Transverse Incidence Skeleton and the Cut-Only Descent Falsifier

Record

Date: 2026-08-14

Status: exact finite (n=8) incidence theorem and falsification of natural Cut-only descent for the free scalar occurrence object. This entry does not define a Grothendieck topology, prove a complete/regular/bounded cd-structure, or identify the dependent route correspondence with a cdh square.

Closed scalar faces

Let (\mathcal T_8) be the 132 triangulations of the octagon. For every noncrossing dissection (S), define the closed support face

[ X_S={T\in\mathcal T_8:S\subseteq T}. ]

Together with the empty object, these faces form the finite base incidence category used in this audit. Exact-core pieces are not substituted for closed at-least-core faces.

The checker enumerates 903 nonempty faces, with associahedral face vector

[ (f_0,f_1,f_2,f_3,f_4,f_5) =(132,330,300,120,20,1). ]

Every one of the 408,156 unordered intersections is exactly

[ X_S\cap X_T= \begin{cases} X_{S\cup T},&S\cup T\text{ noncrossing},\ \varnothing,&S\cup T\text{ crossing}. \end{cases} ]

Thus the closed scalar faces supply a genuine finite Cartesian incidence skeleton.

Transverse square audit

Three previously established transverse families give 2,008 undeformed coordinate squares:

[ 324\ \text{physical/physical},\qquad 1012\ \text{independent scalar/scalar},\qquad 672\ \text{independent scalar/physical}. ]

Saturation along every closed-face inclusion performs 91,488 base-change checks and leaves 17,964 distinct supported squares. All tested squares are pullbacks with monomorphic legs. The typed monic self-intersection identities pass 365,952 checks. The base changes split as

[ 6800\ \text{nondegenerate},\qquad 20400\ \text{degenerate},\qquad 64288\ \text{with an empty leg}. ]

This proves a transverse Cartesian/excision calculus. It does not prove that these squares generate covers.

Exact Cut-only falsifier

Let (\mathcal D_{\rm phys}) be the eight octagon diagonals with opposite endpoint parity and let

[ B_{\rm cut}=\bigcup_{D\in\mathcal D_{\rm phys}}X_{{D}}. ]

The exact support count is

[ |B_{\rm cut}|=128<132=|\mathcal T_8|. ]

The four omitted triangulations contain no physical diagonal. They are exactly the zero-core/contact vertices. Consequently, for the free occurrence module, simultaneous restriction to all physical Cuts has

[ \boxed{ \operatorname{rank}\ker!\left( \mathbb Z[\mathcal T_8] \longrightarrow \bigoplus_{D\in\mathcal D_{\rm phys}}\mathbb Z[X_{{D}}] \right)=4. } ]

More strongly, none of the 2,008 proper transverse face pairs covers the actual triangulation support of its ambient face. Therefore the natural cellular interpretation of these squares is not a Cut-only coverage.

There is an important categorical nuance. In the abstract face poset, the two axes of each tested square have categorical join equal to the ambient face. All 1,813,224 representable square-descent identities pass. One may therefore declare an artificial non-spatial subcanonical topology on that poset. But the physical occurrence object is not separated for it: its contact kernel restricts to zero on every Cut. Sheafification would erase scalar data rather than reconstruct it.

Hence the naive claim

[ \boxed{ \text{transverse physical Cut squares alone define the natural scalar descent site} } ]

is falsified at eight points.

Dependent routes remain transfers

For the representative dependent configuration

[ P={13,35,57},\qquad S={02,04,06},\qquad Q={03,05}, ]

the supports have sizes 5, 4, and 8, while

[ P\cap S=\varnothing,\qquad |P\cap Q|=|S\cap Q|=1. ]

No ordinary scalar pullback square contains the coefficient overlap used by the route-to-cube construction. The pentagon/square coherence is therefore a derived transfer or excess-intersection datum, not a base cdh square.

Corrected descent objective

The smallest faithful replacement for Cut-only descent is a constructible recollement separating the Cut boundary from the locally closed contact sector. Schematically, after the relevant closed/open typing is fixed, one expects a Cousin triangle of the form

[ i_i^!\mathsf J \longrightarrow \mathsf J \longrightarrow j_j^\mathsf J \xrightarrow{\partial_{\rm Cousin}} i_i^!\mathsf J[1], ]

where one term retains the zero-core/contact data and the other is the factorization boundary object. An equivalent recognition topology may use physical Cuts together with ultraviolet/contact boundary data; entry 48’s conservativity theorem already has exactly this logical form.

This correction aligns with entry 89. Its exact boundary-costalk pairing

[ \Phi_{03}^{\rm gr,\partial} ]

cannot be promoted to a full half-object from road data alone, while entry 66’s alternating-conductor chain lift

[ \boldsymbol\sigma_{\rm alt} ]

is the first missing map on the circuit side. The sharp new conjecture is that these are two faces of the same connecting datum:

[ \boxed{ \boldsymbol\sigma_{\rm alt} \ \text{is induced by, or Verdier-dual to, the contact-to-Cut Cousin connecting morphism.} } ]

This is not proved. It is falsified by incompatible support, a different dihedral character, failure of the chain-map identity, or a nonmatching factorization boundary.

Next exact test

Construct the integral cellular pair

[ (K_5,B_{\rm cut}), ]

where (K_5) is the octagon associahedron. Compute the signed relative boundary matrices, Smith normal forms, (D_8)-action, and the Cousin connecting morphism carried by the four zero-core vertices and their incident cells. Then compare its support and character with

[ \chi_N =\operatorname{sgn}_{\rm polarity}\otimes\operatorname{or}(C_3) ]

and entry 66’s alternating conductor. A tautological direct sum on vertex occurrences is insufficient; the test must retain cellular attachments and coefficient maps.

Exact certificate

Run:

rustfmt --check research/voevodsky/check_n8_scalar_cd_site.rs
rustc --edition=2021 -D warnings -O research/voevodsky/check_n8_scalar_cd_site.rs -o "$env:TEMP\\marici-n8-scalar-cd-site.exe"
& "$env:TEMP\\marici-n8-scalar-cd-site.exe"

Certificate SHA-256:

b13e57a630241eaae39fd15392718f2ccd2aa1c3f14349f42ce79d0fe177f8f2

Internal dependencies

  • Entry 31: scalar product-associahedral faces.
  • Entries 32 and 37: transverse physical and mixed Beck–Chevalley maps.
  • Entry 48: Cuts plus ultraviolet boundary data are conservative.
  • Entries 76, 82, and 83: dependent route/cube transfer typing.
  • Entry 88: three-road quotient and crosscap-counit gap.
  • Entry 89: exact boundary-costalk pairing and contact-extension ambiguity.
  • research/voevodsky/context.md.
  • research/voevodsky/check_n8_scalar_cd_site.rs.