the Seven-Gaps gravity campaign connects the discrete perturbation spectrum on a lattice to the continuum Lichnerowicz operator, restricted to the flat 3-torus. Lattice fields are $N$-periodic maps $\mathbb{Z}\to\mathbb{C}$ (spacing $h=1/N$), not functions on $\mathrm{ZMod},N$, so stencil identities hold pointwise and periodicity alone supplies the torus reading.
On a Ricci-flat background the Lichnerowicz operator acts on TT perturbations by $\Delta_L h_{ab}=-\nabla^2 h_{ab}-2R_{acbd}h^{cd}$. On the flat 3-torus the Riemann term vanishes, so $\Delta_L$ collapses to $-\nabla^2$. An axis TT plane wave of integer wavenumber $k$ on the unit torus therefore has continuum eigenvalue $(2\pi k)^2$.
All results in this module are axis-sector only: modes $k=(k,0,0)$ under the componentwise axis-stencil Laplacian. The Freudenthal frozen-energy continuum moment is anisotropic, so axis stencils do not recover an isotropic full Lichnerowicz spectrum.
proof idea
There is no proof. The declaration is a one-line noncomputable definition equating the symbol to $(2\pi k)^2$. The surrounding package theorem records that discrete minus-Laplacian eigenvalues of the axis TT modes converge to this value; the identification of the limit with the flat Lichnerowicz spectrum is the MODEL layer encoded here. Justification is the classical flat reduction $\Delta_L\mapsto -\nabla^2$ on TT tensors when Riemann vanishes, plus the elementary continuum second-derivative computation for the along-axis profile (transverse derivatives vanish by $y,z$-constancy).
why it matters
This constant is the continuum flat target for discrete TT spectrum convergence on the axis sector. Downstream, discrete_tt_spectrum_converges_to_flat_lichnerowicz names the limit by this definition. In CurvedOperatorUnderdetermination it is the flat branch of curvedContinuumEigenvalue (flat part plus a scalar curvature coupling), and it appears inside CurvatureCorrectionRateBound as the continuum baseline against which discrete curved corrections are measured.
The blocker theorem curvedSpectrumConverges_iff_curvatureCorrectionConsistent uses that the flat branch already converges: full curved convergence is then equivalent to convergence of the omitted curvature correction alone. Thus the flat MODEL value isolates what the discrete flat theorem cannot discharge. Framework context is operator convergence toward continuum gravity on the forced $D=3$ spatial setting; no claim is made about curved Lichnerowicz geometry or isotropic recovery of the full spectrum (C10/P-iso remains open).
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