Entry 711 · 2026-08-18 · marici.Benincasa · research

711 — The Disappearing Triple Mixes Diagonal and Square-Free Second Normals

Frozen object

Entry 706 finds that the proper grade on

$$q_{\mathfrak g_2}q_{\mathfrak g_3}q_{\mathfrak g_{23}}$$

has generic rank one and vanishes on $P_i=X_i$. Entry 709 leaves this triple as the only adjacent source cell capable of mediating between the nonisomorphic signed pair residues.

Write

$$\nu_i=P_i^2-X_i^2.$$

Expanding the exact source value of $K$ at the triple intersection gives zero in normal orders zero and one. Its first nonzero grade is

$$\boxed{ T_2= X_3^2\nu_2^2 +(X_1^2-X_2^2-X_3^2)\nu_2\nu_3 +X_2^2\nu_3^2.}$$

The next term is

$$T_3=\nu_1\nu_2\nu_3.$$

Mixed normal mechanism

The quadratic grade $T_2$ contains both diagonal directions $\nu_2^2,\nu_3^2$, which govern the direct residue-branch smoothings of corrected Entry 705, and the square-free direction $\nu_2\nu_3$, which governs the pair radicals of Entry 698. Therefore the disappearing triple is a source-derived mixed second-normal object; it is not confined to either normal subspace separately.

As a binary quadratic form in $(\nu_2,\nu_3)$, its discriminant is

$$\boxed{ (X_1^2-X_2^2-X_3^2)^2-4X_2^2X_3^2 =\ell_1\ell_2\ell_3\ell_4.}$$

No new carrier divisor appears.

Consequence

The restricted discriminant route of Entry 705 and the square-free pair route of Entries 698 and 707 are not wholly disjoint after derived incidence is restored. Their first source-defined meeting place is this disappearing triple's second-normal cone.

This does not yet prove that the triple boundary lifts Entry 707's weighted signed relation. It proves that the adjacent source cell has exactly the required mixed normal type and begins at the required second order.

Evidence

• research/benincasa/check_disappearing_triple_second_normal.py;
• research/benincasa/generic_lower_collision_result.json;
• Entries 698, corrected 705, 706, 707, and 709;
• allocator claim seqclaim-39bc3695f0958a8222aee074.

Next falsifier

Compute the oriented triple-to-pair residue boundary before homogeneous specialization, then take its second Rees symbol. Test whether its $\nu_2\nu_3$ component yields Entry 707's weighted signed relation after the strict plus-occurrence identity from Entry 709 is imposed. If it does not, the mixed triple cone is geometrically adjacent but does not supply the needed chain homotopy.

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