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At equal time, leading loop contributions from a very long-wavelength displacement cancel between perturbative diagrams. This is required by the Equivalence principle, because a coherent long displacement merely translates short-scale structure.

The leading infrared terms cancel in the observable sum:This infrared cancellation in large-scale structure follows from the Equivalence principle: a sufficiently long-wavelength displacement translates short-scale structure without changing an equal-time power spectrum.

The ultraviolet part of $2P_{13}$ is proportional to $k^{2}P(k)$ and depends on the cutoff $\Lambda$. The leading deterministic effective field theory of large-scale structure counterterm has exactly this shape,Its cutoff-dependent part can be chosen aswhich cancels the stated $2P_{13,\mathrm{UV}}$. The $P_{22}$ ultraviolet contribution begins at $k^4$ and, when cutoff sensitive, is absorbed by higher-derivative or stochastic counterterms.

Physically, coarse graining the matter equations does not justify setting the short-scale stress ${\sigma }_{ij}$ to zero. Unresolved multistreaming and nonlinear motion generate an effective pressure and viscosity. Their leading derivative contribution to the Euler equation is proportional to $-c_{\mathrm{eff}}^2{\nabla }_i\delta$, producing the required $k^2\delta$ correction and making long-distance predictions independent of the arbitrary cutoff.

For emission of a soft photon of momentum $q$ from external charged particles, the Soft photon theorem giveswhere ${\eta }_i=+1$ for outgoing and $-1$ for incoming particles. Contracting with $q_{\mu }$ and applying the Ward identity givesFor a nonzero underlying amplitude, total outgoing charge equals total incoming charge. Thus photon gauge redundancy enforces conservation of electric charge.

The analogous Soft graviton theorem for helicity two requires universal coupling to energy-momentum and yields conservation of total four-momentum, the seed of the Equivalence principle. For massless helicity greater than two, the corresponding soft consistency conditions demand conserved higher-rank momentum charges that generic interacting scattering cannot satisfy. Under the assumptions of a Lorentz-invariant local S-matrix with long-range interactions, such particles therefore have no nontrivial coupling; this is the soft-theorem obstruction to interacting massless higher-spin particles.