No Quartic Graviton Dispersion from the Local Covariant Four-Derivative Truncation

This note settles one question about a posited kernel: whether the local covariant four-derivative transverse-traceless kernel ${\mathcal{O}_{\mathrm{TT}}} = \Box\,({c_{\mathrm{EH}}} + \beta\Box)$ deforms the dispersion of the graviton. It does not. The kernel is an assumption, not a consequence of the projective spectral action: a Lorentzian counterpart of that action requires an in-in construction on a closed time path, which is open. Every statement here is conditional on the posit. When ${c_{\mathrm{EH}}} \neq 0$ the two factors are coprime, and the solution space is the direct sum $\ker\Box \oplus \ker({c_{\mathrm{EH}}} + \beta\Box)$: every solution decomposes uniquely into a summand annihilated by $\Box$ and a summand annihilated by ${c_{\mathrm{EH}}} + \beta\Box$. The second summand is a massive sector only when $\beta \neq 0$; for $\beta = 0$ it is trivial and the decomposition degenerates to $\ker\Box$. Wherever both summands are nontrivial, a generic solution satisfies neither factor equation on its own. In units with $c = 1$ the massless summand obeys $\omega^2 = |\vec k|^2$ exactly, with no correction at any order in $|\vec k|$. A quartic deformation of the graviton dispersion is therefore not a consequence of this kernel. It follows that no coefficient of a quartic deformation is induced, hence no mapping of such a coefficient onto the pre-geometric scale $\ell_\chi$, and any bound on $\ell_\chi$ resting on that mapping does not hold. For $\beta \neq 0$ the second factor supplies a helicity-$2$ pole at $m_2^2 = -{c_{\mathrm{EH}}}/\beta$ whose residue is opposite to the massless one. The opposite residue is the ghost character and is independent of the sign of ${c_{\mathrm{EH}}}/\beta$; that sign fixes $m_2^2$ and hence whether the pole is also tachyonic. These are two distinct statements. The transverse-traceless calculation alone does not establish the full massive spin-$2$ multiplet, since the remaining sectors are not treated here; the resemblance to quadratic gravity is a comparison, not an identification. The finite values of ${c_{\mathrm{EH}}}^{\mathrm{ren}}$ and $\beta^{\mathrm{ren}}$, the signed mass squared $m_2^2$, and the presence of a massive scalar from the $R^2$ sector are matching data. Interpretive status. This is a negative result about a posited truncation, and its scope is exactly that. It removes a putative interferometric handle on $\ell_\chi$ and isolates what a genuine one would require — beginning with a Lorentzian construction from which the kernel actually follows.