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  "id": "oai:papers.qnfo.org:discrete-scale-invariance-without-a-universal-factor-why-the-efimov-data-refute-",
  "slug": "discrete-scale-invariance-without-a-universal-factor-why-the-efimov-data-refute-",
  "title": "Discrete Scale Invariance Without a Universal Factor: Why the Efimov Data Refute Factor-2 Log-Periodicity Across Fundamental Physics",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Discrete scale invariance (DSI) — invariance under rescaling by a fixed factor rather than by any factor — is established in few-body quantum physics, Dirac semimetals, and renormalization-group limit cycles. A recurring conjecture holds that the scaling factor itself might be universal, with the dyadic value 2 appearing across unrelated domains. This paper tests that conjecture against the published few-body and condensed-matter evidence. We model each confirmed DSI system as a log-periodic spectrum E_n = E₀ exp(-2π n/s₀) and extract the scaling factor from published values. The three-boson Efimov spectrum has scaling factor $22.694$, which we show corresponds to an RG index s₀ = 2π/ln(22.694) = 2.0124 and a log-periodic period Δ(ln E) = 3.1222, i.e. $4.505$ octaves — a factor of $4.505$ larger than the dyadic period ln 2 = 0.69315. Scaling traps realize continuously tunable factors controlled by the slope of a spatially varying scattering length, and spin-orbit-coupled three-boson systems obey a generalized scaling law in a five-dimensional parameter space, so no single factor is fixed. We conclude that DSI is real and broadly shared, but the specific factor-2 universality hypothesis is excluded by existing data at the level of a factor $11.347$ per spectral step. The practical consequence is a sharp falsification criterion: any claimed universal dyadic periodicity in a physical spectrum must survive comparison against the measured Efimov period of $3.1222$ in ln E, which it cannot if the period is $0.69315$.",
  "abstract_markdown": "Discrete scale invariance (DSI) — invariance under rescaling by a fixed factor rather than by any factor — is established in few-body quantum physics, Dirac semimetals, and renormalization-group limit cycles. A recurring conjecture holds that the scaling factor itself might be universal, with the dyadic value 2 appearing across unrelated domains. This paper tests that conjecture against the published few-body and condensed-matter evidence. We model each confirmed DSI system as a log-periodic spectrum $E_n = E_0 \\exp(-2\\pi n/s_0)$ and extract the scaling factor from published values. The three-boson Efimov spectrum has scaling factor $22.694$, which we show corresponds to an RG index $s_0 = 2\\pi/\\ln(22.694) = 2.0124$ and a log-periodic period $\\Delta(\\ln E) = 3.1222$, i.e. $4.505$ octaves — a factor of $4.505$ larger than the dyadic period $\\ln 2 = 0.69315$. Scaling traps realize continuously tunable factors controlled by the slope of a spatially varying scattering length, and spin-orbit-coupled three-boson systems obey a generalized scaling law in a five-dimensional parameter space, so no single factor is fixed. We conclude that DSI is real and broadly shared, but the specific factor-2 universality hypothesis is excluded by existing data at the level of a factor $11.347$ per spectral step. The practical consequence is a sharp falsification criterion: any claimed universal dyadic periodicity in a physical spectrum must survive comparison against the measured Efimov period of $3.1222$ in $\\ln E$, which it cannot if the period is $0.69315$.",
  "date": "2026-10-21",
  "first_published": "2026-10-21",
  "last_revised": "2026-10-21T00:00:00Z",
  "modified": "2026-10-21T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/discrete-scale-invariance-without-a-universal-factor-why-the-efimov-data-refute-",
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        "name": "Rowan Brad Quni-Gudzinas",
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        "title": "Discrete Scale Invariance Without a Universal Factor: Why the Efimov Data Refute Factor-2 Log-Periodicity Across Fundamental Physics"
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        "description": "Discrete scale invariance (DSI) — invariance under rescaling by a fixed factor rather than by any factor — is established in few-body quantum physics, Dirac semimetals, and renormalization-group limit cycles. A recurring conjecture holds that the scaling factor itself might be universal, with the dyadic value 2 appearing across unrelated domains. This paper tests that conjecture against the published few-body and condensed-matter evidence. We model each confirmed DSI system as a log-periodic spectrum E_n = E₀ exp(-2π n/s₀) and extract the scaling factor from published values. The three-boson Efimov spectrum has scaling factor $22.694$, which we show corresponds to an RG index s₀ = 2π/ln(22.694) = 2.0124 and a log-periodic period Δ(ln E) = 3.1222, i.e. $4.505$ octaves — a factor of $4.505$ larger than the dyadic period ln 2 = 0.69315. Scaling traps realize continuously tunable factors controlled by the slope of a spatially varying scattering length, and spin-orbit-coupled three-boson systems obey a generalized scaling law in a five-dimensional parameter space, so no single factor is fixed. We conclude that DSI is real and broadly shared, but the specific factor-2 universality hypothesis is excluded by existing data at the level of a factor $11.347$ per spectral step. The practical consequence is a sharp falsification criterion: any claimed universal dyadic periodicity in a physical spectrum must survive comparison against the measured Efimov period of $3.1222$ in ln E, which it cannot if the period is $0.69315$.",
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