The Ladder of Stability M: Strict Upper Bound Theorem for Spectral Gap Compression of Random Transfer Operators on Non-Associative Algebras
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2026
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| author | zhou, changzheng zhou, ziqing |
| author_facet | zhou, changzheng zhou, ziqing |
| contents | <p>This paper establishes a strict upper bound theorem for spectral gap com<br>pression of random local transfer operators on finite-dimensional non-associative<br>normed algebras. Let the associator norm of the underlying algebra be δ and the<br>lattice size be N. For the left-right averaged transfer operator defined under three<br>body multiplication gates, when δ > 0 (for sedenions and higher non-associative<br>algebras), there exists a universal constant Cd > 0 depending only on the alge<br>bra dimension such that the intensity-normalized spectral gap ∆int strictly satisfies<br>∆int ≤ Cd · δ2 in the thermodynamic limit N → ∞. This upper bound is inde<br>pendent of system size, indicating that non-associativity imposes a ceiling on the<br>spectral gap determined by the intrinsic algebraic structure, but does not force<br>it to vanish. The core of the proof uses the variational principle in the correct<br>direction: construct a trial state localized on only three lattice sites, quantify the<br>interference deficit under the action of the transfer operator by the associator norm<br>to obtain a lower bound on the expectation value, thereby obtaining a lower bound<br>on the second largest eigenvalue and an upper bound on the spectral gap. On the<br>physical level, the theorem is mapped to the three-body perturbative expansion of<br>the Kitaev honeycomb model, yielding three quantitatively observable predictions:<br>compression of the magnon gap, a crossover behavior in low-temperature thermal<br>transport, and a lengthened time scale for entanglement growth. An independent<br>test scheme based on superconducting qubit arrays is designed. All predictions<br>contain no free fitting parameters; constants are predetermined by the algebra di<br>mension; falsification conditions are declared in advance in Chapter 6, ensuring the<br>theory’s falsifiability.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20269102 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | The Ladder of Stability M: Strict Upper Bound Theorem for Spectral Gap Compression of Random Transfer Operators on Non-Associative Algebras zhou, changzheng zhou, ziqing Non-associative algebra; Associator norm; Spectral gap compres sion; Upper bound theorem; Sedenions; Kitaev model; Quantum simulation <p>This paper establishes a strict upper bound theorem for spectral gap com<br>pression of random local transfer operators on finite-dimensional non-associative<br>normed algebras. Let the associator norm of the underlying algebra be δ and the<br>lattice size be N. For the left-right averaged transfer operator defined under three<br>body multiplication gates, when δ > 0 (for sedenions and higher non-associative<br>algebras), there exists a universal constant Cd > 0 depending only on the alge<br>bra dimension such that the intensity-normalized spectral gap ∆int strictly satisfies<br>∆int ≤ Cd · δ2 in the thermodynamic limit N → ∞. This upper bound is inde<br>pendent of system size, indicating that non-associativity imposes a ceiling on the<br>spectral gap determined by the intrinsic algebraic structure, but does not force<br>it to vanish. The core of the proof uses the variational principle in the correct<br>direction: construct a trial state localized on only three lattice sites, quantify the<br>interference deficit under the action of the transfer operator by the associator norm<br>to obtain a lower bound on the expectation value, thereby obtaining a lower bound<br>on the second largest eigenvalue and an upper bound on the spectral gap. On the<br>physical level, the theorem is mapped to the three-body perturbative expansion of<br>the Kitaev honeycomb model, yielding three quantitatively observable predictions:<br>compression of the magnon gap, a crossover behavior in low-temperature thermal<br>transport, and a lengthened time scale for entanglement growth. An independent<br>test scheme based on superconducting qubit arrays is designed. All predictions<br>contain no free fitting parameters; constants are predetermined by the algebra di<br>mension; falsification conditions are declared in advance in Chapter 6, ensuring the<br>theory’s falsifiability.</p> |
| title | The Ladder of Stability M: Strict Upper Bound Theorem for Spectral Gap Compression of Random Transfer Operators on Non-Associative Algebras |
| topic | Non-associative algebra; Associator norm; Spectral gap compres sion; Upper bound theorem; Sedenions; Kitaev model; Quantum simulation |
| url | https://doi.org/10.5281/zenodo.20269102 |