Exact target

A sharp, intrinsic threshold

Give $\mathbb Z^n$ the Manhattan metric. The conjecture is

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

The bound is optimal: below dimension, the Boolean cube carries nontrivial homology into the lattice complex. The first open sharp cases occur in dimension six at scales six through nine.

Known frontier

Several methods approach the threshold, but none reaches it uniformly

Virk proved the first general theorem at scale$n^2(2n-1)$ and the sharp result through dimension three. Zaremskyimproved the uniform range to$n^2+n-1$. On 6 November 2025, Gupta, Sarkar, and Shukla proved the sharp conjecture through dimension five and dimension-six contractibility from scale ten.

A 26 August 2026 theorem of Hulbert and Zaremskysupplies a new local Morse criterion. They prove the standard-metric result for two-dimensional right-angled Artin groups and predict that cubical dimension is the correct scale in general. The lattice is the canonical abelian flat of that larger prediction.

Intrinsic object

The descending polytope

Let $S$ be an$r$-clique on the word sphere of radius $L$. Its common descending vertices are the lattice points of

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

This polytope is canonical. Its signed half-space inequalities remember the full $\ell^1$geometry without selecting a normal form, a maximal simplex, or a list of sign patterns.

An elementary closure map retracts the Hulbert–Zaremsky descending coface poset onto the Rips complex of these lattice points:

$$D_r(S)\simeq \operatorname{VR}_r\bigl(P_r(S)\cap\mathbb Z^n\bigr).$$

This reduction survives in full poset detail: deleting the non-descending added vertices is an order map below the identity, and its image is exactly the nonempty face poset of the displayed Rips complex.

Attack outcome

Three uniform obstructions stop the route

The signed-height encoding is exact: replacing each lattice vector by initial segments on its signed coordinate rays sends Manhattan distance to symmetric-difference distance. But the stronger structures needed for contraction do not follow.

1. For every $n\ge2$ and$r\ge3$, an orthant carrier can fail the standard$M^\natural$-exchange axiom and need not be a down-set.
2. Closed-orthant subcomplexes can miss the carrier edge$\{e_1,-e_1\}$, so they do not even cover the Rips complex to which a nerve lemma would be applied.
3. For every $n\ge5$ and$r\ge n$, one explicit clique supplies a compatible blocker for each of the$2^{n-2}$ lattice roundings of an edge midpoint. No rounding is adjacent to the whole clique.

The blocker-rounding graph is a complete graph minus a perfect matching, exposing a Boolean collective obstruction rather than an isolated bad rounding. Python and C++20 independently verify the formulas over finite control ranges; the proofs themselves are symbolic.

Selection calibration

Why the route stops here

The target is deliberately narrower than the full RAAG prediction. One fixed cubical lattice, one canonical descending polytope, and one exact Morse translation organize the entire attack. Postnikov's generalized-permutahedron viewpoint suggested the exchange language, but it is credited as an influence rather than silently imported as a theorem.

The hunt's declared gate was to stop if exchange failed symbolically and the orthant nerve could not repair it. Both failures occur. A repair now requires mixed-sign carrier pieces or clique-dependent center sets, which opens the same sort of presentation and residual spaces this program was designed to avoid. Nothing in the audit refutes the Virk–Zaremsky conjecture.

Research record

Packet and sources

• Full hunt packet
• Source ledger
• Hunt evidence archive
• Exact carrier-route audit
• Independent verification archive