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.