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  "slug": "noether-bounded-computation-conservation-laws-as-native-constraints-on-physical-",
  "title": "Noether-Bounded Computation: Conservation Laws as Native Constraints on Physical Processors",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Digital processors that emulate continuous dynamics pay a per-operation rounding tax: each step can violate a conserved quantity of the emulated system, and the violation accumulates with operation count. Whether a physical computing substrate whose operations directly instantiate a continuous symmetry group inherits the corresponding Noether conserved quantities as built-in constraints — and what advantage that inheritance buys in error and energy — has not been formally quantified. This paper derives a Noether-based cost model relating the symmetry group G of an analog processor to bounds on conservation-law drift and error growth. For a b-bit digital emulator we show the worst-case relative drift of a conserved quantity after N steps is Δ Q/Q ≤ N u with u = 2^-b, giving 2.33× 10^-4 at N = 10⁶ steps in 32-bit mantissa arithmetic; for a G-equivariant continuous processor the drift is zero by Noether's theorem up to physical dissipation ω T/Q_res. The model predicts a crossover operation count N^* = (ω T/Q_res)/u ≈ 4.30× 10⁶ steps under stated assumptions, beyond which the symmetry-aligned substrate has strictly lower conservation drift. Each independent conserved quantity also contracts the accessible error-state volume by a factor u. These bounds turn symmetry choice into a designable error-control parameter for analog and neuromorphic accelerator design.",
  "abstract_markdown": "Digital processors that emulate continuous dynamics pay a per-operation rounding tax: each step can violate a conserved quantity of the emulated system, and the violation accumulates with operation count. Whether a physical computing substrate whose operations directly instantiate a continuous symmetry group inherits the corresponding Noether conserved quantities as built-in constraints — and what advantage that inheritance buys in error and energy — has not been formally quantified. This paper derives a Noether-based cost model relating the symmetry group $G$ of an analog processor to bounds on conservation-law drift and error growth. For a $b$-bit digital emulator we show the worst-case relative drift of a conserved quantity after $N$ steps is $\\Delta Q/Q \\le N u$ with $u = 2^{-b}$, giving $2.33\\times 10^{-4}$ at $N = 10^{6}$ steps in 32-bit mantissa arithmetic; for a $G$-equivariant continuous processor the drift is zero by Noether's theorem up to physical dissipation $\\omega T/Q_{\\mathrm{res}}$. The model predicts a crossover operation count $N^{*} = (\\omega T/Q_{\\mathrm{res}})/u \\approx 4.30\\times 10^{6}$ steps under stated assumptions, beyond which the symmetry-aligned substrate has strictly lower conservation drift. Each independent conserved quantity also contracts the accessible error-state volume by a factor $u$. These bounds turn symmetry choice into a designable error-control parameter for analog and neuromorphic accelerator design.",
  "date": "2026-11-13",
  "first_published": "2026-11-13",
  "last_revised": "2026-11-13T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/noether-bounded-computation-conservation-laws-as-native-constraints-on-physical-",
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    "titles": [
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        "title": "Noether-Bounded Computation: Conservation Laws as Native Constraints on Physical Processors"
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