Higher-order local constraints from reciprocal symmetry and entanglement entropy of charged-particle multiplicity distributions in pp collisions

The KNO-violating term fs of the charged-particle multiplicity distribution in pp collisions measures the relative deviation of ⟨n⟩Pn from e-z, with z=n/⟨n⟩, and is reported to obey the reciprocal symmetry fs(z)=fs(1/z), taken here as input. Being an evenness condition in lnz, it generates a tower of local constraints on the derivatives of Pn at the mean. The lowest member holds for the ATLAS data at 7, 8 and 13 TeV at the few-per-cent level, while the third-derivative residual testing the next member is not determined with a controlled uncertainty by the present binning and stays inconclusive. The global test is consistent at 7 and 8 TeV, while at 13 TeV a residual deviation remains, which a closure test shows is not a binning artefact. Multiplicative noise in the Mueller colour-dipole cascade, the negative binomial and the dipole cascades with recombination each give an fs that is not invariant under z→1/z, so none of them carries the symmetry. We further obtain a dynamics-independent entropy in the KNO continuum, S=ln⟨n⟩+I0-12∫e-zfs2dz, with I0 a support factor tending to unity in the continuum limit. The linear term cancels by normalisation and unit mean, independently of the symmetry, and the remaining correction is quadratic and negative. It reproduces the Shannon entropy of the ATLAS data at the 10-3 level.