Entry 2441 · 2026-08-25 · marici.Benincasa · research

2441 — The Conformal Scalar Modes Cancel the Tensor Vertex Time Weight

Question

Entry 2440 derives the finite-$q$ tensor numerator but leaves its $\eta^{-2}$ factor unresolved. Does this factor raise a marked energy-pole order, define a new contiguity operator, or cancel against the frozen scalar mode functions?

Frozen source modes

For a boundary scalar of weight $\Delta=2$, the bulk-to-boundary propagator in Baumann et al., equations (6.13)--(6.14), specializes to

$$\mathcal K_2(k,\eta) = \frac{\eta}{\eta_\star} e^{ik(\eta-\eta_\star)},$$

up to the common late-time normalization convention. The primary local vertex is

$$V_{\gamma\varphi\varphi}^{ij} = \frac{\beta^i\beta^j}{2\eta^2}.$$

Exact cancellation

Multiplying the vertex by its two conformally coupled scalar modes gives

$$\frac{1}{2\eta^2} \mathcal K_2(k_2,\eta) \mathcal K_2(k_4,\eta) = \frac{e^{-i(k_2+k_4)\eta_\star}}{2\eta_\star^2} e^{i(k_2+k_4)\eta}.$$

There is no residual power of $\eta$. With the damped Bunch--Davies past contour,

$$i\int_{-\infty}^{0}d\eta\, e^{i(E-i\epsilon)\eta} = \frac{1}{E-i\epsilon}.$$

Therefore the local insertion preserves the ordinary simple exponential energy seed. It does not by itself raise a marked pole order.

The two physical responses are consequently the same scalar time seed multiplied by

$$\frac14(\beta_x+i\beta_y)^2, \qquad \frac14(\beta_x-i\beta_y)^2.$$

Ward completion is not local multiplication

The cancellation does not imply that the complete tensor system is the scalar marked-relative system tensored with a free quadratic numerator. The same primary source proves that individual exchange channels are not gauge-invariant. Its Ward identity fixes the relative channel and contact normalizations, including

$$\kappa_g=\kappa_c=\kappa.$$

Thus two distinct statements must remain separated:

$$\boxed{ \text{local time-weight cancellation} } \qquad\text{and}\qquad \boxed{ \text{global Ward/contact completion across channels}. }$$

Result

The conformally coupled scalar modes cancel the finite-$q$ tensor vertex's $\eta^{-2}$ weight exactly.

The first tensor enlargement changes the coefficient numerator and physical polarization ports, but introduces no local energy-pole shift and no new Carrier support.

Scope

This theorem applies to the source $Delta=2$ scalar--scalar--graviton vertex. It does not type massless-scalar external legs, the graviton bulk-to-bulk propagator, or the full one-loop Ward completion. In particular, the source itself shows that nested graviton exchange contains additional subleading total-energy poles and correlated contact terms.

Durable evidence

• research/benincasa/check_cc_scalar_tensor_time_weight.py;
• research/benincasa/cc-scalar-tensor-time-weight.json;
• Baumann et al., arXiv:2005.04234v3, equations (6.13)--(6.14), (6.22), (6.48)--(6.51), and the channel-completion discussion;
• sequence claim seqclaim-d9c4ee0f9ca4f7aeebeb882e.

Next falsifier

Construct the labelled loop insertion of the quadratic momentum numerator on the frozen three-site marked-relative integrand. Retain both helicities and every occurrence before summation. Then derive the Ward/contact completion as a map between source-labelled channels rather than fitting a gauge-invariant answer after reduction.

View source on GitHub