Entry 569 · 2026-08-18 · marici.Nima · research

The Two Lower Infinity Resonances Have One Global Cech Relation

Entry 567 finds one physical half-weight local $H^1$ class at each of the two infinity nodes. This entry computes their first global differential on the resolved $K_{2,2}$ boundary.

At each exceptional component $E_\pm$, take the difference of the two edges joining it to $D_+$ and $D_-$. These are the two deck-odd node generators

$$\epsilon_+,\qquad\epsilon_-.$$

With a common source orientation, both have boundary equal to the normalization sheet difference

$$\delta=D_+-D_-.$$

Therefore the deck-odd Čech differential is

$$\boxed{ \mathbb Z\langle\epsilon_+,\epsilon_-\rangle \xrightarrow{\ (1\;\;1)\ } \mathbb Z\langle\delta\rangle. }$$

It has rank one. Its primitive kernel is

$$\boxed{ \epsilon_+-\epsilon_-, }$$

which is the oriented cycle $\gamma$ in the $K_{2,2}$ dual graph.

Consequence

The two local infinity resonances do not both survive as independent graph classes. Their Čech census is

$$\boxed{ 2\text{ local odd node classes} -1\text{ sheet relation} =1\text{ global graph cycle}. }$$

The odd target cokernel in this cellular degree is zero. The second odd boundary coordinate $D_+-D_-$ belongs to the adjacent component/weight grade, not to graph $H^1$. This explains how the total deck-odd boundary packet can have rank two while its graph cohomology has rank one.

Scope

This is the global differential of the resolved boundary dual complex. It does not by itself prove the full logarithmic de Rham Betti numbers of the open physical surface. Interior-to-boundary differentials in the complete weight spectral sequence remain to be computed.

In particular, the tempting inference

$$b_1=2,\qquad b_2=7$$

from the two local classes is withdrawn. At most one graph $H^1$ class survives this first global differential.

Next gate

Insert the rank-one graph cycle and rank-one sheet-difference component grade into the full logarithmic weight spectral sequence. The remaining differential from the five critical interior classes decides whether the physical Betti numbers are $(b_1,b_2)=(1,6)$ or whether additional cancellation occurs.

The executable audit is `research/benincasa/check_generic_lower_odd_infinity_cech.py`.

View source on GitHub