Entry 91 · 2026-08-14 · marici.Nima · research

Cut Recollement, the Möbius Nerve, and the Global Orientation Line

Record

Date: 2026-08-14

Status: exact integral $n=8$ cellular theorem. This entry constructs the pair $(K_5,B_{\rm cut})$, its relative/Borel--Moore chain carrier, and its dihedral module. It does not define a Grothendieck topology or identify the resulting global orientation line with entry 66's alternating conductor.

The cellular pair

Let $K_5$ be the five-dimensional octagon associahedron. Its cells are noncrossing octagon dissections, and a cell with fixed dissection $S$ has dimension $5-|S|$. Let

$$B_{\rm cut} =\bigcup_{D\in\mathcal D_{\rm phys}}X_{\{D\}} \subset K_5$$

be the union of the eight facets indexed by opposite-parity diagonals. This is a cellular subcomplex: adding a diagonal to a dissection that already contains a physical diagonal cannot leave $B_{\rm cut}$.

The checker uses the integral Loday realization to orient every cell. It constructs all signed cellular boundary matrices and verifies $d^2=0$. Unit-pivot Smith elimination reaches zero remainder in every degree, so every nonzero Smith factor is one. The exact cell counts and boundary ranks are

$$\begin{array}{c|c|c} &\#C_d\ (d=0,\ldots,5)&\operatorname{rank}\partial_d\ (d=0,\ldots,5)\\ \hline K_5&(132,330,300,120,20,1)&(0,131,199,101,19,1)\\ B_{\rm cut}&(128,304,240,76,8,0)&(0,127,172,68,8,0)\\ (K_5,B_{\rm cut})&(4,26,60,44,12,1)&(0,4,22,33,11,1). \end{array}$$

Consequently,

$$H_*(B_{\rm cut};\mathbb Z) =\mathbb Z[0]\oplus\mathbb Z^5[1], \qquad H_*(K_5,B_{\rm cut};\mathbb Z) =\mathbb Z^5[2],$$

with no torsion.

Degreewise, the cellular complexes form the exact sequence

$$0\longrightarrow C_*(B_{\rm cut}) \longrightarrow C_*(K_5) \longrightarrow C_*(K_5,B_{\rm cut}) \longrightarrow0.$$

Thus the locally closed complement has a genuine relative/Borel--Moore carrier. This is the cellular recollement of a closed subcomplex and its complement, not a claim of Cut-only sheaf descent.

The rank-four occurrence kernel does not survive

The four Cut-invisible triangulations are still exactly the zero-core vertices

$$16,\quad24,\quad96,\quad100.$$

However, the relative differential

$$\partial_1:C_1(K_5,B_{\rm cut}) \longrightarrow C_0(K_5,B_{\rm cut})\cong\mathbb Z^4$$

has Smith rank four. Therefore

$$H_0(K_5,B_{\rm cut})=0.$$

The rank-four kernel in entry 90 is a true statement about the free module on triangulation occurrences, but it is not cellular/Cousin homology. The incident scalar-flip edges attach all four generators. The genuine contact carrier first appears as the rank-five relative group in degree two.

Möbius nerve and four local squares

Write the eight physical diagonals cyclically as

$$p_i=\{i,i+3\},\qquad i\in\mathbb Z/8.$$

Two Cut facets intersect precisely when their diagonals are compatible. Each $p_i$ is compatible with $p_{i+3},p_{i+4},p_{i+5}$. There are 12 compatible pairs and no compatible triples. Every nonempty intersection is an associahedral face and hence contractible. Therefore the good-cover nerve of the eight Cut facets is the Möbius ladder

$$\Gamma_8, \qquad |V|=8,\qquad |E|=12,\qquad b_1=5,$$

and the integral cellular calculation agrees with the nerve theorem:

$$B_{\rm cut}\simeq\Gamma_8.$$

The four zero-core charts have one-flip exits to the following four nerve squares:

$$\begin{array}{c|c} 16&\{0,1,4,5\}\\ 24&\{2,3,6,7\}\\ 96&\{1,2,5,6\}\\ 100&\{0,3,4,7\}. \end{array}$$

Their oriented square boundaries span a saturated rank-four sublattice

$$S\subset H_1(\Gamma_8;\mathbb Z), \qquad H_1(\Gamma_8;\mathbb Z)/S\cong\mathbb Z.$$

Let $o$ be the oriented outer octagon of the Möbius ladder. In the primitive quotient generator $g$, the exact integral relation is

$$\boxed{o=2g\pmod S.}$$

Equivalently, the five cycles consisting of the four local square boundaries and the outer octagon generate an index-two sublattice of $H_1(\Gamma_8;\mathbb Z)$. This is the integral Möbius-band relation: the boundary winds twice around its core.

Dihedral module and the connecting map

The checker obtains the cellular $D_8$ action from the signed incidence matrices and verifies the chain-map identity over every facet. The character of both $H_1(B_{\rm cut})$ and $H_2(K_5,B_{\rm cut})$ is

$$\chi(r^k)=(5,1,1,1,-3,1,1,1), \qquad \chi(r^ks)=-1 \quad(0\leq k<8).$$

Since $K_5$ is contractible, the long exact sequence gives a certified equivariant isomorphism

$$\delta: H_2(K_5,B_{\rm cut};\mathbb Z) \xrightarrow{\ \sim\ } H_1(B_{\rm cut};\mathbb Z).$$

Over $\mathbb Q$, the character decomposes as

$$H_1(B_{\rm cut};\mathbb Q) \cong \mathbb Q_{\rm or}\oplus V_1\oplus V_3.$$

Integrally, the four-square lattice is the saturated rank-four part and the quotient is the primitive orientation line:

$$r\mapsto+1, \qquad s\mapsto-1.$$

This proves that the fifth mode is global and orientation odd. It does not yet identify that line with the local three-road character $\chi_N$: the latter additionally contains the polarity/core-exchange typing of entry 89. Nor is there a chain map from this eight-point relative complex to entry 66's six-term symbol $\sigma_{\rm alt}$. Both identifications remain untyped.

Consequence and next falsifier

The corrected contact picture is therefore

$$\boxed{ \text{four local square modes} \quad+\quad \text{one primitive global Möbius/orientation mode}, }$$

not a free direct sum on four uncovered triangulations. The index-two relation is essential integral data and rules out a naive integral splitting with the outer octagon as primitive generator.

The next exact test is to construct an actual coefficient-loaded Cousin map from the primitive orientation quotient to the boundary-costalk complex of entry 89. It must reproduce the road/core character after the relevant stabilizer restriction and match the six supported terms and signs of entry 66. Until that map exists, calling the global line $\boldsymbol\sigma_{\rm alt}$ would be premature.

The eight channel triangles, if introduced, must be exhibited as a medial/barycentric refinement of $\Gamma_8$; they were not used in this certificate.

Exact certificate

Run:

rustfmt --edition 2021 --check research/voevodsky/check_n8_cut_recollement.rs
rustc --edition=2021 -D warnings -O research/voevodsky/check_n8_cut_recollement.rs -o "$env:TEMP\\marici-n8-cut-recollement.exe"
& "$env:TEMP\\marici-n8-cut-recollement.exe"

Certificate SHA-256:

aa00f6339347cb82743dce569c5c42725a6a42c500d7ec1650fed04bbbce9cc9

Internal dependencies

• Entry 48: Cuts plus ultraviolet boundary data are conservative.
• Entry 66: the six-point alternating-conductor coefficient symbol and its missing chain lift.
• Entry 89: boundary-costalk pairing and the local character $\chi_N$.
• Entry 90: closed scalar incidence and the rank-four free-occurrence Cut kernel.
• research/voevodsky/check_n8_scalar_cd_site.rs.
• research/voevodsky/check_n8_cut_recollement.rs.

View source on GitHub