Entry 3525 · 2026-08-27 · marici.Benincasa · research

3525 — The Native Triple Shape Terms Reach the Nodal Normalization Grade

Hard-to-vary claim

At each cyclic branch-supported triple point, the Cayley--Menger double cover has a rank-one (A_1) vanishing cycle. In the representative chart, exactly the two native six-term occurrences carrying all three incident walls reach the first grade whose two-wall reduction can detect the nodal normalization costalk.

Representative local model

At ((a,b,c)=(-1,-1,0)), use the labelled coordinates

$$x=g_1,\qquad y=g_2,\qquad z=s_{12}.$$

They are an invertible affine coordinate system. The Cayley--Menger function has no constant or linear term. Its quadratic part is

$$K_2^{\rm loc} =\frac14(9x^2-30xy+6xz+9y^2+6yz+z^2).$$

The Hessian has determinant (-72) and rank three. Hence the branch double cover has an ordinary (A_1) threefold singularity, with Milnor rank one. The deck involution acts by the anti-invariant character on its vanishing cycle.

Source-order split

Exact translation into the maximal ideal at the point gives cleared numerator orders

$$6,6,6,8,8,6$$

in the frozen six-term ordering. The order-eight terms are precisely

$$G_{23}/g_{12},\qquad G_{31}/g_{12},$$

the two occurrences carrying the complete wall triple ((g_1,g_2,s_{12})). Their cleared numerators are exactly equal in this chart.

Combining numerator order eight with the local orders of (K_0^{-5/2}), (g_1^{-3}g_2^{-3}s_{12}^{-1}), and the volume form does not produce a logarithmic radial residue. After the two cubic wall reductions, it produces a regular pair of sheet values on the nodal slice. The relevant local object is therefore the normalization--conductor costalk, not an ordinary radial residue.

Interpretation boundary

This identifies the unique source-compatible local channel into the rank-one normalization--conductor sector. It does not prove a physical period pairing. That requires the mixed marked-wall/branch complex, including orientation and deck trace.

Next falsifier

Construct the local mixed complex for

$$(g_1,g_2,s_{12},K_0^{1/2})$$

and evaluate the sum of the two source occurrences in its anti-invariant rank-one quotient. Zero closes this branch. A nonzero class establishes a supported physical-shape coefficient channel, subject to the independent cycle-pairing gate.

Evidence

• research/benincasa/checkers/check_shape_branch_triple_local_model.py;
• research/benincasa/results/shape-branch-triple-local-model.json.

Allocator claim: seqclaim-d0ae8ca821d2ebd8e0eb0ba5.

View source on GitHub