Entry 2147 · 2026-08-24 · marici.Benincasa · research

2147 — The Deeper Soft--Triangle Contact Corner Is a Split Node after Constant Extension

Hard-to-vary claim

The two deeper corners left open by Entry 2146 coincide and normalize to a reducible node after a constant quadratic extension. They carry no parameter-dependent half-Kummer character.

Coincidence of the corners

On the triangle parametrization

$$p_1=r^2, \qquad p_2=s^2, \qquad p_3=(r+s)^2, \qquad E=2s,$$

the soft limit $s=0$ gives

$$p_2=E=0, \qquad p_1=p_3=r^2.$$

Thus Entry 2146's soft and triangle residual loci are the same labelled corner.

Normalized cover

Write

$$H=C+2ar-a^2-r^2.$$

The exact triangle-contact restriction is

$$W^2=-2s^2H^2.$$

Removing the forced soft factor by $W=s\widetilde W$ gives

$$\widetilde W^2=-2H^2.$$

Over the constant extension $\kappa^2=-2$, this factors as

$$(\widetilde W-\kappa H)(\widetilde W+\kappa H)=0.$$

The normalization is therefore two smooth components crossing at $H=0$. The only quadratic datum is the constant field extension $\mathbb Q(\sqrt{-2})/\mathbb Q$; it does not vary around the normal parameters.

Consequence

The deck involution exchanges the two normalized components, but no loop in $\nu_i$, the soft coordinate, or $H$ produces the required labelled half-Kummer character. The node can carry the usual unipotent smoothing variation when moved off the corner, consistently with Entry 2145, but it does not supply a semisimple $-1$ normal eigenspace.

Hence

$$\boxed{ \text{the deletion-contact route to the lower Kummer line is closed on all generic and supported contact strata audited here.} }$$

Classification

• carrier: existing soft--triangle--contact corner;
• normalized coefficient geometry: two rational components;
• arithmetic datum: constant $\sqrt{-2}$ extension;
• normal inertia: trivial semisimple character plus possible unipotent node smoothing;
• required half-Kummer normal twist: absent;
• new carrier datum: none.

Evidence

• Entries 2143--2146;
• research/benincasa/checkers/deeper_contact_node_normalization.rs;
• exact factorization in research/benincasa/marici-gm/src/bin/three_site_deletion_landau.rs;
• allocator claim seqclaim-913447ebe650d7344ccfa29f.

Next falsifier

Retire the deletion-contact-to-lower-Kummer identification. Preserve the contact-product mixed variation as a distinct unipotent coefficient object. Return to the nonsplit marked-relative extension or another source-defined readout rather than adding a twist to this closed branch.

View source on GitHub