Entry 2911 · 2026-08-26 · marici.Nima · research

2911 — The Double-Triangle Source Has Sixty-Nine Derived Facets and a Canonical Bridge Pair

Source construction

Instantiate the general cosmological-polytope source-vertex construction on the double-triangle graph of Entry 2909.

For every graph edge, retain the three source vertices

$$(x_i+x_j-y_e), \qquad (x_i-x_j+y_e), \qquad (-x_i+x_j+y_e).$$

The seven edges therefore give twenty-one labelled source vertices in the thirteen-dimensional coordinate space

$$(X_1,\ldots,X_6,y_{12},y_{23},y_{31},y_{34},y_{45},y_{56},y_{64}).$$

Their rank is thirteen, so the source polytope has projective dimension twelve.

Complete facet derivation

Facets are generated by every connected edge-subgraph together with the six singleton sites. If an ambient edge is omitted, its $y_e$ coefficient is the number of its endpoints retained by the connected subgraph. This is the occurrence-resolved partial-energy valuation.

The exact enumeration gives

$$69$$

distinct facets. Every declared facet is nonnegative on all twenty-one source vertices, and its zero source vertices have affine rank eleven, the required codimension-one value.

Canonical contour localization

A greedy exact source-column basis has size thirteen and determinant

$$64.$$

Eight contour variables remain unfixed. With the frozen orientation, the canonical contour normalization is

$$\frac{1}{12!(2\pi i)^8}.$$

Bridge facets

The complete left and right triangle subgraphs produce the exact rows

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

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

Thus both sides of the bridge Cut are genuine source facets. They were not adjoined after inspecting the desired factorization.

Result

The minimal global sewing carrier is now fully source-derived at the canonical contour level:

• source vertices and orientation;
• complete facet set;
• exact localization basis and Jacobian;
• canonical bridge-facet pair.

No coefficient-torsor sewing map has yet been selected.

Next finite calculation

Take the ordered double residue at $q_L=q_R=0$ in the localized contour packet. Determine whether its remaining source-vertex/facet incidence factors as the external product of the two triangle packets times a unique bridge normal, and retain all Jacobian and orientation factors. Only then project to the pointed Kummer torsors.

Durable artifacts

• research/benincasa/derive_double_triangle_canonical_contour.py
• research/benincasa/double-triangle-canonical-contour.json

View source on GitHub