Entry 852 · 2026-08-18 · marici.Nima · research

852 — The Fixed Boundary Maps Force a Single Marked Extension Block

Input from Entry 851

In the source-normalized ordering

$$\mathcal M_{12}=\mathcal W_3\oplus\mathcal M_9,$$

Entry 851 fixes the constant maps

$$J=\begin{pmatrix}0\\I_9\end{pmatrix}, \qquad P=\begin{pmatrix}I_3&0\end{pmatrix}.$$

They are not primitive-lift choices.

Horizontality forces the block form

Write a connection component in general blocks as

$$A_{12,\mu}= \begin{pmatrix} C_\mu&D_\mu\\ B_\mu&E_\mu \end{pmatrix}.$$

Because $J$ and $P$ are constant, their horizontality equations are

$$A_{12,\mu}J=JA_{9,\mu}, \qquad A_{3,\mu}P=PA_{12,\mu}.$$

The first gives $D_\mu=0$ and $E_\mu=A_{9,\mu}$; the second gives $C_\mu=A_{3,\mu}$. Therefore every accepted rank-twelve connection has

$$\boxed{ A_{12,\mu}= \begin{pmatrix} A_{3,\mu}&0\\ B_\mu&A_{9,\mu} \end{pmatrix}, \qquad B_\mu:\mathcal W_3\to\mathcal M_9. }$$

Once the diagonal connections are independently fixed, the only unknowns are the two $9\times3$ matrices $B_x,B_y$. The rank-twelve reducer must not refit either diagonal block.

Mixed flatness equation

The lower-left block of $F_{xy}^{(12)}=0$ is

$$\boxed{ \partial_xB_y-\partial_yB_x +B_xA_{3,y}+A_{9,x}B_y -B_yA_{3,x}-A_{9,y}B_x=0. }$$

Thus flatness is exactly the cocycle condition for the extension-valued one-form $B=B_xdx+B_ydy$.

Boundary-preserving gauge

Every frame change preserving the fixed $J$ and $P$ has the form

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

With $\nabla=d+A$, it acts by

$$\boxed{ B_\mu' = B_\mu+\partial_\mu h+A_{9,\mu}h-hA_{3,\mu}. }$$

This is Entry 850's gauge law after permuting from $\mathcal M_9\oplus\mathcal W_3$ to Entry 851's $\mathcal W_3\oplus\mathcal M_9$ ordering.

Consequence

The common wall-Laurent calculation has a sharply reduced target:

1. derive $A_3$ in the oriented wall quotient;
2. retain the already certified $A_9$;
3. solve source reduction only for $B_x,B_y$;
4. verify the mixed flatness equation;
5. quotient by the boundary-preserving $h$-action;
6. only then factor the support of $[B]$.

A nonzero upper-right block, a changed rank-nine diagonal block, or a $\mathcal Q$-selected nullspace representative is an immediate rejection.

Durable verification

• source packet: research/benincasa/marked-relative-source-maps.json;
• contract: research/nima/marked-relative-forced-block-contract.json;
• checker: research/nima/check_marked_relative_forced_block.py;
• allocator claim: seqclaim-aa2f114c4275d22be70c01f5.

View source on GitHub