arXiv subjects

Alex Shvets

Publications and source records attributed to Alex Shvets.

13 recordsLinked to original sources

arXiv2026

Integral magneticity of the level-two K3 packet: CM theta lifts and a 2-isogeny trace contraction

B\"onisch, Duhr, and Maggio introduced three meromorphic modular forms $C_4,C_{6a},C_{6b}$ on $\Gamma_0(2)$, arising from a hypergeometric K3 family, and conjectured that they are magnetic of depths $1,2,2$. Writing \[ C_4=\sum_{n\ge1}c_4(n)q^n,\qquad C_{6a}=\sum_{n\ge1}c_{6a}(n)q^n,\qquad C_{6b}=\sum_{n\ge1}c_{6b}(n)q^n, \] we prove the stronger denominator-one statements \[ \frac{c_4(n)}n,\qquad \frac{c_{6a}(n)}{n^2},\qquad \frac{c_{6b}(n)}{n^2}\in\mathbb Z \qquad(n\ge1). \] The weight-four case is reduced to a termwise binomial divisibility by a hypergeometric change of Hauptmodul. For weight six we identify the two forms with canonical level-two CM forms of discriminants $-8$ and $-4$: \[ f_{3,-8,0,1,1}=-64C_{6a},\qquad f_{3,-4,2,1,1}=32C_{6b}. \] An explicit pair of vector-valued weakly holomorphic forms of weight $-3/2$ then gives the full odd-prime divisibility through the higher-level theta-lift coefficient formula of L\"obrich--Schwagenscheidt. The prime $2$ is treated independently. If $t=(\eta(2\tau)/\eta(\tau))^{24}$, $H=\eta(\tau)^4/\eta(2\tau)^2$, $J=2E_2(2\tau)-E_2(\tau)$, $u=64t$, and $\mathcal T=H^4J$, we prove the infinite-family contraction \[ U_2\bigl(\mathcal Tt\,\mathbb Z_2[[u]]\bigr) \subseteq 2^5\mathcal Tt\,\mathbb Z_2[[u]]. \] Consequently $v_2(c_{6\bullet}(2^rm))\ge5r$, which is stronger than the slope $2r$ required for double magneticity. Thus the complete level-two K3 packet is magnetic with global denominator one.

math.NT↗

arXiv2026

A full $p^{4r}$ supercongruence tower for a level-three symmetric-cube hypergeometric sequence

Let ${}_2F_1(1/3,1/3;1;27z)^3=\sum_{n\ge0}A_nz^n$. We prove the full prime-power supercongruence tower $A_{mp^r}\equiv A_{mp^{r-1}}\pmod{p^{4r}}$ for $p\ge5$, $m\ge1$, $r\ge1$. The level-three modular expansion used below was already recorded by Moy; the contribution here is the depth-preserving modulus $p^{4r}$, extending the previously known depth-one modulus $p^4$ case to all prime powers. The proof is modular. After replacing $A_n$ by $B_n=(-1)^nA_n$, the generating function is realized on $X_0(3)$ by $\sum_{n\ge0}B_nt^n=\eta(\tau)^9/\eta(3\tau)^3$ with $t=\eta(3\tau)^{12}/\eta(\tau)^{12}$. The logarithmic derivative $C=(\sum_{n\ge0}B_nt^n)\,(q/t)\,(dt/dq)$ is the Eisenstein series $3E_{5,\chi_0,\chi_3}$. Lagrange-Buermann gives $B_m=\mathrm{CT}_q(C(q)/t(q)^m)$. The new point is to replace the one-prime Hecke defect by the prime-power defect $T_{p^{s+1}}(C/t^{mp^{s+1}})-T_{p^s}(C/t^{mp^s})$. Its $i=0$ Hecke layer is the sparse Cartier defect controlling $A_{mp^{s+1}}-A_{mp^s}$, while all remaining Hecke layers are divisible by the required power of $p$ by induction. A Fricke involution argument on the two-dimensional space $M_5(\Gamma_0(3),\chi_3)=\mathrm{Span}\{C,tC\}$ then kills the low-order part exactly and the principal part modulo $p^{4(s+1)}$.

math.NT↗

arXiv2026

Split-prime supercongruence at the mixed CM point (1/6, 1/3; 1)

For the mixed CM point (a,b,c) = (1/6, 1/3, 1), define A_n^{mix} := 108^n [z^n] _2F_1(1/6, 1/3; 1; z)^3. For every split prime p >= 7, p == 1 mod 3, and every m >= 1, we prove unconditionally A_{mp}^{mix} == A_m^{mix} mod p^4. The exponent 4 exceeds the generic weight-3 Hodge-gap prediction of 3; the extra factor of p is a CM enhancement attached to j=0. We also establish the matching unconditional inert-prime obstruction (p == 2 mod 3), both as a formal-parameter congruence on the q-side and as a coefficient-level Cartier parity law modulo p. The proof uses the modular realization on Gamma_0(3) with parameter t = u/(1+27u)^2, a Lagrange-Burmann reduction to three Cartier identities Lambda_p(C_mix U_p^l) == 0 mod p^4 for l = 1,2,3, a saturated weak q-expansion lattice on the rigidified stack X_0(3) handling vertical integrality, and a length-three Witt-Cartier pole estimate at the elliptic point P_- driven by mu_3-equivariance of the canonical Frobenius lift.

math.NT↗

arXiv2026

Eta-products, Eichler integrals, and the level-8 Apery limit

We give an independent eta-product derivation of the level-8 Apery limit lim B_n^{(8)}/s_n = (7/32) zeta(3), where s_n = sum_{k=0}^n C(n,k)^2 C(2k,n)^2 and B_n^{(8)} is the rational companion sequence satisfying the same cubic recurrence with initial values B_0^{(8)}=0, B_1^{(8)}=1. This value was identified numerically by Almkvist-van Straten-Zudilin and was proved by Golyshev via Beukers's Atkin-Lehner modular method; it was later recomputed by Golyshev-Kerr-Sasaki in the motivic/normal-function framework. The continued fraction PCF((2n+1)(3n^2+3n+1),-n^6) = 8/(7 zeta(3)) already appears in Batut-Olivier and was later rediscovered by the Ramanujan Machine as conjecture Z1. The contribution of the present paper is an explicit rederivation, in the eta-product normalization, of the already-known level-8 Apery limit. We spell out the eta-product verification of the Wronskian identity, the normalization of the Eichler integral, the residue computation of the Fricke period polynomial, and the elementary continuant conversion.

math.NT↗

arXiv2026

Order-3 pi-formulas, Apery-like kernels, and Clausen functoriality for Conservative Matrix Fields

Raz, Shalyt, Leibtag, Kalisch, Weinbaum, Hadad, and Kaminer recently showed that formulas for $\pi$ can be organized by canonical polynomial recurrences and partially unified by a rank-$2$ Conservative Matrix Field (CMF). We prove that each order-$3$ recurrence explicitly printed in the public Appendix~B.6 of their paper is a shifted summation lift of an explicit order-$2$ kernel, and identify all three kernels: the two $\pi$-kernels are explicit rescalings of the sporadic Ap\'ery-like sequences $A036917$ and $A002895$ (Domb numbers, case~$(\alpha)$), while the Catalan kernel is a hypergeometric twist of the Gauss-square coefficient sequence at $(a,b,c)=(\tfrac12,1,\tfrac32)$. We place these kernels in a unified $\operatorname{Sym}^2$ framework: the first $\pi$-kernel and the Catalan kernel come directly from Gauss-square coefficient sequences, while the Domb kernel is recovered by recasting the classical degree-$3$ Belyi pullback $\phi(x)=108x^2/(1-4x)^3$ and the associated algebraic twist in CMF language. We write an explicit square-gauge matrix for the Gauss CMF, formulate the standard pullback--twist transport in CMF terms, and show that for rank-$2$ objects it is compatible with $\operatorname{Sym}^2$. We further prove an inverse classification: for a fixed $\operatorname{Sym}^2$-type Riemann scheme, the one-parameter family of Fuchsian operators contains a unique $\operatorname{Sym}^2(\mathrm{Gauss})$ point, cut out by the closed-form condition $\lambda_0=2\gamma_1\gamma_2(1-2\alpha)$ on the accessory parameter. Finally, a Belyi-pullback scan over $5040$ configurations produces $11$ additional integer sequences of the form $[x^n]\lambda^n\,{}_2F_1(a,b;c;\phi(x))^2$; we prove their integrality and place them in the same $\operatorname{Sym}^2$-pullback framework.

math.NT↗

arXiv2026

Order drop, Hecke descent, and a mod $p^4$ supercongruence for symmetric-cube hypergeometric coefficients

We prove that the symmetric-cube coefficients $A_n=(-27)^n[z^n]\,_2F_1(1/3,1/3;1;z)^3$ satisfy the supercongruence $A(mp)\equiv A(m) \bmod p^4$ for every prime $p\geq 5$ and every $m\geq 1$. The proof rests on three ingredients: (i) the modular identification $F(t(\tau))=\eta(\tau)^9/\eta(3\tau)^3$ with $t(\tau)=\eta(3\tau)^{12}/\eta(\tau)^{12}$, whose logarithmic derivative is the weight-5 Eisenstein series $C(q)=3E_5(\chi_0,\chi_3)$ on $\Gamma_0(3)$; (ii) exact congruences $c_{mp^r}\equiv c_{mp^{r-1}} \bmod p^{4r}$ for the coefficients of $C$, combined with a Lagrange-Burmann extraction; and (iii) a Hecke descent on weakly holomorphic forms, where the defect is expanded in the two-dimensional space of weight-5 forms on $\Gamma_0(3)$ with character $\chi_3$, spanned by $C$ and $tC$, via a cusp-adapted basis, with the second cusp handled by the Fricke involution $W_3$. As an independent result, we show that the Mao-Tian cubic recurrence drops from order 3 to order 2 at the specialization $(1/3,1/3,1)$.

math.NT↗

arXiv2026

The Domb Ap'ery-limit and a proof of the Ramanujan Machine conjecture Z2

We prove that the ratio $B_n/D_n$ of the Ap\'ery-like sequence $B_n$ to the Domb numbers $D_n$ converges to $(7/24)\zeta(3)$, and that $\sum_{n=1}^{\infty} 64^n/(n^3 D_n D_{n-1}) = (56/3)\zeta(3)$. As a corollary we establish the value $Z_2 = 12/(7\zeta(3))$ conjectured by the Ramanujan Machine project. The proof uses level-6 eta products, Atkin--Lehner involutions, and Eichler integrals of weight-4 modular forms.

math.NT↗

arXiv2026

Anchored Likelihood-Ratio Geometry of Anonymous Shuffle Experiments: Exact Privacy Envelopes and Universal Low-Budget Design

We develop a geometric framework for anonymous shuffle experiments based on an anchored affine likelihood-ratio law: a mean-zero measure on the regular simplex polytope. Every finite-output d-ary channel corresponds, up to refinements, to a unique anchored law, and conversely. On privacy: among all epsilon_0-LDP channels, binary randomized response universally extremizes all convex f-divergences and hockey-stick profiles after shuffling. A rigidity converse shows that saturation of both directed envelopes at finite n forces the binary endpoint law. On design: under the pairwise chi_* budget, we prove exact trace-cap and two-orbit frontier theorems. Every frontier point is realized by a mixture of at most two orbit laws. In the low-budget regime, augmented randomized response is minimax-optimal to the sharp constant over all channels and estimators. Under the raw LDP cap, the problem reduces to subset-selection with explicit optimal subset size. The arguments are self-contained and independent of the author's trilogy.

cs.IT↗

arXiv2026

Growing Alphabets in Canonical Shuffle Experiments: Likelihood-Ratio Laws, Estimation Bounds, and Low-Budget Equivariant Design

We study canonical one-step neighboring shuffle experiments for finite-output epsilon_0-LDP d-ary channels along growing alphabets, with frequency estimation and mechanism design under a pairwise chi-squared budget. The pairwise likelihood-ratio law nu_{ab,d} (pushforward of the row ratio under the null row) is the governing invariant: the canonical shuffled histogram experiment is exactly equivalent to the quotient multinomial experiment generated by nu_{ab,d}. Alphabet growth improves canonical shuffled privacy iff the worst pairwise law collapses to delta_1. We prove a sharp pure-LDP endpoint principle for the pairwise chi-squared, construct full-support obstruction families saturating it, and establish a diluting/persistent dichotomy with explicit finite-n hockey-stick bounds. The worst-case pairwise budget chi_*(W) governs a two-regime Assouad lower bound for arbitrary estimators in the i.i.d. multinomial model. Symmetrization reduces the uniform-point Fisher criterion to permutation-equivariant channels. Calibrated GRR is not optimal; in the low-budget regime, augmented GRR is optimal among permutation-equivariant channels.

cs.IT↗

arXiv2026

Universal Shuffle Asymptotics, Part III: Dominant-Block Quotient Geometry and Hybrid Gaussian--Compound-Poisson Limits in Finite-Alphabet Shuffle Privacy

Part I of this series (arXiv:2602.09029) establishes a sharp Gaussian (LAN/GDP) limit theory for neighboring shuffle experiments in the fixed full-support regime. Part II (arXiv:2603.10073) identifies the first universality-breaking frontier: critical Poisson, Skellam, and multivariate compound-Poisson regimes. The present paper completes the finite-alphabet weak-limit theory by identifying the dominant-block quotient geometry that governs neighboring shuffle experiments. We treat dominant blocks of arbitrary finite size, allow overlap between the dominant output sets under the two neighboring hypotheses, and show that the limiting experiment decomposes according to this geometry: projecting onto the sum of the dominant tangent spaces yields a Gaussian factor, while quotienting by those same tangent spaces isolates a compound-Poisson jump field in the rare block. We also identify the regimes in which this quotient description determines the full privacy-curve, as well as the obstruction that appears when projected jump limits alone do not suffice. Two further sections sharpen the rate picture and the boundary interface: we show that the O(n^{-1/2}) rate for the full hybrid experiment is sharp in general, identify a compatibility condition restoring the O(n^{-1}) rate, and prove a boundary Berry--Esseen theorem giving O(c) Le Cam proximity between the critical Poisson-shift and Gaussian shift experiments as c tends to 0. Together with Parts I--II, this yields a three-regime universality picture and a precise finite-alphabet Levy--Khintchine layer for shuffle privacy.

cs.IT↗

arXiv2026

Universal Shuffle Asymptotics, Part II: Non-Gaussian Limits for Shuffle Privacy -- Poisson, Skellam, and Compound-Poisson Regimes

Part I of this series (arXiv:2602.09029) develops a sharp Gaussian (LAN/GDP) limit theory for neighboring shuffle experiments when the local randomizer is fixed and has full support bounded away from zero. The present paper characterizes the first universality-breaking frontier: critical sequences of increasingly concentrated local randomizers for which classical Lindeberg conditions fail and the shuffle score exhibits rare macroscopic jumps. For shuffled binary randomized response with local privacy $\varepsilon_0 = \varepsilon_0(n)$, we prove experiment-level convergence (in Le Cam distance) to explicit shift limit experiments: a Poisson-shift limit for the canonical neighboring pair when $\exp(\varepsilon_0(n))/n \to c^2$, and a Skellam-shift limit for proportional compositions $k/n \to \pi \in (0,1)$ in the same scaling, including an explicit disappearance of the two-sided $\delta$-floor away from boundary compositions. For general finite alphabets, we introduce a sparse-error critical regime and prove a multivariate compound-Poisson / independent Poisson vector limit for the centered released histogram, yielding a multivariate Poisson-shift experiment and an explicit limiting $(\varepsilon, \delta)$ curve as a multivariate Poisson series. Together with Part I, these results yield a three-regime picture (Gaussian/GDP, critical Poisson/Skellam/compound-Poisson, and super-critical no privacy) under convergent macroscopic scalings.

math.ST↗

arXiv2026

Fixed-Composition Shuffle Asymptotics in the Full-Support Gaussian Regime

We study privacy amplification by shuffling for binary-input local randomizers with a fixed finite output alphabet and full support. For a dataset containing exactly k ones among n users, let T_{n,k} denote the shuffled histogram law. For fixed-composition neighboring shuffled histogram laws in the interior regime, we identify the covariance and Fisher constant governing the neighboring pair (T_{n,k},T_{n,k+1}). For a composition parameter pi in [0,1], the correct covariance is Sigma_pi=(1-pi)Sigma_0+pi Sigma_1 rather than the multinomial covariance of the mixture. With v=W_1-W_0, the resulting constant is I_pi=v^T Sigma_pi^+ v. We prove exact likelihood-ratio identities and a regression decomposition with residual moments E[R^2]=O(n^{-2}) and E[R^4]=O(n^{-4}), uniformly over interior compositions. Consequently, JSD(T_{n,k}||T_{n,k+1})=I_{k/n}/(8n)+O(n^{-2}), and the same constant governs smooth divergence asymptotics. For mu_n=(I_{k/n}/n)^{1/2}, both directed hockey-stick privacy curves at epsilon=t mu_n equal mu_n{phi(t)-t Phi(-t)}+O(n^{-1}) uniformly for t in compact sets. Exact finite-n accounting formulas and fixed-message unbundled specializations are also provided.

cs.IT↗

arXiv2019

2017 Robotic Instrument Segmentation Challenge

In mainstream computer vision and machine learning, public datasets such as ImageNet, COCO and KITTI have helped drive enormous improvements by enabling researchers to understand the strengths and limitations of different algorithms via performance comparison. However, this type of approach has had limited translation to problems in robotic assisted surgery as this field has never established the same level of common datasets and benchmarking methods. In 2015 a sub-challenge was introduced at the EndoVis workshop where a set of robotic images were provided with automatically generated annotations from robot forward kinematics. However, there were issues with this dataset due to the limited background variation, lack of complex motion and inaccuracies in the annotation. In this work we present the results of the 2017 challenge on robotic instrument segmentation which involved 10 teams participating in binary, parts and type based segmentation of articulated da Vinci robotic instruments.

cs.CV↗