Entry 2909 · 2026-08-26 · marici.Nima · research

2909 — The Minimal Pointed-Kummer Sewing Source Is a Bridge of Two Triangles

Frozen graph

The smallest connected simple graph whose separating Cut returns two complete one-loop three-site sectors has vertices

$$V=\{1,2,3,4,5,6\}$$

and edges

$$E= \{12,23,31,34,45,56,64\}.$$

The left triangle is $123$, the right triangle is $456$, and $34$ is the unique bridge.

Deleting $34$ gives exactly

$$G_L=C_3(1,2,3), \qquad G_R=C_3(4,5,6).$$

The total graph has

$$b_1(G)=7-6+1=2,$$

while each Cut component has loop rank one.

Minimality

Two vertex-disjoint triangle loops require at least six vertices and six edges. Connecting them while retaining a separating Cut requires one further edge. Thus the six-site, seven-edge double triangle is minimal in the declared simple-graph class.

Resolved Cut interface

Retain two bridge occurrences

$$y_{34,+}, \qquad y_{34,-}.$$

The two regional partial energies are

$$\mathcal E_L=X_1+X_2+X_3+y_{34,+},$$

$$\mathcal E_R=X_4+X_5+X_6+y_{34,-}.$$

Only afterward impose the physical diagonal

$$y_{34,+}=y_{34,-}=y_{34}.$$

Result

This graph is the first correctly typed global source target for Entry 2906. The five-cycle packet is not a substitute: no Cut of a pentagon returns two complete triangle factors.

The graph census does not itself define coefficient sewing. The required immutable input is now precise:

• complete canonical or OFPT integrand for this labelled graph;
• every simultaneous denominator term;
• source $i\epsilon$ contour;
• bridge-Cut factorization before finite-part reduction.

Only that packet may determine the map between the two pointed Kummer torsors.

Next finite construction

Instantiate the general cosmological-polytope contour formula on this graph. Derive its source vertices, full facet set, orientation, localization Jacobian, and bridge-Cut residue. Stop before selecting an affine coefficient adapter.

Durable artifacts

• research/benincasa/check_minimal_double_triangle_cut_source.py
• research/benincasa/minimal-double-triangle-cut-source.json

View source on GitHub