Sharp lattice-Rips carrier route: exact reductions and stopping audit

Date: 7 September 2026
Mode: ATTACK
Status: exact structural lemmas and exact obstructions; the Virk–Zaremsky conjecture is not resolved

1. Problem and scope

For the standard Manhattan metric on $\mathbb Z^n$, the open Virk–Zaremsky conjecture asks whether

$$\operatorname{VR}_r(\mathbb Z^n)\simeq * \qquad\text{for every }r\ge n.$$

The attack selected in the preceding hunt proposed a specific route. It would turn the descending coface links in the new Hulbert–Zaremsky Morse criterion into integral convex carriers, resolve those carriers by orthants, and contract them by a discrete-polymatroid exchange or a midpoint rounding. This note checks that route at its decisive structural gates.

The outcome is mixed and decisive. The carrier reduction is correct, and there is an exact signed-height model for it. But exchange, the literal orthant nerve, and the natural midpoint-rounding contraction all fail in uniform symbolic families. These failures occur at or below the already known dimensions; they are not artifacts of the first open case.

2. The descending-polytope reduction is exact

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

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

Equivalently, $\Lambda_r(S)=P_r(S)\cap\mathbb Z^n$, where

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

Let $D_r(S)$ denote the geometric realization of the Hulbert–Zaremsky poset of finite sets $S'\supsetneq S$ such that $\operatorname{diam}(S')\le r$ and $\min_{z\in S'}\|z\|_1<L$.

Proposition 2.1 (closure-map reduction). There is a natural homotopy equivalence

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

Proof. Since word lengths are integral, every object $S'$ of the coface poset contains a point of norm at most $L-1$. Define

$$\rho(S')=S\cup\bigl(S'\cap\{z:\|z\|_1\le L-1\}\bigr).$$

This is again an object of the poset, and $\rho(S')\subseteq S'$. The image consists exactly of the sets $S\cup T$, where $T$ is a nonempty simplex of $\operatorname{VR}_r(\Lambda_r(S))$. Indeed, every point of $T$ is within $r$ of $S$, and the points of $T$ are mutually within $r$; the converse follows by taking the lower part of $S'$.

Thus the image is canonically the nonempty face poset of $\operatorname{VR}_r(\Lambda_r(S))$. Its inclusion $i$ satisfies $\rho i=\mathrm{id}$ and $i\rho\le\mathrm{id}$. The order-homotopy lemma retracts the coface poset onto this image, whose realization is the barycentric subdivision of the stated Rips complex. $\square$

This is a useful exact translation, but it does not by itself make the carrier contractible.

3. An exact signed-height model

Let

$$\Omega_n=\{(i,\sigma,k):i\in[n],\ \sigma\in\{+,-\},\ k\ge1\}.$$

Associate to $x\in\mathbb Z^n$ the finite set

$$A(x)=\{(i,\operatorname{sgn}(x_i),k):1\le k\le |x_i|\}.$$

It is an initial segment on at most one of the two signed rays over each coordinate. Directly coordinate by coordinate,

$$|A(x)|=\|x\|_1, \qquad |A(x)\mathbin\triangle A(y)|=\|x-y\|_1.$$

Consequently a flat clique becomes a family of equal-cardinality admissible ray ideals with bounded symmetric differences, and $\Lambda_r(S)$ becomes their common Hamming lens among admissible ray ideals of cardinality at most $L-1$. This exact model survives the audit. What fails is the stronger claim that its orthant restrictions possess the exchange structure needed for the proposed contraction.

4. Uniform failure of discrete-polymatroid exchange

Use the standard $M^\natural$-exchange axiom: if $x_i>y_i$, then either both $x-e_i$ and $y+e_i$ lie in the set, or there is a coordinate $j$ with $x_j<y_j$ for which both exchanged points lie in the set.

Proposition 4.1. For every $n\ge2$ and $r\ge3$, an orthant piece of a descending carrier can fail $M^\natural$-exchange. In particular this happens throughout the threshold regime $r\ge n\ge3$.

Proof. Let

$$s=(-(r-1),1,0,\ldots,0),\qquad S=\{s\},\qquad L=r,$$

and let $C=\Lambda_r(S)\cap\mathbb Z_{\ge0}^n$. Set

$$x=e_1+e_2,\qquad y=0.$$

Then $x,y,x-e_1=e_2\in C$, while

$$\|s-e_1\|_1=r+1,$$

so $y+e_1=e_1\notin C$. Since $x\ge y$ coordinatewise, there is no coordinate $j$ with $x_j<y_j$ to supply the alternative exchange. Thus the axiom fails. The same example shows that $C$ is not even a down-set: $e_1\le x$, but $e_1\notin C$. $\square$

This is a symbolic obstruction, not a low-dimensional exception.

5. Orthant subcomplexes do not cover the Rips complex

The most literal nerve repair also fails before any homotopy calculation.

Proposition 5.1. For every $n\ge2$ and $r\ge3$, the induced Rips subcomplexes on the closed orthant pieces of a descending carrier need not cover the carrier Rips complex.

Proof. Take

$$S=\{(0,\ldots,0,r-1)\},\qquad L=r-1.$$

Both $e_1$ and $-e_1$ have norm at most $L-1=r-2$, and both are at distance $r$ from the point of $S$. Hence they lie in $\Lambda_r(S)$, and $\{e_1,-e_1\}$ is an edge because its diameter is $2\le r$. No closed orthant contains both endpoints. Therefore the orthant-induced subcomplexes cover the vertices but not the simplices, so the nerve lemma cannot be applied to that family. $\square$

A repair would have to introduce mixed-sign carriers. The original orthantwise subset-sum theorem would no longer control those pieces.

6. A Boolean obstruction to midpoint rounding

A second natural route tries to collapse the scale-$r$ Rips complex toward the cubical thickening. For an edge $\{x,y\}$, one rounds its real midpoint to a vertex of the containing unit cube and hopes that one rounding is adjacent to every vertex of any clique containing the edge. The following family disproves that hope at every conjectured scale from dimension five on.

Proposition 6.1 (midpoint blocker cube). Let $n\ge5$ and $r\ge n$. There is an $r$-clique $K\subset\mathbb Z^n$ containing an edge $\{x,y\}$ such that no lattice rounding of $(x+y)/2$ is within distance $r$ of every vertex of $K$.

Construction and proof. Put $q=n-2$, $h=r-n+2$, and

$$x=0, \qquad y=(2,\underbrace{1,\ldots,1}_{q},0).$$

For $u\in\{0,1\}^q$, let $k=|u|$, let $\chi(u)=1$ if $k\ge2$ and $0$ otherwise, and define

$$w_u=(2\chi(u),1-u_1,\ldots,1-u_q,-h).$$

The set

$$K=\{x,y\}\cup\{w_u:u\in\{0,1\}^q\}$$

is an $r$-clique. Indeed,

$$\begin{aligned} d(x,w_u)&=r+2\chi(u)-k\le r,\\ d(y,w_u)&=2(1-\chi(u))+k+h\le r,\\ d(w_u,w_v)&=d_H(u,v)+2|\chi(u)-\chi(v)|\le n\le r. \end{aligned}$$

The midpoint is

$$p=(1,\tfrac12,\ldots,\tfrac12,0),$$

whose $2^q$ lattice roundings are

$$m_v=(1,v_1,\ldots,v_q,0),\qquad v\in\{0,1\}^q.$$

For every $u,v$,

$$d(w_u,m_v)=r+1-d_H(u,v).$$

Thus $w_u$ is at distance $r+1$ from the one rounding $m_u$, while it is within $r$ of all the others. Every midpoint rounding is therefore blocked by a distinct member of the same clique. $\square$

The induced graph on $\{w_u,m_u:u\in\{0,1\}^q\}$ is a complete graph with the perfect matching $\{w_u,m_u\}$ deleted. Its flag complex is the boundary of a $2^q$-dimensional cross-polytope. This induced sphere may be filled by other vertices in the full link; it is not asserted to give a homology class. It does show exactly why a midpoint-only contraction loses collective information: the obstruction has a Boolean family of mutually compatible witnesses even though each witness excludes only one rounding.

7. Computational role

The accompanying Python and C++20 programs independently check the formulas above for finite ranges of $(n,r)$, including the complete adjacency matrix of the midpoint blocker family. The proofs are symbolic and do not depend on those ranges. Additional bounded diagnostics found many dismantlable carrier graphs, including sampled $(n,r)=(6,6)$ carriers, but this is deliberately not promoted as evidence for a theorem.

8. Stopping decision

The selected attack had a declared failure gate: stop if exchange fails in a symbolic family and the orthant nerve does not repair it. Both events occur. The most natural alternative cubical-midpoint contraction also meets a uniform Boolean blocker family.

Continuing from here would require classifying mixed-sign carrier pieces or choosing larger, clique-dependent center sets. That is precisely the growth into presentation and residual spaces that this research program was chosen to avoid. The correct decision is therefore to stop this route. Nothing in this audit refutes the Virk–Zaremsky conjecture, and no new contractibility range is claimed.

9. Sources and credit

• Seth Hulbert and Matthew C. B. Zaremsky, Word length, Morse theory, and Vietoris–Rips complexes, arXiv:2608.25614v1 (26 August 2026): the descending-coface criterion used in Proposition 2.1.
• Raju Kumar Gupta, Sourav Sarkar, and Samir Shukla, On the Vietoris–Rips Complexes of Integer Lattices, arXiv:2511.04238v1: the sharp result through dimension five and the dimension-six scale-ten range.
• Matthew C. B. Zaremsky, Contractible Vietoris–Rips complexes of $\mathbb Z^n$, arXiv:2410.11993v2: the $n^2+n-1$ uniform bound.
• Ziga Virk, Contractibility of the Rips complexes of Integer lattices via local domination, arXiv:2405.09134v4: the original general theorem and sharp-threshold question.
• Alexander Postnikov’s generalized-permutahedron viewpoint motivated the exchange test. It is an intellectual influence, not a formal dependency.

The closure-map packaging and the three obstruction families are application-specific deductions from this attack. They have not been externally reviewed, and no priority claim is made.