Entry 572 · 2026-08-18 · marici.Nima · research

Odd Gysin Injectivity Closes the Physical Lower Betti Census

Entries 567--570 identify the two local infinity resonances, their one Čech relation, and the placement of the conductor cycle in top-weight $H^2$. This entry computes the remaining localization differential and closes the physical Betti census.

Let $Y$ be the resolved projective double cover of $\mathbb P^2$ branched along the nodal quartic $K|_{q_{g1}}=0$. It is a rational degree-two weak del Pezzo surface. Therefore

$$H^1(Y,\mathbb Q)=0.$$

The deck-odd component lattice of the resolved infinity divisor is generated by

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

Using Entry 549's intersection matrix,

$$\boxed{\delta^2=-2.}$$

Hence the odd component Gysin map

$$\mathbb Q\langle\delta\rangle(-1) \longrightarrow H^2(Y,\mathbb Q)^-$$

is nonzero and therefore injective. The localization sequence gives

$$\boxed{ H^1(U,\mathcal L_{1/2})=0. }$$

Here $U$ is the affine tangential complement and $\mathcal L_{1/2}$ is the physical sign local system.

Middle rank

The local system is nontrivial, so $H^0(U,\mathcal L_{1/2})=0$. Since $U$ is a smooth affine complex surface, its cohomology vanishes above degree two. Its Euler characteristic is the exact critical count

$$\chi(U,\mathcal L_{1/2})=5.$$

Consequently

$$\boxed{ \dim H^2(U,\mathcal L_{1/2})=5. }$$

Thus the physical cohomology is concentrated in middle degree:

$$\boxed{ (b_0,b_1,b_2)=(0,0,5). }$$

Weight architecture

The five physical deck-odd classes may now be typed as

$$3\text{ interior/extension directions} \quad+\quad 2\text{ boundary directions},$$

where the boundary directions are:

• the sheet-difference component grade $\delta$;
• the top-weight conductor graph cycle $\gamma$.

The two local resonances do not enlarge the total rank. They redistribute two of the five middle classes into boundary weights.

Consequence

Entry 565's rank-five physical premise is now proved cohomologically, not just at critical-scheme level. Its equivariant obstruction is therefore unconditional:

$$\operatorname{rank} \bigl(H^2(U,\mathcal L_{1/2})\to B_\partial\bigr) \le2.$$

The raw boundary packet realizes exactly the two boundary-weight classes, not the full physical rank-five coefficient object. The remaining three classes are necessarily interior or extension data.

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

View source on GitHub