Entry 2446 · 2026-08-25 · marici.Benincasa · research

2446 — The Physical Tensor Plane Is the Exact Kernel of the Finite-$q$ Ward--Trace Map

Question

Entry 2445 embeds the parity-even tensor numerator into the faithful rank-seven interaction action. What does the source stress-tensor Ward identity require of the physical TT port, and where do its contact terms live?

Primary Ward identity

Baumann et al., equation (4.30), gives

$$z_iq_j\langle T^{ij}_q\varphi_{k_2}\varphi_{k_3}\rangle =-kappa_2(z\mathbin\cdot k_2) \langle\varphi_{k_2+q}\varphi_{k_3}\rangle -\kappa_3(z\mathbin\cdot k_3) \langle\varphi_{k_2}\varphi_{k_3+q}\rangle.$$

The same source derives

$$\kappa_2=\kappa_3=\kappa.$$

Thus the Ward target is a pair of labelled scalar two-point contact terms. It is not another tensor polarization.

Exact finite-$q$ complex

Put $q=(0,0,q)$ and order symmetric-tensor coordinates as

$$(T_{xx},T_{yy},T_{zz},T_{xy},T_{xz},T_{yz}).$$

The divergence map is

$$W(T)=q(T_{xz},T_{yz},T_{zz}).$$

Appending the trace gives a rank-four map from the rank-six symmetric tensor space. Its kernel has rank two and is exactly

$$\boxed{ \ker(W\oplus\operatorname{tr}) = \langle T_{xx}-T_{yy},T_{xy}\rangle. }$$

This is the two-polarization TT plane of Entry 2440. Equivalently,

$$W\Pi_{\rm TT}=0, \qquad \operatorname{tr}\Pi_{\rm TT}=0, \qquad \operatorname{rank}\Pi_{\rm TT}=2.$$

Endpoint lift

For $q\ne0$, the divergence map has a sparse right inverse obtained from

$$T_{xz}=J_x/q, \qquad T_{yz}=J_y/q, \qquad T_{zz}=J_z/q.$$

Its only denominator is $q$. Hence the sole failure of this local endpoint lift is existing tensor-soft support. Different full stress-tensor lifts differ by TT homogeneous solutions and trace/improvement choices, exactly as required by the primary source.

Result

The physical finite-$q$ tensor plane is a strict Ward cocycle; the scalar contact terms live in the complementary longitudinal sector.

Therefore the Ward/contact completion does not modify the TT helicity observer class. It is nevertheless required for a full unprojected stress-tensor object.

The construction is cyclically natural: transporting the labelled tensor site transports the same local complex and preserves the equal coupling.

Scope

This entry types the local Ward differential and endpoint target. It does not compute the loop-corrected scalar two-point contact periods, nor does it construct a global longitudinal lift through the rank-sixty Gauss--Manin system. Those data are unnecessary for the strict TT kernel statement but remain necessary for a complete unprojected tensor correlator.

Durable evidence

• research/benincasa/check_tt_ward_kernel_endpoint_lift.py;
• research/benincasa/tt-ward-kernel-endpoint-lift.json;
• Baumann et al., arXiv:2005.04234v3, equations (4.30)--(4.35);
• Entries 2440 and 2445;
• sequence claim seqclaim-72a9f71aa5f0430031ef0aa7.

Next falsifier

Test Gauss--Manin compatibility of the TT projector and the rank-seven tensor-trace action over the unequal-energy base. Then specialize their supported observer cone at soft, external Gram, Cayley--Menger/Landau, marked-wall, total-energy, and elliptic loci.

View source on GitHub