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Signed spin-network Hamiltonians for SU(2) plaquette chains: an exact factored contraction, a non-gauge underdetermination phenomenon, and its resolution by length-4 data

Lorenzo Mannelli

Published Aug 15, 2026• 0 citations
Publisher

Abstract

We study Kogut--Susskind SU(2) lattice gauge theory on collapsed two-leg plaquette chains in a spin-network (electric) basis with canonical intertwiners fixed in closed form: the Wigner $3j$ tensor with an $\varepsilon$ metric on each head slot, certified by vertex Casimir annihilation. Magnetic plaquette matrix elements factorize in magnitude into per-vertex Racah $6j$ factors (verified to $1.2\times10^{-15}$ through electric truncation $j_{\max}=2$), while their \emph{signs} obey a refined local class law. We report three results. (1) An exact \emph{factored transfer contraction} evaluates any matrix element in $O(10^6)$ flops independent of link labels, giving measured Hamiltonians at any moderate length with no sign table; it reproduces independent DMRG bulk energies to $9\times10^{-16}$. (2) A non-gauge \emph{underdetermination phenomenon}: the complete set of length-$\leq3$ amplitudes leaves a $\mathrm{GF}(2)$ sign freedom that is not pure gauge --- two tables, each exact on all measured length-$\leq3$ data, differ at length $\geq4$ by $O(0.5)$ in ground energy. (3) A \emph{pinning result}: adjoining the (cheap) complete length-4 batteries collapses the freedom to true gauge; the pinned table reproduces independent length-5 and length-6 ground truth to $10^{-14}$--$10^{-13}$. %% ANCHOR: ABSTRACT-R3L5 (one sentence on rung-3 L=5 certification) %% ANCHOR: ABSTRACT-R3L5-END Corrected physics follows: the flux gap rises monotonically in $\lambda=4/g^4$ at every electric truncation, each rung softening the slope, with geometric rung convergence of both gap and bulk energy density. %% ANCHOR: ABSTRACT-EXTRAP (final extrapolation sentence) %% ANCHOR: ABSTRACT-EXTRAP-END All code, data, and machine-verifiable certificates accompany this deposit. The bundle contains the SHA-256 manifest of all 115 files, the seven stage deliverables, source code, and the datasets

DOI
10.5281/zenodo.21955448

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We study Kogut--Susskind SU(2) lattice gauge theory on collapsed two-leg plaquette chains in a spin-network (electric) basis with canonical intertwiners fixed in closed form: the Wigner 3j3j3j tensor with an ε\varepsilonε metric on each head slot, certified by vertex Casimir annihilation. Magnetic plaquette matrix elements factorize in magnitude into per-vertex Racah 6j6j6j factors (verified to 1.2×10−151.2\times10^{-15}1.2×10−15 through electric truncation jmax⁡=2j_{\max}=2jmax​=2), while their \emph{signs} obey a refined local class law. We report three results. (1) An exact \emph{factored transfer contraction} evaluates any matrix element in O(106)O(10^6)O(106) flops independent of link labels, giving measured Hamiltonians at any moderate length with no sign table; it reproduces independent DMRG bulk energies to 9×10−169\times10^{-16}9×10−16. (2) A non-gauge \emph{underdetermination phenomenon}: the complete set of length-≤3\leq3≤3 amplitudes leaves a GF(2)\mathrm{GF}(2)GF(2) sign freedom that is not pure gauge --- two tables, each exact on all measured length-≤3\leq3≤3 data, differ at length ≥4\geq4≥4 by O(0.5)O(0.5)O(0.5) in ground energy. (3) A \emph{pinning result}: adjoining the (cheap) complete length-4 batteries collapses the freedom to true gauge; the pinned table reproduces independent length-5 and length-6 ground truth to 10−1410^{-14}10−14--10−1310^{-13}10−13. %% ANCHOR: ABSTRACT-R3L5 (one sentence on rung-3 L=5 certification) %% ANCHOR: ABSTRACT-R3L5-END Corrected physics follows: the flux gap rises monotonically in λ=4/g4\lambda=4/g^4λ=4/g4 at every electric truncation, each rung softening the slope, with geometric rung convergence of both gap and bulk energy density. %% ANCHOR: ABSTRACT-EXTRAP (final extrapolation sentence) %% ANCHOR: ABSTRACT-EXTRAP-END All code, data, and machine-verifiable certificates accompany this deposit. The bundle contains the SHA-256 manifest of all 115 files, the seven stage deliverables, source code, and the datasets
We construct two quantum error correction codes for pure SU(2) lattice gauge theory in the electric basis truncated at the electric flux jmax=1/2j_{\rm max}=1/2jmax​=1/2, which are applicable on quasi-1D plaquette chains, 2D honeycomb and 3D triamond and hyperhoneycomb lattices. The first code converts Gauss's law at each vertex into a stabilizer while the second only uses half of the vertices and is locally the carbon code. Both codes are able to correct single-qubit errors. The electric and magnetic terms in the SU(2) Hamiltonian are expressed in terms of logical gates in both codes. The logical-gate Hamiltonian in the first code exactly matches the spin Hamiltonian for gauge singlet states found in previous work.
We study an explicit 2^w × 2^w transfer matrix Γ_w(β), the half-spin factor (all link spins ≤ 1/2) of the complete SU(2) transfer operator of an auxiliary open row of w plaquettes, and ask for a spectral gap that is uniform in the width w. (i) At β = 28/25 a Kotecký–Preiss polymer-current certificate gives |λ_k| ≤ e^{−2.90} λ_0 for all k ≥ 1 and all w ≥ 1 (computer-assisted, blind-audited). (ii) The bare current method stops at β ≈ 1.5 although the numerical gap stays between 3.4 and 4.5. (iii) After an exact “dressed” decomposition of Γ_w into connected interval operators, the connected parts obey an all-length geometric bound ‖K̂_L‖ ≤ (2009/50000)(381/2000)^{L−1} at β = 5/2 (exact rational certificate, not audited), with a closed-form decay rate. (iv) Beyond this wall, a time-ordered transfer certificate for the dressed gas with open time boundaries gives |λ_k| ≤ e^{−3.37} λ_0 for every w at β = 5/2, that is 98.8% of the numerical gap 3.4095, and |λ_k| ≤ e^{−5.38} λ_0 at β = 28/25 (computer-assisted, blind-audited except for its inputs from the tail certificate in (iii); one proof gap was found and repaired at no numerical cost). A second route (m ≥ 3.25) is still under audit. All certificates concern the half-spin factor only: with all spins the gap at β = 5/2 is 3.358, 3.312, 3.291 for w = 2, 3, 4, below 3.37 (5.433, 5.428, 5.426 at 28/25; certified by Astra at w = 2 and by our own, not yet audited, engine at w = 3, 4). These ratios follow a nearest-neighbour band law; a single triplet channel carries most of the higher-spin gain and, at β = 5/2, defeats a route-C certificate for the complete row. (v) We record exact structural facts: the both-open stretch of the row is a singlet/triplet mixer with a neutral singlet channel, and positivity of Γ_w forces λ_s ≥ 4p_2² − 3p_3, with physical margin Var(χ_{1/2}) ∼ 6/β². A final part collects observations on Weil positivity for the Riemann zeta function: in the theta-dressed jump formulation the dressed hop rate equals the critical constant 1/2 up to an explicit correction localised where the kernel Φ is not negligible (an exact consequence of W ∗ Φ = 0), so null-function dressings are critical and admit no dilute regime. This indicates why a current-type method does not transfer directly. The Sonin compression of Connes–Consani, the natural candidate ground state with a margin, stops bounding the Weil form on our test banks once the prime 2 enters. We also report finite-size comparator and lane data, experiments on the arithmetic side, and a dictionary between the two problems with explicit open target inequalities. Nothing here is a statement about continuum Yang–Mills theory or a proof of the Riemann hypothesis. Author: Andrei Ursachi (A. Ursachi; Ursachi, Andrei), independent researcher, ORCID 0009-0002-6114-5011. Version 2 (31 October 2026). Supersedes version 1; corrects an error in Definition 1 and extends the results.
We present preliminary numerical evidence for the hypothesis that the Hamiltonian SU(2) gauge theory discretized on a lattice obeys the eigenstate thermalization hypothesis (ETH). To do so we study three approximations: (a) a linear plaquette chain in a reduced Hilbert space limiting the electric field basis to j = 0 , 1 2 , (b) a two-dimensional honeycomb lattice with periodic or closed boundary condition and the same Hilbert space constraint, and (c) a chain of only three plaquettes but such a sufficiently large electric field Hilbert space ( j ≤ 7 2 ) that convergence of all energy eigenvalues in the analyzed energy window is observed. While an unconstrained Hilbert space is required to reach the continuum limit of SU(2) gauge theory, numerical resource constraints do not permit us to realize this requirement for all values of the coupling constant and large lattices. In each of the three studied cases we check first for random matrix theory (RMT) behavior in the eigenenergy spectrum and then analyze the diagonal as well as the off-diagonal matrix elements between energy eigenstates for a few operators. Within current uncertainties all results for (a), (b) and (c) agree with ETH predictions. Furthermore, we find the off-diagonal matrix elements of the electric energy operator exhibit RMT behavior in frequency windows that are small enough in (b) and (c). To unambiguously establish ETH behavior and determine for which class of operators it applies, an extension of our investigations is necessary. Published by the American Physical Society 2024
Volume-Uniform Finite-Volume Bounds with the Representation Cutoff Removed: a Reconstruction and Computer-Assisted Improvement of Creation-Operator Constants We study the Kogut–Susskind Hamiltonian of pure SU(2) and SU(3) lattice gauge theory on three-dimensional cubic boxes, H = aΣ_e L_e + bΣ_p (1 - F_p), with L_e the electric Casimir on link e and F_p = N⁻¹Re tr U_p, at strong coupling (small b/a). An unrefereed preprint by Gauvin adapts the creation-operator method of Bravyi, DiVincenzo and Loss to this setting and states, for SU(3), a physical gap uniform in the volume, with the representation cutoff removed, for b/a ≤ 1/648. We give a self-contained reconstruction of the finite-volume argument, identify its external analytic inputs, and reproduce its constants. We then prove a sequence of refinements of the commutator estimate: a selection rule of Wilson-loop multiplication that restricts admissible creation placements, fractional witness counting, orthogonality of output sectors, Haar-measure bounds for vacuum projections before and after the plaquette multiplication, a two-link sandwich identity, a joint vacuum-row bound, and a size-weighted norm. The finite tuple constants and final sufficient inequalities are checked in exact rational arithmetic by two author implementations and a separate verifier written from the stated definitions. The operator inequalities that justify these arithmetic definitions are proved analytically. On open boxes the physical gap is at least 3a/2 for SU(2) whenever b/a ≤ 2/95, and at least 8a/3 for SU(3) whenever b/a ≤ 2/35 (1/16 with a sharper representation-theoretic bound), uniformly in the box. These sufficient couplings are 3072/95 and 1296/35 times those obtained from the preprint's constants (for SU(2), our transfer of them). The corresponding open-box endpoints in Hamiltonian units are β_H=√(4/95) ≈ 0.205 for SU(2) and β_H=√(6/35) ≈ 0.414 for SU(3), or √(3/16) ≈ 0.433 with the strengthened SU(3) bound. The basic open-box certificates also bound the static fundamental potential below linearly: for static charges at graph distance L ≥ 6, E_{x,y} - E_vac ≥ σ L with σ = 5/16 a for SU(2) at b/a ≤ 2/95 and σ = 5/9 a for SU(3) at b/a ≤ 2/35, uniformly in the box. A variable exclusion fraction and an independent local-decoupling upper bound bracket the first physical excitation energy above the vacuum. On open boxes containing an elementary plaquette, at b/a = 1/5000 this energy lies in [2.990, 3.050] a (SU(2)) and [5.326, 5.390] a (SU(3)). These are strong-coupling statements at fixed lattice spacing, not continuum statements. Files ym_strong_coupling_gaps.pdf: the paper. zenodo_ymgap_v1.0.0_2026-10-23.zip: the certificate bundle: two independent implementations of the certified tuple constants, the verifier, the tamper-test driver, the Section 8 upper-bound program and its exact checker, the number generator, manifests and verifier outputs for all 34 certificate configurations, the complete tuple files, and the paper source with every number generated from these files. Its SHA-256 is in the adjacent .sha256 file; SHA256SUMS inside covers every file. README.md, LICENSE.md: copies of the bundle's front matter. Verification ./replay.sh (Python 3, standard library only; a few minutes) regenerates every tuple file, compares it entry by entry with the second implementation, verifies every manifest in exact rational arithmetic, runs the tamper tests and the exact check of the two-sided bracket upper bounds, and regenerates every number of the paper; it exits non-zero on any failure. Scope: strong-coupling statements for the Kogut–Susskind lattice Hamiltonian at fixed lattice spacing, uniform in the volume, with the representation cutoff removed. They are not continuum statements and do not address the continuum Yang–Mills mass gap problem. Licence: CC BY-NC 4.0 (LICENSE.md), licensed by SmartLedger Solutions. Commercial-use enquiries: labs@smartledger.solutions. No patent licence is granted by this release.
We test the eigenstate thermalization hypothesis (ETH) for 2+12+12+1 dimensional SU(2) lattice gauge theory. By considering the theory on a chain of plaquettes and truncating basis states for link variables at j=1/2j=1/2j=1/2, we can map it onto a quantum spin chain with local interactions and numerically exactly diagonalize the Hamiltonian for reasonably large lattice sizes. We find energy level repulsion in momentum sectors with no remaining discrete symmetry. We study two local observables made up of Wilson loops and calculate their matrix elements in the energy eigenbasis, which are shown consistent with the ETH.
Gauge theories like lattice QCD present a complex problem for quantum simulation. In a basis where the electric part of the Hamiltonian is simple, the magnetic part, generally expressed as a sum over the plaquette operators of the lattice, is quite complicated, producing correlated transitions between several link and site degrees of freedom. We provide an efficient block encoding of the plaquette operator in the irrep basis, a refinement of the electric basis where the internal gauge-variant degrees of freedom are integrated out. The construction removes the plaquette matrix element scaling wall which has been a significant barrier for other approaches in this basis. The algorithm leverages a convenient factorization property of the matrix elements, cheap classical precomputation, and quantum oracles built from lookup tables and programmed rotations.