Entry 2913 · 2026-08-26 · marici.Nima · research

2913 — The Bridge Double Residue Is the Join of Two Triangle Facets and One Vertex

Ordered face

Take the ordered codimension-two residue along

$$q_L=X_1+X_2+X_3+y_{34}=0,$$

then

$$q_R=X_4+X_5+X_6+y_{34}=0.$$

The common zero set inside Entry 2911's twenty-one labelled source vertices contains exactly thirteen vertices.

Canonical partition

The thirteen vertices split without choices into:

1. six left-triangle vertices: types two and three on edges $12,23,31$;
2. the unique type-one bridge vertex on $34$;
3. six right-triangle vertices: types two and three on edges $45,56,64$.

Each six-vertex triangle block has affine rank four, exactly the total-energy facet of the frozen triangle source. The common face has affine rank ten and satisfies

$$10=4+0+4+2,$$

the projective join dimension formula. Hence the face is

$$F_L * v_{34}^{(1)} * F_R.$$

Normalization and orientation

Writing

$$E_L=X_1+X_2+X_3, \qquad E_R=X_4+X_5+X_6,$$

gives

$$q_L=E_L+y_{34}, \qquad q_R=E_R+y_{34}.$$

At fixed $y_{34}$, the normal Jacobian from $(E_L,E_R)$ to $(q_L,q_R)$ is one. The ordered canonical-form residue therefore has unit coefficient. Reversing the two residue operations changes only the derived orientation sign.

Result

At Carrier level, the bridge double residue is canonically the join of the two triangle total-energy facets and one unique bridge normal vertex. The scale ambiguity exhibited in Entry 2906 is absent for the canonical form itself: source residue normalization fixes it to one.

This does not yet prove that the pointed Kummer finite parts sew with unit scale. Their moving-fiber pushforwards and affine origins must still be transported through this residue map.

Next finite calculation

Push the two source-normalized triangle relative cycles through the join residue. Retain the bridge type-one contour variable and compare the composite pointing with $F_L(2)=F_R(2)=0$. Compute any resulting affine mismatch before quotienting the bridge occurrence.

Durable artifacts

• research/benincasa/check_double_triangle_bridge_double_residue.py
• research/benincasa/double-triangle-bridge-double-residue.json

View source on GitHub