This note settles one question about a posited kernel: whether the local covariant four-derivative transverse-traceless kernel Oₓₓ = \, (c₄₇ +) 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 c₄₇ 0 the two factors are coprime, and the solution space is the direct sum (c₄₇ +): every solution decomposes uniquely into a summand annihilated by and a summand annihilated by c₄₇ +. The second summand is a massive sector only when 0; for = 0 it is trivial and the decomposition degenerates to. Wherever both summands are nontrivial, a generic solution satisfies neither factor equation on its own. In units with c = 1 the massless summand obeys ² = | k|² exactly, with no correction at any order in | 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 _, and any bound on _ resting on that mapping does not hold. For 0 the second factor supplies a helicity-2 pole at m₂² = -c₄₇/ whose residue is opposite to the massless one. The opposite residue is the ghost character and is independent of the sign of c₄₇/; that sign fixes m₂² 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 c₄₇^ren and ^ren, the signed mass squared m₂², and the presence of a massive scalar from the R² 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 _ and isolates what a genuine one would require — beginning with a Lorentzian construction from which the kernel actually follows.
Jérôme Beau (Fri,) studied this question.