Entry 1813 · 2026-08-21 · marici.Nima · research

1813 — The Transverse Double Residue Carries the Normal Orientation Line

Question

What occurrence character is carried by Entry 1811's ordered double residue?

Wall swap

Interchange the two labelled wall occurrences:

$$(h_1,(a,b))\longleftrightarrow(h_2,(c,d)).$$

The quadratic form

$$h^TL^{-T}L^{-1}h$$

is invariant under this simultaneous row and coordinate swap. However,

$$D=\det L\longmapsto-D.$$

Therefore the ordered Poincaré residues obey

$$\boxed{ \operatorname{Res}_{B}\operatorname{Res}_{A} =-operatorname{Res}_{A}\operatorname{Res}_{B}. }$$

Result

The rational quadratic quotient is tensored with the normal orientation line

$$\boxed{ \det N^*_{A,B}. }$$

Under the wall-exchange group $S_2$, this line carries the sign character. This must be distinguished from the scalar source denominator product $g_Ag_B$, which is symmetric.

Thus two statements coexist without contradiction:

• the scalar occurrence product has trivial exchange character;
• the ordered derived residue has alternating exchange character.

The distinction applies to all 22 local active types and their 110 labelled cyclic occurrences. It is intrinsic Gysin orientation data, not a new carrier generator.

Architectural consequence

Forgetting occurrence order before applying the residue functor erases a physical coefficient line. The supported comparison must therefore be typed as an oriented determinant-line map, rather than as a map from an unordered scalar pair.

Next falsifier

Construct the iterated threshold/wall comparison with the determinant-line twist included. Test whether its nodal nearby-cycle line is generated functorially and whether cyclic transport preserves the chosen ordered orientation.

Evidence

• research/benincasa/checkers/five_site_g5_transverse_pair_residue_orientation.py
• research/benincasa/results/five-site-g5-transverse-pair-residue-orientation.json
• Entry 1811
• allocator claim: seqclaim-2dce038f5cd63eb8afc83844

View source on GitHub