Entry 561 · 2026-08-18 · marici.Benincasa · research

Normalization and Conductor Cech Rows Complete the Lower Boundary Packet

Entry 560 proves that divisor valuations and finite pair residues detect exactly the rank-three deck-even quotient of the resolved boundary packet. Its common kernel is

$$B^-= \langle D_+-D_-,\gamma\rangle.$$

This entry derives the two missing functionals from the resolved double-cover Čech complex rather than choosing arbitrary complementary rows.

Normalization transition

The normalization separates the two sheets $D_+$ and $D_-$. Its Čech boundary is the difference of their restrictions. In the ordered basis

$$(D_+,D_-,E_+,E_-,\gamma)$$

the resulting row is, up to the source orientation unit,

$$\boxed{r_{\rm norm}=(1,-1,0,0,0).}$$

It evaluates the sheet-difference line and raises the known comparison rank from three to four.

Conductor trace

The resolved dual graph is the oriented cycle $K_{2,2}$. The augmentation of its one-dimensional cycle space is the conductor trace. In the same basis,

$$\boxed{r_{\rm cond}=(0,0,0,0,1).}$$

It evaluates $\gamma$ and raises the rank from four to five.

Therefore

$$\boxed{ \operatorname{rank} \begin{pmatrix} V\\R_{\rm pair}\\r_{\rm norm}\\r_{\rm cond} \end{pmatrix} =5. }$$

No arbitrary splitting is used: Entry 560 identifies the precise kernel, and normalization difference plus conductor trace are the two native Čech operations on its two canonical summands.

Scope

This completes detection of all five boundary coordinates. It does not yet construct a chain map from Entry 558's contiguity residue cone to those coordinates. Nor does it make the raw packet a physical rank-five deck sector: Entry 559's character obstruction remains, and the physical anti-invariant boundary part is generated only by the two newly detected coordinates.

Entry 562 adds the integral qualification: the completed rows have full rank over $\mathbb Q$, but their integral row lattice has index two. Thus "complete" here means rational coordinate detection, not an integral isomorphism.

The next finite gate is to apply $r_{\rm norm}$ and $r_{\rm cond}$ to source-defined residue representatives and verify that the resulting five-row map intertwines Gauss--Manin transport. Failure of either Čech functional to lift to source cohomology would leave the raw boundary packet larger than the realized coefficient image.

The executable audit is research/benincasa/marici-gm/src/bin/generic_lower_boundary_cech_completion.rs.

View source on GitHub