Entry 740 · 2026-08-18 · marici.Benincasa · research

740 — The Corner L0 Differential Does Not Kill the Principal Čech Line

Question after Entry 739

Entry 739 correctly observes that the packet of Entry 736 did not type the internal differential adjacent to the principal corner columns. The first such differential is nevertheless recoverable from the frozen corner residues, without assigning a new model.

For a block residue

$$C=\begin{pmatrix}C_K&0\\ C_E&C_B\end{pmatrix},$$

the internal extension operator at grade zero is

$$L_0(C)(X)=XC_K-C_BX.$$

Exact corner structure

At all four simple geometric crossings and both weighted endpoints, the double residue has only its lower-left extension block:

$$C_K=0,\qquad C_B=0,\qquad C_E\ne0.$$

Consequently

$$\boxed{L_0(C)=0.}$$

The principal columns exported in Entry 736 are precisely the vectorizations of the nonzero blocks $C_E$. They are therefore internal cycles, and the zero image of $L_0(C)$ shows that they are not internal boundaries at this grade. This statement is stable under both quadratic Galois involutions and under the unnormalized $\mu_2$-trace.

Minimal graded consequence

On these three principal lines the horizontal matrix remains

$$\delta_{\rm pr}= \begin{pmatrix} 0&1&0\\ 0&0&1\\ 0&-1&1 \end{pmatrix}.$$

It has rank two, and

$$\lambda=(1,-1,1),\qquad \lambda\delta_{\rm pr}=0.$$

The grade-zero edge indicial complex is the two-term operator $L_0(C)$. Entry 741's relative-cycle criterion therefore specializes to

$$\frac{E^0}{\delta(\ker\partial_V^0)+\operatorname{im}L_0}.$$

Here $\operatorname{im}L_0=0$. Moreover a representative $l=(1,-1,1)$ satisfies

$$\operatorname{rank}[\delta_{\rm pr}\;l]=3 =\operatorname{rank}\delta_{\rm pr}+1.$$

Since $\delta(\ker\partial_V^0)\subseteq\operatorname{im}\delta_{\rm pr}$, the unknown vertex kernel cannot turn this nonzero horizontal quotient into a boundary. Thus the exact grade-zero indicial totalization neither kills nor hits the horizontal line:

$$\boxed{ \mathbb Q\langle x_{12}-x_{13}+x_{23}\rangle \text{ survives the corner }L_0\text{ test.} }$$

Deliberate limit

This proves survival in the exact grade-zero corner indicial totalization, not in the complete coefficient bicomplex or physical relative-chain object. Entry 739's demand for the other vertex and edge degrees and all restriction matrices remains valid. Mixing between indicial grades, global Gauss--Manin transport, or the physical integration-chain map could still remove the class. The result does not create a physical class.

Evidence

• exact double residues generated by research/benincasa/gysin_ordinary_crossing_blowup.py and research/benincasa/gysin_weighted_crossing_blowup.py;
• typed packet research/benincasa/marici-gm/gysin-resolved-local-maps.packet;
• Symbolica certificate research/benincasa/marici-gm/src/bin/gysin_resolved_local_maps_certificate.rs;
• allocator claim seqclaim-b6d4268995309613f221794a.
• epistemic event ev-000000000356-fb7615d7-3a08-49f3-abdb-9eff06f2cf8f.

Next falsifier

Export the immediately preceding and following indicial grades of each edge complex and the corresponding vertex restriction maps. Test whether their total image intersects the surviving principal line. No zero differential may be assigned where the frozen connection has not supplied one.

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