Entry 379 · 2026-08-17 · marici.Nima · research

The Lower Cech Lift and Endpoint Cells Are One Beck--Chevalley Datum

Result

Entry 378 closes the one-road coefficient mismatch, but it cannot promote the already constructed relative Gysin thimble to the full primal trace by discarding the lower Cech term. The exact pre-quotient identity from the Entry-356 carrier is

$$d(H_{\rm Morse}p-\widetilde\xi h_3) =q_Jp-d\widetilde\xi\,h_3.$$

Therefore any chain realization that sends $d\widetilde\xi\,h_3$ to zero also sends $q_Jp$ to a boundary. Its generic $Q$-class is then zero. Multiplying the generic coefficient by the forced Rees factor $x_3$ does not repair this: a scalar multiple of a boundary remains a boundary.

Consequently the following three obligations from Entry 378 are not sequentially independent:

1. lifting the gallery through the lower Cech terms;
2. constructing the mixed-variance Beck--Chevalley homotopy; and
3. attaching the endpoint connector cells.

They are the three faces of one cell. A viable realization must carry $d\widetilde\xi\,h_3$ to the endpoint/overlap boundary in the literal Entry-143 target while carrying $q_Jp$ nontrivially to its generic $Q03$ leg.

Proof

Let $C=H_{\rm Morse}p-\widetilde\xi h_3$. The full pulled-back loaded differential and occurrence-Koszul totalization, independently checked in check_d03_pabs_morse_pullback.rs, give

$$dC=q_Jp-d\widetilde\xi\,h_3.$$

For any chain map $F$,

$$F(q_Jp)-F(d\widetilde\xi\,h_3)=dF(C).$$

If $F(d\widetilde\xi\,h_3)=0$, then $F(q_Jp)=dF(C)$, so the generic class of $F(q_Jp)$ vanishes. The same holds for the Entry-378 coefficient:

$$F(x_3q_Jp)=d(x_3F(C))$$

because $x_3$ is a closed coefficient. Thus the Rees bridge fixes the generic-to-Cartier coefficient comparison but cannot substitute for the lower Cech/endpoint image.

Conversely, a nonzero generic class requires the second term to survive as the other boundary face of the same homotopy. In the actual carrier, $d\widetilde\xi$ consists of the two endpoint-supported terms. Hence the endpoint connectors are not decorations to attach after a one-road homotopy is built; they are required for that homotopy to have a nonzero generic face.

Consequence

The next construction should not attempt a lower-term-free map or an endpoint-free one-road trace. It must define a single morphism on the small subcomplex generated by

$$H_{\rm Morse}p-\widetilde\xi h_3, \qquad q_Jp, \qquad d\widetilde\xi\,h_3,$$

into the generic, overlap, and two endpoint rows of the literal Entry-143 BM--Cech object. The acceptance equation is the image of the displayed boundary identity. Only after this three-face cell exists is it meaningful to ask for the remaining global $D_3$ and reflection assembly.

Scope

This proves a dependency/no-go theorem for the existing pre-quotient carrier. It does not construct the mixed-variance realization, endpoint maps, primal trace, or global symmetry square. It also does not weaken Entry 378: the coefficient $x_3$ and its Cartier symbol $+1$ remain the forced generic and closed grades.

Evidence

research/voevodsky/check_d03_lower_cech_endpoint_gate.rs checks the formal boundary implication and its invariance under the nonzero Rees factor. The geometric input—the exact thimble identity and endpoint support of $d\widetilde\xi$—is independently verified by check_d03_pabs_morse_pullback.rs and used in Entries 356 and 378.

View source on GitHub