The full-character induction has an essential order. First construct the full native character, then transfer it to a higher multiplicity, add the actual correction on its forced domain, and extend at the next maximal native parameter field.
On a scalar \(u\in U_F^1\), multiplicity \(p\) transfer gives
The correction is trivial on the scalar intersection by trace zero in the odd case, or integral trace and the depth inequality in the dyadic case. The forced deep character therefore agrees with the imposed trivial scalar character on their intersection. A finite abelian unit-character extension supplies the full next character.
For an odd dyadic critical level \(r\), put \(m=\lfloor r/2\rfloor\). The endpoint satisfies \(2(m+1)-r=1\), making the additive cross term lie in the character's radical.
Scope
Checking a leading trace alone does not check shallow forced values. The earlier fixed-field central argument failed on an actual scalar unit; the recursive native-field proof repairs that problem.