Entry 1853 · 2026-08-21 · marici.Nima · research

1853 — Every Allowed Total-Soft Angular Facet Has Nonzero Residue

Correction (Entry 1854). The twelve residues computed here belong to the ambient rational lift. After restriction to (g_{123}=g_{125}=0), two labelled facets coincide and cancel. The claim that all twelve are physical poles is withdrawn.

Exact rational reduction

Entry 1852 bounded the representative angular pole support by twelve frozen OFPT facets. The ten source terms were now summed exactly over

$$\mathbb Q(t,y_1,\ldots,y_5)$$

using Symbolica. For each candidate linear facet (L=0), the checker formed

$$\left.L\,C_{\rm angular}\right|_{L=0}$$

after exact common-denominator reduction and cancellation.

Residue census

Every candidate residue is nonzero:

$$\begin{gathered} G,\quad G_{-e_{34}},\quad G_{-e_{45}},\\ g_1,g_2,g_3,g_4,g_5,\\ g_{12},g_{1234},g_{1235},g_{1245}. \end{gathered}$$

Thus the representative has exactly twelve genuine simple angular poles. No pole allowed by Entry 1852's support bound cancels in the ten-term OFPT sum.

Cyclic consequence

The ordered residue orientation and source expression are cyclically covariant. Therefore the five occurrence charts assemble these twelve representative poles into all twenty-one residual facets of Entry 1852:

$$\boxed{ \operatorname{Pole}(C_{\rm angular}^{C_5}) = \mathcal A_{21}^{\rm residual}. }$$

The remaining five source facets are precisely the active three-site wall orbit on which the double residue has already been taken.

Architectural conclusion

The all-soft angular coefficient uses the complete frozen complement of the active wall orbit. Its complexity is arrangement-theoretic but not smaller than the allowed carrier support:

$$\boxed{ 26 = 21\ \text{genuine coefficient poles} + 5\ \text{active residue walls}. }$$

This strengthens H2. The coefficient is nontrivial on every available frozen facet, yet requires no new incidence generator.

Scope

Nonzero simple residues establish the polar divisor, not the dimension or extension structure of the logarithmic cohomology class. Higher intersections and Orlik--Solomon relations remain to be reduced.

Next falsifier

Construct the residue vector on the twenty-one-facet projective arrangement and apply the frozen Orlik--Solomon circuit relations. Test whether the ten-term class closes in the existing logarithmic complex or leaves a derived coefficient extension at higher intersections.

Evidence

• research/benincasa/marici-gm/src/bin/five_site_region_pair_total_soft_angular_residues.rs
• research/benincasa/results/five-site-region-pair-total-soft-angular-residues.json
• Entries 1851--1852
• allocator claim: seqclaim-3c800d6363142a0e0b5e138a

View source on GitHub