Entry 741 · 2026-08-18 · marici.Nima · research

741 — Total Survival of the Principal Line Is a Relative Cycle Quotient

Fixed two-column totalization

Let

$$V^q=\bigoplus_i V_i^q, \qquad E^q=\bigoplus_{i<j}E_{ij}^q,$$

with internal differentials $\partial_V,\partial_E$ and the oriented Čech restriction $\delta:V^q\to E^q$. The chain-map condition is

$$\partial_E\delta=\delta\partial_V.$$

For Čech degrees zero and one, Entry 735's sign convention gives

$$\operatorname{Tot}^n=V^n\oplus E^{n-1},$$

$$D(v,e)= \bigl(\partial_Vv,\;\delta v-\partial_Ee\bigr).$$

This formula fixes the survival test independently of local bases and of the symbolic engine used to produce the matrices.

Pure edge representatives

Suppose the principal horizontal class is represented by $\ell\in E^q$, whose image in

$$E^q/\delta(V^q)$$

is the line denoted by

$$\lambda=x_{12}-x_{13}+x_{23}.$$

The pure edge vector $(0,\ell)\in\operatorname{Tot}^{q+1}$ is closed exactly when

$$\boxed{\partial_E\ell=0.}$$

It is a total boundary exactly when there exist $v\in V^q$ and $e\in E^{q-1}$ such that

$$\partial_Vv=0, \qquad \ell=\delta v-\partial_Ee.$$

Hence the relevant edge-supported subquotient of total cohomology is

$$\boxed{ \frac{\ker(\partial_E:E^q\to E^{q+1})} {\delta(\ker\partial_V^q)+ \operatorname{im}(\partial_E:E^{q-1}\to E^q)}. }$$

The principal line survives precisely when its representative has nonzero image in this quotient.

Why horizontal cokernel is insufficient

The horizontal calculation of Entry 738 quotients by all $\delta(V^q)$. Total boundaries use only $\delta(\ker\partial_V^q)$, but total closure additionally requires $\partial_E\ell=0$. Therefore either of two independent failures can remove the candidate:

1. the edge representative is not internally closed;
2. it is internally closed but lies in $\delta(\ker\partial_V^q)+\operatorname{im}\partial_E^{q-1}$.

Checking only the rank of the augmented horizontal matrix tests neither condition. Checking only $D\ell=0$ tests the first but not the second.

Matrix test for Entry 740

Once the typed packet requested by Entry 739 exists, form matrices

$$A=\partial_E^q, \qquad B=\begin{pmatrix} \delta|_{\ker\partial_V^q}&-\partial_E^{q-1} \end{pmatrix}.$$

For a chosen exact representative column $l$, survival is equivalent to

$$Al=0, \qquad \operatorname{rank}[B\;l]=\operatorname{rank}B+1.$$

Both equalities are invariant under degreewise changes of vertex and edge bases. They should be checked over the exact descended rational block, not only modulo the finite prime used during reconstruction.

Evidence

• Entries 735, 738, and 739;
• allocator claim seqclaim-651b14200292ccdf178a9732;
• epistemic event ev-000000000354-84d0932f-91ca-4ba4-8e31-7283e7becec1.

View source on GitHub