Entry 2146 · 2026-08-24 · marici.Benincasa · research

2146 — The Soft and Triangle Contact Restrictions Split the Cayley--Menger Cover

Hard-to-vary claim

The two existing support components excluded from Entry 2145 do not supply the half-Kummer normal character at their generic contact intersections. Instead, the restricted Cayley--Menger polynomial becomes a perfect square.

Soft contact intersection

For the component contact with focal separation $p_2$, impose

$$p_2=0, \qquad E=0.$$

The exact restricted polynomial is

$$\boxed{ K|_{q,;p_2=E=0} =-2a^2(p_1-p_3)^2. }$$

Away from the deeper coordinate locus $p_1=p_3$, the double cover splits into two rational sheets. The residue has an ordinary logarithmic pole in $a$, not a new square-root character in the contact normal.

Triangle contact intersection

Parametrize a generic component of the triangle wall by

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

and impose the contact condition $E=2s$. Exact substitution gives

$$\boxed{ K|_{q,;\Lambda=0,;E=2s} =-2s^2 \left(C+2ar-a^2-r^2\right)^2. }$$

For $s\ne0$, this cover likewise splits. Any sign attached to choosing $s=\sqrt{p_2}$ is the already resolved momentum-polarity sheet; it is not a half-Kummer character in $\nu_2=p_2-E^2/4$.

Consequence

The generic contact, soft-contact, and triangle-contact routes now agree:

$$\boxed{ \text{none supplies the }(-1,-1)\text{ normal character required by [Entry 2144](/research/ledger/20260824-2144-the-rank-one-mixed-lines-lie-in-different-monodromy-eigenspaces).} }$$

On the supported strata the reason is stronger than generic unipotence: the branch cover itself splits after restriction.

Scope

The conclusion remains generic along each support component. It does not classify the deeper intersections

$$p_2=E=0, p_1=p_3,$$

or

$$s=0, C+2ar-a^2-r^2=0.$$

Those are existing coordinate-soft/triangle corners, not new carrier divisors.

Evidence

• Entry 2136 and Entries 2143--2145;
• exact assertions in research/benincasa/marici-gm/src/bin/three_site_deletion_landau.rs;
• allocator claim seqclaim-506710ecf17f5d0d6f3ad026.

Next falsifier

Resolve the two displayed deeper corners and compute the normalized cover and residue inertia. Accept a bridge to the lower Kummer line only if a labelled half-character survives normalization and is compatible with the physical sheet orientation.

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