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.
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.
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 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 ε 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 1.2×10−15 through electric truncation jmax=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) flops independent of link labels, giving measured Hamiltonians at any moderate length with no sign table; it reproduces independent DMRG bulk energies to 9×10−16. (2) A non-gauge \emph{underdetermination phenomenon}: the complete set of length-≤3 amplitudes leaves a GF(2) sign freedom that is not pure gauge --- two tables, each exact on all measured length-≤3 data, differ at length ≥4 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 λ=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
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.
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.
For a real symmetric matrix H and distinct vertices a,b, we classify exponents k for which Hk has perfect state transfer (PST) from a to b. If their supported eigenvalues are integer multiples of a common positive number, every odd exponent reduces to H and every positive even exponent reduces to H2. We determine the minimum transfer times using a greatest common divisor of supported spectral differences. For rational symmetric matrices, symmetry of the source vertex support about zero implies the odd-power equivalence without a commensurability assumption; this includes all bipartite graphs. If the source vertex supports zero, PST under one positive even power implies PST under every positive even power. For a symmetric three-point quadratic spectrum whose outer projection signs agree and differ from the central sign, a nonzero rational shift leaves exactly one PST exponent. We classify all adjacency powers of hypercubes, cycles, and Johnson graphs, and all adjacency squares of paths. In particular, the adjacency matrix of P7 has PST from vertex 2 to vertex 6 only at exponent 2.