Width-uniform transfer-matrix gap certificates for the half-spin factor of a lattice gauge toy row: audited computer-assisted bounds at strong coupling and beyond the strong-coupling wall, with structural notes on Weil positivity
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 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.
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 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.
This record contains the full author preprint (12 pages), its mathematical supplement (17 pages), and the complete article archive with LaTeX sources, the version note, SHA-256 manifests and the licence. The two PDFs are also provided as individual files for direct reading. Version v3 clarifies the Haar normalization convention in section 9.1 of the mathematical supplement (equation 45). The unnormalised transform H has H*H = 2I and Tr(HQH*) = 2Tr(Q); the equivalent orthogonal transform is H/sqrt(2). This agrees with the existing root/detail identity and exact verifier. The main article, verification code, Lean files, matrix certificates and stated constants are unchanged. See the publication version note for the precise scope of this correction. AbstractWe prove that the nearest-neighbor spin-1/2 antiferromagnetic Heisenberg model on the infinite Kagome lattice is gapless in the bulk GNS sense. The argument constructs six explicit local preimages and their complete infinite-lattice Hamiltonian commutators. Two rigorous 17-spin positivity certificates, each decomposed into 31 nontrivial swap-character blocks and nine SU(2) multiplicity blocks, give a state-uniform off-kernel mass floor. Exact operator transport on the two genuine outward Kagome continuations propagates the same physical family through a finite stationary interface. A root/detail/current decomposition, a local current–boundary identity, and state-uniform norm estimates yield a horizon-independent Dirichlet-energy bound. With harmonic weights, for every infinite-volume ground state ω and every R ≥ 1 the same finite family satisfies M_R(ω) ≥ (15/32) ∑_{k=1}^{R} 1/k and E_R(ω) ≤ 761429817528146602401/25088000. Any positive GNS gap would force the second quantity to diverge with the first. Hence every admissible ground-state representation has zero gap, and the declared bulk gap is zero. All finite operator identities and matrix inequalities are checked from exact algebraic data, with rigorous ball arithmetic used for the large positive-definiteness tests. Lean 4 verifies the arbitrary-horizon harmonic implication and the final variational contradiction. The proof uses neither finite-size gap extrapolation nor a spectral cutoff. Companion public verification code, Lean 4 formalization and exact certificates: https://doi.org/10.5281/zenodo.23675708 Distribution and reuse are governed by the Public Scientific Evaluation and Reproducibility Terms included with this record. The deposit excludes the author's private discovery engine and proprietary know-how.
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 construct Lean-kernel-checked spectral certificates for rational definite pencils representing the periodic SU(2) Kogut–Susskind ladder with link spins truncated at `j ≤ 1`. The construction uses an unnormalised spin-network basis with a positive diagonal Gram matrix. At unit coupling and circumferences `N = 3, 4`, the certificates enclose both centre-sector ground energies and give strict lower bounds for both sector first-excited energies. They identify a simple charge-0 ground level and a simple charge-1 first-excited level of the full pencil, with full spectral gaps in `[0.865089, 0.865122]` and `[1.183111, 1.183144]`, respectively. The gap therefore increases strictly between these two sizes. Compared with the independently certified `j ≤ ½` gaps at the same sizes and coupling, it decreases strictly at both sizes, while the first excitation remains in the nontrivial centre sector. The `N = 4` sector dimensions are 579 and 434. Their certificates use separately kernel-checked arithmetic segments composed by proved chain lemmas, with measured segment-process memory near 1.6 GiB; the final aggregation has a larger memory requirement. The final theorems use only `propext`, `Classical.choice` and `Quot.sound`. The spectral certificates are kernel-checked statements about the supplied pencils. Their interpretation as statements about the truncated Hamiltonian is established by a hybrid verification: kernel-checked finite identities, paper proofs of the spin-network identification, and independent exact-rational verification of the selection-eligible magnetic contractions, with vanishing elsewhere established by the selection-rule proof and stored-support certificates. The full identification chain is not kernel-checked. No cutoff-convergence or untruncated-gap conclusion is asserted. This bundle contains the manuscript draft, the 57445-module Lean 4 acceptance closure (acceptance command `lake build ChristMath.J1Accept`), the exact pencils and local tables, the emitters, the independent Rust implementation, and the fresh-process axiom audit.
We study transfer operators built from the deterministically computed Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) four-simplex vertex amplitude (sl2cfoam-next) at boundary spins 2j = 2-16. Three results. (i) An alternating-sign involution P on the four-valent intertwiner recoupling basis is an exact symmetry of the melonic transfer operator, [T,P]=0 at machine precision at every gate tested: the transfer superselects into two parity sectors, which we identify with the two orientation branches of the vertex asymptotics (the ``cosine problem'' structure). At half-integer boundary spin the leading eigenvalue is an exactly degenerate parity doublet; at integer spin the subleading eigenvalue is exactly the top of the opposite parity sector. (ii) Consequently the slow closing of the transfer spectral gap at large spin, mu* = (lambda_2/lambda_1)^2 -> 1 - which we exhibit across the full gate range - is not a renormalization-group instability: it is the splitting of a superselected symmetry doublet, and one composition step acts on it exactly as mu* mu*^2, i.e. strictly gap-opening. On the clean (integer-spin) data the splitting closes as a power law, g = 1-mu* ~ j^-p with p = 3.4-3.5 at the accessible gates and drifting toward 7/2, rather than exponentially; earlier exponential appearances are a crossover effect. This yields two falsifiable predictions: the tail exponent should be independent of the Barbero-Immirzi parameter gamma (while the prefactor is strongly gamma-dependent), and orientation-fixing vertex proposals (proper vertex, causal vertices) should remove the doublet altogether - a simple leading eigenvalue at half-integer spin and no closing gap. (iii) At every gate tested, within each parity sector the transfer is entrywise positive in its Perron gauge, and we establish certified contraction bounds via the Birkhoff-Hopf theorem with outward-rounded interval enclosures: across a gamma [0.1,5], j 1,32,2 grid, all 25 sector certificates close with projective contraction ratio kappa <= 0.9404, near-tight at the infrared gate (kappa^2 = 0.5087 against a measured mu* = 0.5066), together with certified perturbation balls. The results characterize precisely what orientation-fixing modifications of the EPRL amplitude would change, and supply a portable certified-spectral method for spin-foam transfer operators.