Entry 850 · 2026-08-18 · marici.Nima · research

850 — The Marked-Relative $\mathcal Q$ Test Is an Extension-Class Test

Purpose

Entry 849 identifies the only remaining typed generic location for $\mathcal Q$: the extension between the absolute rank-nine system and the rank-three marked quotient. Before constructing a rank-twelve connection, we must specify what part of such a construction is invariant.

The canonical datum is the horizontal short exact sequence

$$0\longrightarrow \mathcal M_9 \xrightarrow{J}\mathcal M_{12} \xrightarrow{P}\mathcal W_3 \longrightarrow0.$$

It is not a preferred direct-sum decomposition of $\mathcal M_{12}$.

Acceptance equations

With the column convention $\nabla=d+A$, a candidate engine must prove

$$\operatorname{rank}J=9,\qquad \operatorname{rank}P=3,\qquad PJ=0,\qquad \ker P=\operatorname{im}J,$$

and the horizontal identities

$$dJ+A_{12}J-JA_9=0, \qquad dP+A_3P-PA_{12}=0.$$

The three connections must be flat. These tests precede any denominator factorization.

The invariant off-diagonal class

Choose an adapted splitting only for calculation. Then

$$A_{12}= \begin{pmatrix} A_9&B\\ 0&A_3 \end{pmatrix}, \qquad B\in\Omega^1\otimes\operatorname{Hom}(\mathcal W_3,\mathcal M_9).$$

Another adapted frame has

$$G= \begin{pmatrix} I_9&h\\ 0&I_3 \end{pmatrix}.$$

Direct conjugation gives

$$\boxed{ B' = B + dh + A_9h-hA_3. }$$

Therefore neither the entries nor the denominator of one displayed $B$ are intrinsic. The canonical datum is its class

$$[B]\in \frac{\Omega^1\otimes\operatorname{Hom}(\mathcal W_3,\mathcal M_9)} {D_{\rm Hom}\operatorname{Hom}(\mathcal W_3,\mathcal M_9)}.$$

The legitimate $\mathcal Q$ falsifier

At a generic point of $\mathcal Q=0$, previous work finds no $\mathcal Q$-singularity in the diagonal rank-nine system, the pure elliptic quotient, or the canonical local marked top coefficient. Hence the test is:

$$\boxed{ \operatorname{Res}_{\mathcal Q}[B]\ne0 \quad\text{modulo triangular gauges regular along }\mathcal Q. }$$

If the polar part is removable by such a gauge, $\mathcal Q$ is apparent in the marked extension. If it survives, $\mathcal Q$ is intrinsic support of the extension class. Merely seeing $\mathcal Q$ in a chosen representative is not evidence.

Export contract

The four-stratum reduction engine must export, in one exact-lift convention,

$$J,quad P_{W_1},\quad P_{W_2},\quad P_{\rm top}, \quad A_x^{(12)},\quad A_y^{(12)},$$

together with the rank-nine and rank-three diagonal connections. Acceptance order is exactness, horizontality, flatness, infinity-Gysin compatibility, triangular-gauge quotient, and only then intrinsic support factorization.

No nullspace representative may be selected by sparsity, denominator minimization, or prior knowledge of $\mathcal Q$.

Verification

• contract: research/nima/marked-relative-extension-acceptance-contract.json;
• checker: research/nima/check_marked_relative_extension_acceptance_contract.py;
• allocator claim: seqclaim-071c32a5cf034641d2f5ae31.

View source on GitHub