slug: ostrovskii-shcherbakov-conjecture-62-refutation bibkey: ostrovskii2025amplitude doi: 10.48550/arXiv.2508.13554 url: https://arxiv.org/abs/2508.13554v2 triage: theorem motivation_gids:

• D5/S3/Analytic/Interpolation/SelfConjugateGridSchurRatioRefutation.result

Refutation of Ostrovskii–Shcherbakov Conjecture 6.2

Problem

Ostrovskii and Shcherbakov, Amplitude maximization in stable systems, Schur positivity, and some conjectures on polynomial interpolation, arXiv:2508.13554v2 [math.CV], state on page 15:

For all $0\le k<n\le t$ and self-conjugate $z_{1:n}\in {\mathbb{D}}^n$, $|Q_{t,n,k}(z_{1:n}+1_n)|\le 1$. Moreover, if the grid additionally satisfies $\Re (z_{1:n})\in {\mathbb{R}}_{+}^n$, then $|s_{(t-n|n-k-1)}(z_{1:n})| \le \binom{t}{n} |e_{n-k}(z_{1:n})|$.

The same paper labels this Conjecture 6.2 equivalent to Conjecture 6.1. The formal claim retains both clauses, the self-conjugacy conditions, the closed unit disk, and the natural inequalities k<n<=t.

Motivation

The conjecture is a 2025 externally named open problem. Issue #11514 preregistered its source, full quantifiers, witness, tier and bounded literature check before implementation. The Library note Library/Analytic/ostrovskii2025amplitude.md records the arXiv locator, DOI and the source definitions of self-conjugacy, Q and the tableau Schur functions.

Gap

The preregistered literature check found arXiv:2508.13554v2 to be the latest version and found no subsequent settlement in the checked arXiv author and abstract searches, MathDB problem records, Semantic Scholar citation record, or the repository search. The one citing work reachable in that check concerns a difference-equation part of the source; its treatment of Section 6 remains ASSUMED-UNVERIFIED. These bounded readings do not establish exhaustive worldwide literature coverage or publication priority.

Route

For the hook shape $\lambda =(a+1,1^b)$, the Schur polynomial is the sum of tableau monomials over semistandard Young tableaux of shape $\lambda$. Regrouping first by content and then by permutations of that content gives the source’s Kostka expansion $s_{\lambda }={\sum }_{\mu }{K}_{\lambda \mu }{m}_{\mu }$: $K_{\lambda \mu }$ counts tableaux of shape $\lambda$ and content $\mu$, and $m_{\mu }$ sums the distinct monomials with that exponent partition. Specializing to the finite alphabet Fin n relabels $1,\dots ,n$ by $0,\dots ,n-1$ and sets the remaining variables to zero. Thus

$$s_{\lambda }(z)=\sum _{T\in {\mathrm{SSYT}}_n(\lambda )}\prod _{u\in \lambda }{z}_{T(u)}=\sum _{\mu :\ell (\mu )\le n}{K}_{\lambda \mu }{m}_{\mu }(z).$$

In schurHook, a tableau has corner c, an arm multiset of cardinality $a$ with entries at least c, and a leg subset of cardinality $b$ with entries strictly greater than c. Sorting the arm and leg gives the weak row and strict column, counting each tableau once. This is the finite specialization of the source’s expansion described in Library/Analytic/ostrovskii2025amplitude.md. The other Lean definitions specialize the elementary and complete homogeneous symmetric sums to Fin n, then define the alternating ratio Q and the conjunction of the two conjectured bounds.

For the refutation choose $(t,n,k)=(3,2,1)$ and $z=(-9/10+(2/5)i,-9/10-(2/5)i)$. The entries are a conjugate pair, have squared norm $97/100$, and have no real values. After adding the pointwise numeral $1$, the finite sums give $e_1=1/5$, $s_{(0|0)}=1/5$ and $s_{(1|0)}=-13/100$. The resulting ratio is $Q(3,2,1,z+1)=73/60$, whose norm is greater than one.

The second-clause witness recorded in #11514 also has $(t,n,k)=(3,2,1)$, but uses $z=(1/10+(9/10)i,1/10-(9/10)i)$. It is a nonreal conjugate pair, with $|z_i{|}^2=41/50\le 1$ and $\Re z_i=1/10\ge 0$. Here $e_1(z)=1/5$, $e_2(z)=41/50$, and the one-row hook gives $s_{(1|0)}(z)=h_2(z)=e_1(z{)}^2-e_2(z)=-39/50$. Hence

$$|s_{(1|0)}(z)|=|h_2(z)|=39/50>3/5=\left(\genfrac{}{}{0ex}{}{3}{2}\right)|e_1(z)|.$$

[computed: python3 fractions.Fraction, complex multiplication on rational real/imaginary pairs; squared norms $41/50$, $h_2=-39/50$, right-hand side $3/5$, strict gap $9/50$.] This is exact arithmetic evidence for failure of the second clause. The single Lean settlement uses the first-clause witness and does not separately formalize this second calculation.

Falsifier

A proof of the formal universal claim would falsify this refutation. The witness calculation would also fail if the pair were outside the closed disk, were not self-conjugate, or produced a denominator different from the stated nonzero value. The second conjectured clause is retained in the claim but is not needed for this counterexample.

Evidence

The canonical source is D5/S3/Analytic/Interpolation/SelfConjugateGridSchurRatioRefutation.lean. Its public declarations are the definitions selfConjugate, e, h, schurHook, Q, claim, and the theorem result : ¬ claim. The freeze state pins statement identity sha256:0db8847f194177b39d4133cf0557f6dce02ca1edeaf96dcf8f8ff837cb6be821; the Freeze event is sha256:30bb2e0d7cd063200d27cd082fd37251a325cd7f73d4a59ed2bd283e07c698d5. The event has no prerequisite frozen node. The result is kernel-checked with the standard axioms propext, Classical.choice and Quot.sound; no new axiom, sorry or native_decide is used.

Triage

Tier 1 externally named conjecture; resolution refuted. The failure mechanism is the complex conjugate pair in the unit disk: after the shift, the imaginary parts make the degree-two complete homogeneous value negative, so the first Schur-ratio bound exceeds one. The public result is bind-only finite evaluation under the open-problem-resolution admission basis; the settlement is the new information that this named conjecture has a concrete counterexample.

What the settlement shows

• Proved: the first clause fails at $(t,n,k)=(3,2,1)$ for the displayed self-conjugate pair.
• Computed: the second clause fails at the same parameters for $z=(1/10\pm (9/10)i)$, including its nonnegative-real-part premise, by the exact rational calculation above. This is not a second Lean settlement.
• Open: no universal corrected bound, no maximal admissible radius, and no proof for all real self-conjugate grids is claimed here. The t=n identity and the k=0 neighboring cases are not formalized or independently computed by this delivery.
• Boundary: the source’s equivalence transfers the refutation to its Conjecture 6.1 reading; source results that use Conjecture 6.2 as an assumption require a replacement hypothesis or a restricted domain.

ASSUMED-UNVERIFIED

The bounded literature check does not establish that no later publication settles the conjecture. The cited difference-equation paper’s treatment of Section 6 was not verified. The source’s analytic equivalence to Conjecture 6.1 is quoted from the paper rather than formalized in this module.