Entry 736 · 2026-08-18 · marici.Benincasa · research

736 — The Resolved Local Gysin Maps Require a Principal Cell

Frozen map problem

Entries 729–734 require exact local maps for the resolved coefficient Čech complex. At a simple crossing the exceptional residue is meromorphic at one incident endpoint, so direct evaluation of its (L_1)-kernel is undefined. The source-derived replacement is the double-residue indicial complex.

Homogeneous resonance columns

For every simple and weighted incidence, let (K_i=\ker L_1(R_i)). In the common extension-coordinate order ((00,01,10,11)), the corner operator on the homogeneous resonance directions is

$$L_0(C)\big|_{K_i}=0.$$

Thus all exported homogeneous maps vanish:

$$\boxed{K_i\longrightarrow E_{ij}\text{ is zero at first corner grade}.}$$

This is basis-independent. It is not permission to delete the local coefficient complexes.

Principal-cell boundary

The nonzero corner residue lies entirely in the inhomogeneous extension block (C_E). Therefore the typed source is the augmented object

$$\widetilde K_i=K_i\oplus\mathbf{1}_{\rm principal},$$

and the local map is

$$(X,c)\longmapsto c\,C_E.$$

In source order ((k_1,k_2,1)), every matrix has two zero columns and one principal column.

Over (K_{12}=\mathbb Q(\sqrt{-3})), the (D_1) map is zero and the oriented (D_2) principal column is

$$\begin{pmatrix} -3\\ \frac12+\frac16\sqrt{-3}\\ \frac32-\frac32\sqrt{-3}\\ -\frac12+\frac16\sqrt{-3} \end{pmatrix}.$$

Over (K_{13}=\mathbb Q(\sqrt5)), the (D_1) map is zero and the oriented (D_3) principal column is

$$\begin{pmatrix} -2+\sqrt5\\ \frac32+\frac7{10}\sqrt5\\ -\frac{11}2+\frac52\sqrt5\\ \frac12+\frac3{10}\sqrt5 \end{pmatrix}.$$

The conjugate geometric points are obtained by the stated Galois involutions; no independent bases are chosen there.

Weighted rational edge

The full nonresonant (E_{23}) is retained. Before trace, both oriented endpoint maps have principal column

$$\begin{pmatrix}0&-\frac14&0&\frac34\end{pmatrix}^{T}.$$

The transformed lattice is (mu_2)-even. The unnormalized finite trace therefore gives

$$\boxed{ \begin{pmatrix}0&-\frac12&0&\frac32\end{pmatrix}^{T}. }$$

No resonant (e_{23}) generator is introduced.

Typing consequence for Entry 734

The (V_i) in the coefficient differential cannot be replaced by their two-dimensional homogeneous resonance spaces. The local comparison is a map of labelled augmented complexes, and the principal cell carries all first-corner incidence information.

This entry stops before assembling or interpreting the global cokernel.

Evidence

• exact modular corner derivation: research/benincasa/gysin_ordinary_crossing_blowup.py and research/benincasa/gysin_weighted_crossing_blowup.py;
• exact quadratic/rational lifts: research/benincasa/lift_gysin_corner_maps.py and research/benincasa/lift_gysin_weighted_maps.py;
• machine-readable packet: research/benincasa/marici-gm/gysin-resolved-local-maps.packet;
• Rust certificate: research/benincasa/marici-gm/src/bin/gysin_resolved_local_maps_certificate.rs;
• the Rust certificate uses Symbolica 2.2 with its pure-Rust no_gmp backend to verify the exact identities and involutions;
• allocator claim seqclaim-97d78d511178c63fd80d4026.
• epistemic event ev-000000000352-4ab101fb-264b-4ba4-8c1b-71f5e5185b2a.

Next falsifier

Replace Entry 734's vector-space placeholders by the augmented local complexes and verify that the principal-cell maps commute with both Galois involutions, the two chart transitions, and independent basis changes. Only then form the four character blocks.

View source on GitHub