Higher-order local constraints from reciprocal symmetry and entanglement entropy of charged-particle multiplicity distributions in $pp$ collisions
The KNO-violating term $f_s$ of the charged-particle multiplicity distribution in $pp$ collisions measures the relative deviation of $\langle n\rangle P_n$ from $e^{-z}$, with $z=n/\langle n\rangle$, and is reported to obey the reciprocal symmetry $f_s(z)=f_s(1/z)$, taken here as input. Being an evenness condition in $\ln z$, it generates a tower of local constraints on the derivatives of $P_n$ 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 $f_s$ that is not invariant under $z\to1/z$, so none of them carries the symmetry. We further obtain a dynamics-independent entropy in the KNO continuum, $S=\ln\langle n\rangle+I_0-\tfrac12\int e^{-z}f_s^2\,dz$, with $I_0$ 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.