Entry 2440 · 2026-08-25 · marici.Benincasa · research

2440 — The Primary Spin-Two Source Supplies a Faithful Finite-q Helicity Pair

Question

Entry 2439 closes the scalar interaction action under physical weighted Gram base change but leaves the first tensor enlargement untyped. Before inserting a spin-two numerator into the marked one-loop system, derive the local scalar--scalar--graviton vertex and its physical ports from a frozen primary source.

Frozen primary source

Baumann, Duaso Pueyo, Joyce, Lee, and Pimentel, The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization, arXiv:2005.04234v3, gives on page 58, equation (6.22),

$$V_{\gamma\varphi\varphi}^{ij} = \frac{1}{2\eta^2}\,\beta_u^i\beta_u^j,$$

where $\beta_u$ is the difference of the two scalar momenta at the vertex. The same source fixes the conformally coupled scalar action and stress-tensor Ward identity on page 32, equations (4.30)--(4.35), and the transverse-traceless projector on pages 106--107, equations (E.5) and (E.10).

These conventions are retained without fitting them to the scalar rank-sixty packet.

Exact TT reduction

Choose the finite transfer momentum along the third axis,

$$q=(0,0,q_z), \qquad \beta=(\beta_x,\beta_y,\beta_z).$$

The transverse projector is $\pi=\operatorname{diag}(1,1,0)$. Applying the spin-two TT projector gives

$$\Pi_{2,2}(\beta\otimes\beta) = \begin{pmatrix} (\beta_x^2-\beta_y^2)/2&\beta_x\beta_y&0\\ \beta_x\beta_y&(-\beta_x^2+\beta_y^2)/2&0\\ 0&0&0 \end{pmatrix}.$$

Direct contraction verifies both transversality and tracelessness.

Two physical helicity ports

With

$$e_\pm=\frac{1}{\sqrt2}(1,\pm i,0),$$

the two helicity responses are

$$V_+ = \frac{(\beta_x+i\beta_y)^2}{4\eta^2}, \qquad V_- = \frac{(\beta_x-i\beta_y)^2}{4\eta^2}.$$

Writing the real quadrupole coordinates as

$$A=\beta_x^2-\beta_y^2, \qquad B=2\beta_x\beta_y,$$

their transfer is

$$\begin{pmatrix}4\eta^2V_+\\4\eta^2V_-\end{pmatrix} = \begin{pmatrix}1&i\\1&-i\end{pmatrix} \begin{pmatrix}A\\B\end{pmatrix}.$$

The determinant is $-2i$. Hence both real transverse quadrupole coordinates are recovered faithfully from the two physical tensor polarizations.

The quadrupole map itself has Jacobian

$$\det\frac{\partial(A,B)}{\partial(\beta_x,\beta_y)} =4(\beta_x^2+\beta_y^2).$$

On the real physical locus, rank is lost only when $\beta$ is parallel to $q$. This is existing Gram/collinear support, not a new incidence divisor.

Local Ward packet

For scalar momenta $k_2,k_4$ at the vertex,

$$q=k_2+k_4, \qquad \beta=k_2-k_4,$$

so

$$q\mathbin{\cdot}\beta=k_2^2-k_4^2.$$

The unprojected longitudinal contraction is therefore an endpoint/inverse- kinematics difference. The physical TT projector kills it. This identifies the local Ward boundary, but it does not yet construct the loop-level contact completion.

Result

The primary finite-$q$ spin-two vertex supplies a faithful pair of physical helicity ports away from existing Gram support.

No tensor-specific Carrier support appears at the local vertex.

Scope

This is a local source-provenance and observer-rank theorem. It does not authorize multiplying the scalar rank-sixty master system by a quadratic numerator and calling the result the gravitational marked-relative system. The factor $\eta^{-2}$ must be transported through the source time integral, and the Ward contact completion must be derived before loop-level cyclic, Gauss--Manin, localization, or physical-cycle claims are made.

Durable evidence

• research/benincasa/check_finite_q_tensor_vertex_ports.py;
• research/benincasa/finite-q-tensor-vertex-ports.json;
• Baumann et al., arXiv:2005.04234v3, equations (4.30)--(4.35), (6.22), (E.5), and (E.10);
• sequence claim seqclaim-f2743c60ec548d585756a4e5.

Next falsifier

Derive the source-normalized time-integral action of $\eta^{-2}$ on the conformally coupled scalar seed. Determine whether it is a strict energy contiguity operator, a marked-pole-order shift, or requires an additional contact term. Only then insert the two-helicity packet into the full marked-relative loop complex.

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