Entry 96 · 2026-08-14 · marici.Nima · research

Factorization-Marked Normal-Crossing Span and the Pair-Local Relation Obstruction

Record

Date: 2026-08-14

Status: exact finite-nonresonant theorem for the $D=03$ marked coefficient span, together with a pair-local no-go theorem for the $\Delta$ relation.

Scope: factorization-marked six-point scalar geometry, the occurrence cosheaf of entries 83 and 86, and the facewise Pochhammer--Cousin complex of entry 38. This is a boundary-costalk result. It is not yet the bivariant trace to a primal circuit tag or the full scalar total-specialization map.

Claim

The physical face $F_{03}$ is the actual product square

$$F_{03}=K_4\times K_4$$

with vertices

$$v_{00}=x_0x_3,\qquad v_{10}=x_1x_3,\qquad v_{01}=x_0x_4,\qquad v_{11}=x_1x_4.$$

Entry 86's sink marks select the two coordinate edges

$$Z_0=\{03,02\}=[v_{00},v_{01}], \qquad Z_3=\{03,35\}=[v_{00},v_{10}].$$

Their fiber product in the factorization-marked scalar face category is the single actual triangulation

$$\boxed{ W_{03}=Z_0\times_{F_{03}}Z_3 =\{03,02,35\}=v_{00}.}$$

Thus the correspondence requested in entry 95 exists canonically:

$$\boxed{Z_0\longleftarrow W_{03}\longrightarrow Z_3.}$$

The fixed marks are essential. After forgetting them, both $v_{00}$ and $v_{11}$ are saturated common lower cells.

On the universal normal torus, put

$$u_0=q_0-1,\qquad u_3=q_3-1.$$

The two pullbacks to $W_{03}$ are the independent primitive characters

$$u_0\longmapsto(1,0), \qquad u_3\longmapsto(0,1).$$

They form a regular sequence. The minimal support-preserving middle coefficient is therefore the oriented two-variable Koszul complex

$$\boxed{ K(u_0,u_3)=K(u_0)\otimes K(u_3), \qquad \operatorname{or}=h_0\wedge h_3.}$$

Rank zero erases support, while rank one identifies the two normal divisors up to a Laurent unit and destroys the bifiltration. No common rank-one target character is introduced.

Because $Z_0$ and $Z_3$ are transverse coordinate faces, entry 38 applies without a new coefficient specialization. It gives the occurrence-decorated PC face tubes, normal Koszul factors, Cousin maps, and the ambient $F_{03}$ top-cell coherence.

Orient the four square edges by

$$a:v_{00}\to v_{10},\quad b:v_{01}\to v_{11},\quad c:v_{00}\to v_{01},\quad d:v_{10}\to v_{11}.$$

The marked span supports only $a$ and $c$. The unique integral supported primitive with the entry-86 endpoint boundary is

$$\boxed{H_{03}^{\rm mark}=a-c, \qquad \partial H_{03}^{\rm mark}=v_{10}-v_{01}.}$$

The alternative $b-d$ passes through the unmarked corner $v_{11}$. The two representatives remain derived-equivalent because

$$(a-c)-(b-d)=a-b-c+d=\partial[F_{03}].$$

This removes the strict lower-Cousin ambiguity left in entry 95. The output is nevertheless typed on the road costalk. With entry 89's Laurent duality it is the cocycle

$$d_1^\vee\otimes\chi_N,$$

not the primal tag $d_1$.

There is also a sharp pair-local obstruction. The two relevant conductor columns obey

$$K_{\rm alt}(u_0)=-d_1, \qquad K_{\rm alt}(u_3)=+d_1.$$

Every degree-one image of this pair is therefore contained in $\mathbb Z d_1$, while

$$\Delta=d_0+d_1+d_2\notin\mathbb Z d_1.$$

Consequently:

$$\boxed{ \text{the single }(u_0,u_3)\text{ span cannot realize the }\Delta \text{ relation in a chain map}.}$$

The relation can only be tested after the three already-existing pairs

$$(u_2,u_5)\to d_0, \qquad (u_0,u_3)\to d_1, \qquad (u_1,u_4)\to d_2$$

have been assembled with one separately typed relation object.

Evidence

The actual scalar-face and occurrence census is

research/voevodsky/check_d03_factorization_marked_span.rs.

SHA-256:

1c7ff7d8e3d3fbb11042929efbe45e27f75d7facfbc532b5c1a7347f54c8c337

The universal-character, Koszul, minimality, and pair-local relation audit is

research/voevodsky/check_d03_minimal_normal_torus_span.rs.

SHA-256:

99fd0571bb61075fe7a44913fa3b1311ea633436f5aac2fa4159a5102a23907d

Reproduce both with:

$sources = @(
  "research/voevodsky/check_d03_factorization_marked_span.rs",
  "research/voevodsky/check_d03_minimal_normal_torus_span.rs"
)
foreach ($src in $sources) {
  rustfmt --edition 2021 --check $src
  $exe = Join-Path $env:TEMP ((Split-Path $src -LeafBase) + ".exe")
  rustc --edition=2021 -D warnings -O $src -o $exe
  & $exe | ConvertFrom-Json | Out-Null
}

The primary audit reran both programs successfully. Inherited inputs are entries 38, 83, 86, 89, 94, and 95 and their cited certificates.

Boundary

• The PC statement is on entry 38's finite nonresonant domain. Resonant nearby-cycle extension is not claimed.
• The factorization marks are genuine extra scalar-geometric data. The bare one-parameter amplitude family does not select $v_{00}$.
• The theorem constructs a supported road-costalk diagram and its dual cocycle. It does not identify $d_1\cong d_1^\vee$.
• The ambient square top cell proves homotopy between the two Cousin paths; it does not supply a circuit relation generator.
• The no-go result is pair-local. Entry 95's complete three-pair carrier fold to $\Delta$ remains valid.
• No rational splitting, new transition map, fitted differential, or new generator has been used.

Consequence

The occurrence-loaded problem now separates into two coherent stages.

First construct a bivariant pairing

$$\boxed{ \Theta_1^{\rm PC}: \mathcal S_1^{\rm mark}\boxtimes\mathcal Q_{03}^{\rm PC} \longrightarrow\mathbf1_{\chi_N},}$$

where $\mathcal S_1^{\rm mark}$ is the supported diagram $Z_0\leftarrow W_{03}\to Z_3$ and $\mathcal Q_{03}^{\rm PC}$ is the road costalk. Its associated grade must be entry 89's unit Laurent pairing and its boundary must be entry 86's marked endpoint counit. Currying would give the desired local primal-tag trace

$$\operatorname{Tr}_1^{\rm PC}: \mathcal S_1^{\rm mark} \longrightarrow \mathbb D(\mathcal Q_{03}^{\rm PC})\otimes\chi_N =:\mathcal T_1^{\rm PC}.$$

Only after constructing the rotated $\mathcal T_0^{\rm PC}$ and $\mathcal T_2^{\rm PC}$ should one test a combined relation cell

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

The next falsifier is therefore local and precise: prove or disprove the bivariant PC pairing $\Theta_1^{\rm PC}$ with the two independent normal characters, all lower Cousin terms, twist reversal, and the entry-86 endpoint normalization retained.

View source on GitHub