HUNT packet: the sharp Rips threshold for integer lattices

Date: 7 September 2026
Mode: HUNT
Status: selected for attack; no theorem claim

Contents

Exact question

Give $\mathbb Z^n$ its standard word metric

$d_1(x,y)=\sum_{i=1}^n |x_i-y_i|.$

For an integer scale $r$ , the Vietoris–Rips complex $\operatorname{VR}_r(\mathbb Z^n)$ has a simplex on every finite set of diameter at most $r$ . Prove or refute the sharp-threshold conjecture

$\boxed{ \operatorname{VR}_r(\mathbb Z^n)\text{ is contractible for every }r\ge n. }$

The threshold is optimal: for $r<n$ , the cubical subspace $\{0,1\}^n$ contributes nontrivial homology that injects into the lattice Rips complex.

Known frontier

Virk proved contractibility for $r\ge n^2(2n-1)$ , and the sharp statement for $n\le 3$ , using local crushing and local domination.
Zaremsky improved the uniform range to $r\ge n^2+n-1$ using Bestvina–Brady discrete Morse theory.
Gupta, Sarkar, and Shukla proved the sharp conjecture for $n\le 5$ and proved $\operatorname{VR}_r(\mathbb Z^6)$ contractible for $r\ge10$ . Thus the first unresolved sharp cases are $(n,r)=(6,6),(6,7),(6,8),(6,9)$ .
Hulbert and Zaremsky introduced a word-length Morse criterion on 26 August
1. It asks for contractibility of a canonical descending coface poset for each finite constant-length clique. They prove the corresponding standard-metric result for every two-dimensional right-angled Artin group and predict the scale should equal cubical dimension in general.
CAT(0) cubical groups are known to admit some generating set with a contractible Rips complex through Helly-graph methods. That does not settle the standard $\ell^1$ word metric used here.

The intrinsic object

Let $S\subset\mathbb Z^n$ be a finite $r$ -clique on the word sphere $\|s\|_1=L$ . Define its common descending set

$\Lambda_r(S)= \left\{z\in\mathbb Z^n: \|z\|_1\le L-1, \ \|z-s\|_1\le r\text{ for every }s\in S \right\}.$

This is not a chosen presentation of $S$ . It is the set of lattice points of the canonical convex polytope

$P_r(S)=(L-1)B_1\cap\bigcap_{s\in S}(s+rB_1).$

Equivalently, it is cut out by the signed inequalities

$\varepsilon\cdot z\le L-1, \qquad \varepsilon\cdot(z-s)\le r \quad(\varepsilon\in\{\pm1\}^n, s\in S).$

There is an elementary exact reduction behind the proposed attack. If $D_r(S)$ is the Hulbert–Zaremsky descending coface poset, intersecting a coface with the strict lower word ball gives a closure-map retraction

$D_r(S)\simeq \operatorname{VR}_r(\Lambda_r(S)).$

Indeed, every descending coface contains a nonempty $r$ -clique in $\Lambda_r(S)$ , every such clique extends $S$ , and deleting the additional non-descending vertices is an order-preserving map below the identity. This turns the open Morse obligation into a question about one canonical integral polytope.

Why this target is worth pursuing

The problem is elementary to state, sharp, and recognized. It is also the abelian flat inside the new right-angled Artin group prediction, so a genuinely dimension-free mechanism would matter beyond the lattice case.

More importantly for target selection, the attack does not begin by listing maximal simplices, sign patterns, or low-dimensional residual cases. The object $P_r(S)$ exists before any coordinate presentation is chosen. Success would be a structural translation from convex lattice geometry to the contractibility of the Morse link.

Proposed structural bridge

Resolve $P_r(S)$ by orthants. In a fixed orthant its defining inequalities become subset-sum inequalities. Test the following carrier principle.

Descending carrier principle. For every flat $r$ -clique $S$ and $r\ge n$ , the orthant pieces of $P_r(S)\cap\mathbb Z^n$ admit canonical coordinate-shaving maps. These maps dismantle $\operatorname{VR}_r(\Lambda_r(S))$ , and their nonempty intersections are controlled by the same signed-inequality polytope.

The likely language is discrete convexity: after the correct translation, an orthant carrier may be an integral polymatroid or an $M$ -convex set. If so, the exchange axiom supplies the shaving move, while the signed-face nerve glues the orthants without a case census. Generalized permutahedra are an influence on this formulation, not a formal dependency or a claim that $P_r(S)$ is already known to lie in that class.

Concrete first attack

Prove the closure-map equivalence $D_r(S)\simeq\operatorname{VR}_r(\Lambda_r(S))$ in full poset detail.
In one orthant, rewrite the signed $\ell^1$ -constraints as a set function on coordinate subsets. Decide whether the integer points satisfy the symmetric exchange axiom of a discrete polymatroid.
If they do, construct a canonical norm-decreasing exchange and prove that it dominates the vertex being shaved in the Rips graph.
Prove that the orthant subcomplexes and every nonempty intersection are contractible; apply the nerve lemma to the full descending link.
Feed the result into the Hulbert–Zaremsky Morse criterion. Only then try to extend the carrier from the cubical flat $\mathbb Z^n$ to a general RAAG.

The first decisive question is therefore exact:

$\boxed{ \text{Do the orthant lattice points of }P_r(S)\text{ satisfy a uniform exchange law strong enough to give a Rips domination map?} }$

Failure certificate

Stop this route if the exchange law fails for a symbolic family and the orthant nerve does not repair it. Record the smallest parametric obstruction, not a catalogue of exceptional cliques. A finite-dimensional verification, including the known $n\le5$ cases, is a control and not evidence for a dimension-free proof.

Alternatives considered

The full RAAG threshold prediction was too broad for a first attack. The lattice flat isolates the canonical obstruction without normal-form choices.
Bier-sphere universality and explicit nonpolytopality immediately lead to realization-space gadgets and configuration searches.
A recent regular-black-hole selection problem remains dependent on extra physical prescriptions, repeating the contact-data ambiguity that stopped the JT-gravity route.
A new Cartan-convexity truncation question has an appealing intrinsic geometry but a less settled frontier and several unresolved boundary conventions.

Sources and credit

Seth Hulbert and Matthew C. B. Zaremsky, Word length, Morse theory, and Vietoris–Rips complexes, arXiv:2608.25614v1, submitted 26 August 2026: https://arxiv.org/html/2608.25614v1
Raju Kumar Gupta, Sourav Sarkar, and Samir Shukla, On the Vietoris–Rips Complexes of Integer Lattices, arXiv:2511.04238v1, submitted 6 November 2025: https://arxiv.org/html/2511.04238v1
Matthew C. B. Zaremsky, Contractible Vietoris–Rips complexes of $\mathbb Z^n$ , arXiv:2410.11993v2, revised 12 August 2025: https://arxiv.org/abs/2410.11993
Žiga Virk, Contractibility of the Rips complexes of Integer lattices via local domination, arXiv:2405.09134v4, revised 26 July 2025: https://arxiv.org/abs/2405.09134
Jérémie Chalopin, Victor Chepoi, Anthony Genevois, Hiroshi Hirai, and Damian Osajda, Helly groups, Geometry & Topology 29 (2025): https://msp.org/gt/2025/29-1/gt-v29-n1-p01-s.pdf

The conjecture belongs to Virk and Zaremsky, with the current low-dimensional frontier due to Gupta–Sarkar–Shukla. The descending-poset reduction uses the Morse setup of Hulbert–Zaremsky. Postnikov’s generalized-permutahedron viewpoint influences the proposed carrier language but is not used as an established theorem here.