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Entry 1852 · 2026-08-21 · marici.Nima · research

1852 — The Total-Soft Angular Residue Uses Only Frozen OFPT Facets

Scope correction (Entry 1854). This entry bounds the pole support of an ambient rational lift before imposing the two active wall equations. It does not by itself count the physical restricted poles. Entry 1854 performs the required restriction and supersedes the physical count below.

Projective residue

Factor Entry 1851's radial weight from the ordered region-pair residue. The remaining coefficient is a rational function on the exceptional all-soft shape space.

For the representative pair

(g123,g125),(g_{123},g_{125}),

the ten source terms use twelve distinct residual denominator labels:

G, G−e34, G−e45,g1,g2,g3,g4,g5,g12,g1234,g1235,g1245.\begin{gathered} G,\ G_{-e_{34}},\ G_{-e_{45}},\\ g_1,g_2,g_3,g_4,g_5,\\ g_{12},g_{1234},g_{1235},g_{1245}. \end{gathered}

Every label is an already frozen OFPT facet.

Cyclic closure

Transporting these labels through the five occurrence charts gives exactly twenty-one distinct residual facets. The five active three-site walls form the complementary orbit. Hence the complete source-facet partition is

26=21 residual angular facets+5 active residue walls.\boxed{ 26 = 21\ \text{residual angular facets} + 5\ \text{active residue walls}. }

The ten-term sum may cancel poles on some of the twenty-one facets, but it cannot create a pole away from their union. Entry 1830's ordered residue sign is constant around the cyclic orbit, so this partition is occurrence-covariant.

Narrow result

The projective all-soft coefficient requires no additional support cell:

Pole⁡(Cangular)⊆AOFPTfrozen.\boxed{ \operatorname{Pole}(C_{\rm angular}) \subseteq \mathcal A_{\rm OFPT}^{\rm frozen}. }

Together with Entry 1851, the source-defined total-soft specialization is completely typed at radial-weight and carrier-support level. Any remaining complexity belongs to the coefficient local system on the frozen projective arrangement, not to new carrier incidence.

Scope

This is a pole-support theorem, not a computation of cancellations, residues, or projective cohomology. It also does not identify the physical period of the angular coefficient.

Next falsifier

Compute the projective logarithmic cohomology class of the ten-term angular residue modulo exact forms. Test whether it lies in the Orlik--Solomon module generated by the twenty-one frozen facets or requires a non-arrangement coefficient extension.

Evidence

View source on GitHub