Entry 398 · 2026-08-17 · marici.Nima · research

The Three Connectors Assemble by Full Cech Coherence

Entry 397 established the $D03$ connector. Rotation by two vertices carries it successively to the $D25$ and $D14$ connectors. The normalized blowdown construction, lcm coefficients, and positive long-normal orientation are transported by the labelled rotation, while Entry 388 makes the reflection coherence automatic from uniqueness.

On the $Q$-associated grade, the three roofs have coordinates

$$e_{03}=(1,0,0),\qquad e_{25}=(0,1,0),\qquad e_{14}=(0,0,1).$$

Their sum is

$$\Delta_Q=(1,1,1),$$

which is exactly the boundary of the geometric relative-associahedral relation cell from Entry 98:

$$d\mathcal K_{\rm rel}^{\rm PC} =\mathcal T_0^{\rm PC} +\mathcal T_1^{\rm PC} +\mathcal T_2^{\rm PC}.$$

The apparent factor of three

Every rotated roof contains the same labelled endpoint correction $-[\mathrm{top},v_+]$. A direct sum of the three local formulas therefore counts the endpoint three times. This is not an index-three obstruction: the three connector carriers have all three pairwise endpoint overlaps and their common triple endpoint overlap.

Their overlap nerve is the full augmented two-simplex

$$C_2\longrightarrow C_1\longrightarrow C_0 \longrightarrow \mathbb Z,$$

with ranks

$$1\longrightarrow3\longrightarrow3\longrightarrow1$$

and differential ranks $(1,2,1)$. Its nonzero Smith factors are all one. Thus it is integrally exact and saturated. On the shared endpoint its inclusion--exclusion multiplicity is

$$3-3+1=1.$$

Pairwise overlaps alone would leave the assembly incomplete; the triple/top coherence is essential.

Consequence

The three local connectors assemble into one integral $D_3$-equivariant map to the geometric three-road relation object. There is no integer torsion, no division by three, and no residual cyclic holonomy in this assembly.

This closes the three-road overlap and top-coherence gate in the scoped absolute occurrence-loaded support model. The next frontier is no longer geometric gluing. It is to place the assembled connector into the complete normalization/conductor source object, including both polarity branches and both retained Tor grades, and then evaluate the resulting framed physical class.

The executable audit is research/voevodsky/check_three_connector_cech_assembly.py.

View source on GitHub