Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.16781

A Bethe-Ansatz Source of Early Token Correlations: A marginal-preserving construction for parallel proposals

Abraham Chachamovits

Published Oct 12, 2026• 0 citations
arXivHTMLPDF

Abstract

Parallel token proposals need a joint law, not merely accurate token marginals. We construct coordinate-Bethe-ansatz laws from the periodic XXZ chain and compare them with their maximum-entropy pairwise projections at identical one- and two-site moments. In an eight-site, three-excitation sector, symmetry makes the full probability-law difference one-dimensional, determined by a contiguous three-site moment $δ$. Along the Bethe branch, this gap changes sign and vanishes exactly at $Δ=(3+\sqrt3)/2$, where the eigenstate law becomes pairwise. A nonlinear two-token readout preserves separate token marginals yet exposes a distance $2|δ|$; a direct three-token readout preserves all one- and two-token marginals yet exposes $4|δ|$. At $Δ=1.5$ these deterministic binary distances are $0.02654573$ and $0.05309145$. We give an exact target-alignment formula and prove a strict speculative accepted-prefix advantage throughout an unrestricted open neighborhood of a constructed target. These are finite-law existence results, not measured language-model or wall-clock gains.

DOI
10.48550/arxiv.2610.16781

Related papers

Four-Token Correlations from Bethe Eigenstates: Certified Source Families and Controlled Proposal Tests

Abraham Chachamovits

Finite coordinate-Bethe solutions in three XXZ sectors supply strictly positive four-bit laws through marginal-preserving residual transfer. For a ten-site, four-excitation sector and twelve-site sectors with four or five excitations, we certify four affinely independent laws per sector. All twelve laws share every one-site and two-site marginal; each sector spans a three-dimensional simplex. Interval root and maximum-entropy projection inclusions certify positivity and nonzero determinants. An exact signed-law formula transfers the two inequivalent triple moments and four-site moment through positionwise token channels; distinct channel pairs preserve dimension. A target constructed from the source under identity emission demonstrates a strict acceptance effect. An exploratory initial-block test on SmolLM2-135M selects one source vertex and loses to a matched-budget classical search; an exact-arithmetic cover bounds every continuous mixture of that original simplex below the classical control under the archived emissions. A separately frozen follow-up compares 495 expanded Bethe and 495 classical candidates at target-generated contexts in two model settings, including a 1.7B target with a target-specific continued 135M draft. On 64 fresh held-out questions per setting, the late-context Bethe-classical differences are 0.000204 and 0.000143 accepted proposal tokens/block; both adjusted 97.5% descriptive intervals cross zero and neither preset practical criterion is met. All 767 context records are arithmetically audited. The certified construction therefore broadens the available higher-order sources without establishing a practical Bethe or decoding-speed advantage. A retrospective linear-programming analysis gives numerical upper estimates of about 0.0140 and 0.0113 additional tokens/block over the archived references for any joint proposal sharing their one- and two-token marginals on the saved development supports. These estimates are distinct from the exact-arithmetic mixture certificate.

•Oct 25, 2026•0 citations

The Bethe ansatz for the six-vertex and XXZ models: An exposition

Hugo Duminil‐Copin, Maxime Gagnebin, Matan Harel +2 more

In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi $ and energy $\Lambda $, which satisfy $V\psi =\Lambda \psi $, where $V$ is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights $a=b=1$ and $c>0$. We also show that the same vector $\psi $ satisfies $H\psi =E\psi $, where $H$ is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value $E$ computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [5] that amounts to proving that the random-cluster model on $\mathbb{Z}^{2}$ with cluster weight $q>4$ exhibits a first-order phase transition.

arXiv:1611.09909•Probability Surveys•Jan 1, 2018•16 citations

Derivation of the Angular Correlation Law and Construction of Joint Probabilities for Quantum Entanglement

Hailing Sheng

This paper derives the angular correlation law and constructs joint probabilities for quantum entanglement within the All-Domain Framework (ADF). It presents explicit mathematical results, readout exclusion results, and experimental analysis procedures. Main results Angular correlation law. Starting from the invariant graph d = Kx and the Riccati compatibility condition C + EK − KA − KBK = 0, the paper derives the cross-moment identity Qxd = QxKᵀ. On the anti-relational isotropic branch K = −I and Qx = sI, the normalized continuous correlation is exactly ρ(a,b) = −a·b. Joint-probability construction. A deterministic measurable partition of the maintenance circle is constructed with four outcome-region measures μ(RAB) = (1 − ABc)/4. The construction gives unbiased local marginals and ⟨AB⟩ = −c. Under the identification c = a·b, it reproduces the angular correlation and the standard CHSH value 2√2. Readout exclusion results. Seven candidate mechanisms are audited under their stated assumptions. The results identify why passive isotropic averaging, local hard thresholds, and the examined dynamical mechanisms do not independently generate the required four-sector outcome law. Experimental analysis methods. The paper provides graph reconstruction from measured cross moments, anti-relational residual tests, independent isotropy tests, and continuous-correlation reconstruction. Calibration is separated from testing against the target correlations. Synthetic numerical verification. The reported tests demonstrate branch discrimination, graph-inversion convergence, and sensitivity to noise. A hard-sign negative control identifies a systematic discrepancy between local threshold readout and the target joint probabilities. The continuous-correlation derivation and joint-partition existence theorem are established mathematical results under the specified conditions. Physical analyzer-setting transduction and dynamical generation of individual binary outcomes remain explicitly identified interfaces for further work. The numerical results use synthetic data. This illustrated edition contains 20 pages, including six vector figures.

•Sep 26, 2026•0 citations

Uniqueness of the Bethe ansatz of theXXZ chain and statistical interaction

Dong Mi, Yang Fuzhong, Buoyuan Hou

International Journal of Theoretical Physics•May 1, 1997•0 citations

Fault-Secure Target Identifiability Under Sparse Diagnostic Corruption

Matthew Riley

This paper develops exact target-specific identifiability criteria for linear diagnostic systems subject to sparse arbitrary corruption. The unknown parameter may remain only partially observable: the objective is not full-state reconstruction, but recovery of a prescribed protected target. Pairwise sparse-fault confusability is therefore treated as a quotient problem, yielding an exact kernel and rank criterion for whether the target remains well defined under every admitted fault support. For a finite hostile support family, the paper proves a sharp hardware law: the minimum number of unrestricted clean scalar augmentations equals the largest target-changing residual rank exposed by any one admissible support. If the new clean bank must also survive f complete erasures, the minimum increases by exactly f. A single coded augmentation can therefore protect an entire finite support library without reconstructing the full mechanism. Structural identifiability and finite-noise robustness nevertheless separate sharply. For K equally spaced hostile residual lines in ℝ², one scalar measurement remains structurally sufficient for every finite K, while the optimal worst-support gain is sin(π/(2K)) and the corresponding Gaussian repetition burden grows as Θ(K²). For dense hostile families, every sub-full-dimensional readout has zero uniform singular margin; for the complete Grassmannian, full target-dimensional readout becomes structurally necessary. A quotient-first generalized singular margin then profiles nuisance, calibration and fault directions before target normalization. The results are finite-dimensional linear geometry. Quantum Hamiltonian identification is one application, but no automatic physical realizability of the mathematically optimal measurements is assumed.

•Sep 18, 2026•0 citations

Thermal Quantum Correlations in a Two-Qubit XXZ Heisenberg Chain with Dzyaloshinskii–Moriya, Kaplan–Shekhtman–Entin–Wohlman–Aharony, and Asymmetric In-Plane Anisotropy Interactions

Mounaim Hamdoune, Mohammed Daoud A

Journal of Superconductivity and Novel Magnetism•Aug 1, 2026•0 citations
Parallel token proposals need a joint law, not merely accurate token marginals. We construct coordinate-Bethe-ansatz laws from the periodic XXZ chain and compare them with their maximum-entropy pairwise projections at identical one- and two-site moments. In an eight-site, three-excitation sector, symmetry makes the full probability-law difference one-dimensional, determined by a contiguous three-site moment δδδ. Along the Bethe branch, this gap changes sign and vanishes exactly at Δ=(3+3)/2Δ=(3+\sqrt3)/2Δ=(3+3​)/2, where the eigenstate law becomes pairwise. A nonlinear two-token readout preserves separate token marginals yet exposes a distance 2∣δ∣2|δ|2∣δ∣; a direct three-token readout preserves all one- and two-token marginals yet exposes 4∣δ∣4|δ|4∣δ∣. At Δ=1.5Δ=1.5Δ=1.5 these deterministic binary distances are 0.026545730.026545730.02654573 and 0.053091450.053091450.05309145. We give an exact target-alignment formula and prove a strict speculative accepted-prefix advantage throughout an unrestricted open neighborhood of a constructed target. These are finite-law existence results, not measured language-model or wall-clock gains.
Finite coordinate-Bethe solutions in three XXZ sectors supply strictly positive four-bit laws through marginal-preserving residual transfer. For a ten-site, four-excitation sector and twelve-site sectors with four or five excitations, we certify four affinely independent laws per sector. All twelve laws share every one-site and two-site marginal; each sector spans a three-dimensional simplex. Interval root and maximum-entropy projection inclusions certify positivity and nonzero determinants. An exact signed-law formula transfers the two inequivalent triple moments and four-site moment through positionwise token channels; distinct channel pairs preserve dimension. A target constructed from the source under identity emission demonstrates a strict acceptance effect. An exploratory initial-block test on SmolLM2-135M selects one source vertex and loses to a matched-budget classical search; an exact-arithmetic cover bounds every continuous mixture of that original simplex below the classical control under the archived emissions. A separately frozen follow-up compares 495 expanded Bethe and 495 classical candidates at target-generated contexts in two model settings, including a 1.7B target with a target-specific continued 135M draft. On 64 fresh held-out questions per setting, the late-context Bethe-classical differences are 0.000204 and 0.000143 accepted proposal tokens/block; both adjusted 97.5% descriptive intervals cross zero and neither preset practical criterion is met. All 767 context records are arithmetically audited. The certified construction therefore broadens the available higher-order sources without establishing a practical Bethe or decoding-speed advantage. A retrospective linear-programming analysis gives numerical upper estimates of about 0.0140 and 0.0113 additional tokens/block over the archived references for any joint proposal sharing their one- and two-token marginals on the saved development supports. These estimates are distinct from the exact-arithmetic mixture certificate.
In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi $ and energy $\Lambda $, which satisfy $V\psi =\Lambda \psi $, where VVV is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights a=b=1a=b=1a=b=1 and c>0c>0c>0. We also show that the same vector $\psi $ satisfies $H\psi =E\psi $, where HHH is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value EEE computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [5] that amounts to proving that the random-cluster model on Z2\mathbb{Z}^{2}Z2 with cluster weight q>4q>4q>4 exhibits a first-order phase transition.
This paper derives the angular correlation law and constructs joint probabilities for quantum entanglement within the All-Domain Framework (ADF). It presents explicit mathematical results, readout exclusion results, and experimental analysis procedures. Main results Angular correlation law. Starting from the invariant graph d = Kx and the Riccati compatibility condition C + EK − KA − KBK = 0, the paper derives the cross-moment identity Qxd = QxKᵀ. On the anti-relational isotropic branch K = −I and Qx = sI, the normalized continuous correlation is exactly ρ(a,b) = −a·b. Joint-probability construction. A deterministic measurable partition of the maintenance circle is constructed with four outcome-region measures μ(RAB) = (1 − ABc)/4. The construction gives unbiased local marginals and ⟨AB⟩ = −c. Under the identification c = a·b, it reproduces the angular correlation and the standard CHSH value 2√2. Readout exclusion results. Seven candidate mechanisms are audited under their stated assumptions. The results identify why passive isotropic averaging, local hard thresholds, and the examined dynamical mechanisms do not independently generate the required four-sector outcome law. Experimental analysis methods. The paper provides graph reconstruction from measured cross moments, anti-relational residual tests, independent isotropy tests, and continuous-correlation reconstruction. Calibration is separated from testing against the target correlations. Synthetic numerical verification. The reported tests demonstrate branch discrimination, graph-inversion convergence, and sensitivity to noise. A hard-sign negative control identifies a systematic discrepancy between local threshold readout and the target joint probabilities. The continuous-correlation derivation and joint-partition existence theorem are established mathematical results under the specified conditions. Physical analyzer-setting transduction and dynamical generation of individual binary outcomes remain explicitly identified interfaces for further work. The numerical results use synthetic data. This illustrated edition contains 20 pages, including six vector figures.
This paper develops exact target-specific identifiability criteria for linear diagnostic systems subject to sparse arbitrary corruption. The unknown parameter may remain only partially observable: the objective is not full-state reconstruction, but recovery of a prescribed protected target. Pairwise sparse-fault confusability is therefore treated as a quotient problem, yielding an exact kernel and rank criterion for whether the target remains well defined under every admitted fault support. For a finite hostile support family, the paper proves a sharp hardware law: the minimum number of unrestricted clean scalar augmentations equals the largest target-changing residual rank exposed by any one admissible support. If the new clean bank must also survive f complete erasures, the minimum increases by exactly f. A single coded augmentation can therefore protect an entire finite support library without reconstructing the full mechanism. Structural identifiability and finite-noise robustness nevertheless separate sharply. For K equally spaced hostile residual lines in ℝ², one scalar measurement remains structurally sufficient for every finite K, while the optimal worst-support gain is sin(π/(2K)) and the corresponding Gaussian repetition burden grows as Θ(K²). For dense hostile families, every sub-full-dimensional readout has zero uniform singular margin; for the complete Grassmannian, full target-dimensional readout becomes structurally necessary. A quotient-first generalized singular margin then profiles nuisance, calibration and fault directions before target normalization. The results are finite-dimensional linear geometry. Quantum Hamiltonian identification is one application, but no automatic physical realizability of the mathematically optimal measurements is assumed.