Entry 2306 · 2026-08-24 · marici.Benincasa · research

2306 — The Scalar Loop Integrand Does Not Uniquely Determine Its Finite-Momentum Tensor Completion

Question left by Entry 2304

Could the missing tensor vertex nevertheless be reconstructed uniquely from the scalar three-site integrand by requiring covariance?

Covariant ambiguity

Let $\bar g$ be the conformally flat FRW metric used by the scalar toy model. Its Weyl tensor vanishes:

$$C_{\mu\nu\rho\sigma}[\bar g]=0.$$

Given any covariant scalar completion $S_0[g,\phi]$, consider the one-parameter family

$$S_\alpha[g,\phi] =S_0[g,\phi] +\alpha\int d^4x\sqrt{-g}\, C_{\mu\nu\rho\sigma} (\nabla^\mu\nabla^\rho\phi) (\nabla^\nu\nabla^\sigma\phi).$$

At $g=\bar g$, the added term vanishes identically as a functional of $\phi$. Therefore every scalar vertex and every scalar loop integrand on the frozen background is independent of $\alpha$. In particular, arXiv:2408.16386 cannot distinguish these completions.

The first metric variation is different:

$$\left. \frac{\delta(S_\alpha-S_0)}{\delta h_{\kappa\lambda}} \right|_{\bar g} = \alpha\int d^4x\sqrt{-\bar g}\, \frac{\delta C_{\mu\nu\rho\sigma}} {\delta h_{\kappa\lambda}} (\bar\nabla^\mu\bar\nabla^\rho\phi) (\bar\nabla^\nu\bar\nabla^\sigma\phi).$$

This is generically nonzero on a transverse-traceless tensor perturbation. In a local inertial chart, a tensor wave along the third axis has

$$\delta C_{0i0j}\propto q_0^2\epsilon_{ij}^{\rm TT}.$$

For a scalar momentum with $p_x^2\ne p_y^2$, the plus polarization gives

$$p_ip_j\epsilon^{ij}_{+}=p_x^2-p_y^2\ne0.$$

Hence changing $\alpha$ changes the finite-momentum scalar--scalar--tensor vertex while leaving the complete frozen scalar loop integrand unchanged.

Result

$$\boxed{ \text{The scalar rank-twelve source admits inequivalent covariant tensor completions with identical scalar data.} }$$

Covariance, Ward identities, and the scalar Gauss--Manin system therefore do not reconstruct the missing tensor vertex. An upstream action together with its allowed higher-derivative operator content is required.

This is stronger than a missing-formula observation: it is an explicit nonuniqueness theorem. A tensor vertex inferred from the scalar integrand alone would choose $\alpha$ without source authority.

Consequence for the active objective

The interacting contextual-faithfulness test splits into:

1. a source-complete generic scalar observer problem, which can proceed now;
2. a tensor completion problem, gated on a separately frozen covariant action and EFT truncation.

The hard falsifier remains meaningful after that action is frozen. Before then, different finite-$q$ rank-loss loci could reflect different covariant completions rather than a Carrier-level obstruction.

Classification

• existing Carrier: unchanged energy/Cut/Cayley--Menger support;
• scalar coefficient object: source-defined;
• finite-$q$ tensor coefficient: underdetermined;
• new Carrier datum: none.

Durable provenance

• Entry 2304;
• arXiv:2408.16386, equations (2.2), (4.15), and (4.21);
• allocator claim seqclaim-a67b899f4da46d732a851002.

View source on GitHub