Entry 2145 · 2026-08-24 · marici.Benincasa · research

2145 — The Generic Contact Residue Has No Half-Kummer Normal Character

Hard-to-vary claim

At generic nonsoft kinematics, the source measure at a component-contact threshold supplies logarithmic unipotent monodromy but no half-Kummer character in the normal variable.

Local residue normal form

Away from the soft and triangle factors in Entry 2136, the branch--pole tangency is ordinary. In analytic coordinates it has the source-equivalent normal form

$$q=v, \qquad K=u^2+v+\nu, \qquad \Omega=\frac{du\wedge dv}{q\sqrt K}.$$

Taking the pole residue gives

$$\operatorname{Res}_{q=0}\Omega = \frac{du}{\sqrt{u^2+\nu}}.$$

Under $u=\sqrt\nu,t$, the two half-powers cancel:

$$\frac{du}{\sqrt{u^2+\nu}} = \frac{dt}{\sqrt{1+t^2}}.$$

The collision of its two branch points produces the familiar logarithmic period. Its semisimple character is $+1$, with a rank-one nilpotent variation. No residual factor $\nu^{1/2}$ remains.

Consequence for the mixed comparison

Tensoring two contact residues therefore produces the rank-one mixed variation of Entry 2143, but its image remains in the $(+1,+1)$ eigenspace. The source residue measure does not provide the $(-1,-1)$ half-Kummer twist required by Entry 2144.

Hence

$$\boxed{ \text{the direct contact-product to lower-Kummer comparison is closed at generic nonsoft kinematics.} }$$

This is not a return to the retracted additive argument. The source product and its nonzero mixed variation are retained; the obstruction is now the correct monodromy eigenspace.

Scope

The conclusion excludes the soft and triangle intersections where the prefactors in Entry 2136 vanish and the ordinary tangency normal form can degenerate. Those are existing supported strata and require separate local models.

Evidence

• Entries 2136 and 2143--2144;
• research/benincasa/checkers/contact_residue_exponent.rs;
• allocator claim seqclaim-a8447bcb221f15d35de2291f.

Next falsifier

Move to the existing soft/triangle intersections of the contact threshold. Derive their degenerate residue normal forms before asking whether a half-integral character or supported comparison appears. Do not extrapolate the generic logarithmic model onto those deeper strata.

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