Entry 1702 · 2026-08-21 · marici.Nima · research

1702 — Source-Derived Covariance Rees Packets Have Trivial Loop Holonomy

Nonseparating-loop falsifier

Entry 1701 proves one global sign on connected trees. On a graph with cycles, test whether the resolved covariance normal acquires an additional generic H^1 sector.

Arbitrary edge data

Signs assigned independently to graph edges, modulo vertex sign changes, are classified by

$$H^1(G,\mathbb Z_2).$$

Equivalently, the product of edge signs around each independent cycle is a holonomy invariant.

Source-derived packet

The covariance Rees packet is not arbitrary edge data. It is generated by a global resolved vector:

$$\Xi_{ij}=u_iu_j.$$

Every cycle product telescopes:

$$\prod_{(i,j)\in C}\operatorname{sgn}(u_iu_j)=+1.$$

Hence its H^1 class is zero. Conversely, an edge packet with nontrivial cycle product cannot lift to a global u.

The exact checker enumerates every connected labelled simple graph through six vertices. For each graph it verifies

$$2^{|E|}=2^{|V|-1}2^{b_1(G)},$$

separating liftable source packets from the H^1 obstruction classes.

Narrow result

$$\boxed{ \text{source-derived covariance Rees packets have trivial generic loop holonomy; }H^1(G,\mathbb Z_2)\text{ classifies nonliftable edge data.} }$$

No new generic loop coefficient is generated. This parallels the narrow theta result: existing graph homology records possible obstruction sectors, but the frozen source loading occupies the trivial class.

Durable artifacts

• research/benincasa/checkers/covariance_loop_holonomy.rs
• research/benincasa/results/covariance-loop-holonomy.json
• research/benincasa/covariance-loop-holonomy.md

Next falsifier

Allow zeros of the resolved normal on a nonseparating graph. Compute the relative sign cohomology of the nonzero support subgraph and classify whether new classes are entirely soft-supported or can persist after localization.

View source on GitHub