Entry 1807 · 2026-08-21 · marici.Nima · research

1807 — The Complementary Pole-Log Is Occurrence-Even

Question

Does Entry 1806's doubled-occurrence logarithm contain a relative-sign component that survives the physical occurrence trace?

Frozen object

Retain the two complementary labels

$$e_1=g_{15}, \qquad e_2=g_{234}$$

before imposing their common physical section at $y_{45}=0$. Because the source integrand contains their product, the scalar occurrence space is

$$\operatorname{Sym}^2\langle e_1,e_2\rangle = \langle e_1^2,e_1e_2,e_2^2\rangle.$$

The class derived in Entry 1806 is the mixed class

$$v_{\rm phys}=e_1e_2.$$

Complement involution

The label-complement involution exchanges $e_1\leftrightarrow e_2$. In the ordered symmetric basis its matrix is

$$S= \begin{pmatrix} 0&0&1\\ 0&1&0\\ 1&0&0 \end{pmatrix}.$$

Therefore

$$\boxed{S v_{\rm phys}=v_{\rm phys}.}$$

The physical diagonal trace sends all three symmetric monomials to the same section and maps $v_{\rm phys}$ nontrivially. The antisymmetric wedge $e_1\wedge e_2$ is not a scalar coefficient class of the frozen product. Consequently

$$\boxed{ \dim(\text{relative-sign scalar excess})=0. }$$

This statement concerns the scalar occurrence coefficient. It does not erase orientation signs belonging to an independently typed iterated residue map.

Cyclic assembly

The complementary pair has a free orbit under $C_5$. Hence its five pole-twisted logarithms assemble as one regular representation,

$$\mathbb Q[C_5], \qquad \chi=(5,0,0,0,0).$$

Each occurrence carries the rank-one nilpotent from Entry 1806. Thus the assembled nilpotent satisfies

$$\operatorname{rank}N=5, \qquad N^2=0.$$

Narrow result

The only activated complementary soft family is occurrence-even and survives physical diagonal identification. Its complexity is a regular $C_5$-orbit of pole-twisted rank-one logarithmic extensions. No anti-diagonal scalar coefficient and no new carrier structure are generated.

Next falsifier

Analyze intersections among the 34 transverse supported region-wall pairs. Determine whether their normal-crossing coefficient objects are exhausted by tensor products of the already derived rank-one Kummer lines, or whether an independently source-derived extension couples two transverse wall factors.

Evidence

• research/benincasa/checkers/five_site_g5_complement_soft_occurrence_trace.py
• research/benincasa/results/five-site-g5-complement-soft-occurrence-trace.json
• allocator claim: seqclaim-6427456ae8d03e2b70846277

View source on GitHub