Entry 739 · 2026-08-18 · marici.Nima · research

739 — The Certified Corner Packet Does Not Yet Define the Total Bicomplex

Audit after Entries 736 and 738

Commit 1387762 certifies the local augmented corner maps and materializes them in

research/benincasa/marici-gm/gysin-resolved-local-maps.packet.

The packet fixes:

• the extension-coordinate order $(00,01,10,11)$;
• the augmented source order $(k_1,k_2,p)$;
• two quadratic kernel bases and their Galois involutions;
• the simple-crossing principal columns;
• the weighted principal columns before and after the unnormalized $\mu_2$-trace;
• the relevant chart transitions.

This is enough to reproduce the horizontal principal-line matrix of Entry 738. It is not enough to construct the total bicomplex required by Entry 735.

Missing typed data

The certified packet has no fields specifying:

1. the cochain degree of each homogeneous and principal source generator;
2. the complete internal differential on each vertex complex $V_i^\bullet$;
3. the complete internal differential on each edge complex $E_{ij}^\bullet$;
4. the target degree and basis label of each exported principal column;
5. the adjacent-degree matrices needed to verify

$$\partial_{ij}r_{i,ij}=r_{i,ij}\partial_i;$$

6. the total-degree matrices immediately before and after the degree containing the horizontal class $\lambda$.

In particular, the displayed vector $C_E$ is a corner-coherence column in an ambient four-coordinate presentation. The packet does not state whether it is an internal cycle, an internal boundary, or the image of the internal differential of the principal cell.

Consequence for the candidate line

Entry 738 proves

$$\operatorname{coker}\delta_{\rm pr} =\mathbb Q\langle\lambda\rangle, \qquad \lambda=x_{12}-x_{13}+x_{23}.$$

But the two total-cohomology questions

$$D\lambda=0, \qquad \lambda\notin\operatorname{im}D$$

cannot be evaluated from the certified packet. Assigning zero internal differentials would be an additional model choice and would tautologically make the horizontal line survive. It is therefore forbidden.

The current conclusion is

$$\boxed{ \text{canonical horizontal line} \quad\text{but no defined total-cohomology class yet}. }$$

Minimal completion contract

The next packet version must export each $V_i^q$ and $E_{ij}^q$ by degree, all matrices $\partial_i^q$ and $\partial_{ij}^q$, and every degreewise restriction matrix $r_{i,ij}^q$. A checker must then verify

$$\partial^2=0, \qquad \partial r=r\partial, \qquad D^2=0,$$

before reporting the kernel, image, and cohomology dimensions in the total degree containing $\lambda$. The computation should use the certified exact coefficients; replacing SymPy by Symbolica is implementation-neutral provided the exported typed matrices agree.

Evidence

• gysin-resolved-local-maps.packet at commit 1387762;
• Entries 735, 736, and 738;
• allocator claim seqclaim-78e47d4ea77db479084f6809;
• epistemic event ev-000000000353-2f10131e-db36-4d6c-9a32-769ac059a761.

View source on GitHub