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  "slug": "a-noether-type-correspondence-for-physical-computing-continuous-symmetry-as-a-co",
  "title": "A Noether-Type Correspondence for Physical Computing: Continuous Symmetry as a Conserved Computational Resource in Hybrid Intelligence",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Physical computing substrates—resonator networks, cyclic mechanical structures, and engineered Hamiltonian systems—promise order-of-magnitude energy savings over digital inference, but no existing principle tells a designer which physical structure to build so that its symmetry does computational work. This paper addresses that gap. We formalize a correspondence between a continuous SO(2)/U(1) symmetry of a physical computing substrate and a conserved computational quantity: a sector label (harmonic index) that the substrate dynamics cannot mix, analogous to Noether's theorem relating continuous symmetries to conservation laws. Exploiting the conservation law, we derive a block-diagonalization cost model in which an N_s-fold cyclic substrate reduces a dense spectral problem of size M=N_s m from c M³ to N_s c m³ operations, a speedup of N_s² (computed: 64× for N_s=8, m=100), and a parameter-sharing reduction of factor m for cyclic layers (computed: 65,536→256 parameters at m=256). Against the Landauer bit energy k_B Tln 2 = 2.87×10^-21 J at T=300 K, we frame the gains as bits-per-joule headroom under the accounting conventions of thermodynamic epiplexity. The result gives hybrid-system designers a concrete selection rule—maximize exploitable continuous symmetry—and defines the smallest falsifying test: a symmetry-constrained autoencoder whose parameter count and reconstruction fidelity are compared against an unconstrained baseline.",
  "abstract_markdown": "Physical computing substrates—resonator networks, cyclic mechanical structures, and engineered Hamiltonian systems—promise order-of-magnitude energy savings over digital inference, but no existing principle tells a designer which physical structure to build so that its symmetry does computational work. This paper addresses that gap. We formalize a correspondence between a continuous $SO(2)/U(1)$ symmetry of a physical computing substrate and a conserved computational quantity: a sector label (harmonic index) that the substrate dynamics cannot mix, analogous to Noether's theorem relating continuous symmetries to conservation laws. Exploiting the conservation law, we derive a block-diagonalization cost model in which an $N_s$-fold cyclic substrate reduces a dense spectral problem of size $M=N_s m$ from $c\\,M^3$ to $N_s\\,c\\,m^3$ operations, a speedup of $N_s^2$ (computed: $64\\times$ for $N_s=8$, $m=100$), and a parameter-sharing reduction of factor $m$ for cyclic layers (computed: $65{,}536\\to256$ parameters at $m=256$). Against the Landauer bit energy $k_B T\\ln 2 = 2.87\\times10^{-21}\\,\\mathrm{J}$ at $T=300\\,\\mathrm{K}$, we frame the gains as bits-per-joule headroom under the accounting conventions of thermodynamic epiplexity. The result gives hybrid-system designers a concrete selection rule—maximize exploitable continuous symmetry—and defines the smallest falsifying test: a symmetry-constrained autoencoder whose parameter count and reconstruction fidelity are compared against an unconstrained baseline.",
  "date": "2026-11-13",
  "first_published": "2026-11-13",
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  "url": "https://papers.qnfo.org/papers/a-noether-type-correspondence-for-physical-computing-continuous-symmetry-as-a-co",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "A Noether-Type Correspondence for Physical Computing: Continuous Symmetry as a Conserved Computational Resource in Hybrid Intelligence"
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        "description": "Physical computing substrates—resonator networks, cyclic mechanical structures, and engineered Hamiltonian systems—promise order-of-magnitude energy savings over digital inference, but no existing principle tells a designer which physical structure to build so that its symmetry does computational work. This paper addresses that gap. We formalize a correspondence between a continuous SO(2)/U(1) symmetry of a physical computing substrate and a conserved computational quantity: a sector label (harmonic index) that the substrate dynamics cannot mix, analogous to Noether's theorem relating continuous symmetries to conservation laws. Exploiting the conservation law, we derive a block-diagonalization cost model in which an N_s-fold cyclic substrate reduces a dense spectral problem of size M=N_s m from c M³ to N_s c m³ operations, a speedup of N_s² (computed: 64× for N_s=8, m=100), and a parameter-sharing reduction of factor m for cyclic layers (computed: 65,536→256 parameters at m=256). Against the Landauer bit energy k_B Tln 2 = 2.87×10^-21 J at T=300 K, we frame the gains as bits-per-joule headroom under the accounting conventions of thermodynamic epiplexity. The result gives hybrid-system designers a concrete selection rule—maximize exploitable continuous symmetry—and defines the smallest falsifying test: a symmetry-constrained autoencoder whose parameter count and reconstruction fidelity are compared against an unconstrained baseline.",
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