Entry 2449 · 2026-08-25 · marici.Benincasa · research

2449 — The Soft Tensor Port Is the First Scalar Momentum Score of the Ward Endpoint

Question

Entry 2448 closes the generic Gram wall away from soft support. At tensor momentum $q=0$, however, Entry 2446's longitudinal endpoint lift contains an apparent $1/q$ pole and the TT direction is undefined. Is this a new soft--Gram obstruction?

Ward endpoint expansion

Let the scalar two-point coefficient be isotropic,

$$\langle\varphi_k\varphi_{-k}\rangle=F(k),$$

and put

$$q=\epsilon n, \qquad k_2=k, \qquad k_3=-k-q.$$

The primary Ward identity of Entry 2446 has endpoint target

$$J_i(q) =-kappa \left[ k_iF(|k+q|)+(-k_i-q_i)F(k) \right].$$

Its exact first soft expansion is

$$\boxed{ J_i(q) =-kappa\epsilon \left[ k_i\frac{F'(k)}k(k\mathbin\cdot n)-n_iF(k) \right] +O(\epsilon^2). }$$

The zeroth grade vanishes. Therefore Entry 2446's apparent $1/q$ endpoint lift is removable on the source Ward target.

Soft tensor lift

A symmetric lift of the first grade is

$$T^{\rm soft}_{ij} =-kappa \left[ \frac{F'(k)}k k_i k_j-F(k)\delta_{ij} \right].$$

Contracting with $n_j$ reproduces the Ward target. TT projection relative to $n$ kills the isotropic $F(k)\delta_{ij}$ term exactly, leaving

$$\boxed{ \Pi_{\rm TT}T^{\rm soft} =-kappa\frac{F'(k)}k \Pi_{\rm TT}(k\otimes k). }$$

Thus the soft tensor port is the first radial momentum score of the scalar two-point coefficient.

For the conformally coupled scalar normalization $F(k)=k$, one has $F'(k)=1$. In a frame with soft direction $n=\hat z$ and hard momentum in the $(x,z)$ plane, the plus response is

$$-\frac{\kappa k_x^2}{2k}.$$

It is nonzero generically and vanishes only when the hard momentum is parallel to the soft direction, an existing collinear Gram locus.

Result

$$\boxed{ \text{the leading soft tensor class is recovered by the existing scalar momentum-score/Ward endpoint port.} }$$

There is no new soft--Gram Carrier support and no unresolved $1/q$ residue at this grade.

Scope

This is the universal leading soft theorem. It does not fix subleading soft orders or the finite renormalized part of a loop-corrected scalar two-point function. Those require their own source normalization and cannot be inferred from $F(k)=k$.

Durable evidence

• research/benincasa/check_soft_tensor_ward_score_recovery.py;
• research/benincasa/soft-tensor-ward-score-recovery.json;
• Baumann et al., arXiv:2005.04234v3, equation (4.30);
• Entries 2446 and 2448;
• sequence claim seqclaim-73c928c064d14c6bb3a983cb.

Next falsifier

Transport the surviving tensor trace and the Ward endpoint score through the marked localization sequence and total-energy nearby cycle. Then test their compatibility with the elliptic degeneration and existing Landau support.

View source on GitHub