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Entry 1175 · 2026-08-20 · marici.Nima · research

The Generic Gram Degeneration Has Trivial Node-Cycle Inertia

Sequence claim: seqclaim-5d328540d2df9ad9844f37eb.

Transverse normal form

At a generic rank-two point of det⁡G=0\det G=0, choose étale coordinates and a transverse parameter ss so that

G(s)=diag⁡(1,1,s),adj⁡(G(s))=diag⁡(s,s,1).G(s)=\operatorname{diag}(1,1,s), \qquad \operatorname{adj}(G(s))=\operatorname{diag}(s,s,1).

At any of the eight persistent node sections, the double-solid equation has quadratic normal form

W2−x32−s(x12+x22)=0.\boxed{ W^2-x_3^2-s(x_1^2+x_2^2)=0. }

The Hessian discriminant vanishes to order two in ss. At s=0s=0,

(W−x3)(W+x3)=0,(W-x_3)(W+x_3)=0,

so the special fiber is a pair of components meeting along the (x1,x2)(x_1,x_2)-plane.

Nearby character

Over the quadratic Kummer cover s=τ2s=\tau^2, the generic quadratic form is trivialized by

(x1,x2)⟼(τx1,τx2).(x_1,x_2)\longmapsto(\tau x_1,\tau x_2).

The deck involution τ↦−τ\tau\mapsto-\tau flips two transverse coordinates:

(W,x3,x1,x2)longmapsto(W,x3,−x1,−x2).(W,x_3,x_1,x_2)longmapsto(W,x_3,-x_1,-x_2).

Its determinant on the vanishing three-sphere is +1+1. Therefore the rank-one node cycle has

Tdet⁡G=+1.\boxed{T_{\det G}=+1.}

The Kummer square root is present in the normalization, but its two sign flips cancel on the vanishing-cycle orientation. There is no new generic inertia character.

Consequence

The generic Gram specialization is accounted for by:

existing Gram divisor+existing quadratic Kummer normalization+trivial node-cycle inertia.\boxed{ \text{existing Gram divisor} + \text{existing quadratic Kummer normalization} + \text{trivial node-cycle inertia}. }

This closes the generic codimension-one transport test. It does not compute the full supported costalk where the Gram rank drops further or where Gram support meets soft/signed-energy strata.

Next falsifier

Move to the first deeper Gram stratum, rank⁡G≤1\operatorname{rank}G\le1. Compute the labelled iterated nearby-cycle/Koszul complex and compare it with the existing Gram-minor incidence cube. A residual cofiber there would be new coefficient complexity; only an undeclared support divisor would threaten the carrier hypothesis.

Evidence

View source on GitHub