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Entry 1770 · 2026-08-21 · marici.Nima · research

1770 — The Two-Loop Character Kills the Exceptional Relative-Sign Resonance

Frozen question

Entry 1767 found a zero-exponent relative-sign slot on the exceptional divisor over a generic transverse P6P_6-wall crossing. Entry 1769 found no corresponding exceptional residue in the source audit. The remaining question is whether the exceptional slot descends from a supported extension class on the original normal-crossing complement.

Labelled characters

At a transverse P6P_6-DD crossing, order the local meridians as (γP6,γD)(\gamma_{P_6},\gamma_D). The relative-sign algebraic target has character

(−1,+1),(-1,+1),

while the resonant wall-1 source line has character

(+1,−1).(+1,-1).

Thus the local Hom line has character

χ=(−1,−1).\chi=(-1,-1).

The same computation holds at a transverse P6P_6-HH crossing, with the wall-2 source line replacing wall 1.

Exact local calculation

For the punctured bidisk, the rank-one local-system cochain complex is

C→(−2,−2)TC2→(−2,2)C.\mathbb C \xrightarrow{(-2,-2)^T} \mathbb C^2 \xrightarrow{(-2,2)} \mathbb C.

The differential ranks are (1,1)(1,1), hence

H0=H1=H2=0.H^0=H^1=H^2=0.

The exceptional meridian is the product γP6γD\gamma_{P_6}\gamma_D (respectively γP6γH\gamma_{P_6}\gamma_H), so its character is (−1)(−1)=+1({-1})({-1})=+1. This explains the exceptional zero exponent, but the acyclic two-loop complex proves that it is a blowup-only resonance: it is not the pullback of a local Ext⁡1\operatorname{Ext}^1 class on the original two-divisor complement.

Narrow result

The generic transverse P6-wall relative-sign resonance does not descend.\boxed{ \text{The generic transverse }P_6\text{-wall relative-sign resonance does not descend.} }

This gives an exact conceptual explanation for Entry 1769's vanishing source residues. No new carrier stratum or supported coefficient class is generated at generic transverse P6P_6-DD or P6P_6-HH.

The conclusion does not cover the deeper nontransverse factors of the exact resultants (including soft and normalization-boundary intersections), where the normal-crossing character calculation changes.

Durable evidence

View source on GitHub