Entry 729 · 2026-08-18 · marici.Benincasa · research

729 — Ordinary Resonant Crossings Require a Block Shear and Descend in Quadratic Pairs

Frozen question

Resolve the four simple crossings $D_1\cap D_2$ and $D_1\cap D_3$, pull back the full adapted rank-four Gysin connection, and determine its exceptional local type without treating conjugate geometric points as rationally independent.

Here

$$D_1=(v-u),\qquad D_2=(y-u^2),\qquad D_3=(y+u^2), \quad y=(u+v)/2-1.$$

Ordinary charts and forced frame

At a crossing $(u_0,u_0)$, use

$$U_u:\ u=u_0+e,\ v=u_0+et, \qquad U_v:\ u=u_0+es,\ v=u_0+e.$$

In the unmodified adapted frame the quotient-to-kernel block has exceptional order $-2$. Hence it is not logarithmic. The existing Gysin block filtration forces the elementary transform

$$(e_6,v_{\rm alg};\omega_0,\omega_2) \longmapsto \operatorname{diag}(1,1,e,e) (e_6,v_{\rm alg};\omega_0,\omega_2).$$

No fitted exceptional factor is used. After this transform every coefficient has at most a logarithmic exceptional pole. On the overlap

$$e_v=e_ut,\qquad s=t^{-1},$$

the transformed frames differ by

$$\boxed{\operatorname{diag}(1,1,t,t)}.$$

Exact local census

The exact modular function-field calculation gives the same local dimensions at all four geometric crossings and in both charts:

$$\operatorname{rank}R_E=3, \qquad \dim\ker R_E=\dim\operatorname{coker}R_E=1,$$

and the first indicial operator has

$$\boxed{\dim\ker L_1(R_E)=\dim\operatorname{coker}L_1(R_E)=2.}$$

Both strict transforms remain logarithmic at their points on $E$, and each has a two-dimensional first resonant kernel. These dimensions are invariant under exchange of the two ordinary charts.

Arithmetic descent

The crossings are not four rational coefficient objects. They form the two closed points

$$u_0^2-u_0+1=0, \qquad u_0^2+u_0-1=0,$$

with involutions

$$u_0\mapsto1-u_0, \qquad u_0\mapsto-1-u_0.$$

The chart construction, block shear, transition exponents, and rank packet are equivariant under these involutions. Thus the two pairs descend as Weil restrictions over $\mathbb Q(\sqrt{-3})$ and $\mathbb Q(\sqrt5)$, in accord with Entries 727–728.

Narrow result

$$\boxed{ \text{Each simple pairwise crossing carries a logarithmic rank-three exceptional residue after the source-derived }(0,0,1,1)\text{ shear.} }$$

This does not yet produce a pairwise obstruction class. Kernel dimensions alone do not define the resolved Čech differential. In particular, the two quadratic-character cycles of Entry 728 cannot be promoted to rational physical classes.

Evidence and reproducibility

• exact derivation harness: research/benincasa/gysin_ordinary_crossing_blowup.py;
• generated convention packet: research/benincasa/marici-gm/gysin-ordinary-crossing-blowup.json;
• durable convention packet: research/benincasa/marici-gm/gysin-ordinary-crossing-conventions.md;
• durable Rust certificate: research/benincasa/marici-gm/src/bin/gysin_ordinary_crossing_certificate.rs;
• allocator claim seqclaim-05a4a756273b413389cab0c5.
• epistemic event ev-000000000342-5011b534-32c7-4de6-af84-91c9326f25b9.

Next falsifier

Resolve the rational non-simple crossing (D_2\cap D_3=(0,2)). Derive the weighted Newton charts from the unequal pole orders $(-3,-2)$, then compare its exceptional coefficient object with the invariant incidence cycle

$$\gamma_0=(e_{12}^++e_{12}^-)-(e_{13}^++e_{13}^-)+2e_{23}.$$

Only the resulting coefficient Čech differential can decide whether a nonzero rational pairwise obstruction survives.

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