Ambient saturation of a differential ideal

Exact statement

Let $I$ be a minimal nonzero connected-supergroup-invariant two-sided ideal of a finite algebra $A$ , with $I^2=I$ . If $J\triangleleft I$ is nonzero and invariant under every superderivation of $I$ , then $AJA=I$ and $I=IJI\subseteq J$ ; hence $I$ is differentiably simple.

StatusProved and hostile-audited within FPRD Lab

External reviewNo external or specialist review is recorded; the proof has passed a hostile internal audit.

Context

A derivation-stable ideal of I need not itself be an ambient A-ideal. The saturation AJA is the intrinsic bridge that repairs this mismatch.

Hypotheses and scope

The base field is algebraically closed of characteristic zero.
The acting algebraic supergroup is connected.
I is a minimal nonzero invariant two-sided ideal and I squared equals I.

Proof or evidence

Connected integration makes J invariant. The nonzero ambient ideal AJA lies in I, so minimality gives AJA=I. Multiplying on both sides by I yields I=IJI contained in J.

Ambient saturation proof →

Verification notes

This lemma is the repaired step found by the internal audit. The candidate proof's direct appeal to minimality was rejected because J need not be an A-ideal.

Limitations

The lemma does not classify derivations or invariant ideals.
Its use of connected integration depends on finite rational supergroup representations in characteristic zero.

Open work

Seek specialist review of the connected-integration and saturation step.