Selective normality and algebraic rigidity of multiplicative traces

A completely multiplicative sign function may be changed so that two quadratic traces remain identical at every index while a third becomes normal. We classify this freedom for the traces $n^2+1$, $n^2+2$ and $n^2-2$. After two traces and a finite prefix have been protected, a monic irreducible polynomial can be made normal from every prescribed source exactly when the field generated by one of its roots contains neither protected quadratic field. In the complementary case, every value of the polynomial is already fixed. A single decidable set of primes of relative density zero supports a continuous, relatively computable completion treating all admissible polynomials jointly, with uniform bounds for each fixed observation family. The proof combines weighted root counts with a sieve for moments whose order increases with the observation scale, followed by a selection using exact finite normalisers. We also give a sufficient condition on prime availability, explicit sparser reservoirs, and a divergent reservoir on which some sources have no completion with a convergent ordinary mean, despite convergence of all fixed word laws in probability. For larger protected families, the obstruction is governed by odd products of quadratic square classes. An explicit example distinguishes certificates for every numerical value from a rational norm identity in the variable.