Gate N-Ins and the inherited kernel

Gate N-Ins is the classical insertion interface through order $\kappa$. The seed is $F_t=⟨t_D,h\otimes h\otimes \varphi ⟩$. The option that sets $\varphi =0$ is rejected, because that specialization kills $F_t$. The completion space is the minimal sector $(h,\varphi ,c;h^{\star },{\varphi }^{\star },c^{\star })$, with

seed $I_t{|}_{{\Phi }^{\star }=0}=F_t$, and no new inverse operators.

The primary constructive definition of $t_D$ is quoted from the predecessors and is not redefined. In the companion transcription the relevant XVI-I labels are eq:t-test and prop:Ft; the science volume's name prop:test is that predecessor statement, not a new label. XVI-H thm:trep is the equivalent polarized form. XII-D thm:root and lem:root-kernel supply the compact root. XVI-K retains the construction data and does not supply a new closed-form kernel. Component tensors of the cubic source remain implicit. They are not a single numerically expanded kernel. Sha256 strings printed beside those files in the science volume are provenance pointers. They are not a byte-identity claim for the copies registered here, and they are not a new premise.

On the invariant free shell $Ah=0$ and $(\square +m^2)\varphi =0$, with compact ghost supported in the plateau where ${\sigma }_{\epsilon }=1$, the science volume records the reduction

The cubic-variation step cites XI-E thm:cubicvar and is re-checked pointwise. It is not re-proved. The nilpotence step cites XI-B lem:nilpotence as a written derivation. Neither citation is a declared input of a new theorem.