Generalized Weave
Generalized Weave is a Cyclic Lifting of a substitution tiling. It treats a standard substitution rule as a “host” or “carrier” for an $N$-fold topological cover. By cloning the base rule $N$ times and assigning modular “routing offsets” to child tiles, it generates a new configuration of states without altering the underlying geometry.
Applied to any whole-tile substitution rule, the weave is descriptive; it generates a non-MLD (Mutually Local Derivable) configuration where the information is stored in the global hierarchy, effectively “painting” a signal onto the standard tileset (e.g., $W(P3, N)$).
The Generalized Weave reveals that every valid whole-tile substitution rule is actually the “ground state” of a vast, $N^M$ vector space:
$N$: The number of states (the “sheets” of the cover).
$M$: The number of child tiles in the substitution definition.
The Routing Vector: A vector $k \in \mathbb{Z}_N^M$ where each element defines the phase shift for a specific child tile relative to its parent.
The Reduction to “Prime Weaves”. While the mathematical possibilities are infinite ($N^M$), the search space is refined by two filters:
Connectivity: Ensuring the routing vector creates a fully “woven” set rather than isolated sub-systems.
Isomorphism: Deduplicating patterns that are mere relabelings or geometric rotations of one another to find the unique topological species.
The Non-MLD Property: A defining characteristic of the Generalized Weave is that the resulting tilings are pairwise non-MLD. Despite sharing an identical geometric base, the phase-state of any given tile cannot be locally derived from its neighbors. Each valid routing vector $k$ generates a fundamentally distinct topological cover, ensuring that the “Woven” information remains a global property of the substitution hierarchy.