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.