# Exact source and accepted target

Frozen identity: OWR-734-007, record 30000082, ulamai/UnsolvedMath v1.6.0 commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e (dataset namespace: ulamai/UnsolvedMath). The upstream extraction speaks about a single complex, but the original definition is for a strongly hereditary family with one common polynomial constant.

Authoritative source: Gil Kalai, Combinatorial expectations from commutative algebra, in Combinatorial Commutative Algebra, Oberwolfach Report 32/2004, pp1729--1734, DOI 10.4171/OWR/2004/32. Definitions 1--2 and Conjecture 3 are on printed p1730. The standalone/preprint layout puts them on p1726; do not confuse pagination with a different problem. The frozen metadata labels the report 2005, but the report itself is 32/2004.

Question: Does algebraic shifting preserve a uniform polynomial bound on the sum of ordinary Betti numbers when induced subcomplexes and links are included? No fixed upper bound on simplicial dimension is imposed. We take rational coefficients, standard exterior/symmetric algebraic shifts, and a single constant throughout each family. The output family must be closed under the same two operations. Hell, Tverberg-type theorems and the Fractional Helly property (2006), pp129--131, Definition5.29/Conjecture5.31/Observation5.32, makes this closure requirement explicit.

Target resolved by complete_proof.md: one full negative answer to that unrestricted preservation assertion. An input family of minimum polynomial exponent exactly1 yields shifted closures with no finite polynomial exponent, for either algebraic shift over Q. Neither the d-Leray theorem nor any fixed-simplicial-dimension modification is contradicted. No assertion is made about adjacent source records.


# Algebraic shifting can destroy every polynomial HVC bound

Work over Q. Write b(K)=sum_{i>=0} dim H_i(K;Q). For positive vertex counts, a strongly hereditary family has polynomial HVC exponent d if b(K)<A f_0(K)^d for a common A. Strongly hereditary means closed under links of all faces and induced subcomplexes, up to relabeling. The empty-face complex is retained in these closures; its ordinary b is zero, and the bound is imposed for positive vertex counts (equivalently use max(1,f_0) in the bound). This harmless convention avoids the literal strict inequality 0<0 at zero vertices.

## Theorem

There is a strongly hereditary family C of minimum polynomial HVC exponent exactly 1, with constant A=3, such that the strongly hereditary closure of its exterior algebraic shifts has no finite polynomial HVC exponent. The same assertion holds for symmetric algebraic shifting over Q.

More precisely, let K_r be the boundary of the r-dimensional cross-polytope, and let Delta_r be either shift. For r>=2 and 0<=s<=r-1, the link L_{r,s}=lk_{Delta_r}([s]) has 2r-s vertices and

    dim Htilde_{r-s-1}(L_{r,s};Q) = binom(r,s).

This refutes the unrestricted HVC-preservation assertion in Kalai, Oberwolfach Report 32/2004, printed p1730, Conjecture 3, the precise source of frozen OWR-734-007. No bounded-simplicial-dimension variant is settled here.

## 1. Input family and its uniform bound

Let D_m denote m isolated vertices, with only the empty face and singleton faces. Let J_{r,t}=(D_2)^{*r}*Sigma_t for r,t>=0, where Sigma_t is the simplex on t vertices and Sigma_0={empty face} is the join identity. C consists of all J_{r,t}, all D_m, and their relabelings.

An induced subcomplex of J_{r,t} is a join of factors containing either two, one, or zero selected vertices from each D_2 factor and a simplex from Sigma_t. Absorb all one-point factors into the simplex factor. The result is another J_{r',t'}. A face uses at most one vertex from each D_2 factor. Taking its link removes every such used factor and deletes the chosen vertices of the simplex factor, again yielding J_{r',t'}. Links and induced subcomplexes of D_m are discrete complexes or {empty face}. Hence C is strongly hereditary.

If t>0 then J_{r,t} is contractible. If t=0 and r>=1 then J_{r,0} is the sphere S^{r-1}. Its total ordinary Betti number is 2, including the case r=1 (two points). The zero-vertex case has b=0. Thus b(J_{r,t})<=2 and b(D_m)=m. For every member on n>=1 vertices, b(K)<=max(2,n)<3n. No exponent e<1 can work for all D_m because m/m^e=m^{1-e} tends to infinity. The minimum exponent is exactly 1.

## 2. Standard shifting input

Both exterior and symmetric algebraic shifting over Q produce shifted complexes and preserve the f-vector. For a Cohen--Macaulay complex, either shift is pure of the same dimension. These standard properties are recorded in Murai, Algebraic shifting of cyclic polytopes and stacked polytopes, Lemma 1.2 of the full paper (Lemma 1.3 of the FPSAC 2006 version). We use the standard reverse-lexicographic algebraic shifts; no statement about arbitrary initial ideals or combinatorial shifts is needed.

K_r=J_{r,0} is a simplicial sphere and hence Cohen--Macaulay over Q. The face enumerator is F_{K_r}(x)=(1+2x)^r. Consequently

    h_{K_r}(t)=(1-t)^r F_{K_r}(t/(1-t))=(1+t)^r.

Every Delta_r is therefore a pure shifted (r-1)-complex on 2r vertices with h_i(Delta_r)=binom(r,i).

## 3. Facet partition for pure shifted complexes

Let Delta be a pure shifted complex with facets of size r on [N]. Shifted means replacing any vertex in a face by a smaller missing vertex produces a face. For a facet F define s(F) to be the length of the largest initial interval [s] contained in F, and R(F)=F\[s(F)]. Then the intervals [R(F),F] partition the faces of Delta.

Indeed, assign any face A to the lexicographically first facet F containing A. If A omitted j in R(F), then j>s(F)+1 and s(F)+1 is absent from F. Shift j down to s(F)+1. The resulting facet precedes F and still contains A, a contradiction. Thus R(F) is contained in A. Conversely, if R(F) is contained in A and A is contained in F, no lexicographically smaller facet G can contain A. At the first element of the symmetric difference, such a G would have to contain an element absent from F. Every element absent from F exceeds s(F); all elements of F outside [s(F)] already belong to A and must remain in G. Because G and F have the same size, G would have to omit an element of [s(F)], producing a smaller element of F\G than every element of G\F, contradicting G<F. This proves the partition.

Each interval contributes x^{r-s(F)}(1+x)^{s(F)} to F_Delta(x). Multiplying F_Delta(t/(1-t)) by (1-t)^r gives

    h_Delta(t) = sum_{F facet} t^{r-s(F)}.

In particular, the number of facets containing [s] and omitting s+1 is h_{r-s}(Delta). This is the classical facet interpretation due to Kalai (Murai, full paper Lemma 2.1; FPSAC Lemma 1.4). The preceding argument supplies a direct proof and makes no novelty claim for this identity.

## 4. Homology of initial links

For 0<=s<=r-1 the complex L=lk_Delta([s]) is pure of dimension q=r-s-1 and is shifted on [s+1,N]. All N-s of those vertices occur. To see this, start with a facet containing a given vertex v outside [s]; there are enough other positions (r-1>=s) to shift vertices into every missing element of [s] without removing v. This supplies a facet containing [s] union {v}.

For completeness, the top reduced Betti number of a pure shifted q-complex L equals the number of facets avoiding its least vertex a. For each such facet T, the formal boundary z_T=partial([a]*[T]) is a q-cycle in L. Although [a]*[T] is not a face, all faces in its boundary lie in L by shiftedness. The cycles are independent since T appears in its own z_T with coefficient 1 and in none of the other cycles. They span: subtract these multiples from any top cycle to leave a cycle supported only on facets containing a. For each remaining facet {a} union U, the coefficient of U in the boundary is its coefficient up to sign, since no other facet containing a has U as its opposite face. All coefficients vanish. There are no (q+1)-chains. For q=0 the same argument uses the augmented boundary and the cycles [v]-[a].

Apply this with a=s+1. Facets of the link avoiding a correspond exactly to facets of Delta containing [s] and omitting s+1. Therefore

    dim Htilde_{r-s-1}(lk_Delta([s]);Q)=h_{r-s}(Delta).

For Delta_r the right side is binom(r,r-s)=binom(r,s), proving the claimed identity.

## 5. No polynomial bound after shifting

Take s=floor(r/2). The link has m_r=2r-floor(r/2)<=2r vertices and top reduced Betti number binom(r,floor(r/2)). The largest of the r+1 binomial coefficients is at least their average, so

    b(L_{r,s}) >= binom(r,floor(r/2)) >= 2^r/(r+1).

For any fixed e>=0 and B>0, 2^r/((r+1)(2r)^e) tends to infinity. For example, the ratio of successive terms tends to 2 and is eventually greater than 3/2. Thus b(L_{r,s}) exceeds B m_r^e for all sufficiently large r. Every strongly hereditary family containing all Delta_r must contain these links. It follows that neither shifted closure has a finite polynomial HVC exponent. This is an infinite-family proof, independent of any finite tests.

## Scope and interpretation

The common constant is essential: allowing constants to depend on each finite complex makes the notion vacuous. Hell's 2006 thesis, Definition 5.29, Conjecture 5.31 and Observation 5.32, explicitly discusses the strongly hereditary closure of the shifted family; we use exactly that interpretation. Ordinary homology of K_r itself is preserved, but links need not preserve a uniform bound. The example uses unbounded simplicial dimension and does not contradict preservation of a fixed d-Leray property. We make no closure claim for adjacent fractional-Helly or nerve conjectures. The bounded primary-source search located no prior instance of this HVC counterexample; it does not establish absolute priority.


# Self-audit of the HVC counterexample

The originating system checked the argument; this is not independent review or proof-assistant verification.

- Source quantifiers: the original family definition has one common A; Hell explicitly requires adding links and induced subcomplexes after shifting. The construction refutes even preservation of *any* polynomial exponent, so it does not depend on preserving exactly the original A.
- HVC exponent versus simplicial dimension: input exponent is1, whereas cross-polytope dimensions r tend to infinity. There is no fixed simplicial-dimension assumption in the source. This distinction must remain in all notices and metadata.
- Strong heredity: an induced subset of a two-point factor has0,1,or2 vertices. A nonempty face in such a factor removes the whole factor on taking a link. Singleton factors combine into the simplex. D_m links are empty-face complexes. Union of these two closed families is closed.
- Exact input exponent: b(D_m)=m rules out every real exponent below1; all other inputs have b<=2. Constant3 works uniformly for positive vertex counts. The empty-face convention is stated because the source's strict bound at n=0 would be impossible for every strongly hereditary family.
- Shifting hypotheses: Q has characteristic0, so both standard shifts apply. K_r is a sphere, hence Cohen--Macaulay. Purity of the shifted result is a cited standard theorem, not inferred merely from homology preservation. The f-vector gives its h-vector.
- Facet identity: the index is h_{r-s}, not h_s in general. Cross-polytope symmetry makes these equal only in the final application. The independent checker uses nonsymmetric h-vectors too. The lexicographic partition proof explicitly proves both directions.
- Link vertices: for s<=r-1 every remaining vertex can be retained while filling the first s positions by shifts; hence the count is exactly2r-s. Purity and shiftedness of the link are preserved for these initial faces.
- Chain argument: partial(a*T) is a formal boundary, not a simplex in L. Its boundary faces lie in L. There are no higher-dimensional chains. Distinct missing-a facets supply independent coefficients; the opposite-face argument proves spanning. Reduced augmentation handles q=0.
- Infinite growth: central binomial lower bound and a successive-ratio limit prove domination of every fixed polynomial. Finite examples and boundary ranks are diagnostic support only.
- Negative controls: the unshifted square violates the link/h identity, and an intentionally wrong h-index is detected on its shifted version.
- Novelty: Kalai's h-vector facet formula, standard shifting properties, and shifted homology are credited. The claimed contribution is their cross-polytope application to the unrestricted HVC-family conjecture. Bounded search is not absolute-priority certification.
- Publication scope: complete negative answer to OWR-734-007; no full source-report, fractional-Helly, nerve, or bounded-dimensional-variant closure claim.


# Primary-source review, 27 October 2026

1. Gil Kalai, Combinatorial expectations from commutative algebra, Oberwolfach Report32/2004, pp1729--1734; Definitions1--2/Conjecture3, p1730. https://doi.org/10.4171/OWR/2004/32 . The authentic report PDF was reused with its SHA-256 checked. It defines a common polynomial bound for a strongly hereditary family and imposes no fixed simplicial-dimension bound.
2. Stephan Hell, Tverberg-type theorems and the Fractional Helly property, TU Berlin thesis, 2006, Section5.5, particularly pp129--131, Definition5.29/Conjecture5.31/Observation5.32. https://doi.org/10.14279/depositonce-1464 . Original PDF: https://depositonce.tu-berlin.de/bitstreams/548fce06-a942-44e7-a3d0-34bc98e094eb/download . The thesis explicitly says to add induced subcomplexes and links to the shifted family when determining its HVC dimension. Its finite two-disjoint-edges example demonstrates failure of closure, not the unbounded polynomial growth proved here. The conjecture is still presented as a conjecture in this source.
3. Satoshi Murai, Algebraic shifting of cyclic polytopes and stacked polytopes, Discrete Mathematics307(14)(2007),1707--1721, https://doi.org/10.1016/j.disc.2006.09.018 . Author copy https://s-murai.w.waseda.jp/papers/Cyclic.pdf was read through the public web tool. Full-paper Lemma1.2 records Cohen--Macaulay/purity equivalence for exterior and symmetric shifts; Lemma2.1 gives the shifted facet/h-vector interpretation, attributed to Kalai. The FPSAC2006 version at https://garsia.math.yorku.ca/fpsac06/papers/10.pdf was downloaded and visually inspected; corresponding statements are Lemmas1.3--1.4 on p5. No novelty is claimed for these standard properties.
4. Hell's semester report WS05/06, https://www.inf.fu-berlin.de/graduate-programs/cgc/Berichte/WS05-06/hell.pdf , documents his HVC investigation; no cross-polytope disproof appears there.
5. Goaoc, Holmsen and Patakova, Intersection patterns in spaces with a forbidden homological minor, Forum of Mathematics Sigma (2026), https://doi.org/10.1017/fms.2026.10248 , Section7: its homological-shatter-function conjecture concerns families in a fixed ambient space and low homological degrees. It is not the algebraic-shifting conjecture refuted here; do not conflate them.

Targeted searches included exact HVC/homological-VC phrases with algebraic shifting, counterexample, cross-polytope, links and strongly hereditary; and title/DOI checks for the dependencies. These located the above source chain and current related work but no prior instance of the present family obstruction. This is a bounded novelty search, not evidence of absolute priority or a proof that no unpublished example exists.

Contribution claimed: applying the established shifted facet formula to cross-polytope boundaries produces exponentially large initial-link homology despite a strongly hereditary input family with exact polynomial exponent1. The accompanying elementary proofs are explanatory, not novelty claims for the known h-vector or shifted-homology formulas.

Source acquisition failures were retained: Python TLS validation for the thesis and HTTP403 responses for two author-hosted PDFs. The thesis was then downloaded successfully with system curl's ordinary certificate verification, without insecure flags. The full Murai dependency is also backed by the downloaded primary FPSAC version. No third-party source PDFs or web-page text will be included in the public manuscript archive.