Theorem: Conditional free-fiber-preserving quantum quotient

For Definition def:quantuminput, the quotient

has a well-defined nilpotent induced derivation precisely when

Under the stated free-fiber identities,

with the identity on free generators. If additionally

the quotient has no $\kappa$-torsion.

These conditions do not prove causal factorization, renormalized time ordering, positivity, or existence of the intended analytical representation.

Proof. Ideal preservation is exactly representative independence. The second condition is exactly square-zero after quotienting. The free-fiber statement is the third isomorphism theorem applied to $\mathfrak{T}/(\hat{J}+\kappa \mathfrak{T})$. The saturation condition is precisely the assertion that $\kappa [F]=0$ implies $[F]=0$. The additional analytical and physical properties do not occur among these algebraic hypotheses.