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Certified Spectral Gaps for SU(3) Lattice Gauge Hamiltonians on One, Two, and Three Cubes

Ken Umeno

Published Oct 25, 2026• 0 citations
Publisher

Abstract

We give computer-assisted, two-sided spectral-gap bounds for the untruncated SU(3) lattice gauge Hamiltonian on open chains of one, two, and three elementary cubes. All vertices satisfy the neutral Gauss law, and all elementary plaquettes, including shared interior faces, enter the Hamiltonian. The normalization is K = T + 3F − M/2, with C2(3) = 4/3. Complete electric-energy windows of dimensions 141, 531, and 1205, exact Haar Gram matrices, and certified complementary-space estimates yield gap enclosures [2.915605103, 4.978456040], [0.777213074, 5.394807087], and [0.328406186, 5.692597307]. For three cubes, retaining the complete Wilson output resolved into 20 electric-energy layers closes the Schur inertia test. We quantify the energy-window ceiling, prove an unconditional exponential dimension obstruction for complete absolute electric windows, and obtain positive gap bounds throughout 0.45 ≤ β ≤ 0.502 for Kβ = T + β(6F − M). Larger-window rational trial vectors sharpen both endpoints; an additional certified minimal-truncation benchmark and independent Haar audits accompany the proof. The results concern finite lattices; no thermodynamic or continuum mass gap is inferred.

DOI
10.5281/zenodo.23641689

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quant-pharXiv:2511.13721•Physical review. D/Physical review. D.•Nov 18, 2025•1 citation
We give computer-assisted, two-sided spectral-gap bounds for the untruncated SU(3) lattice gauge Hamiltonian on open chains of one, two, and three elementary cubes. All vertices satisfy the neutral Gauss law, and all elementary plaquettes, including shared interior faces, enter the Hamiltonian. The normalization is K = T + 3F − M/2, with C2(3) = 4/3. Complete electric-energy windows of dimensions 141, 531, and 1205, exact Haar Gram matrices, and certified complementary-space estimates yield gap enclosures [2.915605103, 4.978456040], [0.777213074, 5.394807087], and [0.328406186, 5.692597307]. For three cubes, retaining the complete Wilson output resolved into 20 electric-energy layers closes the Schur inertia test. We quantify the energy-window ceiling, prove an unconditional exponential dimension obstruction for complete absolute electric windows, and obtain positive gap bounds throughout 0.45 ≤ β ≤ 0.502 for Kβ = T + β(6F − M). Larger-window rational trial vectors sharpen both endpoints; an additional certified minimal-truncation benchmark and independent Haar audits accompany the proof. The results concern finite lattices; no thermodynamic or continuum mass gap is inferred.
We give a computer-assisted proof that the first twelve excitations of the neutral SU(3) lattice gauge Hamiltonian on a single cube form two isolated six-dimensional spectral subspaces. Their representations under spatial symmetry and charge conjugation are A+ ⊕ E+ ⊕ T + and A− ⊕ E− ⊕ T − . Complete electric- 1g g 1u 1u u 1g energy windows, rational Ritz bounds, and full Wilson-output Gram matrices yield certified eigenvalue counts for the untruncated operator. The main theorems use interior thresholds with larger comparison-form margins and give separations of at least 0.78482728 between the clusters and 0.72784037 above the second cluster. A separate refinement improves these values to 0.89782728 and 0.85184037. The neutral gap lies in [4.415605103, 4.978456040]. All multiplicities are retained, and selected matrix entries are audited with an independent Haar-integration algorithm. No ordering of the irreducible components within either cluster, volume-uniform gap, or continuum limit is asserted.
We construct complete Gauss cutoffs and certify spectral gaps for finite SU(3) lattice gauge Hamiltonians with positive electric coefficients and Wilson potentials. Each cutoff retains all allowed edge representations and invariant tensors. A separate finite extension retains every one-step Wilson output and determines the exact boundary Gram matrix. Decomposing these outputs by excess-edge sets gives local overlap and electric-energy- relative bounds. Under explicit uniform local hypotheses, they imply a volume-independent O(N−2) norm-resolvent error for the Hamiltonian before vacuum subtraction. A quadratic- form Schur argument combines the finite data with an analytic bound on the entire omitted space to certify the original untruncated operator. For two adjacent plaquettes, the complete cutoff has dimension nine and its full-output extension has dimension thirty-five. Exact rational arithmetic gives eigenvalue intervals, a gap lower bound 154/39 at coupling 1/2, and the bound 3/5 throughout [1/2, 5/6]. We also give an exact weighted-cycle formula for the electric gap and an explicit positive interacting gap on each fixed graph. As a representation result, a common Cauchy-relative Perron–Frobenius embedding transports the heat semigroup, Gauss constraints, and ordered observables. Its integer parameter K does not determine a physical gap law in log K. Conditional limit statements identify the additional vacuum, reconstruction, and uniform infrared estimates required for a four-dimensional Yang–Mills theory.
We study the untruncated SU(3) lattice gauge Hamiltonian on open chains of elementary cubes, with fixed transverse plaquette coupling and variable side coupling s. A product of exact transverse plaquette ground states defines a nonnegative relative-excitation operator Dn. For every fixed 0 < s ≤ 1, we prove that the interacting vacuum has Dn-expectation at least c(s)n, with an explicit positive rational function c. Consequently, complete global relative-energy windows used with an affine complementary-space lower bound require linearly growing thresholds and exponentially growing dimensions. In the same model, we prove a spectral gap uniform in the chain length for 0 ≤ s ≤ 77/2350080: both the full kinematic Hamiltonian and its neutral Gauss restriction have gap at least 77/765. The proof combines an all-charge single-plaquette gap bound with a self-contained creation- operator contraction and excitation-resolvent estimate on infinite-dimensional site spaces. All smallness constants are explicit. The two results distinguish the cost of a fixed-reference window certificate from the presence of a physical spectral gap. They complement the finite-volume certificates at s = 1 in G1B. No uniform statement at s = 1, growing three-dimensional cross-section, or continuum limit is asserted.
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 give explicit sufficient conditions for a spectral gap uniform in the number of sites of a three-dimensional SU(3) Hamiltonian lattice, partitioned into disjoint cube blocks and free bridge links. Using previously certified one-cube spectral and comparison bounds, we obtain a dimensionless gap at least 2/9, on the full kinematic Hilbert space and hence on its Gauss-invariant subspace, for inter-block plaquette strength 0 ≤ s ≤ 1/6144. The point s = 1 is the isotropic Kogut–Susskind lattice at β = 1/2 in our Hamiltonian convention. The argument combines a gauge-Casimir comparison, a product-vacuum fixed-point construction, an excitation resolvent criterion, and orthogonal-output estimates with constants independent of volume. An isolated two-cube, two-bridge block has a full kinematic gap at least 1/3 at s = 1. For a four-face interface we retain all signed entries of an 18-vector output Gram matrix, reducing a specified residual upper bound from 0.160048 to 0.146438. Explicit bounded shears improve the derivative on the one-edge charged input subspace and eliminate its columns in an origin-level excitation test. These shears act trivially on the physical neutral sector; neither they nor the interface Gram establish a full-input contraction at s = 1. The particular derivative majorant used for the global theorem cannot certify s ≥ 1/4860; this is not a no-go result for other methods. The scope of the inherited certificates and the remaining thermodynamic and continuum questions are stated separately.
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.