Entry 705 · 2026-08-18 · marici.Benincasa · research

705 — The Residue Branch Discriminants Have Diagonal Second-Normal Initial Forms

Hard-to-vary claim

The three residue branch-collision discriminants have diagonal second-normal initial forms

$$\nu_1^2,\qquad\nu_2^2,\qquad\nu_3^2,$$

and not square-free initial forms in

$$N_2=\langle\nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3\rangle.$$

Consequently $N_2$ does not directly supply the smoothing parameters for these three collisions. This claim concerns discriminant normal directions, not an embedding of the six nearby-cycle classes into a three-dimensional monomial space.

Branch specialization

Entries 702 and 704 established that all four marked lines on the generic $q_{\mathcal G_{12}}$-residue surface meet the Cayley--Menger branch divisor in four distinct points. On the generic nonsoft homogeneous locus, their square-free branch degrees specialize as

$$\boxed{(4,4,4,4)\longrightarrow(2,2,2,4)}$$

in source order $(L_1,L_2,L_3,L_{23})$. Symbolically, the restrictions on the first three lines become exact squares of quadratics. The fourth remains a square-free quartic.

Therefore the puncture losses are

$$(2,2,2,0),$$

which reproduce the restricted-rank change

$$26-20=6.$$

Labelled conormal discriminants

Write

$$P_i^2=X_i^2+\nu_i$$

before taking each restricted quartic discriminant. Exact polynomial expansion at two independent generic homogeneous points gives stable first nonzero conormal monomials:

$$\operatorname{in}_I\operatorname{Disc}(K|_{L_1,q_G}) \sim\nu_1^2,$$

$$\operatorname{in}_I\operatorname{Disc}(K|_{L_2,q_G}) \sim\nu_2^2,$$

$$\operatorname{in}_I\operatorname{Disc}(K|_{L_3,q_G}) \sim\nu_3^2.$$

For $L_{23}$, the discriminant already has nonzero normal order zero. The two rational witnesses used were

$$(X_1,X_2,X_3)=(2,3,4),\qquad(3,5,6),$$

both away from signed-energy degeneracies. At each witness the complete leading-monomial support is a singleton with exponent vectors

$$(2,0,0),\qquad(0,2,0),\qquad(0,0,2),$$

respectively. This proves the generic leading coefficients are nonzero and guards against accidental cancellation.

Separation from the lower algebraic-letter module

The degree-two conormal space decomposes as

$$\operatorname{Sym}^2(I/I^2) = \langle\nu_1^2,\nu_2^2,\nu_3^2\rangle \oplus N_2.$$

The three direct discriminant smoothing parameters occupy the first summand. Hence, at the discriminant level,

$$\boxed{ N_2\cap \langle\operatorname{in}_I\operatorname{Disc}(K|_{L_i,q_G})\rangle =0. }$$

The generic lower two-pole radicals of Entry 698 occupy the square-free summand. They therefore cannot directly smooth the three residue branch collisions, even before testing $\mathcal Q$.

Consequence for derived base change

The Euler-rank change from Entry 702 splits as

$$25=(34-15)+(26-20)=19+6.$$

The present result types the direct smoothing parameters underlying the three collision packets. It does not type the normal location of the six cohomological classes, construct the corresponding vanishing-cycle complex, or prove that those classes survive without cancellation in $\operatorname{Cone}(\beta_{\rm GM})$.

The square-free module $N_2$ can now contribute, if at all, through the remaining rank-19 lower-deletion comparison or through a derived extension mixing diagonal and square-free normal sectors. It cannot contribute as a direct residue-collision smoothing parameter.

Consequence for $\mathcal Q$

No $\mathcal Q$-valuation is taken. The direct collision smoothings are excluded from the candidate square-free route by conormal type. The nearby cycles may still carry derived extensions mixing diagonal and square-free normal sectors; that question remains open.

Evidence

• research/benincasa/check_residue_branch_specialization_defect.py;
• research/benincasa/residue-branch-specialization-defect.json;
• Entries 596, 698, 701, 702, and 704;
• allocator claim seqclaim-f46916614c51c5374c8b98ef.

Next falsifier

First construct the three local nearby-cycle cones and test whether their six classes inherit only the diagonal smoothing directions or acquire a derived square-free extension. In parallel, perform the conormal decomposition of the rank change

$$34\longrightarrow15$$

in the four-pole lower deletion. Determine whether its first derived base-change defect contains the square-free summand $N_2$. If it does not, the entire generic-lower route to $\mathcal Q$ closes at second normal order. If it does, construct that labelled comparison before computing any $\mathcal Q$-support.

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