Homshift: Symbolic Dynamics & Higher-Spin Shifts

• Homshift in symbolic dynamics is a d-dimensional shift of finite type defined by graph homomorphisms from Z^d to a finite graph, encapsulating nearest-neighbor constraints and mixing behavior.
• The associated topological structures, including square covers and square groups, provide a framework to analyze and decide key dynamical properties and reveal a dichotomy in block-gluing rates.
• In higher-spin theory, Homshift refers to shifted homotopy operators used in perturbative Vasiliev-type equations to control auxiliary spinor variables and define cubic vertices.

Searching arXiv for recent and foundational papers on “homshift” and related work. “Homshift” denotes two distinct technical notions in contemporary arXiv literature. In symbolic dynamics, a homshift is a $d$-dimensional shift of finite type obtained as the space of graph homomorphisms from the grid graph $\mathbb Z^d$ to a finite connected undirected graph $G$; this notion is central to the study of multidimensional SFTs, mixing scales, graph covers, and algorithmic decidability (Chandgotia et al., 28 Jul 2025). In higher-spin theory, “Homshift” is also used as shorthand for shifted homotopy operators in perturbative Vasiliev-type equations, where homotopy shifts in auxiliary spinor variables and in the argument of $\omega(Y)$ parametrize admissible cohomology representatives and the resulting cubic vertices (Tarusov et al., 2022). The two usages are mathematically unrelated, but both concern how constrained local data are extended globally: in one case to lattice configurations, in the other to solutions of $d_Z$-equations in unfolded dynamics.

1. Homshift in symbolic dynamics

For a finite, connected, undirected graph $G = (V(G), E(G))$ and an integer $d \ge 1$, the $d$-dimensional homshift associated with $G$ is the set of graph homomorphisms from the grid graph $\mathbb Z^d$ to $\mathbb Z^d$0 (Chandgotia et al., 28 Jul 2025). Equivalently, it is the subshift

$\mathbb Z^d$1

defined by nearest-neighbor constraints,

$\mathbb Z^d$2

This is a nearest-neighbor shift of finite type: local forbidden patterns are exactly pairs of symbols not joined by an edge of $\mathbb Z^d$3, checked on adjacent lattice sites (Chandgotia et al., 28 Jul 2025).

Canonical examples include proper $\mathbb Z^d$4-colorings, obtained by taking $\mathbb Z^d$5 without loops; the hard-square model, encoded by a graph with two vertices, an edge between them, and a loop on one vertex; and classes such as Lipschitz height functions, clock models, and many classical lattice spin systems (Chandgotia et al., 28 Jul 2025). The formulation is graph-theoretic but already captures a large portion of the standard nearest-neighbor lattice repertoire.

Several coarse structural properties are unusually tractable in this setting. Non-emptiness is trivial as soon as $\mathbb Z^d$6 has at least one edge. Pattern extension is decidable because a pattern extends to a global configuration iff it extends to a rectangle, reducing the extension problem to a finite check. Entropy is computable from the homshift’s description, as a consequence of transfer-matrix methods in strips (Chandgotia et al., 28 Jul 2025). Topological transitivity and topological mixing are also characterized directly by graph structure: $\mathbb Z^d$7 is transitive iff $\mathbb Z^d$8 is connected, and it is mixing iff $\mathbb Z^d$9 is connected and not bipartite (Chandgotia et al., 28 Jul 2025).

This cluster of decidability results initially suggested that homshifts might avoid the usual undecidability phenomena that pervade general multidimensional SFTs. The later undecidability results for finer mixing scales are significant precisely because they arise inside a class whose coarser properties remain comparatively tame (Chandgotia et al., 28 Jul 2025).

2. Quantitative mixing and block-gluing classes

Block-gluing is a quantitative mixing property for multidimensional shifts. Informally, any two admissible patterns on distant shapes can be simultaneously embedded in a single global configuration once their supports are far enough apart, with the required separation controlled by a scale function $G$0 (Chandgotia et al., 28 Jul 2025). For finite shapes $G$1, their lattice separation is

$G$2

A subshift $G$3 is $G$4-block gluing if there exists $G$5 such that for all $G$6, whenever $G$7 are globally admissible patterns supported on finite shapes of diameter at most $G$8 and $G$9, there exists $\omega(Y)$0 extending $\omega(Y)$1 and $\omega(Y)$2 simultaneously, with no phase shift (Chandgotia et al., 28 Jul 2025). A phased variant permits a bounded translation $\omega(Y)$3 with $\omega(Y)$4, producing the notion of $\omega(Y)$5-phased block gluing.

The notation $\omega(Y)$6-phased block gluing means that the relevant separation function belongs to $\omega(Y)$7 (Chandgotia et al., 28 Jul 2025). Two standard facts organize the homshift case. First, for any finite undirected $\omega(Y)$8 and any $\omega(Y)$9, $d_Z$0 is $d_Z$1-phased block gluing. Second, if $d_Z$2 is not bipartite, then $d_Z$3-block gluing is equivalent to $d_Z$4-phased block gluing; otherwise only the phased version can hold (Chandgotia et al., 28 Jul 2025). This parity obstruction is fundamental: in bipartite graphs, exact alignment can fail even when a bounded phase shift restores compatibility.

For two-dimensional homshifts, Gangloff–Hellouin de Menibus–Oprocha established a sharp dichotomy: $d_Z$5 is either $d_Z$6-phased block gluing or $d_Z$7-phased block gluing, with no intermediate asymptotic growth rate (Chandgotia et al., 28 Jul 2025, Gangloff et al., 2022). This dichotomy is unusually rigid for a quantitative mixing invariant. It implies that, in dimension two, the long-range gluing behavior of homshifts is not merely bounded between linear and logarithmic scales; it falls exactly into one of those two classes.

A related strip-graph formulation clarifies the mechanism. If $d_Z$8 denotes the graph whose vertices are length-$d_Z$9 walks in $G = (V(G), E(G))$0 and whose edges connect pointwise adjacent walks, then the diameter growth of $G = (V(G), E(G))$1 is either $G = (V(G), E(G))$2 or $G = (V(G), E(G))$3, with the same criterion that governs the block-gluing rate (Chandgotia et al., 28 Jul 2025). This diameter controls strip mixing across width and relates to spectral bounds used in entropy computations. A plausible implication is that block-gluing in this setting is best understood not as a purely combinatorial extension property, but as a coarse geometric invariant of graph-lift structure.

3. Square covers, square groups, and topological interpretation

The topological analysis of homshifts centers on the square cover and the square group (Chandgotia et al., 28 Jul 2025). Let $G = (V(G), E(G))$4 be connected. Its universal cover $G = (V(G), E(G))$5 has as vertices all non-backtracking walks in $G = (V(G), E(G))$6 from a fixed base vertex, with edges connecting walks differing by a single step of extension or removal; the projection $G = (V(G), E(G))$7 sends a walk to its terminal vertex (Chandgotia et al., 28 Jul 2025). This is a graph covering map, and deck transformations correspond to the free and transitive action of $G = (V(G), E(G))$8 on each fiber.

A square in $G = (V(G), E(G))$9 is a non-backtracking cycle of length $d \ge 1$0. Let $d \ge 1$1 be the normal subgroup of $d \ge 1$2 generated by all conjugates $d \ge 1$3, where $d \ge 1$4 is a non-backtracking walk and $d \ge 1$5 is a square (Chandgotia et al., 28 Jul 2025). The square group is then

$d \ge 1$6

This quotient kills all squares, together with their conjugates, in the fundamental group. The associated square cover is

$d \ge 1$7

a regular cover of $d \ge 1$8 with covering map $d \ge 1$9 (Chandgotia et al., 28 Jul 2025).

The main structural correspondences are direct. The square group $d$0 is finite if and only if the square cover $d$1 is finite. A cover $d$2 lifts every square of $d$3 to a square iff $d$4; equivalently, $d$5 is the largest cover to which every square in $d$6 has a square lift (Chandgotia et al., 28 Jul 2025). This yields a configuration lifting criterion: $d$7 lifts to $d$8 for every configuration, with prescribed origin lift, iff $d$9.

There is also a useful square-equivalence interpretation. Two non-backtracking walks with the same endpoints are square-equivalent if their difference lies in $G$0; equivalently, one can pass from one to the other by finitely many “differ-by-a-square” moves (Chandgotia et al., 28 Jul 2025). Then $G$1 is the quotient of $G$2 by square-equivalence classes of non-backtracking walks.

This topological recasting explains why the block-gluing dichotomy can be read off from a covering invariant. In dimension two, $G$3 is $G$4-phased block gluing iff the square cover is infinite, and it is $G$5-phased block gluing iff the square cover is finite (Chandgotia et al., 28 Jul 2025, Gangloff et al., 2022). The symbolic-dynamical invariant is therefore controlled by a quotient of the fundamental group.

4. Undecidability of block-gluing classes

The central result of “Undecidability of the block gluing classes of homshifts” is that the $G$6 versus $G$7 block-gluing dichotomy for two-dimensional homshifts is undecidable (Chandgotia et al., 28 Jul 2025). More precisely, given a finite connected undirected graph $G$8, it is not algorithmically decidable whether the associated homshift $G$9 is $\mathbb Z^d$0-block gluing, and likewise not decidable whether it is $\mathbb Z^d$1-block gluing (Chandgotia et al., 28 Jul 2025).

The reduction proceeds through three ingredients. First, regular covers of $\mathbb Z^d$2 correspond to normal subgroups of $\mathbb Z^d$3, and $\mathbb Z^d$4 corresponds to $\mathbb Z^d$5. Second, the GHO23 dichotomy identifies the block-gluing class of $\mathbb Z^d$6 with finiteness or infiniteness of $\mathbb Z^d$7 (Chandgotia et al., 28 Jul 2025, Gangloff et al., 2022). Third, every finitely presented group $\mathbb Z^d$8 arises as the square group $\mathbb Z^d$9 of some algorithmically constructible finite, connected, undirected graph $\mathbb Z^d$00 (Chandgotia et al., 28 Jul 2025). Since the finiteness problem for finitely presented groups is undecidable by the Adian–Rabin theorem, deciding the block-gluing class would decide group finiteness, which is impossible (Chandgotia et al., 28 Jul 2025).

The argument is dimension-specific. The undecidability theorem is established for $\mathbb Z^d$01, and the paper does not claim undecidability for block-gluing when $\mathbb Z^d$02 (Chandgotia et al., 28 Jul 2025). Many of the relevant lifting arguments, strip approximations, and the dichotomy itself are two-dimensional.

The paper emphasizes that this source of undecidability is “completely different” from classical undecidability proofs for general multidimensional SFTs (Chandgotia et al., 28 Jul 2025). Berger- or Kari-type constructions encode universal computation into anisotropic local constraints. Homshifts are too symmetric for that style of simulation: the constraints are graph-adjacency and isotropic. Here undecidability comes instead from elementary algebraic topology, covering theory, and group-theoretic reductions. This contrast is conceptually important. It suggests that homshifts evade computational universality at the local rule level while still inheriting algorithmic hardness from global topological invariants.

5. Examples, edge cases, and open directions

Several examples illustrate how the square group governs mixing behavior. If $\mathbb Z^d$03 is a tree, then $\mathbb Z^d$04 is trivial, so $\mathbb Z^d$05 is trivial, the square cover is finite, and $\mathbb Z^d$06 is $\mathbb Z^d$07-phased block gluing; if $\mathbb Z^d$08 is made non-bipartite by adding a self-loop, then one obtains $\mathbb Z^d$09-block gluing (Chandgotia et al., 28 Jul 2025). For $\mathbb Z^d$10, every cycle is square-decomposable, so the square group is trivial and the square cover finite. For $\mathbb Z^d$11 with $\mathbb Z^d$12, there are no squares, $\mathbb Z^d$13, $\mathbb Z^d$14 is infinite, and $\mathbb Z^d$15 is $\mathbb Z^d$16-phased block gluing (Chandgotia et al., 28 Jul 2025).

For complete graphs $\mathbb Z^d$17 with $\mathbb Z^d$18, the situation is more intricate. These graphs are non-bipartite and have many squares, but the square group depends on the particular combinatorics of squares in $\mathbb Z^d$19 (Chandgotia et al., 28 Jul 2025). The dichotomy still applies, but determining which side occurs requires understanding whether $\mathbb Z^d$20 is finite.

Dimension one behaves differently. In $\mathbb Z^d$21, a homshift is a nearest-neighbor Markov chain on $\mathbb Z^d$22, and mixing properties are classical and decidable; the undecidability phenomena arise in $\mathbb Z^d$23 through square geometry and strip growth (Chandgotia et al., 28 Jul 2025). This suggests that the relevant complexity threshold is not merely “higher-dimensionality” in the abstract, but specifically the availability of two-dimensional square combinatorics.

The current open problems are narrowly targeted. It remains open whether strong irreducibility is decidable for $\mathbb Z^d$24 (Chandgotia et al., 28 Jul 2025). It also remains open whether there exists a two-dimensional homshift that is $\mathbb Z^d$25-block gluing but not $\mathbb Z^d$26-block gluing, whether analogous undecidability holds in dimensions $\mathbb Z^d$27, and how square-cover methods extend to homshifts on other Cayley graphs (Chandgotia et al., 28 Jul 2025). The paper also mentions a continuous analogue involving $\mathbb Z^d$28-Lipschitz maps into a compact Riemannian manifold and a homotopy-like filling distance $\mathbb Z^d$29, raising the question of sublinear but superlogarithmic filling behavior (Chandgotia et al., 28 Jul 2025). These are not established results, but they indicate that the square-group perspective may generalize beyond the combinatorics of finite graphs.

6. “Homshift” as shifted homotopy in higher-spin theory

A second, unrelated usage appears in higher-spin theory, where “Homshift” refers to shifted homotopy operators in the analysis of linearized higher-spin equations in arbitrary higher-spin backgrounds (Tarusov et al., 2022). The setting is unfolded higher-spin dynamics in four dimensions, formulated in terms of regular functions of auxiliary Sp(4) spinor variables $\mathbb Z^d$30, auxiliary oscillators $\mathbb Z^d$31, and Klein operators. At first order, the master equations reduce to differential equations in $\mathbb Z^d$32-space of the form

$\mathbb Z^d$33

to be solved by homotopy techniques (Tarusov et al., 2022).

The conventional choice uses an unshifted contracting homotopy, corresponding to a vanishing homotopy shift vector $\mathbb Z^d$34 (Tarusov et al., 2022). More generally, one defines a contracting homotopy $\mathbb Z^d$35 and cohomology projector $\mathbb Z^d$36 satisfying

$\mathbb Z^d$37

The central innovation is to allow the homotopy shift $\mathbb Z^d$38 to include constant or background-dependent pieces proportional to $\mathbb Z^d$39, to the “derivatives” $\mathbb Z^d$40, and to the argument of $\mathbb Z^d$41 entering through homotopy integrals (Tarusov et al., 2022). In the paper’s terminology, this is “Homshift.”

The admissible class preserving the proper form of the free higher-spin equations is sharply constrained. Lorentz covariance forces constant spinors $\mathbb Z^d$42 and $\mathbb Z^d$43 to vanish. The $\mathbb Z^d$44-shifts $\mathbb Z^d$45 in $\mathbb Z^d$46 must vanish. Within each homotopy step, the $\mathbb Z^d$47- and $\mathbb Z^d$48-shift coefficients must be equal, so the allowed shifts are uniform in $\mathbb Z^d$49:

$\mathbb Z^d$50

with $\mathbb Z^d$51 and $\mathbb Z^d$52 allowed to differ between the $\mathbb Z^d$53 and $\mathbb Z^d$54 steps (Tarusov et al., 2022). This is called the relaxed uniform shift. In AdS$\mathbb Z^d$55, it preserves the First On-Shell Theorem in both physical and topological sectors and produces a one-parameter family of vertices, parametrized by $\mathbb Z^d$56, that contains both the conventional homotopy case $\mathbb Z^d$57 and the uniform-shift case $\mathbb Z^d$58 (Tarusov et al., 2022).

A separate result concerns shifts by the argument of $\mathbb Z^d$59. When the homotopy shift for $\mathbb Z^d$60 contains a term $\mathbb Z^d$61, with $\mathbb Z^d$62 the homotopy parameter, this shifts the evaluation point along the homotopy line in $\mathbb Z^d$63-space and effectively shifts the $\mathbb Z^d$64-arguments of the $\mathbb Z^d$65-factors through star-exchange identities (Tarusov et al., 2022). However, pure $\mathbb Z^d$66-argument shifts do not affect $\mathbb Z^d$67 nor the cubic vertices at first order: the resulting vertices coincide with those from the conventional homotopy (Tarusov et al., 2022). The paper interprets this as a limitation on the extent to which such shifts can be used to improve locality or alter holographic couplings at first order.

This higher-spin usage of “Homshift” is unrelated to graph homomorphism shifts, despite the shared name. The common linguistic element is the use of homotopy or homomorphism-based structure to control extension problems, but the mathematical frameworks—unfolded gauge theory versus multidimensional symbolic dynamics—are disjoint.

7. Terminological ambiguity and research significance

The coexistence of these two meanings makes “homshift” a context-sensitive term. In symbolic dynamics, it names a class of SFTs defined by graph homomorphisms from $\mathbb Z^d$68 to a finite graph (Chandgotia et al., 28 Jul 2025). In higher-spin theory, it denotes shifted homotopy analysis, especially the relaxed uniform $\mathbb Z^d$69-shift and $\mathbb Z^d$70-argument shifts in perturbative solutions of Vasiliev-type equations (Tarusov et al., 2022). Conflating them would be a category error: the first is a dynamical system on lattice configurations, the second a choice of homological resolution scheme in an auxiliary spinor-oscillator algebra.

Within symbolic dynamics, the importance of homshifts lies in the contrast between tractable coarse properties and undecidable fine mixing scales. Non-emptiness, extension, entropy, transitivity, and mixing remain decidable, yet the distinction between $\mathbb Z^d$71 and $\mathbb Z^d$72 block-gluing is undecidable in dimension two (Chandgotia et al., 28 Jul 2025). This places homshifts at a precise boundary between structurally rigid and algorithmically wild behavior. The topological route to undecidability, via square covers and square groups, further connects symbolic dynamics to covering theory and finitely presented groups rather than to computational universality.

Within higher-spin theory, the significance of Homshift is more local and perturbative. It clarifies scheme dependence in homotopy resolutions, identifies the admissible class of shifts compatible with the First On-Shell Theorem, and shows that pure shifts by the argument of $\mathbb Z^d$73 are inert at first order (Tarusov et al., 2022). This suggests that any attempt to use shifted homotopies to optimize spin-locality or modify effective couplings must rely on the relaxed uniform $\mathbb Z^d$74 class or on higher-order effects rather than on pure $\mathbb Z^d$75-argument shifts.

Taken together, these usages show how the same lexical form can designate two specialized constructions in different subfields: one grounded in graph homomorphisms and multidimensional SFTs, the other in contracting homotopies for higher-spin master-field equations.