Arithmetic screening of rational PMNS columns from finite residual symmetries

Precision measurements of lepton mixing sharpen tests of finite residual symmetries. Without group reconstruction, we derive a local \(p\)-adic exclusion certificate for columns of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix with rational squared moduli. Two prime conditions on reduced reflection weights exclude trace integrality for the column reflection times any nonscalar finite-order diagonal charged-lepton residual. For three nonzero, nondegenerate Majorana masses, no neutrino residual chain, including generalized charge conjugation and parity (CP) transformations, in a finite group containing such a charged-lepton residual can then fix these squared moduli through symmetries alone. Columns not excluded need not be realizable. For rational squared moduli all below \(\tfrac{1}{2}\), an archimedean bound and Kronecker's theorem show that only trimaximal and quarter-entry columns pass the reflection test. The certificate covers columns with a squared modulus above \(\tfrac{1}{2}\) when its prime conditions hold. A bounded data-derived second-column survey records the local and reflection-test verdicts separately; finite generator pairs found there need not predict a unique column. For the target squared moduli \((\tfrac{11}{36},\tfrac{25}{64},\tfrac{175}{576})\), extensions of the certificate exclude finite realizations of the column-isolating linear family and the specified antiunitary family with symmetric unitary part and root-of-unity spectral ratio. We exclude rigid stabilizer-eigenline alignments of the target column at \(\tau=\omega=e^{i\frac{2}{3}\pi}\) for seven triplet types at modular levels \(3,4,5\) in the canonical \(T\)-diagonal charged-lepton basis. Generic values of \(\tau\) and coupling-selected directions are not covered.