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

The KNO-violating term fₛ f s of the charged-particle multiplicity distribution in pp collisions measures the relative deviation of n Pₙ ⟨ n ⟩ P n from e⁻ᶻ e - z , with z=n/ n z = n / ⟨ n ⟩ , and is reported to obey the reciprocal symmetry fₛ(z)=fₛ(1/z) f s ( z ) = f s ( 1 / z ) , taken here as input. Being an evenness condition in ln z ln z , it generates a tower of local constraints on the derivatives of Pₙ 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 fₛ f s that is not invariant under z→ 1/z z → 1 / z , so none of them carries the symmetry. We further obtain a dynamics-independent entropy in the KNO continuum, S=ln n +I₀-1/2∫ e⁻ᶻfₛ²\,dz S = ln ⟨ n ⟩ + I 0 - 1 2 ∫ e - z f s 2 d z , with <jats:tex-mat