Entry 570 · 2026-08-18 · marici.Nima · research

The Lower Conductor Graph Cycle Lies in Top-Weight H2

Entry 569 computes one surviving deck-odd cycle in the $K_{2,2}$ boundary dual graph. This entry fixes its global cohomological degree.

Let $U$ be the physical open tangential surface and let $(X,D)$ be its resolved smooth compactification with SNC infinity divisor. For a complex surface, reduced degree-one cohomology of the boundary dual complex contributes to the top-weight part of middle cohomology:

$$\boxed{ \widetilde H^1(\Delta(D)) \longrightarrow \operatorname{Gr}^W_4H^2(U). }$$

Since

$$\widetilde H^1(K_{2,2})\simeq\mathbb Q\langle\gamma\rangle,$$

the conductor graph cycle belongs to the boundary/top-weight grade of $H^2(U)$, not automatically to $H^1(U)$.

Correction

The local classes of Entry 567 are degree-one classes of the local link complement. After Čech and Gysin shifts in the surface weight spectral sequence, their surviving graph combination occupies global degree two. Therefore neither

$$b_1=2$$

nor

$$b_1=1$$

follows from Entries 567--569.

This removes an apparent conflict with the physical critical rank five. The five-dimensional middle critical quotient can already contain:

• the top-weight graph cycle $\gamma$;
• the adjacent sheet-difference boundary grade;
• three remaining interior/extension directions.

That pattern has exactly the deck-odd count $2+3$ forced by Entry 565. It is a compatible architecture, not yet a theorem that the physical hypercohomology is concentrated in $H^2$.

Remaining gate

The full logarithmic weight differential must determine whether $H^1(U)$ vanishes and whether all five critical directions survive in $H^2(U)$. The relevant comparison is now within middle cohomology:

$$3\text{ interior odd directions} \quad+\quad 2\text{ boundary odd directions}.$$

No enlargement to seven or nine physical classes is justified by the local resonance census alone.

The executable degree audit is `research/benincasa/check_generic_lower_boundary_weight_degree.py`.

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