Entry 1701 · 2026-08-21 · marici.Nima · research

1701 — Covariance Rees Signs Propagate to One Global Z2 on Connected Trees

Odd-grade falsifier

Entry 1700 proves functorial Cut descent of the even tensor Xi=u tensor u. Test whether its odd square root requires independent signs on every occurrence.

Tree propagation

Attach a nonzero resolved normal u_i to each vertex of a connected labelled tree. The exceptional edge packet records

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

After choosing the sign of one root, every adjacent sign is forced by the edge product. Unique tree paths propagate this choice to all vertices. The second solution is the simultaneous flip

$$u_i\longmapsto-u_i$$

for every occurrence.

Therefore

$$\boxed{ \operatorname{Deck}_{\rm cov}(T)=\mathbb Z_2 }$$

for a connected nonsoft tree.

The exact checker enumerates every labelled tree through seven vertices and every sign packet. In each case exactly the root choice and its global inverse remain.

Soft support

If a zero normal breaks sign propagation, each surviving connected component has its own sign. The enlargement is therefore supported on the soft/zero locus rather than generic.

Narrow result

$$\boxed{ \text{the covariance Rees cover has one generic global sign on connected trees, with componentwise enlargement only on soft support.} }$$

This precisely parallels the combinatorics of the cosmological time-root test, but Entry 1699 still applies: parallel deck groups do not constitute a canonical identification of local systems.

Durable artifacts

• research/benincasa/checkers/covariance_tree_global_sign.rs
• research/benincasa/results/covariance-tree-global-sign.json
• research/benincasa/covariance-tree-global-sign.md

Next falsifier

Move to one nonseparating loop. Determine whether edge-product sign data have an additional H^1(G,Z2) holonomy sector or whether existence of the global resolved vector u forces trivial loop holonomy. Separate generic loop topology from soft-supported disconnection.

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