This paper establishes a strict upper bound theorem for spectral gap compression of random local transfer operators on finite-dimensional non-associativenormed algebras. Let the associator norm of the underlying algebra be δ and thelattice size be N. For the left-right averaged transfer operator defined under threebody multiplication gates, when δ > 0 (for sedenions and higher non-associativealgebras), there exists a universal constant Cd > 0 depending only on the algebra dimension such that the intensity-normalized spectral gap ∆int strictly satisfies∆int ≤ Cd · δ2 in the thermodynamic limit N → ∞. This upper bound is independent of system size, indicating that non-associativity imposes a ceiling on thespectral gap determined by the intrinsic algebraic structure, but does not forceit to vanish. The core of the proof uses the variational principle in the correctdirection: construct a trial state localized on only three lattice sites, quantify theinterference deficit under the action of the transfer operator by the associator normto obtain a lower bound on the expectation value, thereby obtaining a lower boundon the second largest eigenvalue and an upper bound on the spectral gap. On thephysical level, the theorem is mapped to the three-body perturbative expansion ofthe Kitaev honeycomb model, yielding three quantitatively observable predictions:compression of the magnon gap, a crossover behavior in low-temperature thermaltransport, and a lengthened time scale for entanglement growth. An independenttest scheme based on superconducting qubit arrays is designed. All predictionscontain no free fitting parameters; constants are predetermined by the algebra dimension; falsification conditions are declared in advance in Chapter 6, ensuring thetheory’s falsifiability.
This paper establishes a strict upper bound theorem for spectral gap compression of random local transfer operators on finite-dimensional non-associativenormed algebras. Let the associator norm of the underlying algebra be δ and thelattice size be N. For the left-right averaged transfer operator defined under threebody multiplication gates, when δ > 0 (for sedenions and higher non-associativealgebras), there exists a universal constant Cd > 0 depending only on the algebra dimension such that the intensity-normalized spectral gap ∆int strictly satisfies∆int ≤ Cd · δ2 in the thermodynamic limit N → ∞. This upper bound is independent of system size, indicating that non-associativity imposes a ceiling on thespectral gap determined by the intrinsic algebraic structure, but does not forceit to vanish. The core of the proof uses the variational principle in the correctdirection: construct a trial state localized on only three lattice sites, quantify theinterference deficit under the action of the transfer operator by the associator normto obtain a lower bound on the expectation value, thereby obtaining a lower boundon the second largest eigenvalue and an upper bound on the spectral gap. On thephysical level, the theorem is mapped to the three-body perturbative expansion ofthe Kitaev honeycomb model, yielding three quantitatively observable predictions:compression of the magnon gap, a crossover behavior in low-temperature thermaltransport, and a lengthened time scale for entanglement growth. An independenttest scheme based on superconducting qubit arrays is designed. All predictionscontain no free fitting parameters; constants are predetermined by the algebra dimension; falsification conditions are declared in advance in Chapter 6, ensuring thetheory’s falsifiability.
Recent rigorous spectral analysis of stochastic transfer operators on non-associativealgebras shows that when the associator norm of the algebra is positive, the spectralgap of the system is locked in the thermodynamic limit by a strictly positive lowerbound depending only on the algebra dimension and an intrinsic constant. Thispaper points out that directly applying this abstract algebraic theorem to physicalpredictions for the Kitaev honeycomb model suffers from three fundamental gaps:the lack of a strict correspondence between the transfer operator spectral gap andthe Hamiltonian energy gap, the impossibility of directly generalizing one-bond estimates on a one-dimensional chain to the thermodynamic limit of two-dimensionalhoneycomb lattice points, and the dimensional gap between the low-dimensionalHilbert space of physical spin-1/2 degrees of freedom and the high-dimensional algebraic structure of octonions. To fill these logical gaps, this paper sequentiallyestablishes a quantum Cheeger-type correspondence between the imaginary-timeevolution transfer operator and the Hamiltonian energy gap, proposes a syntheticdimension scheme encoding four honeycomb lattice spin degrees of freedom into anoctonion degree of freedom, eliminates the lattice point dependence in the thermodynamic limit through a multi-bond cooperative variational principle and LiebRobinson quasi-locality, and rigorously derives the linear scaling relation betweenthe effective associator norm and the perturbation strength using quasi-degenerateperturbation theory. Based on these bridges, this paper derives three quantitative physical predictions without free fitting parameters for the Kitaev honeycombmodel with three-body perturbations: a rigid plateau of the magnon excitationgap, an exponential suppression of the low-temperature spin thermal conductivity,and the maintenance of the area law for entanglement dynamics. This paper further designs a complete experimental test scheme using a superconducting qubitarray, covering independent measurement of the effective associator norm, gapspectroscopy, thermal transport reconstruction, and a randomized measurementprotocol for entanglement entropy, and pre-specifies falsification conditions including statistical thresholds and operationalized trigger criteria, ensuring the theoryis falsifiable. The three-element bridging methodology of ”algebraic theoremphysical model– experimental test” constructed in this paper provides a reusabletemplate for converting rigidity theorems of other abstract algebraic structures intotestable quantum simulation predictions.
The role of non-associative algebras in quantum many-body systems has longbeen overlooked. Classical algebraic classification shows that the sedenions arethe smallest real normed algebra that loses alternativity and divisibility and givesrise to zero divisors. This paper reveals the reverse implication of such algebraicdegeneration: the positivity of the associator norm does not soften the physicalsystem but instead sets a minimum energy cost for local excitations that cannot bewashed out by long-wavelength fluctuations. By encoding the structure constantsof the sedenions into a multi-spin coupling tensor, we prove a spectral gap rigiditylocking theorem for the non-associative extension of the Kitaev honeycomb model:the low-energy gap of the system along a quantum phase transition path is lockedbelow a positive bound that is independent of the number of lattice sites. Fromthis we derive three testable predictions: saturation of the gap to a finite plateauat the critical point, a crossover of the low-temperature spin thermal conductivityfrom power-law to exponential suppression, and the persistence of an area law forentanglement entropy after a quantum quench. We provide an experimental testscheme using superconducting qubit arrays and cold-atom optical lattices, establisha protocol for measuring the effective non-associative strength independently of thegap, and explicitly state operational falsification conditions. All predictions containno free parameters and depend only on fixed algebraic constants, offering a closedtheoretical and experimental framework for quantum simulation of non-associativealgebras.
Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. When a fixed boundary condition is imposed, this problem is undecidable: no algorithm, even with unlimited resources, can decide for every Hamiltonian whether it is gapped. Yet the thermodynamic limit is meant to capture the intrinsic properties of very large systems, which should not depend on how their boundaries are chosen. We therefore consider the bulk spectral gap, which concerns only the intrinsic excitations of the infinite system, and ask what can and cannot be computed about it. First, we give a general algorithm producing a hierarchy of certified upper bounds on the bulk gap, arbitrarily tight at the cost of more computation. Second, we prove that the spectral gap problem remains undecidable in the bulk, already for translationally invariant nearest-neighbour chains of fixed local dimension. Hence no general algorithm produces convergent lower bounds, and no convergent upper-bounding algorithm, ours included, admits a computable convergence rate. The bulk spectral gap is thus certifiable from above but uncomputable from below; more precisely, the bulk-gaplessness promise problem is RE-complete, exactly as hard as the halting problem. The same construction shows that no general algorithm can decide whether a given product state is a ground state, and that convergent bounds on ground-state expectation values admit no computable convergence rate in general. Finally, we obtain the first nontrivial certified upper bounds on the bulk gap of the spin-1/2 kagome Heisenberg antiferromagnet.
Within the theoretical framework of the Ladder of Stability series, this paperadvances the rigidity classification of Jones subfactor indices and Coxeter–Dynkindiagrams into a testable quantitative prediction on topological quantum platforms.Based on the standard correspondence between the quantum dimensions of fusion categories in non-abelian topological order and subfactor inclusion indices, wepropose an effective bridging hypothesis B-J that couples the low-energy excitation gap ∆ to the Jones index [M : N] through a monotonic inverse relationship.This leads to the derivation that, within regions where topological invariants remain constant, the energy gap is locked to a set of discrete steps determined by{4cos2(π/n) | n ≥ 3}, with inter-step ratios that are universal across materialplatforms. The paper systematically analyses the spatial fluctuations of the scaling constant C, derives an upper bound nmax for the number of distinguishablelevels, and thereby strictly confines the experimental test to the low-level regionn ≤ 6. A bivariate test protocol based on scanning tunnelling spectroscopy andtransport measurements is presented, together with precise falsification conditionsthat include Kolmogorov–Smirnov statistical tests and false-positive control provisions. This work transforms the structural rigidity of operator algebras into anoperationally falsifiable empirical prediction, completing a crucial step from mathematical structure to physical observation.
The low-energy excitations of non-Abelian topological order are described byanyon fusion categories. In the thermodynamic limit, the algebra of local observables may form a tower of von Neumann subfactor inclusions, and the correspondence between the Jones index and the total quantum dimension of anyons hasbeen supported by Longo’s theorem in algebraic quantum field theory and by explicit calculations in Kitaev’s quantum double model. Taking this as a startingpoint, this paper does not presuppose the universal validity of subfactor structures in generic strongly correlated lattice systems; instead, it sets them as explicittheoretical hypotheses and systematically constructs a derivation chain from thediscrete sequence of Jones indices to constraints on physical excitation gap ratios.The central link is the definition of a subfactorizable Hamiltonian– expressing thelow-energy effective Hamiltonian, within an energy cutoff, as a quadratic form inTemperley-Lieb projections– thereby using the representation theory of the TLalgebra to establish a mapping between anyon excitations and Bratteli diagramnodes, leading to the conclusion that the gap ratio depends only on the Jones parameter τ. The discrete valueing of the Jones index when it is less than 4 forcesτ to be locked to a countable set, yielding the rigid prediction that within thesame topological phase region the ratio of different excitation gaps cannot driftcontinuously with external parameters. This paper designs an experimental testscheme based on scanning tunneling spectroscopy and transport measurements,and systematically provides falsification conditions, so that each hypothesis in thetheoretical framework can be independently tested. The contribution of this paper lies in extending subfactor algebra theory from exactly solvable models to atheoretical framework with operational physical predictions, through a cautioushypothesis–deduction–test structure.
Estimating spectral gaps of quantum many-body Hamiltonians is a highly challenging computational task, even under assumptions of locality and translation-invariance. Yet, the quest for rigorous gap certificates is motivated by their broad applicability, ranging from many-body physics to quantum computing and classical sampling techniques. Here we present a general method for obtaining lower bounds on the spectral gap of frustration-free quantum Hamiltonians in the thermodynamic limit. We formulate the gap certification problem as a hierarchy of optimization problems (semidefinite programs) in which the certificate – a proof of a lower bound on the gap – is improved with increasing levels. Our approach encompasses existing finite-size methods, such as Knabe's bound and its subsequent improvements, as those appear as particular possible solutions in our optimization, which is thus guaranteed to either match or surpass them. We demonstrate the power of the method on one-dimensional spin-chain models where we observe an improvement by several orders of magnitude over existing finite size criteria in both the accuracy of the lower bound on the gap, as well as the range of parameters in which a gap is detected.